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Theorem dvtan 33460
Description: Derivative of tangent. (Contributed by Brendan Leahy, 7-Aug-2018.)
Assertion
Ref Expression
dvtan |- ( CC _D tan ) = ( x e. dom tan |-> ( ( cos ` x ) ^ -u 2 ) )
Proof of Theorem dvtan
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 df-tan 14802 . . . 4 |- tan = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) / ( cos ` x ) ) )
2 cnvimass 5485 . . . . . . . . 9 |- ( `' cos " ( CC \ { 0 } ) ) C_ dom cos
3 cosf 14855 . . . . . . . . . 10 |- cos : CC --> CC
43fdmi 6052 . . . . . . . . 9 |- dom cos = CC
52, 4sseqtri 3637 . . . . . . . 8 |- ( `' cos " ( CC \ { 0 } ) ) C_ CC
65sseli 3599 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> x e. CC )
76sincld 14860 . . . . . 6 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( sin ` x ) e. CC )
86coscld 14861 . . . . . 6 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( cos ` x ) e. CC )
9 ffn 6045 . . . . . . . 8 |- ( cos : CC --> CC -> cos Fn CC )
10 elpreima 6337 . . . . . . . 8 |- ( cos Fn CC -> ( x e. ( `' cos " ( CC \ { 0 } ) ) <-> ( x e. CC /\ ( cos ` x ) e. ( CC \ { 0 } ) ) ) )
113, 9, 10mp2b 10 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) <-> ( x e. CC /\ ( cos ` x ) e. ( CC \ { 0 } ) ) )
12 eldifsni 4320 . . . . . . . 8 |- ( ( cos ` x ) e. ( CC \ { 0 } ) -> ( cos ` x ) =/= 0 )
1312adantl 482 . . . . . . 7 |- ( ( x e. CC /\ ( cos ` x ) e. ( CC \ { 0 } ) ) -> ( cos ` x ) =/= 0 )
1411, 13sylbi 207 . . . . . 6 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( cos ` x ) =/= 0 )
157, 8, 14divrecd 10804 . . . . 5 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( sin ` x ) / ( cos ` x ) ) = ( ( sin ` x ) x. ( 1 / ( cos ` x ) ) ) )
1615mpteq2ia 4740 . . . 4 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) / ( cos ` x ) ) ) = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) x. ( 1 / ( cos ` x ) ) ) )
171, 16eqtri 2644 . . 3 |- tan = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) x. ( 1 / ( cos ` x ) ) ) )
1817oveq2i 6661 . 2 |- ( CC _D tan ) = ( CC _D ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) x. ( 1 / ( cos ` x ) ) ) ) )
19 cnelprrecn 10029 . . . . 5 |- CC e. { RR , CC }
2019a1i 11 . . . 4 |- ( T. -> CC e. { RR , CC } )
21 difss 3737 . . . . . . . . 9 |- ( CC \ { 0 } ) C_ CC
22 imass2 5501 . . . . . . . . 9 |- ( ( CC \ { 0 } ) C_ CC -> ( `' cos " ( CC \ { 0 } ) ) C_ ( `' cos " CC ) )
2321, 22ax-mp 5 . . . . . . . 8 |- ( `' cos " ( CC \ { 0 } ) ) C_ ( `' cos " CC )
24 fimacnv 6347 . . . . . . . . 9 |- ( cos : CC --> CC -> ( `' cos " CC ) = CC )
253, 24ax-mp 5 . . . . . . . 8 |- ( `' cos " CC ) = CC
2623, 25sseqtri 3637 . . . . . . 7 |- ( `' cos " ( CC \ { 0 } ) ) C_ CC
2726sseli 3599 . . . . . 6 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> x e. CC )
2827sincld 14860 . . . . 5 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( sin ` x ) e. CC )
2928adantl 482 . . . 4 |- ( ( T. /\ x e. ( `' cos " ( CC \ { 0 } ) ) ) -> ( sin ` x ) e. CC )
