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Theorem cos01gt0 7419
Description: The cosine of a positive real number less than or equal to 1 is positive. (Contributed by Paul Chapman, 19-Jan-2008.)
Assertion
Ref Expression
cos01gt0 |- (A e. (0(,]1) -> 0 < (cos` A))
Proof of Theorem cos01gt0
StepHypRef Expression
1 0re 5412 . . . . . . . . . . 11 |- 0 e. RR
2 1re 5407 . . . . . . . . . . 11 |- 1 e. RR
3 elioc2t 6322 . . . . . . . . . . 11 |- ((0 e. RR /\ 1 e. RR) -> (A e. (0(,]1) <-> (A e. RR /\ 0 < A /\ A <_ 1)))
41, 2, 3mp2an 695 . . . . . . . . . 10 |- (A e. (0(,]1) <-> (A e. RR /\ 0 < A /\ A <_ 1))
54biimp 151 . . . . . . . . 9 |- (A e. (0(,]1) -> (A e. RR /\ 0 < A /\ A <_ 1))
653simp1d 792 . . . . . . . 8 |- (A e. (0(,]1) -> A e. RR)
7 resqclt 6552 . . . . . . . 8 |- (A e. RR -> (A^2) e. RR)
86, 7syl 10 . . . . . . 7 |- (A e. (0(,]1) -> (A^2) e. RR)
98recnd 5287 . . . . . 6 |- (A e. (0(,]1) -> (A^2) e. CC)
10 2cn 5927 . . . . . . 7 |- 2 e. CC
11 3re 5928 . . . . . . . 8 |- 3 e. RR
1211recn 5286 . . . . . . 7 |- 3 e. CC
13 3nn 5947 . . . . . . . . 9 |- 3 e. NN
1413nnne0 5899 . . . . . . . 8 |- 3 =/= 0
15 div12t 5707 . . . . . . . 8 |- (((2 e. CC /\ (A^2) e. CC /\ 3 e. CC) /\ 3 =/= 0) -> (2 x. ((A^2) / 3)) = ((A^2) x. (2 / 3)))
1614, 15mpan2 694 . . . . . . 7 |- ((2 e. CC /\ (A^2) e. CC /\ 3 e. CC) -> (2 x. ((A^2) / 3)) = ((A^2) x. (2 / 3)))
1710, 12, 16mp3an13 904 . . . . . 6 |- ((A^2) e. CC -> (2 x. ((A^2) / 3)) = ((A^2) x. (2 / 3)))
189, 17syl 10 . . . . 5 |- (A e. (0(,]1) -> (2 x. ((A^2) / 3)) = ((A^2) x. (2 / 3)))
19 2re 5926 . . . . . . . . 9 |- 2 e. RR
2019, 11, 14redivcl 5754 . . . . . . . 8 |- (2 / 3) e. RR
2119ltp1 5769 . . . . . . . . . . . . 13 |- 2 < (2 + 1)
22 df-3 5918 . . . . . . . . . . . . 13 |- 3 = (2 + 1)
2321, 22breqtrr 2630 . . . . . . . . . . . 12 |- 2 < 3
24 3pos 5938 . . . . . . . . . . . . 13 |- 0 < 3
2519, 11, 11, 24ltdiv1i 5779 . . . . . . . . . . . 12 |- (2 < 3 <-> (2 / 3) < (3 / 3))
2623, 25mpbi 189 . . . . . . . . . . 11 |- (2 / 3) < (3 / 3)
2712, 14divid 5726 . . . . . . . . . . 11 |- (3 / 3) = 1
2826, 27breqtr 2628 . . . . . . . . . 10 |- (2 / 3) < 1
29 ltmul2t 5787 . . . . . . . . . 10 |- ((((2 / 3) e. RR /\ 1 e. RR /\ (A^2) e. RR) /\ 0 < (A^2)) -> ((2 / 3) < 1 <-> ((A^2) x. (2 / 3)) < ((A^2) x. 1)))
3028, 29mpbii 193 . . . . . . . . 9 |- ((((2 / 3) e. RR /\ 1 e. RR /\ (A^2) e. RR) /\ 0 < (A^2)) -> ((A^2) x. (2 / 3)) < ((A^2) x. 1))
3130ex 373 . . . . . . . 8 |- (((2 / 3) e. RR /\ 1 e. RR /\ (A^2) e. RR) -> (0 < (A^2) -> ((A^2) x. (2 / 3)) < ((A^2) x. 1)))
3220, 2, 31mp3an12 903 . . . . . . 7 |- ((A^2) e. RR -> (0 < (A^2) -> ((A^2) x. (2 / 3)) < ((A^2) x. 1)))
33 2nn0 6062 . . . . . . . . . 10 |- 2 e. NN0
34 expgt0t 6520 . . . . . . . . . 10 |- ((A e. RR /\ 2 e. NN0 /\ 0 < A) -> 0 < (A^2))
3533, 34mp3an2 901 . . . . . . . . 9 |- ((A e. RR /\ 0 < A) -> 0 < (A^2))
36353adant3 797 . . . . . . . 8 |- ((A e. RR /\ 0 < A /\ A <_ 1) -> 0 < (A^2))
374, 36sylbi 199 . . . . . . 7 |- (A e. (0(,]1) -> 0 < (A^2))
