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Theorem dvtan 31893
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 14061 . . . 4  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  /  ( cos `  x
) ) )
2 cnvimass 5143 . . . . . . . . 9  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  dom  cos
3 cosf 14115 . . . . . . . . . 10  |-  cos : CC
--> CC
43fdmi 5687 . . . . . . . . 9  |-  dom  cos  =  CC
52, 4sseqtri 3432 . . . . . . . 8  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  CC
65sseli 3396 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  x  e.  CC )
76sincld 14120 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( sin `  x
)  e.  CC )
86coscld 14121 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  e.  CC )
9 ffn 5682 . . . . . . . 8  |-  ( cos
: CC --> CC  ->  cos 
Fn  CC )
10 elpreima 5954 . . . . . . . 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 4062 . . . . . . . 8  |-  ( ( cos `  x )  e.  ( CC  \  { 0 } )  ->  ( cos `  x
)  =/=  0 )
1312adantl 467 . . . . . . 7  |-  ( ( x  e.  CC  /\  ( cos `  x )  e.  ( CC  \  { 0 } ) )  ->  ( cos `  x )  =/=  0
)
1411, 13sylbi 198 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  =/=  0 )
157, 8, 14divrecd 10330 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x )  /  ( cos `  x ) )  =  ( ( sin `  x )  x.  (
1  /  ( cos `  x ) ) ) )
1615mpteq2ia 4442 . . . 4  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  /  ( cos `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( ( sin `  x )  x.  ( 1  / 
( cos `  x
) ) ) )
171, 16eqtri 2444 . . 3  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  x.  ( 1  / 
( cos `  x
) ) ) )
1817oveq2i 6253 . 2  |-  ( CC 
_D  tan )  =  ( CC  _D  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  x.  ( 1  /  ( cos `  x
) ) ) ) )
19 cnelprrecn 9576 . . . . 5  |-  CC  e.  { RR ,  CC }
2019a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
21 difss 3528 . . . . . . . . 9  |-  ( CC 
\  { 0 } )  C_  CC
22 imass2 5159 . . . . . . . . 9  |-  ( ( CC  \  { 0 } )  C_  CC  ->  ( `' cos " ( CC  \  { 0 } ) )  C_  ( `' cos " CC ) )
2321, 22ax-mp 5 . . . . . . . 8  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  ( `' cos " CC )
24 fimacnv 5964 . . . . . . . . 9  |-  ( cos
: CC --> CC  ->  ( `' cos " CC )  =  CC )
253, 24ax-mp 5 . . . . . . . 8  |-  ( `' cos " CC )  =  CC
2623, 25sseqtri 3432 . . . . . . 7  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  CC
2726sseli 3396 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  x  e.  CC )
2827sincld 14120 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( sin `  x
)  e.  CC )
2928adantl 467 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( sin `  x
)  e.  CC )
308adantl 467 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( cos `  x
)  e.  CC )
31 sincl 14116 . . . . . 6  |-  ( x  e.  CC  ->  ( sin `  x )  e.  CC )
3231adantl 467 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  ( sin `  x )  e.  CC )
33 coscl 14117 . . . . . 6  |-  ( x  e.  CC  ->  ( cos `  x )  e.  CC )
3433adantl 467 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  ( cos `  x )  e.  CC )
35 dvsin 22869 . . . . . 6  |-  ( CC 
_D  sin )  =  cos
36 sinf 14114 . . . . . . . . 9  |-  sin : CC
--> CC
3736a1i 11 . . . . . . . 8  |-  ( T. 
->  sin : CC --> CC )
3837feqmptd 5871 . . . . . . 7  |-  ( T. 
->  sin  =  ( x  e.  CC  |->  ( sin `  x ) ) )
3938oveq2d 6258 . . . . . 6  |-  ( T. 
->  ( CC  _D  sin )  =  ( CC  _D  ( x  e.  CC  |->  ( sin `  x ) ) ) )
403a1i 11 . . . . . . 7  |-  ( T. 
->  cos : CC --> CC )
4140feqmptd 5871 . . . . . 6  |-  ( T. 