308adantl 482 . . . 4 |- ( ( T. /\ x e. ( `' cos " ( CC \ { 0 } ) ) ) -> ( cos ` x ) e. CC )
31 sincl 14856 . . . . . 6 |- ( x e. CC -> ( sin ` x ) e. CC )
3231adantl 482 . . . . 5 |- ( ( T. /\ x e. CC ) -> ( sin ` x ) e. CC )
33 coscl 14857 . . . . . 6 |- ( x e. CC -> ( cos ` x ) e. CC )
3433adantl 482 . . . . 5 |- ( ( T. /\ x e. CC ) -> ( cos ` x ) e. CC )
35 dvsin 23745 . . . . . 6 |- ( CC _D sin ) = cos
36 sinf 14854 . . . . . . . . 9 |- sin : CC --> CC
3736a1i 11 . . . . . . . 8 |- ( T. -> sin : CC --> CC )
3837feqmptd 6249 . . . . . . 7 |- ( T. -> sin = ( x e. CC |-> ( sin ` x ) ) )
3938oveq2d 6666 . . . . . 6 |- ( T. -> ( CC _D sin ) = ( CC _D ( x e. CC |-> ( sin ` x ) ) ) )
403a1i 11 . . . . . . 7 |- ( T. -> cos : CC --> CC )
4140feqmptd 6249 . . . . . 6 |- ( T. -> cos = ( x e. CC |-> ( cos ` x ) ) )
4235, 39, 413eqtr3a 2680 . . . . 5 |- ( T. -> ( CC _D ( x e. CC |-> ( sin ` x ) ) ) = ( x e. CC |-> ( cos ` x ) ) )
4326a1i 11 . . . . 5 |- ( T. -> ( `' cos " ( CC \ { 0 } ) ) C_ CC )
44 eqid 2622 . . . . . . . 8 |- ( TopOpen ` ℂfld ) = ( TopOpen ` ℂfld )
4544cnfldtopon 22586 . . . . . . 7 |- ( TopOpen ` ℂfld ) e. (TopOn ` CC )
4645toponunii 20721 . . . . . . . 8 |- CC = U. ( TopOpen ` ℂfld )
4746restid 16094 . . . . . . 7 |- ( ( TopOpen ` ℂfld ) e. (TopOn ` CC ) -> ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld ) )
4845, 47ax-mp 5 . . . . . 6 |- ( ( TopOpen ` ℂfld ) ↾t CC ) = ( TopOpen ` ℂfld )
4948eqcomi 2631 . . . . 5 |- ( TopOpen ` ℂfld ) = ( ( TopOpen ` ℂfld ) ↾t CC )
50 dvtanlem 33459 . . . . . 6 |- ( `' cos " ( CC \ { 0 } ) ) e. ( TopOpen ` ℂfld )
5150a1i 11 . . . . 5 |- ( T. -> ( `' cos " ( CC \ { 0 } ) ) e. ( TopOpen ` ℂfld ) )
5220, 32, 34, 42, 43, 49, 44, 51dvmptres 23726 . . . 4 |- ( T. -> ( CC _D ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( sin ` x ) ) ) = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( cos ` x ) ) )
538, 14reccld 10794 . . . . 5 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( 1 / ( cos ` x ) ) e. CC )
5453adantl 482 . . . 4 |- ( ( T. /\ x e. ( `' cos " ( CC \ { 0 } ) ) ) -> ( 1 / ( cos ` x ) ) e. CC )
55 ovexd 6680 . . . 4 |- ( ( T. /\ x e. ( `' cos " ( CC \ { 0 } ) ) ) -> ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) e. _V )
5611simprbi 480 . . . . . 6 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( cos ` x ) e. ( CC \ { 0 } ) )
5756adantl 482 . . . . 5 |- ( ( T. /\ x e. ( `' cos " ( CC \ { 0 } ) ) ) -> ( cos ` x ) e. ( CC \ { 0 } ) )
5829negcld 10379 . . . . 5 |- ( ( T. /\ x e. ( `' cos " ( CC \ { 0 } ) ) ) -> -u ( sin ` x ) e. CC )
59 eldifi 3732 . . . . . . 7 |- ( y e. ( CC \ { 0 } ) -> y e. CC )
60 eldifsni 4320 . . . . . . 7 |- ( y e. ( CC \ { 0 } ) -> y =/= 0 )
6159, 60reccld 10794 . . . . . 6 |- ( y e. ( CC \ { 0 } ) -> ( 1 / y ) e. CC )
6261adantl 482 . . . . 5 |- ( ( T. /\ y e. ( CC \ { 0 } ) ) -> ( 1 / y ) e. CC )
63 negex 10279 . . . . . 6 |- -u ( 1 / ( y ^ 2 ) ) e. _V
6463a1i 11 . . . . 5 |- ( ( T. /\ y e. ( CC \ { 0 } ) ) -> -u ( 1 / ( y ^ 2 ) ) e. _V )
6532negcld 10379 . . . . . 6 |- ( ( T. /\ x e. CC ) -> -u ( sin ` x ) e. CC )
