Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  dvtan Structured version   Visualization version   Unicode version

Theorem dvtan 32056
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 14202 . . . 4  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  /  ( cos `  x
) ) )
2 cnvimass 5194 . . . . . . . . 9  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  dom  cos
3 cosf 14256 . . . . . . . . . 10  |-  cos : CC
--> CC
43fdmi 5746 . . . . . . . . 9  |-  dom  cos  =  CC
52, 4sseqtri 3450 . . . . . . . 8  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  CC
65sseli 3414 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  x  e.  CC )
76sincld 14261 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( sin `  x
)  e.  CC )
86coscld 14262 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  e.  CC )
9 ffn 5739 . . . . . . . 8  |-  ( cos
: CC --> CC  ->  cos 
Fn  CC )
10 elpreima 6017 . . . . . . . 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 4089 . . . . . . . 8  |-  ( ( cos `  x )  e.  ( CC  \  { 0 } )  ->  ( cos `  x
)  =/=  0 )
1312adantl 473 . . . . . . 7  |-  ( ( x  e.  CC  /\  ( cos `  x )  e.  ( CC  \  { 0 } ) )  ->  ( cos `  x )  =/=  0
)
1411, 13sylbi 200 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  =/=  0 )
157, 8, 14divrecd 10408 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x )  /  ( cos `  x ) )  =  ( ( sin `  x )  x.  (
1  /  ( cos `  x ) ) ) )
1615mpteq2ia 4478 . . . 4  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  /  ( cos `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( ( sin `  x )  x.  ( 1  / 
( cos `  x
) ) ) )
171, 16eqtri 2493 . . 3  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  x.  ( 1  / 
( cos `  x
) ) ) )
1817oveq2i 6319 . 2  |-  ( CC 
_D  tan )  =  ( CC  _D  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  x.  ( 1  /  ( cos `  x
) ) ) ) )
19 cnelprrecn 9650 . . . . 5  |-  CC  e.  { RR ,  CC }
2019a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
21 difss 3549 . . . . . . . . 9  |-  ( CC 
\  { 0 } )  C_  CC
22 imass2 5210 . . . . . . . . 9  |-  ( ( CC  \  { 0 } )  C_  CC  ->  ( `' cos " ( CC  \  { 0 } ) )  C_  ( `' cos " CC ) )
2321, 22ax-mp 5 . . . . . . . 8  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  ( `' cos " CC )
24 fimacnv 6027 . . . . . . . . 9  |-  ( cos
: CC --> CC  ->  ( `' cos " CC )  =  CC )
253, 24ax-mp 5 . . . . . . . 8  |-  ( `' cos " CC )  =  CC
2623, 25sseqtri 3450 . . . . . . 7  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  CC
2726sseli 3414 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  x  e.  CC )
2827sincld 14261 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( sin `  x
)  e.  CC )
2928adantl 473 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( sin `  x
)  e.  CC )
308adantl 473 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( cos `  x
)  e.  CC )
31 sincl 14257 . . . . . 6  |-  ( x  e.  CC  ->  ( sin `  x )  e.  CC )
3231adantl 473 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  ( sin `  x )  e.  CC )
33 coscl 14258 . . . . . 6  |-  ( x  e.  CC  ->  ( cos `  x )  e.  CC )
3433adantl 473 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  ( cos `  x )  e.  CC )
35 dvsin 23013 . . . . . 6  |-  ( CC 
_D  sin )  =  cos
36 sinf 14255 . . . . . . . . 9  |-  sin : CC
--> CC
3736a1i 11 . . . . . . . 8  |-  ( T. 
->  sin : CC --> CC )
3837feqmptd 5932 . . . . . . 7  |-  ( T. 
->  sin  =  ( x  e.  CC  |->  ( sin `  x ) ) )
3938oveq2d 6324 . . . . . 6  |-  ( T. 
->  ( CC  _D  sin )  =  ( CC  _D  ( x  e.  CC  |->  ( sin `  x ) ) ) )
403a1i 11 . . . . . . 7  |-  ( T. 
->  cos : CC --> CC )
4140feqmptd 5932 . . . . . 6  |-  ( T. 
