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Theorem hoadddirt 9647
Description: Scalar product reverse distributive law for Hilbert space operators.
Assertion
Ref Expression
hoadddirt |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) .op T) = ((A .op T) +op (B .op T)))
Proof of Theorem hoadddirt
StepHypRef Expression
1 ax-hvdistr2 8800 . . . . . . . . 9 |- ((A e. CC /\ B e. CC /\ (T` x) e. H~) -> ((A + B) .h (T` x)) = ((A .h (T` x)) +h (B .h (T` x))))
2 ffvelrn 3799 . . . . . . . . 9 |- ((T:H~-->H~ /\ x e. H~) -> (T` x) e. H~)
31, 2syl3an3 859 . . . . . . . 8 |- ((A e. CC /\ B e. CC /\ (T:H~-->H~ /\ x e. H~)) -> ((A + B) .h (T` x)) = ((A .h (T` x)) +h (B .h (T` x))))
433exp 830 . . . . . . 7 |- (A e. CC -> (B e. CC -> ((T:H~-->H~ /\ x e. H~) -> ((A + B) .h (T` x)) = ((A .h (T` x)) +h (B .h (T` x))))))
54exp4a 378 . . . . . 6 |- (A e. CC -> (B e. CC -> (T:H~-->H~ -> (x e. H~ -> ((A + B) .h (T` x)) = ((A .h (T` x)) +h (B .h (T` x)))))))
653imp1 844 . . . . 5 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A + B) .h (T` x)) = ((A .h (T` x)) +h (B .h (T` x))))
7 homvalt 9435 . . . . . . 7 |- (((A + B) e. CC /\ T:H~-->H~ /\ x e. H~) -> (((A + B) .op T)` x) = ((A + B) .h (T` x)))
873expa 831 . . . . . 6 |- ((((A + B) e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A + B) .op T)` x) = ((A + B) .h (T` x)))
9 axaddcl 5243 . . . . . . . 8 |- ((A e. CC /\ B e. CC) -> (A + B) e. CC)
109anim1i 334 . . . . . . 7 |- (((A e. CC /\ B e. CC) /\ T:H~-->H~) -> ((A + B) e. CC /\ T:H~-->H~))
11103impa 826 . . . . . 6 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) e. CC /\ T:H~-->H~))
128, 11sylan 448 . . . . 5 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A + B) .op T)` x) = ((A + B) .h (T` x)))
13 homvalt 9435 . . . . . . . 8 |- ((A e. CC /\ T:H~-->H~ /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
14133expa 831 . . . . . . 7 |- (((A e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
15143adantl2 802 . . . . . 6 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((A .op T)` x) = (A .h (T` x)))
16 homvalt 9435 . . . . . . . 8 |- ((B e. CC /\ T:H~-->H~ /\ x e. H~) -> ((B .op T)` x) = (B .h (T` x)))
17163expa 831 . . . . . . 7 |- (((B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((B .op T)` x) = (B .h (T` x)))
18173adantl1 801 . . . . . 6 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> ((B .op T)` x) = (B .h (T` x)))
1915, 18opreq12d 3963 . . . . 5 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A .op T)` x) +h ((B .op T)` x)) = ((A .h (T` x)) +h (B .h (T` x))))
206, 12, 193eqtr4d 1509 . . . 4 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A + B) .op T)` x) = (((A .op T)` x) +h ((B .op T)` x)))
21 hosvalt 9433 . . . . . 6 |- (((A .op T):H~-->H~ /\ (B .op T):H~-->H~ /\ x e. H~) -> (((A .op T) +op (B .op T))` x) = (((A .op T)` x) +h ((B .op T)` x)))
22213expa 831 . . . . 5 |- ((((A .op T):H~-->H~ /\ (B .op T):H~-->H~) /\ x e. H~) -> (((A .op T) +op (B .op T))` x) = (((A .op T)` x) +h ((B .op T)` x)))
23 homulclt 9602 . . . . . . 7 |- ((A e. CC /\ T:H~-->H~) -> (A .op T):H~-->H~)
24 homulclt 9602 . . . . . . 7 |- ((B e. CC /\ T:H~-->H~) -> (B .op T):H~-->H~)
2523, 24anim12i 333 . . . . . 6 |- (((A e. CC /\ T:H~-->H~) /\ (B e. CC /\ T:H~-->H~)) -> ((A .op T):H~-->H~ /\ (B .op T):H~-->H~))
26253impdir 878 . . . . 5 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A .op T):H~-->H~ /\ (B .op T):H~-->H~))
2722, 26sylan 448 . . . 4 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A .op T) +op (B .op T))` x) = (((A .op T)` x) +h ((B .op T)` x)))
2820, 27eqtr4d 1502 . . 3 |- (((A e. CC /\ B e. CC /\ T:H~-->H~) /\ x e. H~) -> (((A + B) .op T)` x) = (((A .op T) +op (B .op T))` x))
2928r19.21aiva 1706 . 2 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> A.x e. H~ (((A + B) .op T)` x) = (((A .op T) +op (B .op T))` x))
30 hoeqt 9603 . . 3 |- ((((A + B) .op T):H~-->H~ /\ ((A .op T) +op (B .op T)):H~-->H~) -> (A.x e. H~ (((A + B) .op T)` x) = (((A .op T) +op (B .op T))` x) <-> ((A + B) .op T) = ((A .op T) +op (B .op T))))
31 homulclt 9602 . . . . 5 |- (((A + B) e. CC /\ T:H~-->H~) -> ((A + B) .op T):H~-->H~)
3231, 9sylan 448 . . . 4 |- (((A e. CC /\ B e. CC) /\ T:H~-->H~) -> ((A + B) .op T):H~-->H~)
33323impa 826 . . 3 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) .op T):H~-->H~)
34 hoaddclt 9601 . . . . 5 |- (((A .op T):H~-->H~ /\ (B .op T):H~-->H~) -> ((A .op T) +op (B .op T)):H~-->H~)
3534, 23, 24syl2an 454 . . . 4 |- (((A e. CC /\ T:H~-->H~) /\ (B e. CC /\ T:H~-->H~)) -> ((A .op T) +op (B .op T)):H~-->H~)
36353impdir 878 . . 3 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A .op T) +op (B .op T)):H~-->H~)
3730, 33, 36sylanc 471 . 2 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> (A.x e. H~ (((A + B) .op T)` x) = (((A .op T) +op (B .op T))` x) <-> ((A + B) .op T) = ((A .op T) +op (B .op T))))
3829, 37mpbid 195 1 |- ((A e. CC /\ B e. CC /\ T:H~-->H~) -> ((A + B) .op T) = ((A .op T) +op (B .op T)))
Colors of variables: wff set class
Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637  -->wf 3168  ` cfv 3172  (class class class)co 3948  CCcc 5204   + caddc 5209  H~chil 8727   +h cva 8728   .h csm 8729   +op chos 8746   .op chot 8747
This theorem is referenced by:  ho2timest 9662
This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857  ax-inf2 4597  ax-hilex 8790  ax-hfvadd 8791  ax-hfvmul 8796  ax-hvdistr2 8800
This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-reu 1643  df-rab 1644  df-v 1803  df-sbc 1932  df-csb 1992  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-int 2524  df-iun 2558  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-id 2824  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944  df-om 3122  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-fv 3188  df-rdg 3917  df-opr 3950  df-oprab 3951  df-1st 4063  df-2nd 4064  df-1o 4117  df-oadd 4119  df-omul 4120  df-er 4245  df-ec 4247  df-qs 4250  df-map 4308  df-ni