66 dvcos 23746 . . . . . . 7 |- ( CC _D cos ) = ( x e. CC |-> -u ( sin ` x ) )
6741oveq2d 6666 . . . . . . 7 |- ( T. -> ( CC _D cos ) = ( CC _D ( x e. CC |-> ( cos ` x ) ) ) )
6866, 67syl5reqr 2671 . . . . . 6 |- ( T. -> ( CC _D ( x e. CC |-> ( cos ` x ) ) ) = ( x e. CC |-> -u ( sin ` x ) ) )
6920, 34, 65, 68, 43, 49, 44, 51dvmptres 23726 . . . . 5 |- ( T. -> ( CC _D ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( cos ` x ) ) ) = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> -u ( sin ` x ) ) )
70 ax-1cn 9994 . . . . . 6 |- 1 e. CC
71 dvrec 23718 . . . . . 6 |- ( 1 e. CC -> ( CC _D ( y e. ( CC \ { 0 } ) |-> ( 1 / y ) ) ) = ( y e. ( CC \ { 0 } ) |-> -u ( 1 / ( y ^ 2 ) ) ) )
7270, 71mp1i 13 . . . . 5 |- ( T. -> ( CC _D ( y e. ( CC \ { 0 } ) |-> ( 1 / y ) ) ) = ( y e. ( CC \ { 0 } ) |-> -u ( 1 / ( y ^ 2 ) ) ) )
73 oveq2 6658 . . . . 5 |- ( y = ( cos ` x ) -> ( 1 / y ) = ( 1 / ( cos ` x ) ) )
74 oveq1 6657 . . . . . . 7 |- ( y = ( cos ` x ) -> ( y ^ 2 ) = ( ( cos ` x ) ^ 2 ) )
7574oveq2d 6666 . . . . . 6 |- ( y = ( cos ` x ) -> ( 1 / ( y ^ 2 ) ) = ( 1 / ( ( cos ` x ) ^ 2 ) ) )
7675negeqd 10275 . . . . 5 |- ( y = ( cos ` x ) -> -u ( 1 / ( y ^ 2 ) ) = -u ( 1 / ( ( cos ` x ) ^ 2 ) ) )
7720, 20, 57, 58, 62, 64, 69, 72, 73, 76dvmptco 23735 . . . 4 |- ( T. -> ( CC _D ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( 1 / ( cos ` x ) ) ) ) = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) ) )
7820, 29, 30, 52, 54, 55, 77dvmptmul 23724 . . 3 |- ( T. -> ( CC _D ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) x. ( 1 / ( cos ` x ) ) ) ) ) = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( ( cos ` x ) x. ( 1 / ( cos ` x ) ) ) + ( ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) x. ( sin ` x ) ) ) ) )
7978trud 1493 . 2 |- ( CC _D ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( sin ` x ) x. ( 1 / ( cos ` x ) ) ) ) ) = ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( ( cos ` x ) x. ( 1 / ( cos ` x ) ) ) + ( ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) x. ( sin ` x ) ) ) )
80 ovex 6678 . . . . 5 |- ( ( sin ` x ) / ( cos ` x ) ) e. _V
8180, 1dmmpti 6023 . . . 4 |- dom tan = ( `' cos " ( CC \ { 0 } ) )
8281eqcomi 2631 . . 3 |- ( `' cos " ( CC \ { 0 } ) ) = dom tan
838sqcld 13006 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( cos ` x ) ^ 2 ) e. CC )
847sqcld 13006 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( sin ` x ) ^ 2 ) e. CC )
85 sqne0 12930 . . . . . . . . 9 |- ( ( cos ` x ) e. CC -> ( ( ( cos ` x ) ^ 2 ) =/= 0 <-> ( cos ` x ) =/= 0 ) )
868, 85syl 17 . . . . . . . 8 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( cos ` x ) ^ 2 ) =/= 0 <-> ( cos ` x ) =/= 0 ) )
8714, 86mpbird 247 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( cos ` x ) ^ 2 ) =/= 0 )
8883, 84, 83, 87divdird 10839 . . . . . 6 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( ( cos ` x ) ^ 2 ) + ( ( sin ` x ) ^ 2 ) ) / ( ( cos ` x ) ^ 2 ) ) = ( ( ( ( cos ` x ) ^ 2 ) / ( ( cos ` x ) ^ 2 ) ) + ( ( ( sin ` x ) ^ 2 ) / ( ( cos ` x ) ^ 2 ) ) ) )