->  cos  =  ( x  e.  CC  |->  ( cos `  x ) ) )
4235, 39, 413eqtr3a 2529 . . . . 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 2471 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4544cnfldtopon 21881 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4645toponunii 20024 . . . . . . . 8  |-  CC  =  U. ( TopOpen ` fld )
4746restid 15410 . . . . . . 7  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
4845, 47ax-mp 5 . . . . . 6  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4948eqcomi 2480 . . . . 5  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
50 dvtanlem 32054 . . . . . 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 22996 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( sin `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( cos `  x ) ) )
538, 14reccld 10398 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( 1  / 
( cos `  x
) )  e.  CC )
5453adantl 473 . . . 4  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( 1  /  ( cos `  x ) )  e.  CC )
55 ovex 6336 . . . . 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 471 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  e.  ( CC 
\  { 0 } ) )
5857adantl 473 . . . . 5  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  -> 
( cos `  x
)  e.  ( CC 
\  { 0 } ) )
5929negcld 9992 . . . . 5  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  ->  -u ( sin `  x
)  e.  CC )
60 eldifi 3544 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  e.  CC )
61 eldifsni 4089 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  =/=  0
)
6260, 61reccld 10398 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  ->  ( 1  / 
y )  e.  CC )
6362adantl 473 . . . . 5  |-  ( ( T.  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( 1  / 
y )  e.  CC )
64 negex 9893 . . . . . 6  |-  -u (
1  /  ( y ^ 2 ) )  e.  _V
6564a1i 11 . . . . 5  |-  ( ( T.  /\  y  e.  ( CC  \  {
0 } ) )  ->  -u ( 1  / 
( y ^ 2 ) )  e.  _V )
6632negcld 9992 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  e.  CC )
67 dvcos 23014 . . . . . . 7  |-  ( CC 
_D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) )
6841oveq2d 6324 . . . . . . 7  |-  ( T. 
->  ( CC  _D  cos )  =  ( CC  _D  ( x  e.  CC  |->  ( cos `  x ) ) ) )
6967, 68syl5reqr 2520 . . . . . 6  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( cos `  x ) ) )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )
7020, 34, 66, 69, 43, 49, 44, 51dvmptres 22996 . . . . 5  |-  ( T. 
->  ( CC  _D  (
x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( cos `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  -u ( sin `  x ) ) )
71 ax-1cn 9615 . . . . . 6  |-  1  e.  CC
72 dvrec 22988 . . . . . 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 6316 . . . . 5  |-  ( y  =  ( cos `  x
)  ->  ( 1  /  y )  =  ( 1  /  ( cos `  x ) ) )
75 oveq1 6315 . . . . . . 7  |-  ( y  =  ( cos `  x
)  ->  ( y ^ 2 )  =  ( ( cos `  x
) ^ 2 ) )
7675oveq2d 6324 . . . . . 6  |-  ( y  =  ( cos `  x
)  ->  ( 1  /  ( y ^
2 ) )  =  ( 1  /  (
( cos `  x
) ^ 2 ) ) )
7776negeqd 9889 . . . . 5  |-  ( y  =  ( cos `  x
)  ->  -u ( 1  /  ( y ^
2 ) )  = 
-u ( 1  / 
( ( cos `  x
) ^ 2 ) ) )
7820, 20, 58, 59, 63, 65, 70, 73, 74, 77dvmptco 23005 . . . 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 22994 . . 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 1461 . 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 6336 . . . . 5  |-  ( ( sin `  x )  /  ( cos `  x
) )  e.  _V
8281, 1dmmpti 5717 . . . 4  |-  dom  tan  =  ( `' cos " ( CC  \  {
0 } ) )
8382eqcomi 2480 . . 3  |-  ( `' cos " ( CC 
\  { 0 } ) )  =  dom  tan
848sqcld 12452 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ 2 )  e.  CC )
857sqcld 12452 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x ) ^ 2 )  e.  CC )
86 sqne0 12379 . . . . . . . . 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 240 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ 2 )  =/=  0 )
8984, 85, 84, 88divdird 10443 . . . . . 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 9853 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  +  ( ( sin `  x
) ^ 2 ) )  =  ( ( ( sin `  x
) ^ 2 )  +  ( ( cos `  x ) ^ 2 ) ) )
91 sincossq 14307 . . . . . . . . 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 2505 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  +  ( ( sin `  x
) ^ 2 ) )  =  1 )
9493oveq1d 6323 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( ( cos `  x
) ^ 2 )  +  ( ( sin `  x ) ^ 2 ) )  /  (
( cos `  x
) ^ 2 ) )  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
9589, 94eqtr3d 2507 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( ( cos `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) )  +  ( ( ( sin `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) ) )  =  ( 1  /  (
( cos `  x
) ^ 2 ) ) )
968, 14recidd 10400 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x )  x.  (
1  /  ( cos `  x ) ) )  =  1 )
