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Theorem dvtan 28351
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 13353 . . . 4  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  /  ( cos `  x
) ) )
2 cnvimass 5186 . . . . . . . . 9  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  dom  cos
3 cosf 13405 . . . . . . . . . 10  |-  cos : CC
--> CC
43fdmi 5561 . . . . . . . . 9  |-  dom  cos  =  CC
52, 4sseqtri 3385 . . . . . . . 8  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  CC
65sseli 3349 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  x  e.  CC )
76sincld 13410 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( sin `  x
)  e.  CC )
86coscld 13411 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  e.  CC )
9 ffn 5556 . . . . . . . 8  |-  ( cos
: CC --> CC  ->  cos 
Fn  CC )
10 elpreima 5820 . . . . . . . 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 3998 . . . . . . . 8  |-  ( ( cos `  x )  e.  ( CC  \  { 0 } )  ->  ( cos `  x
)  =/=  0 )
1312adantl 463 . . . . . . 7  |-  ( ( x  e.  CC  /\  ( cos `  x )  e.  ( CC  \  { 0 } ) )  ->  ( cos `  x )  =/=  0
)
1411, 13sylbi 195 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  =/=  0 )
157, 8, 14divrecd 10106 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x )  /  ( cos `  x ) )  =  ( ( sin `  x )  x.  (
1  /  ( cos `  x ) ) ) )
1615mpteq2ia 4371 . . . 4  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  /  ( cos `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( ( sin `  x )  x.  ( 1  / 
( cos `  x
) ) ) )
171, 16eqtri 2461 . . 3  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  x.  ( 1  / 
( cos `  x
) ) ) )
1817oveq2i 6101 . 2  |-  ( CC 
_D  tan )  =  ( CC  _D  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  x.  ( 1  /  ( cos `  x
) ) ) ) )
19 cnelprrecn 9371 . . . . 5  |-  CC  e.  { RR ,  CC }
2019a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
21 difss 3480 . . . . . . . . 9  |-  ( CC 
\  { 0 } )  C_  CC
22 imass2 5201 . . . . . . . . 9  |-  ( ( CC  \  { 0 } )  C_  CC  ->  ( `' cos " ( CC  \  { 0 } ) )  C_  ( `' cos " CC ) )
2321, 22ax-mp 5 . . . . . . . 8  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  ( `' cos " CC )
24 fimacnv 5832 . . . . . . . . 9  |-  ( cos
: CC --> CC  ->  ( `' cos " CC )  =  CC )
253, 24ax-mp 5 . . . . . . . 8  |-  ( `' cos " CC )  =  CC
2623, 25sseqtri 3385 . . . . . . 7  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  CC
2726sseli 3349 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  x  e.  CC )
2827sincld 13410 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( sin `  x
)  e.  CC )
2928adantl 463 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( sin `  x
)  e.  CC )
308adantl 463 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( cos `  x
)  e.  CC )
31 sincl 13406 . . . . . 6  |-  ( x  e.  CC  ->  ( sin `  x )  e.  CC )
3231adantl 463 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  ( sin `  x )  e.  CC )
33 coscl 13407 . . . . . 6  |-  ( x  e.  CC  ->  ( cos `  x )  e.  CC )
3433adantl 463 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  ( cos `  x )  e.  CC )
35 dvsin 21354 . . . . . 6  |-  ( CC 
_D  sin )  =  cos
36 sinf 13404 . . . . . . . . 9  |-  sin : CC
--> CC
3736a1i 11 . . . . . . . 8  |-  ( T. 
->  sin : CC --> CC )
3837feqmptd 5741 . . . . . . 7  |-  ( T. 
->  sin  =  ( x  e.  CC  |->  ( sin `  x ) ) )
3938oveq2d 6106 . . . . . 6  |-  ( T. 
->  ( CC  _D  sin )  =  ( CC  _D  ( x  e.  CC  |->  ( sin `  x ) ) ) )
403a1i 11 . . . . . . 7  |-  ( T. 
->  cos : CC --> CC )
4140feqmptd 5741 . . . . . 6  |-  ( T. 