8983, 84addcomd 10238 . . . . . . . 8 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( cos ` x ) ^ 2 ) + ( ( sin ` x ) ^ 2 ) ) = ( ( ( sin ` x ) ^ 2 ) + ( ( cos ` x ) ^ 2 ) ) )
90 sincossq 14906 . . . . . . . . 9 |- ( x e. CC -> ( ( ( sin ` x ) ^ 2 ) + ( ( cos ` x ) ^ 2 ) ) = 1 )
916, 90syl 17 . . . . . . . 8 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( sin ` x ) ^ 2 ) + ( ( cos ` x ) ^ 2 ) ) = 1 )
9289, 91eqtrd 2656 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( cos ` x ) ^ 2 ) + ( ( sin ` x ) ^ 2 ) ) = 1 )
9392oveq1d 6665 . . . . . 6 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( ( cos ` x ) ^ 2 ) + ( ( sin ` x ) ^ 2 ) ) / ( ( cos ` x ) ^ 2 ) ) = ( 1 / ( ( cos ` x ) ^ 2 ) ) )
9488, 93eqtr3d 2658 . . . . 5 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( ( cos ` x ) ^ 2 ) / ( ( cos ` x ) ^ 2 ) ) + ( ( ( sin ` x ) ^ 2 ) / ( ( cos ` x ) ^ 2 ) ) ) = ( 1 / ( ( cos ` x ) ^ 2 ) ) )
958, 14recidd 10796 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( cos ` x ) x. ( 1 / ( cos ` x ) ) ) = 1 )
9683, 87dividd 10799 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( cos ` x ) ^ 2 ) / ( ( cos ` x ) ^ 2 ) ) = 1 )
9795, 96eqtr4d 2659 . . . . . 6 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( cos ` x ) x. ( 1 / ( cos ` x ) ) ) = ( ( ( cos ` x ) ^ 2 ) / ( ( cos ` x ) ^ 2 ) ) )
987, 7, 83, 87div23d 10838 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( sin ` x ) x. ( sin ` x ) ) / ( ( cos ` x ) ^ 2 ) ) = ( ( ( sin ` x ) / ( ( cos ` x ) ^ 2 ) ) x. ( sin ` x ) ) )
997sqvald 13005 . . . . . . . 8 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( sin ` x ) ^ 2 ) = ( ( sin ` x ) x. ( sin ` x ) ) )
10099oveq1d 6665 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( sin ` x ) ^ 2 ) / ( ( cos ` x ) ^ 2 ) ) = ( ( ( sin ` x ) x. ( sin ` x ) ) / ( ( cos ` x ) ^ 2 ) ) )
10183, 87reccld 10794 . . . . . . . . . 10 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( 1 / ( ( cos ` x ) ^ 2 ) ) e. CC )
102101, 7mul2negd 10485 . . . . . . . . 9 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) = ( ( 1 / ( ( cos ` x ) ^ 2 ) ) x. ( sin ` x ) ) )
1037, 83, 87divrec2d 10805 . . . . . . . . 9 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( sin ` x ) / ( ( cos ` x ) ^ 2 ) ) = ( ( 1 / ( ( cos ` x ) ^ 2 ) ) x. ( sin ` x ) ) )
104102, 103eqtr4d 2659 . . . . . . . 8 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) = ( ( sin ` x ) / ( ( cos ` x ) ^ 2 ) ) )
105104oveq1d 6665 . . . . . . 7 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) x. ( sin ` x ) ) = ( ( ( sin ` x ) / ( ( cos ` x ) ^ 2 ) ) x. ( sin ` x ) ) )
10698, 100, 1053eqtr4rd 2667 . . . . . 6 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) x. ( sin ` x ) ) = ( ( ( sin ` x ) ^ 2 ) / ( ( cos ` x ) ^ 2 ) ) )
10797, 106oveq12d 6668 . . . . 5 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( cos ` x ) x. ( 1 / ( cos ` x ) ) ) + ( ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) x. ( sin ` x ) ) ) = ( ( ( ( cos ` x ) ^ 2 ) / ( ( cos ` x ) ^ 2 ) ) + ( ( ( sin ` x ) ^ 2 ) / ( ( cos ` x ) ^ 2 ) ) ) )