9784, 88dividd 10403 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) )  =  1 )
9896, 97eqtr4d 2508 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x )  x.  (
1  /  ( cos `  x ) ) )  =  ( ( ( cos `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) ) )
997, 7, 84, 88div23d 10442 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x )  x.  ( sin `  x
) )  /  (
( cos `  x
) ^ 2 ) )  =  ( ( ( sin `  x
)  /  ( ( cos `  x ) ^ 2 ) )  x.  ( sin `  x
) ) )
1007sqvald 12451 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x ) ^ 2 )  =  ( ( sin `  x )  x.  ( sin `  x
) ) )
101100oveq1d 6323 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) )  =  ( ( ( sin `  x
)  x.  ( sin `  x ) )  / 
( ( cos `  x
) ^ 2 ) ) )
10284, 88reccld 10398 . . . . . . . . . 10  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( 1  / 
( ( cos `  x
) ^ 2 ) )  e.  CC )
103102, 7mul2negd 10094 . . . . . . . . 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 10409 . . . . . . . . 9  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x )  /  (
( cos `  x
) ^ 2 ) )  =  ( ( 1  /  ( ( cos `  x ) ^ 2 ) )  x.  ( sin `  x
) ) )
105103, 104eqtr4d 2508 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( -u (
1  /  ( ( cos `  x ) ^ 2 ) )  x.  -u ( sin `  x
) )  =  ( ( sin `  x
)  /  ( ( cos `  x ) ^ 2 ) ) )
106105oveq1d 6323 . . . . . . 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 2516 . . . . . 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 6326 . . . . 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 10910 . . . . . 6  |-  2  e.  NN0
110 expneg 12318 . . . . . 6  |-  ( ( ( cos `  x
)  e.  CC  /\  2  e.  NN0 )  -> 
( ( cos `  x
) ^ -u 2
)  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
1118, 109, 110sylancl 675 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ -u 2
)  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
11295, 108, 1113eqtr4d 2515 . . . 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 2766 . . 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 4475 . . 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 686 . 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 2497 1  |-  ( CC 
_D  tan )  =  ( x  e.  dom  tan  |->  ( ( cos `  x
) ^ -u 2
) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    /\ wa 376    = wceq 1452   T. wtru 1453    e. wcel 1904    =/= wne 2641   A.wral 2756   _Vcvv 3031    \ cdif 3387    C_ wss 3390   {csn 3959   {cpr 3961    |-> cmpt 4454   `'ccnv 4838   dom cdm 4839   "cima 4842    Fn wfn 5584   -->wf 5585   ` cfv 5589  (class class class)co 6308   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    x. cmul 9562   -ucneg 9881    / cdiv 10291   2c2 10681   NN0cn0 10893   ^cexp 12310   sincsin 14193   cosccos 14194   tanctan 14195   ↾t crest 15397   TopOpenctopn 15398  ℂfldccnfld 19047  TopOnctopon 19995    _D cdv 22897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-inf2 8164  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635  ax-addf 9636  ax-mulf 9637
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-supp 6934  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-pm 7493  df-ixp 7541  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-fsupp 7902  df-fi 7943  df-sup 7974  df-inf 7975  df-oi 8043  df-card 8391  df-cda 8616  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-div 10292  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-n0 10894  df-z 10962  df-dec 11075  df-uz 11183  df-q 11288  df-rp 11326  df-xneg 11432  df-xadd 11433  df-xmul 11434  df-ico 11666  df-icc 11667  df-fz 11811  df-fzo 11943  df-fl 12061  df-seq 12252  df-exp 12311  df-fac 12498  df-bc 12526  df-hash 12554  df-shft 13207  df-cj 13239  df-re 13240  df-im 13241  df-sqrt 13375  df-abs 13376  df-limsup 13603  df-clim 13629  df-rlim 13630  df-sum 13830  df-ef 14198  df-sin 14200  df-cos 14201  df-tan 14202  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-ress 15206  df-plusg 15281  df-mulr 15282  df-starv 15283  df-sca 15284  df-vsca 15285  df-ip 15286  df-tset 15287  df-ple 15288  df-ds 15290  df-unif 15291  df-hom 15292  df-cco 15293  df-rest 15399  df-topn 15400  df-0g 15418  df-gsum 15419  df-topgen 15420  df-pt 15421  df-prds 15424  df-xrs 15478  df-qtop 15484  df-imas 15485  df-xps 15488  df-mre 15570  df-mrc 15571  df-acs 15573  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-submnd 16661  df-mulg 16754  df-cntz 17049  df-cmn 17510  df-psmet 19039  df-xmet 19040  df-met 19041  df-bl 19042  df-mopn 19043  df-fbas 19044  df-fg 19045  df-cnfld 19048  df-top 19998  df-bases 19999  df-topon 20000  df-topsp 20001  df-cld 20111  df-ntr 20112  df-cls 20113  df-nei 20191  df-lp 20229  df-perf 20230  df-cn 20320  df-cnp 20321  df-t1 20407  df-haus 20408  df-tx 20654  df-hmeo 20847  df-fil 20939  df-fm 21031  df-flim 21032  df-flf 21033  df-xms 21413  df-ms 21414  df-tms 21415  df-cncf 21988  df-limc 22900  df-dv 22901
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator