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Theorem dvtan 30308
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 13892 . . . 4  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  /  ( cos `  x
) ) )
2 cnvimass 5345 . . . . . . . . 9  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  dom  cos
3 cosf 13945 . . . . . . . . . 10  |-  cos : CC
--> CC
43fdmi 5718 . . . . . . . . 9  |-  dom  cos  =  CC
52, 4sseqtri 3521 . . . . . . . 8  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  CC
65sseli 3485 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  x  e.  CC )
76sincld 13950 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( sin `  x
)  e.  CC )
86coscld 13951 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  e.  CC )
9 ffn 5713 . . . . . . . 8  |-  ( cos
: CC --> CC  ->  cos 
Fn  CC )
10 elpreima 5983 . . . . . . . 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 4142 . . . . . . . 8  |-  ( ( cos `  x )  e.  ( CC  \  { 0 } )  ->  ( cos `  x
)  =/=  0 )
1312adantl 464 . . . . . . 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 10319 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x )  /  ( cos `  x ) )  =  ( ( sin `  x )  x.  (
1  /  ( cos `  x ) ) ) )
1615mpteq2ia 4521 . . . 4  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  /  ( cos `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( ( sin `  x )  x.  ( 1  / 
( cos `  x
) ) ) )
171, 16eqtri 2483 . . 3  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  x.  ( 1  / 
( cos `  x
) ) ) )
1817oveq2i 6281 . 2  |-  ( CC 
_D  tan )  =  ( CC  _D  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  x.  ( 1  /  ( cos `  x
) ) ) ) )
19 cnelprrecn 9574 . . . . 5  |-  CC  e.  { RR ,  CC }
2019a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
21 difss 3617 . . . . . . . . 9  |-  ( CC 
\  { 0 } )  C_  CC
22 imass2 5360 . . . . . . . . 9  |-  ( ( CC  \  { 0 } )  C_  CC  ->  ( `' cos " ( CC  \  { 0 } ) )  C_  ( `' cos " CC ) )
2321, 22ax-mp 5 . . . . . . . 8  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  ( `' cos " CC )
24 fimacnv 5995 . . . . . . . . 9  |-  ( cos
: CC --> CC  ->  ( `' cos " CC )  =  CC )
253, 24ax-mp 5 . . . . . . . 8  |-  ( `' cos " CC )  =  CC
2623, 25sseqtri 3521 . . . . . . 7  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  CC
2726sseli 3485 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  x  e.  CC )
2827sincld 13950 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( sin `  x
)  e.  CC )
2928adantl 464 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( sin `  x
)  e.  CC )
308adantl 464 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( cos `  x
)  e.  CC )
31 sincl 13946 . . . . . 6  |-  ( x  e.  CC  ->  ( sin `  x )  e.  CC )
3231adantl 464 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  ( sin `  x )  e.  CC )
33 coscl 13947 . . . . . 6  |-  ( x  e.  CC  ->  ( cos `  x )  e.  CC )
3433adantl 464 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  ( cos `  x )  e.  CC )
35 dvsin 22552 . . . . . 6  |-  ( CC 
_D  sin )  =  cos
36 sinf 13944 . . . . . . . . 9  |-  sin : CC
--> CC
3736a1i 11 . . . . . . . 8  |-  ( T. 
->  sin : CC --> CC )
3837feqmptd 5901 . . . . . . 7  |-  ( T. 
->  sin  =  ( x  e.  CC  |->  ( sin `  x ) ) )
3938oveq2d 6286 . . . . . 6  |-  ( T. 
->  ( CC  _D  sin )  =  ( CC  _D  ( x  e.  CC  |->  ( sin `  x ) ) ) )
403a1i 11 . . . . . . 7  |-  ( T. 
->  cos : CC --> CC )
4140feqmptd 5901 . . . . . 6  |-  ( T. 
->  cos  =  ( x  e.  CC  |->  ( cos `  x ) ) )
4235, 39, 413eqtr3a 2519 . . . . 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 2454 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4544cnfldtopon 21459 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4645toponunii 19603 . . . . . . . 8  |-  CC  =  U. ( TopOpen ` fld )
4746restid 14926 . . . . . . 7  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
4845, 47ax-mp 5 . . . . . 6  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4948eqcomi 2467 . . . . 5  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
50 dvtanlem 30307 . . . . . 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 22535 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( sin `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( cos `  x ) ) )
538, 14reccld 10309 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( 1  / 
( cos `  x
) )  e.  CC )
5453adantl 464 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( 1  /  ( cos `  x ) )  e.  CC )
55 ovex 6298 . . . . 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 462 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  e.  ( CC 
\  { 0 } ) )
5857adantl 464 . . . . 5  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( cos `  x
)  e.  ( CC 
\  { 0 } ) )
5929negcld 9909 . . . . 5  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  ->  -u ( sin `  x
)  e.  CC )
60 eldifi 3612 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  e.  CC )
61 eldifsni 4142 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  =/=  0
)
6260, 61reccld 10309 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  ->  ( 1  / 
y )  e.  CC )
6362adantl 464 . . . . 5  |-  ( ( T.  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( 1  / 
y )  e.  CC )
64 negex 9809 . . . . . 6  |-  -u (
1  /  ( y ^ 2 ) )  e.  _V
6564a1i 11 . . . . 5  |-  ( ( T.  /\  y  e.  ( CC  \  {
0 } ) )  ->  -u ( 1  / 
( y ^ 2 ) )  e.  _V )
6632negcld 9909 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  e.  CC )
67 dvcos 22553 . . . . . . 7  |-  ( CC 
_D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) )
6841oveq2d 6286 . . . . . . 7  |-  ( T. 