->  cos  =  ( x  e.  CC  |->  ( cos `  x ) ) )
4235, 39, 413eqtr3a 2497 . . . . 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 2441 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4544cnfldtopon 20262 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4645toponunii 18437 . . . . . . . 8  |-  CC  =  U. ( TopOpen ` fld )
4746restid 14368 . . . . . . 7  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
4845, 47ax-mp 5 . . . . . 6  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4948eqcomi 2445 . . . . 5  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
50 dvtanlem 28350 . . . . . 6  |-  ( `' cos " ( CC 
\  { 0 } ) )  e.  (
TopOpen ` fld )
5150a1i 11 . . . . 5  |-  ( T. 
->  ( `' cos " ( CC  \  { 0 } ) )  e.  (
TopOpen ` fld ) )
5220, 32, 34, 42, 43, 49, 44, 51dvmptres 21337 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( sin `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( cos `  x ) ) )
538, 14reccld 10096 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( 1  / 
( cos `  x
) )  e.  CC )
5453adantl 463 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( 1  /  ( cos `  x ) )  e.  CC )
55 ovex 6115 . . . . 5  |-  ( -u ( 1  /  (
( cos `  x
) ^ 2 ) )  x.  -u ( sin `  x ) )  e.  _V
5655a1i 11 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( -u ( 1  / 
( ( cos `  x
) ^ 2 ) )  x.  -u ( sin `  x ) )  e.  _V )
5711simprbi 461 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  e.  ( CC 
\  { 0 } ) )
5857adantl 463 . . . . 5  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( cos `  x
)  e.  ( CC 
\  { 0 } ) )
5929negcld 9702 . . . . 5  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  ->  -u ( sin `  x
)  e.  CC )
60 eldifi 3475 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  e.  CC )
61 eldifsni 3998 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  =/=  0
)
6260, 61reccld 10096 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  ->  ( 1  / 
y )  e.  CC )
6362adantl 463 . . . . 5  |-  ( ( T.  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( 1  / 
y )  e.  CC )
64 negex 9604 . . . . . 6  |-  -u (
1  /  ( y ^ 2 ) )  e.  _V
6564a1i 11 . . . . 5  |-  ( ( T.  /\  y  e.  ( CC  \  {
0 } ) )  ->  -u ( 1  / 
( y ^ 2 ) )  e.  _V )
6632negcld 9702 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  e.  CC )
67 dvcos 21355 . . . . . . 7  |-  ( CC 
_D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) )
6841oveq2d 6106 . . . . . . 7  |-  ( T. 
->  ( CC  _D  cos )  =  ( CC  _D  ( x  e.  CC  |->  ( cos `  x ) ) ) )
6967, 68syl5reqr 2488 . . . . . 6  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( cos `  x ) ) )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )
7020, 34, 66, 69, 43, 49, 44, 51dvmptres 21337 . . . . 5  |-  ( T. 
->  ( CC  _D  (
x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( cos `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  -u ( sin `  x ) ) )
71 ax-1cn 9336 . . . . . 6  |-  1  e.  CC
72 dvrec 21329 . . . . . 6  |-  ( 1  e.  CC  ->  ( CC  _D  ( y  e.  ( CC  \  {
0 } )  |->  ( 1  /  y ) ) )  =  ( y  e.  ( CC 
\  { 0 } )  |->  -u ( 1  / 
( y ^ 2 ) ) ) )
7371, 72mp1i 12 . . . . 5  |-  ( T. 