108 2nn0 11309 . . . . . 6 |- 2 e. NN0
109 expneg 12868 . . . . . 6 |- ( ( ( cos ` x ) e. CC /\ 2 e. NN0 ) -> ( ( cos ` x ) ^ -u 2 ) = ( 1 / ( ( cos ` x ) ^ 2 ) ) )
1108, 108, 109sylancl 694 . . . . 5 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( cos ` x ) ^ -u 2 ) = ( 1 / ( ( cos ` x ) ^ 2 ) ) )
11194, 107, 1103eqtr4d 2666 . . . 4 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) -> ( ( ( cos ` x ) x. ( 1 / ( cos ` x ) ) ) + ( ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) x. ( sin ` x ) ) ) = ( ( cos ` x ) ^ -u 2 ) )
112111rgen 2922 . . 3 |- A. x e. ( `' cos " ( CC \ { 0 } ) ) ( ( ( cos ` x ) x. ( 1 / ( cos ` x ) ) ) + ( ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) x. ( sin ` x ) ) ) = ( ( cos ` x ) ^ -u 2 )
113 mpteq12 4736 . . 3 |- ( ( ( `' cos " ( CC \ { 0 } ) ) = dom tan /\ A. x e. ( `' cos " ( CC \ { 0 } ) ) ( ( ( cos ` x ) x. ( 1 / ( cos ` x ) ) ) + ( ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) x. ( sin ` x ) ) ) = ( ( cos ` x ) ^ -u 2 ) ) -> ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( ( cos ` x ) x. ( 1 / ( cos ` x ) ) ) + ( ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) x. ( sin ` x ) ) ) ) = ( x e. dom tan |-> ( ( cos ` x ) ^ -u 2 ) ) )
11482, 112, 113mp2an 708 . 2 |- ( x e. ( `' cos " ( CC \ { 0 } ) ) |-> ( ( ( cos ` x ) x. ( 1 / ( cos ` x ) ) ) + ( ( -u ( 1 / ( ( cos ` x ) ^ 2 ) ) x. -u ( sin ` x ) ) x. ( sin ` x ) ) ) ) = ( x e. dom tan |-> ( ( cos ` x ) ^ -u 2 ) )
11518, 79, 1143eqtri 2648 1 |- ( CC _D tan ) = ( x e. dom tan |-> ( ( cos ` x ) ^ -u 2 ) )
Colors of variables: wff setvar class
Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   T. wtru 1484   e. wcel 1990   =/= wne 2794   A.wral 2912   _Vcvv 3200   \ cdif 3571   C_ wss 3574   {csn 4177   {cpr 4179   |-> cmpt 4729   `'ccnv 5113   dom cdm 5114   "cima 5117   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   -ucneg 10267   / cdiv 10684   2c2 11070   NN0cn0 11292   ^cexp 12860   sincsin 14794   cosccos 14795   tanctan 14796   ↾t crest 16081   TopOpenctopn 16082  ℂfldccnfld 19746  TopOnctopon 20715   _D cdv 23627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-addf 10015  ax-mulf 10016
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ico 12181  df-icc 12182  df-fz 12327  df-fzo 12466  df-fl 12593  df-seq 12802  df-exp 12861  df-fac 13061  df-bc 13090  df-hash 13118  df-shft 13807  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-sum 14417  df-ef 14798  df-sin 14800  df-cos 14801  df-tan 14802  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-fbas 19743  df-fg 19744  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-ntr 20824  df-cls 20825  df-nei 20902  df-lp 20940  df-perf 20941  df-cn 21031  df-cnp 21032  df-t1 21118  df-haus 21119  df-tx 21365  df-hmeo 21558  df-fil 21650  df-fm 21742  df-flim 21743  df-flf 21744  df-xms 22125  df-ms 22126  df-tms 22127  df-cncf 22681  df-limc 23630  df-dv 23631
This theorem is referenced by: (None)
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