->  ( CC  _D  cos )  =  ( CC  _D  ( x  e.  CC  |->  ( cos `  x ) ) ) )
6967, 68syl5reqr 2510 . . . . . 6  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( cos `  x ) ) )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )
7020, 34, 66, 69, 43, 49, 44, 51dvmptres 22535 . . . . 5  |-  ( T. 
->  ( CC  _D  (
x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( cos `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  -u ( sin `  x ) ) )
71 ax-1cn 9539 . . . . . 6  |-  1  e.  CC
72 dvrec 22527 . . . . . 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 6278 . . . . 5  |-  ( y  =  ( cos `  x
)  ->  ( 1  /  y )  =  ( 1  /  ( cos `  x ) ) )
75 oveq1 6277 . . . . . . 7  |-  ( y  =  ( cos `  x
)  ->  ( y ^ 2 )  =  ( ( cos `  x
) ^ 2 ) )
7675oveq2d 6286 . . . . . 6  |-  ( y  =  ( cos `  x
)  ->  ( 1  /  ( y ^
2 ) )  =  ( 1  /  (
( cos `  x
) ^ 2 ) ) )
7776negeqd 9805 . . . . 5  |-  ( y  =  ( cos `  x
)  ->  -u ( 1  /  ( y ^
2 ) )  = 
-u ( 1  / 
( ( cos `  x
) ^ 2 ) ) )
7820, 20, 58, 59, 63, 65, 70, 73, 74, 77dvmptco 22544 . . . 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 22533 . . 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 1407 . 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 6298 . . . . 5  |-  ( ( sin `  x )  /  ( cos `  x
) )  e.  _V
8281, 1dmmpti 5692 . . . 4  |-  dom  tan  =  ( `' cos " ( CC  \  {
0 } ) )
8382eqcomi 2467 . . 3  |-  ( `' cos " ( CC 
\  { 0 } ) )  =  dom  tan
848sqcld 12293 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ 2 )  e.  CC )
857sqcld 12293 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x ) ^ 2 )  e.  CC )
86 sqne0 12219 . . . . . . . . 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 10354 . . . . . 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 9771 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  +  ( ( sin `  x
) ^ 2 ) )  =  ( ( ( sin `  x
) ^ 2 )  +  ( ( cos `  x ) ^ 2 ) ) )
91 sincossq 13996 . . . . . . . . 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 2495 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  +  ( ( sin `  x
) ^ 2 ) )  =  1 )
9493oveq1d 6285 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( ( cos `  x
) ^ 2 )  +  ( ( sin `  x ) ^ 2 ) )  /  (
( cos `  x
) ^ 2 ) )  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
9589, 94eqtr3d 2497 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( ( cos `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) )  +  ( ( ( sin `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) ) )  =  ( 1  /  (
( cos `  x
) ^ 2 ) ) )
968, 14recidd 10311 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x )  x.  (
1  /  ( cos `  x ) ) )  =  1 )
9784, 88dividd 10314 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) )  =  1 )
9896, 97eqtr4d 2498 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x )  x.  (
1  /  ( cos `  x ) ) )  =  ( ( ( cos `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) ) )
997, 7, 84, 88div23d 10353 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x )  x.  ( sin `  x
) )  /  (
( cos `  x
) ^ 2 ) )  =  ( ( ( sin `  x
)  /  ( ( cos `  x ) ^ 2 ) )  x.  ( sin `  x
) ) )
1007sqvald 12292 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x ) ^ 2 )  =  ( ( sin `  x )  x.  ( sin `  x
) ) )
101100oveq1d 6285 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) )  =  ( ( ( sin `  x
)  x.  ( sin `  x ) )  / 
( ( cos `  x
) ^ 2 ) ) )
10284, 88reccld 10309 . . . . . . . . . 10  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( 1  / 
( ( cos `  x
) ^ 2 ) )  e.  CC )
103102, 7mul2negd 10007 . . . . . . . . 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 10320 . . . . . . . . 9  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x )  /  (
( cos `  x
) ^ 2 ) )  =  ( ( 1  /  ( ( cos `  x ) ^ 2 ) )  x.  ( sin `  x
) ) )
105103, 104eqtr4d 2498 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( -u (
1  /  ( ( cos `  x ) ^ 2 ) )  x.  -u ( sin `  x
) )  =  ( ( sin `  x
)  /  ( ( cos `  x ) ^ 2 ) ) )
