HomeHome Metamath Proof Explorer < Previous   Next >
Related theorems
Unicode version
Theorem cnring 8099
Description: The set of complex numbers is a (unital) ring. (Contributed by Steve Rodriguez, 2-Feb-2007.)
Assertion
Ref Expression
cnring |- <. + , x. >. e. Ring
Proof of Theorem cnring
StepHypRef Expression
1 mulex 5290 . . 3 |- x. e. V
2 cnaddabl 8063 . . . . . 6 |- + e. Abel
3 ablgrp 8038 . . . . . 6 |- ( + e. Abel -> + e. Grp)
42, 3ax-mp 7 . . . . 5 |- + e. Grp
5 axaddopr 5237 . . . . 5 |- + :(CC X. CC)-->CC
64, 5grprnOLD 7991 . . . 4 |- CC = ran +
76isring 8078 . . 3 |- ( x. e. V -> (<. + , x. >. e. Ring <-> (( + e. Abel /\ x. :(CC X. CC)-->CC) /\ (A.x e. CC A.y e. CC A.z e. CC (((x x. y) x. z) = (x x. (y x. z)) /\ (x x. (y + z)) = ((x x. y) + (x x. z)) /\ ((x + y) x. z) = ((x x. z) + (y x. z))) /\ E.x e. CC A.y e. CC ((y x. x) = y /\ (x x. y) = y)))))
81, 7ax-mp 7 . 2 |- (<. + , x. >. e. Ring <-> (( + e. Abel /\ x. :(CC X. CC)-->CC) /\ (A.x e. CC A.y e. CC A.z e. CC (((x x. y) x. z) = (x x. (y x. z)) /\ (x x. (y + z)) = ((x x. y) + (x x. z)) /\ ((x + y) x. z) = ((x x. z) + (y x. z))) /\ E.x e. CC A.y e. CC ((y x. x) = y /\ (x x. y) = y))))
9 axmulopr 5238 . . 3 |- x. :(CC X. CC)-->CC
102, 9pm3.2i 285 . 2 |- ( + e. Abel /\ x. :(CC X. CC)-->CC)
11 axmulass 5250 . . . . 5 |- ((x e. CC /\ y e. CC /\ z e. CC) -> ((x x. y) x. z) = (x x. (y x. z)))
12 axdistr 5251 . . . . 5 |- ((x e. CC /\ y e. CC /\ z e. CC) -> (x x. (y + z)) = ((x x. y) + (x x. z)))
13 adddirt 5291 . . . . 5 |- ((x e. CC /\ y e. CC /\ z e. CC) -> ((x + y) x. z) = ((x x. z) + (y x. z)))
1411, 12, 133jca 817 . . . 4 |- ((x e. CC /\ y e. CC /\ z e. CC) -> (((x x. y) x. z) = (x x. (y x. z)) /\ (x x. (y + z)) = ((x x. y) + (x x. z)) /\ ((x + y) x. z) = ((x x. z) + (y x. z))))
1514rgen3 1716 . . 3 |- A.x e. CC A.y e. CC A.z e. CC (((x x. y) x. z) = (x x. (y x. z)) /\ (x x. (y + z)) = ((x x. y) + (x x. z)) /\ ((x + y) x. z) = ((x x. z) + (y x. z)))
16 ax1cn 5241 . . . 4 |- 1 e. CC
17 ax1id 5254 . . . . . 6 |- (y e. CC -> (y x. 1) = y)
18 mulid2t 5389 . . . . . 6 |- (y e. CC -> (1 x. y) = y)
1917, 18jca 288 . . . . 5 |- (y e. CC -> ((y x. 1) = y /\ (1 x. y) = y))
2019rgen 1690 . . . 4 |- A.y e. CC ((y x. 1) = y /\ (1 x. y) = y)
21 opreq2 3954 . . . . . . . 8 |- (x = 1 -> (y x. x) = (y x. 1))
2221eqeq1d 1475 . . . . . . 7 |- (x = 1 -> ((y x. x) = y <-> (y x. 1) = y))
23 opreq1 3953 . . . . . . . 8 |- (x = 1 -> (x x. y) = (1 x. y))
2423eqeq1d 1475 . . . . . . 7 |- (x = 1 -> ((x x. y) = y <-> (1 x. y) = y))
2522, 24anbi12d 626 . . . . . 6 |- (x = 1 -> (((y x. x) = y /\ (x x. y) = y) <-> ((y x. 1) = y /\ (1 x. y) = y)))
2625ralbidv 1655 . . . . 5 |- (x = 1 -> (A.y e. CC ((y x. x) = y /\ (x x. y) = y) <-> A.y e. CC ((y x. 1) = y /\ (1 x. y) = y)))
2726rcla4ev 1868 . . . 4 |- ((1 e. CC /\ A.y e. CC ((y x. 1) = y /\ (1 x. y) = y)) -> E.x e. CC A.y e. CC ((y x. x) = y /\ (x x. y) = y))
2816, 20, 27mp2an 695 . . 3 |- E.x e. CC A.y e. CC ((y x. x) = y /\ (x x. y) = y)
2915, 28pm3.2i 285 . 2 |- (A.x e. CC A.y e. CC A.z e. CC (((x x. y) x. z) = (x x. (y x. z)) /\ (x x. (y + z)) = ((x x. y) + (x x. z)) /\ ((x + y) x. z) = ((x x. z) + (y x. z))) /\ E.x e. CC A.y e. CC ((y x. x) = y /\ (x x. y) = y))
308, 10, 29mpbir2an 728 1 |- <. + , x. >. e. Ring
Colors of variables: wff set class
Syntax hints:   <-> wb 146   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  A.wral 1637  E.wrex 1638  Vcvv 1802  <.cop 2401   X. cxp 3158  -->wf 3168  (class class class)co 3948  CCcc 5204  1c1 5207   + caddc 5209   x. cmul 5211  Grpcgr 7967  Abelcabl 8035  Ringcring 8076
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
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-fo 3186  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-ni 4972  df-pli 4973  df-mi 4974  df-lti 4975  df-plpq 5007  df-mpq 5008  df-enq 5009  df-nq 5010  df-plq 5011  df-mq 5012  df-rq 5013  df-ltq 5014  df-1q 5015  df-np 5058  df-1p 5059  df-plp 5060  df-mp 5061  df-ltp 5062  df-plpr 5136  df-mpr 5137  df-enr 5138  df-nr 5139  df-plr 5140  df-mr 5141  df-0r 5143  df-1r 5144  df-m1r 5145  df-c 5212  df-0 5213  df-1 5214  df-i 5215  df-r 5216  df-plus 5217  df-mul 5218  df-sub 5328  df-neg 5330  df-grp 7971  df-abl 8036  df-ring 8077
Copyright terms: Public domain