3832, 8, 37sylc 68 . . . . . 6 |- (A e. (0(,]1) -> ((A^2) x. (2 / 3)) < ((A^2) x. 1))
39 ax1id 5254 . . . . . . 7 |- ((A^2) e. CC -> ((A^2) x. 1) = (A^2))
409, 39syl 10 . . . . . 6 |- (A e. (0(,]1) -> ((A^2) x. 1) = (A^2))
4138, 40breqtrd 2629 . . . . 5 |- (A e. (0(,]1) -> ((A^2) x. (2 / 3)) < (A^2))
4218, 41eqbrtrd 2625 . . . 4 |- (A e. (0(,]1) -> (2 x. ((A^2) / 3)) < (A^2))
43 le2sqit 6563 . . . . . . . . 9 |- (((A e. RR /\ 0 <_ A) /\ (1 e. RR /\ A <_ 1)) -> (A^2) <_ (1^2))
442, 43mpanr1 707 . . . . . . . 8 |- (((A e. RR /\ 0 <_ A) /\ A <_ 1) -> (A^2) <_ (1^2))
45 ltlet 5493 . . . . . . . . . 10 |- ((0 e. RR /\ A e. RR) -> (0 < A -> 0 <_ A))
461, 45mpan 693 . . . . . . . . 9 |- (A e. RR -> (0 < A -> 0 <_ A))
4746imdistani 443 . . . . . . . 8 |- ((A e. RR /\ 0 < A) -> (A e. RR /\ 0 <_ A))
4844, 47sylan 448 . . . . . . 7 |- (((A e. RR /\ 0 < A) /\ A <_ 1) -> (A^2) <_ (1^2))
49483impa 826 . . . . . 6 |- ((A e. RR /\ 0 < A /\ A <_ 1) -> (A^2) <_ (1^2))
504, 49sylbi 199 . . . . 5 |- (A e. (0(,]1) -> (A^2) <_ (1^2))
51 sq1 6568 . . . . 5 |- (1^2) = 1
5250, 51syl6breq 2644 . . . 4 |- (A e. (0(,]1) -> (A^2) <_ 1)
53 ltletrt 5497 . . . . . 6 |- (((2 x. ((A^2) / 3)) e. RR /\ (A^2) e. RR /\ 1 e. RR) -> (((2 x. ((A^2) / 3)) < (A^2) /\ (A^2) <_ 1) -> (2 x. ((A^2) / 3)) < 1))
542, 53mp3an3 902 . . . . 5 |- (((2 x. ((A^2) / 3)) e. RR /\ (A^2) e. RR) -> (((2 x. ((A^2) / 3)) < (A^2) /\ (A^2) <_ 1) -> (2 x. ((A^2) / 3)) < 1))
55 redivclt 5756 . . . . . . . 8 |- (((A^2) e. RR /\ 3 e. RR /\ 3 =/= 0) -> ((A^2) / 3) e. RR)
5611, 14, 55mp3an23 905 . . . . . . 7 |- ((A^2) e. RR -> ((A^2) / 3) e. RR)
578, 56syl 10 . . . . . 6 |- (A e. (0(,]1) -> ((A^2) / 3) e. RR)
58 axmulrcl 5246 . . . . . . 7 |- ((2 e. RR /\ ((A^2) / 3) e. RR) -> (2 x. ((A^2) / 3)) e. RR)
5919, 58mpan 693 . . . . . 6 |- (((A^2) / 3) e. RR -> (2 x. ((A^2) / 3)) e. RR)
6057, 59syl 10 . . . . 5 |- (A e. (0(,]1) -> (2 x. ((A^2) / 3)) e. RR)
6154, 60, 8sylanc 471 . . . 4 |- (A e. (0(,]1) -> (((2 x. ((A^2) / 3)) < (A^2) /\ (A^2) <_ 1) -> (2 x. ((A^2) / 3)) < 1))
6242, 52, 61mp2and 701 . . 3 |- (A e. (0(,]1) -> (2 x. ((A^2) / 3)) < 1)
63 posdift 5627 . . . . 5 |- (((2 x. ((A^2) / 3)) e. RR /\ 1 e. RR) -> ((2 x. ((A^2) / 3)) < 1 <-> 0 < (1 - (2 x. ((A^2) / 3)))))
642, 63mpan2 694 . . . 4 |- ((2 x. ((A^2) / 3)) e. RR -> ((2 x. ((A^2) / 3)) < 1 <-> 0 < (1 - (2 x. ((A^2) / 3)))))
6560, 64syl 10 . . 3 |- (A e. (0(,]1) -> ((2 x. ((A^2) / 3)) < 1 <-> 0 < (1 - (2 x. ((A^2) / 3)))))
6662, 65mpbid 195 . 2 |- (A e. (0(,]1) -> 0 < (1 - (2 x. ((A^2) / 3))))
67 cos01bnd 7415 . . 3 |- (A e. (0(,]1) -> ((1 - (2 x. ((A^2) / 3))) < (cos` A) /\ (cos` A) < (1 - ((A^2) / 3))))
6867pm3.26d 321 . 2 |- (A e. (0(,]1) -> (1 - (2 x. ((A^2) / 3))) < (cos` A))
69 axlttrn 5476 . . . 4 |- ((0 e. RR /\ (1 - (2 x. ((A^2) / 3))) e. RR /\ (cos` A) e. RR) -> ((0 < (1 - (2 x. ((A^2) / 3))) /\ (1 - (2 x. ((A^2) / 3))) < (cos` A)) -> 0 < (cos` A)))
701, 69mp3an1 900 . . 3 |- (((1 - (2 x. ((A^2) / 3))) e. RR /\ (cos` A) e. RR) -> ((0 < (1 - (2 x. ((A^2) / 3))) /\ (1 - (2 x. ((A^2) / 3))) < (cos` A))