106105oveq1d 6285 . . . . . . 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 2506 . . . . . 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 6288 . . . . 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 10808 . . . . . 6  |-  2  e.  NN0
110 expneg 12159 . . . . . 6  |-  ( ( ( cos `  x
)  e.  CC  /\  2  e.  NN0 )  -> 
( ( cos `  x
) ^ -u 2
)  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
1118, 109, 110sylancl 660 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ -u 2
)  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
11295, 108, 1113eqtr4d 2505 . . . 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 2814 . . 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 4518 . . 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 670 . 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 2487 1  |-  ( CC 
_D  tan )  =  ( x  e.  dom  tan  |->  ( ( cos `  x
) ^ -u 2
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398   T. wtru 1399    e. wcel 1823    =/= wne 2649   A.wral 2804   _Vcvv 3106    \ cdif 3458    C_ wss 3461   {csn 4016   {cpr 4018    |-> cmpt 4497   `'ccnv 4987   dom cdm 4988   "cima 4991    Fn wfn 5565   -->wf 5566   ` cfv 5570  (class class class)co 6270   CCcc 9479   RRcr 9480   0cc0 9481   1c1 9482    + caddc 9484    x. cmul 9486   -ucneg 9797    / cdiv 10202   2c2 10581   NN0cn0 10791   ^cexp 12151   sincsin 13884   cosccos 13885   tanctan 13886   ↾t crest 14913   TopOpenctopn 14914  ℂfldccnfld 18618  TopOnctopon 19565    _D cdv 22436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-inf2 8049  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558  ax-pre-sup 9559  ax-addf 9560  ax-mulf 9561
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-fal 1404  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-iin 4318  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-se 4828  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-isom 5579  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-of 6513  df-om 6674  df-1st 6773  df-2nd 6774  df-supp 6892  df-recs 7034  df-rdg 7068  df-1o 7122  df-2o 7123  df-oadd 7126  df-er 7303  df-map 7414  df-pm 7415  df-ixp 7463  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-fsupp 7822  df-fi 7863  df-sup 7893  df-oi 7927  df-card 8311  df-cda 8539  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-div 10203  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-n0 10792  df-z 10861  df-dec 10977  df-uz 11083  df-q 11184  df-rp 11222  df-xneg 11321  df-xadd 11322  df-xmul 11323  df-ioo 11536  df-ioc 11537  df-ico 11538  df-icc 11539  df-fz 11676  df-fzo 11800  df-fl 11910  df-mod 11979  df-seq 12093  df-exp 12152  df-fac 12339  df-bc 12366  df-hash 12391  df-shft 12985  df-cj 13017  df-re 13018  df-im 13019  df-sqrt 13153  df-abs 13154  df-limsup 13379  df-clim 13396  df-rlim 13397  df-sum 13594  df-ef 13888  df-sin 13890  df-cos 13891  df-tan 13892  df-pi 13893  df-struct 14721  df-ndx 14722  df-slot 14723  df-base 14724  df-sets 14725  df-ress 14726  df-plusg 14800  df-mulr 14801  df-starv 14802  df-sca 14803  df-vsca 14804  df-ip 14805  df-tset 14806  df-ple 14807  df-ds 14809  df-unif 14810  df-hom 14811  df-cco 14812  df-rest 14915  df-topn 14916  df-0g 14934  df-gsum 14935  df-topgen 14936  df-pt 14937  df-prds 14940  df-xrs 14994  df-qtop 14999  df-imas 15000  df-xps 15002  df-mre 15078  df-mrc 15079  df-acs 15081  df-mgm 16074  df-sgrp 16113  df-mnd 16123  df-submnd 16169  df-mulg 16262  df-cntz 16557  df-cmn 17002  df-psmet 18609  df-xmet 18610  df-met 18611  df-bl 18612  df-mopn 18613  df-fbas 18614  df-fg 18615  df-cnfld 18619  df-top 19569  df-bases 19571  df-topon 19572  df-topsp 19573  df-cld 19690  df-ntr 19691  df-cls 19692  df-nei 19769  df-lp 19807  df-perf 19808  df-cn 19898  df-cnp 19899  df-t1 19985  df-haus 19986  df-tx 20232  df-hmeo 20425  df-fil 20516  df-fm 20608  df-flim 20609  df-flf 20610  df-xms 20992  df-ms 20993  df-tms 20994  df-cncf 21551  df-limc 22439  df-dv 22440
This theorem is referenced by: (None)
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