->  ( CC  _D  (
y  e.  ( CC 
\  { 0 } )  |->  ( 1  / 
y ) ) )  =  ( y  e.  ( CC  \  {
0 } )  |->  -u ( 1  /  (
y ^ 2 ) ) ) )
74 oveq2 6098 . . . . 5  |-  ( y  =  ( cos `  x
)  ->  ( 1  /  y )  =  ( 1  /  ( cos `  x ) ) )
75 oveq1 6097 . . . . . . 7  |-  ( y  =  ( cos `  x
)  ->  ( y ^ 2 )  =  ( ( cos `  x
) ^ 2 ) )
7675oveq2d 6106 . . . . . 6  |-  ( y  =  ( cos `  x
)  ->  ( 1  /  ( y ^
2 ) )  =  ( 1  /  (
( cos `  x
) ^ 2 ) ) )
7776negeqd 9600 . . . . 5  |-  ( y  =  ( cos `  x
)  ->  -u ( 1  /  ( y ^
2 ) )  = 
-u ( 1  / 
( ( cos `  x
) ^ 2 ) ) )
7820, 20, 58, 59, 63, 65, 70, 73, 74, 77dvmptco 21346 . . . 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 ) ) ) )
7920, 29, 30, 52, 54, 56, 78dvmptmul 21335 . . 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
) ) ) ) )
8079trud 1373 . 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
) ) ) )
81 ovex 6115 . . . . 5  |-  ( ( sin `  x )  /  ( cos `  x
) )  e.  _V
8281, 1dmmpti 5537 . . . 4  |-  dom  tan  =  ( `' cos " ( CC  \  {
0 } ) )
8382eqcomi 2445 . . 3  |-  ( `' cos " ( CC 
\  { 0 } ) )  =  dom  tan
848sqcld 12002 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ 2 )  e.  CC )
857sqcld 12002 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x ) ^ 2 )  e.  CC )
86 sqne0 11928 . . . . . . . . 9  |-  ( ( cos `  x )  e.  CC  ->  (
( ( cos `  x
) ^ 2 )  =/=  0  <->  ( cos `  x )  =/=  0
) )
878, 86syl 16 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  =/=  0  <->  ( cos `  x
)  =/=  0 ) )
8814, 87mpbird 232 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ 2 )  =/=  0 )
8984, 85, 84, 88divdird 10141 . . . . . 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 ) ) ) )
9084, 85addcomd 9567 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  +  ( ( sin `  x
) ^ 2 ) )  =  ( ( ( sin `  x
) ^ 2 )  +  ( ( cos `  x ) ^ 2 ) ) )
91 sincossq 13456 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
( ( sin `  x
) ^ 2 )  +  ( ( cos `  x ) ^ 2 ) )  =  1 )
926, 91syl 16 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x ) ^ 2 )  +  ( ( cos `  x
) ^ 2 ) )  =  1 )
9390, 92eqtrd 2473 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  +  ( ( sin `  x
) ^ 2 ) )  =  1 )
9493oveq1d 6105 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( ( cos `  x
) ^ 2 )  +  ( ( sin `  x ) ^ 2 ) )  /  (
( cos `  x
) ^ 2 ) )  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
9589, 94eqtr3d 2475 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( ( cos `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) )  +  ( ( ( sin `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) ) )  =  ( 1  /  (
( cos `  x
) ^ 2 ) ) )
968, 14recidd 10098 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x )  x.  (
1  /  ( cos `  x ) ) )  =  1 )
9784, 88dividd 10101 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) )  =  1 )
9896, 97eqtr4d 2476 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x )  x.  (
1  /  ( cos `  x ) ) )  =  ( ( ( cos `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) ) )
997, 7, 84, 88div23d 10140 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x )  x.  ( sin `  x
) )  /  (
( cos `  x
) ^ 2 ) )  =  ( ( ( sin `  x
)  /  ( ( cos `  x ) ^ 2 ) )  x.  ( sin `  x
) ) )
1007sqvald 12001 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x ) ^ 2 )  =  ( ( sin `  x )  x.  ( sin `  x
) ) )
101100oveq1d 6105 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) )  =  ( ( ( sin `  x
)  x.  ( sin `  x ) )  / 
( ( cos `  x
) ^ 2 ) ) )
10284, 88reccld 10096 . . . . . . . . . 10  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( 1  / 
( ( cos `  x
) ^ 2 ) )  e.  CC )
103102, 7mul2negd 9795 . . . . . . . . 9  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( -u (
1  /  ( ( cos `  x ) ^ 2 ) )  x.  -u ( sin `  x
) )  =  ( ( 1  /  (
( cos `  x
) ^ 2 ) )  x.  ( sin `  x ) ) )
1047, 84, 88divrec2d 10107 . . . . . . . . 9  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x )  /  (
( cos `  x
) ^ 2 ) )  =  ( ( 1  /  ( ( cos `  x ) ^ 2 ) )  x.  ( sin `  x
) ) )
105103, 104eqtr4d 2476 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( -u (
1  /  ( ( cos `  x ) ^ 2 ) )  x.  -u ( sin `  x
) )  =  ( ( sin `  x
)  /  ( ( cos `  x ) ^ 2 ) ) )