->  cos  =  ( x  e.  CC  |->  ( cos `  x ) ) )
4235, 39, 413eqtr3a 2480 . . . . 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 2422 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4544cnfldtopon 21738 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4645toponunii 19882 . . . . . . . 8  |-  CC  =  U. ( TopOpen ` fld )
4746restid 15268 . . . . . . 7  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
4845, 47ax-mp 5 . . . . . 6  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4948eqcomi 2431 . . . . 5  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
50 dvtanlem 31891 . . . . . 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 22852 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( sin `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( cos `  x ) ) )
538, 14reccld 10320 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( 1  / 
( cos `  x
) )  e.  CC )
5453adantl 467 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( 1  /  ( cos `  x ) )  e.  CC )
55 ovex 6270 . . . . 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 465 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  e.  ( CC 
\  { 0 } ) )
5857adantl 467 . . . . 5  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( cos `  x
)  e.  ( CC 
\  { 0 } ) )
5929negcld 9917 . . . . 5  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  ->  -u ( sin `  x
)  e.  CC )
60 eldifi 3523 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  e.  CC )
61 eldifsni 4062 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  =/=  0
)
6260, 61reccld 10320 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  ->  ( 1  / 
y )  e.  CC )
6362adantl 467 . . . . 5  |-  ( ( T.  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( 1  / 
y )  e.  CC )
64 negex 9817 . . . . . 6  |-  -u (
1  /  ( y ^ 2 ) )  e.  _V
6564a1i 11 . . . . 5  |-  ( ( T.  /\  y  e.  ( CC  \  {
0 } ) )  ->  -u ( 1  / 
( y ^ 2 ) )  e.  _V )
6632negcld 9917 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  e.  CC )
67 dvcos 22870 . . . . . . 7  |-  ( CC 
_D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) )
6841oveq2d 6258 . . . . . . 7  |-  ( T. 
->  ( CC  _D  cos )  =  ( CC  _D  ( x  e.  CC  |->  ( cos `  x ) ) ) )
6967, 68syl5reqr 2471 . . . . . 6  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( cos `  x ) ) )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )
7020, 34, 66, 69, 43, 49, 44, 51dvmptres 22852 . . . . 5  |-  ( T. 
->  ( CC  _D  (
x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( cos `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  -u ( sin `  x ) ) )
71 ax-1cn 9541 . . . . . 6  |-  1  e.  CC
72 dvrec 22844 . . . . . 6  |-  ( 1  e.  CC  ->  ( CC  _D  ( y  e.  ( CC  \  {
0 } )  |->  ( 1  /  y ) ) )  =  ( y  e.  ( CC 
\  { 0 } )  |->  -u ( 1  / 
( y ^ 2 ) ) ) )
7371, 72mp1i 13 . . . . 5  |-  ( T. 
->  ( CC  _D  (
y  e.  ( CC 
\  { 0 } )  |->  ( 1  / 
y ) ) )  =  ( y  e.  ( CC  \  {
0 } )  |->  -u ( 1  /  (
y ^ 2 ) ) ) )
74 oveq2 6250 . . . . 5  |-  ( y  =  ( cos `  x
)  ->  ( 1  /  y )  =  ( 1  /  ( cos `  x ) ) )
75 oveq1 6249 . . . . . . 7  |-  ( y  =  ( cos `  x
)  ->  ( y ^ 2 )  =  ( ( cos `  x
) ^ 2 ) )
7675oveq2d 6258 . . . . . 6  |-  ( y  =  ( cos `  x
)  ->  ( 1  /  ( y ^
2 ) )  =  ( 1  /  (
( cos `  x
) ^ 2 ) ) )
7776negeqd 9813 . . . . 5  |-  ( y  =  ( cos `  x
)  ->  -u ( 1  /  ( y ^
2 ) )  = 
-u ( 1  / 
( ( cos `  x
) ^ 2 ) ) )
7820, 20, 58, 59, 63, 65, 70, 73, 74, 77dvmptco 22861 . . . 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 22850 . . 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 1446 . 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 6270 . . . . 5  |-  ( ( sin `  x )  /  ( cos `  x
) )  e.  _V
8281, 1dmmpti 5661 . . . 4  |-  dom  tan  =  ( `' cos " ( CC  \  {
0 } ) )
8382eqcomi 2431 . . 3  |-  ( `' cos " ( CC 
\  { 0 } ) )  =  dom  tan
848sqcld 12357 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ 2 )  e.  CC )
857sqcld 12357 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x ) ^ 2 )  e.  CC )
86 sqne0 12284 . . . . . . . . 9  |-  ( ( cos `  x )  e.  CC  ->  (
( ( cos `  x
) ^ 2 )  =/=  0  <->  ( cos `  x )  =/=  0
) )
878, 86syl 17 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  =/=  0  <->  ( cos `  x
)  =/=  0 ) )
8814, 87mpbird 235 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ 2 )  =/=  0 )
8984, 85, 84, 88divdird 10365 . . . . . 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 9779 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  +  ( ( sin `  x
) ^ 2 ) )  =  ( ( ( sin `  x
) ^ 2 )  +  ( ( cos `  x ) ^ 2 ) ) )
91 sincossq 14166 . . . . . . . . 9  |-  ( x  e.  CC  ->  (
( ( sin `  x
) ^ 2 )  +  ( ( cos `  x ) ^ 2 ) )  =  1 )
926, 91syl 17 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x ) ^ 2 )  +  ( ( cos `  x
) ^ 2 ) )  =  1 )
9390, 92eqtrd 2456 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  +  ( ( sin `  x
) ^ 2 ) )  =  1 )
9493oveq1d 6257 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( ( cos `  x
) ^ 2 )  +  ( ( sin `  x ) ^ 2 ) )  /  (
( cos `  x
) ^ 2 ) )  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
9589, 94eqtr3d 2458 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( ( cos `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) )  +  ( ( ( sin `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) ) )  =  ( 1  /  (
( cos `  x
) ^ 2 ) ) )
968, 14recidd 10322 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x )  x.  (
1  /  ( cos `  x ) ) )  =  1 )
9784, 88dividd 10325 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) )  =  1 )
9896, 97eqtr4d 2459 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x )  x.  (
1  /  ( cos `  x ) ) )  =  ( ( ( cos `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) ) )
997, 7, 84, 88div23d 10364 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x )  x.  ( sin `  x
) )  /  (
( cos `  x
) ^ 2 ) )  =  ( ( ( sin `  x
)  /  ( ( cos `  x ) ^ 2 ) )  x.  ( sin `  x
) ) )
1007sqvald 12356 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x ) ^ 2 )  =  ( ( sin `  x )  x.  ( sin `  x
) ) )
101100oveq1d 6257 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) )  =  ( ( ( sin `  x
)  x.  ( sin `  x ) )  / 
( ( cos `  x
) ^ 2 ) ) )
10284, 88reccld 10320 . . . . . . . . . 10  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( 1  / 
( ( cos `  x
) ^ 2 ) )  e.  CC )
103102, 7mul2negd 10017 . . . . . . . . 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 10331 . . . . . . . . 9  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x )  /  (
( cos `  x
) ^ 2 ) )  =  ( ( 1  /  ( ( cos `  x ) ^ 2 ) )  x.  ( sin `  x
) ) )
105103, 104eqtr4d 2459 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( -u (
1  /  ( ( cos `  x ) ^ 2 ) )  x.  -u ( sin `  x
) )  =  ( ( sin `  x
)  /  ( ( cos `  x ) ^ 2 ) ) )
106105oveq1d 6257 . . . . . . 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 2467 . . . . . 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 6260 . . . . 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 10830 . . . . . 6  |-  2  e.  NN0
110 expneg 12223 . . . . . 6  |-  ( ( ( cos `  x
)  e.  CC  /\  2  e.  NN0 )  -> 
( ( cos `  x
) ^ -u 2
)  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
1118, 109, 110sylancl 666 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ -u 2
)  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
11295, 108, 1113eqtr4d 2466 . . . 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 2718 . . 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 4439 . . 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 676 . 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 2448 1  |-  ( CC 
_D  tan )  =  ( x  e.  dom  tan  |->  ( ( cos `  x
) ^ -u 2
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 187    /\ wa 370    = wceq 1437   T. wtru 1438    e. wcel 1872    =/= wne 2593   A.wral 2708   _Vcvv 3016    \ cdif 3369    C_ wss 3372   {csn 3934   {cpr 3936    |-> cmpt 4418   `'ccnv 4788   dom cdm 4789   "cima 4792    Fn wfn 5532   -->wf 5533   ` cfv 5537  (class class class)co 6242   CCcc 9481   RRcr 9482   0cc0 9483   1c1 9484    + caddc 9486    x. cmul 9488   -ucneg 9805    / cdiv 10213   2c2 10603   NN0cn0 10813   ^cexp 12215   sincsin 14052   cosccos 14053   tanctan 14054   ↾t crest 15255   TopOpenctopn 15256  ℂfldccnfld 18906  TopOnctopon 19853    _D cdv 22753
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2058  ax-ext 2402  ax-rep 4472  ax-sep 4482  ax-nul 4491  ax-pow 4538  ax-pr 4596  ax-un 6534  ax-inf2 8092  ax-cnex 9539  ax-resscn 9540  ax-1cn 9541  ax-icn 9542  ax-addcl 9543  ax-addrcl 9544  ax-mulcl 9545  ax-mulrcl 9546  ax-mulcom 9547  ax-addass 9548  ax-mulass 9549  ax-distr 9550  ax-i2m1 9551  ax-1ne0 9552  ax-1rid 9553  ax-rnegex 9554  ax-rrecex 9555  ax-cnre 9556  ax-pre-lttri 9557  ax-pre-lttrn 9558  ax-pre-ltadd 9559  ax-pre-mulgt0 9560  ax-pre-sup 9561  ax-addf 9562  ax-mulf 9563
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2274  df-mo 2275  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2552  df-ne 2595  df-nel 2596  df-ral 2713  df-rex 2714  df-reu 2715  df-rmo 2716  df-rab 2717  df-v 3018  df-sbc 3236  df-csb 3332  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-pss 3388  df-nul 3698  df-if 3848  df-pw 3919  df-sn 3935  df-pr 3937  df-tp 3939  df-op 3941  df-uni 4156  df-int 4192  df-iun 4237  df-iin 4238  df-br 4360  df-opab 4419  df-mpt 4420  df-tr 4455  df-eprel 4700  df-id 4704  df-po 4710  df-so 4711  df-fr 4748  df-se 4749  df-we 4750  df-xp 4795  df-rel 4796  df-cnv 4797  df-co 4798  df-dm 4799  df-rn 4800  df-res 4801  df-ima 4802  df-pred 5335  df-ord 5381  df-on 5382  df-lim 5383  df-suc 5384  df-iota 5501  df-fun 5539  df-fn 5540  df-f 5541  df-f1 5542  df-fo 5543  df-f1o 5544  df-fv 5545  df-isom 5546  df-riota 6204  df-ov 6245  df-oprab 6246  df-mpt2 6247  df-of 6482  df-om 6644  df-1st 6744  df-2nd 6745  df-supp 6863  df-wrecs 6976  df-recs 7038  df-rdg 7076  df-1o 7130  df-2o 7131  df-oadd 7134  df-er 7311  df-map 7422  df-pm 7423  df-ixp 7471  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-fsupp 7830  df-fi 7871  df-sup 7902  df-inf 7903  df-oi 7971  df-card 8318  df-cda 8542  df-pnf 9621  df-mnf 9622  df-xr 9623  df-ltxr 9624  df-le 9625  df-sub 9806  df-neg 9807  df-div 10214  df-nn 10554  df-2 10612  df-3 10613  df-4 10614  df-5 10615  df-6 10616  df-7 10617  df-8 10618  df-9 10619  df-10 10620  df-n0 10814  df-z 10882  df-dec 10996  df-uz 11104  df-q 11209  df-rp 11247  df-xneg 11353  df-xadd 11354  df-xmul 11355  df-ico 11585  df-icc 11586  df-fz 11729  df-fzo 11860  df-fl 11971  df-seq 12157  df-exp 12216  df-fac 12403  df-bc 12431  df-hash 12459  df-shft 13067  df-cj 13099  df-re 13100  df-im 13101  df-sqrt 13235  df-abs 13236  df-limsup 13462  df-clim 13488  df-rlim 13489  df-sum 13689  df-ef 14057  df-sin 14059  df-cos 14060  df-tan 14061  df-struct 15059  df-ndx 15060  df-slot 15061  df-base 15062  df-sets 15063  df-ress 15064  df-plusg 15139  df-mulr 15140  df-starv 15141  df-sca 15142  df-vsca 15143  df-ip 15144  df-tset 15145  df-ple 15146  df-ds 15148  df-unif 15149  df-hom 15150  df-cco 15151  df-rest 15257  df-topn 15258  df-0g 15276  df-gsum 15277  df-topgen 15278  df-pt 15279  df-prds 15282  df-xrs 15336  df-qtop 15342  df-imas 15343  df-xps 15346  df-mre 15428  df-mrc 15429  df-acs 15431  df-mgm 16424  df-sgrp 16463  df-mnd 16473  df-submnd 16519  df-mulg 16612  df-cntz 16907  df-cmn 17368  df-psmet 18898  df-xmet 18899  df-met 18900  df-bl 18901  df-mopn 18902  df-fbas 18903  df-fg 18904  df-cnfld 18907  df-top 19856  df-bases 19857  df-topon 19858  df-topsp 19859  df-cld 19969  df-ntr 19970  df-cls 19971  df-nei 20049  df-lp 20087  df-perf 20088  df-cn 20178  df-cnp 20179  df-t1 20265  df-haus 20266  df-tx 20512  df-hmeo 20705  df-fil 20796  df-fm 20888  df-flim 20889  df-flf 20890  df-xms 21270  df-ms 21271  df-tms 21272  df-cncf 21845  df-limc 22756  df-dv 22757
This theorem is referenced by: (None)
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