106105oveq1d 6105 . . . . . . 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
) ) )
10799, 101, 1063eqtr4rd 2484 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( -u ( 1  /  (
( cos `  x
) ^ 2 ) )  x.  -u ( sin `  x ) )  x.  ( sin `  x
) )  =  ( ( ( sin `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) ) )
10898, 107oveq12d 6108 . . . . 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 ) ) ) )
109 2nn0 10592 . . . . . 6  |-  2  e.  NN0
110 expneg 11869 . . . . . 6  |-  ( ( ( cos `  x
)  e.  CC  /\  2  e.  NN0 )  -> 
( ( cos `  x
) ^ -u 2
)  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
1118, 109, 110sylancl 657 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ -u 2
)  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
11295, 108, 1113eqtr4d 2483 . . . 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 ) )
113112rgen 2779 . . 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 )
114 mpteq12 4368 . . 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
) ) )
11583, 113, 114mp2an 667 . 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
) )
11618, 80, 1153eqtri 2465 1  |-  ( CC 
_D  tan )  =  ( x  e.  dom  tan  |->  ( ( cos `  x
) ^ -u 2
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1364   T. wtru 1365    e. wcel 1761    =/= wne 2604   A.wral 2713   _Vcvv 2970    \ cdif 3322    C_ wss 3325   {csn 3874   {cpr 3876    e. cmpt 4347   `'ccnv 4835   dom cdm 4836   "cima 4839    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090   CCcc 9276   RRcr 9277   0cc0 9278   1c1 9279    + caddc 9281    x. cmul 9283   -ucneg 9592    / cdiv 9989   2c2 10367   NN0cn0 10575   ^cexp 11861   sincsin 13345   cosccos 13346   tanctan 13347   ↾t crest 14355   TopOpenctopn 14356  ℂfldccnfld 17718  TopOnctopon 18399    _D cdv 21238
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-inf2 7843  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355  ax-pre-sup 9356  ax-addf 9357  ax-mulf 9358
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-int 4126  df-iun 4170  df-iin 4171  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-se 4676  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-isom 5424  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-of 6319  df-om 6476  df-1st 6576  df-2nd 6577  df-supp 6690  df-recs 6828  df-rdg 6862  df-1o 6916  df-2o 6917  df-oadd 6920  df-er 7097  df-map 7212  df-pm 7213  df-ixp 7260  df-en 7307  df-dom 7308  df-sdom 7309  df-fin 7310  df-fsupp 7617  df-fi 7657  df-sup 7687  df-oi 7720  df-card 8105  df-cda 8333  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-div 9990  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-n0 10576  df-z 10643  df-dec 10752  df-uz 10858  df-q 10950  df-rp 10988  df-xneg 11085  df-xadd 11086  df-xmul 11087  df-ioo 11300  df-ioc 11301  df-ico 11302  df-icc 11303  df-fz 11434  df-fzo 11545  df-fl 11638  df-mod 11705  df-seq 11803  df-exp 11862  df-fac 12048  df-bc 12075  df-hash 12100  df-shft 12552  df-cj 12584  df-re 12585  df-im 12586  df-sqr 12720  df-abs 12721  df-limsup 12945  df-clim 12962  df-rlim 12963  df-sum 13160  df-ef 13349  df-sin 13351  df-cos 13352  df-tan 13353  df-pi 13354  df-struct 14172  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-ress 14177  df-plusg 14247  df-mulr 14248  df-starv 14249  df-sca 14250  df-vsca 14251  df-ip 14252  df-tset 14253  df-ple 14254  df-ds 14256  df-unif 14257  df-hom 14258  df-cco 14259  df-rest 14357  df-topn 14358  df-0g 14376  df-gsum 14377  df-topgen 14378  df-pt 14379  df-prds 14382  df-xrs 14436  df-qtop 14441  df-imas 14442  df-xps 14444  df-mre 14520  df-mrc 14521  df-acs 14523  df-mnd 15411  df-submnd 15461  df-mulg 15541  df-cntz 15828  df-cmn 16272  df-psmet 17709  df-xmet 17710  df-met 17711  df-bl 17712  df-mopn 17713  df-fbas 17714  df-fg 17715  df-cnfld 17719  df-top 18403  df-bases 18405  df-topon 18406  df-topsp 18407  df-cld 18523  df-ntr 18524  df-cls 18525  df-nei 18602  df-lp 18640  df-perf 18641  df-cn 18731  df-cnp 18732  df-t1 18818  df-haus 18819  df-tx 19035  df-hmeo 19228  df-fil 19319  df-fm 19411  df-flim 19412  df-flf 19413  df-xms 19795  df-ms 19796  df-tms 19797  df-cncf 20354  df-limc 21241  df-dv 21242
This theorem is referenced by: (None)
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