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

Theorem dvtan 31696
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 14103 . . . 4  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  /  ( cos `  x
) ) )
2 cnvimass 5208 . . . . . . . . 9  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  dom  cos
3 cosf 14157 . . . . . . . . . 10  |-  cos : CC
--> CC
43fdmi 5751 . . . . . . . . 9  |-  dom  cos  =  CC
52, 4sseqtri 3502 . . . . . . . 8  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  CC
65sseli 3466 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  x  e.  CC )
76sincld 14162 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( sin `  x
)  e.  CC )
86coscld 14163 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( cos `  x
)  e.  CC )
9 ffn 5746 . . . . . . . 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 4129 . . . . . . . 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 10385 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x )  /  ( cos `  x ) )  =  ( ( sin `  x )  x.  (
1  /  ( cos `  x ) ) ) )
1615mpteq2ia 4508 . . . 4  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  /  ( cos `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( ( sin `  x )  x.  ( 1  / 
( cos `  x
) ) ) )
171, 16eqtri 2458 . . 3  |-  tan  =  ( x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( ( sin `  x )  x.  ( 1  / 
( cos `  x
) ) ) )
1817oveq2i 6316 . 2  |-  ( CC 
_D  tan )  =  ( CC  _D  ( x  e.  ( `' cos " ( CC  \  {
0 } ) ) 
|->  ( ( sin `  x
)  x.  ( 1  /  ( cos `  x
) ) ) ) )
19 cnelprrecn 9631 . . . . 5  |-  CC  e.  { RR ,  CC }
2019a1i 11 . . . 4  |-  ( T. 
->  CC  e.  { RR ,  CC } )
21 difss 3598 . . . . . . . . 9  |-  ( CC 
\  { 0 } )  C_  CC
22 imass2 5224 . . . . . . . . 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 3502 . . . . . . 7  |-  ( `' cos " ( CC 
\  { 0 } ) )  C_  CC
2726sseli 3466 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  x  e.  CC )
2827sincld 14162 . . . . 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 14158 . . . . . 6  |-  ( x  e.  CC  ->  ( sin `  x )  e.  CC )
3231adantl 467 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  ( sin `  x )  e.  CC )
33 coscl 14159 . . . . . 6  |-  ( x  e.  CC  ->  ( cos `  x )  e.  CC )
3433adantl 467 . . . . 5  |-  ( ( T.  /\  x  e.  CC )  ->  ( cos `  x )  e.  CC )
35 dvsin 22811 . . . . . 6  |-  ( CC 
_D  sin )  =  cos
36 sinf 14156 . . . . . . . . 9  |-  sin : CC
--> CC
3736a1i 11 . . . . . . . 8  |-  ( T. 
->  sin : CC --> CC )
3837feqmptd 5934 . . . . . . 7  |-  ( T. 
->  sin  =  ( x  e.  CC  |->  ( sin `  x ) ) )
3938oveq2d 6321 . . . . . 6  |-  ( T. 
->  ( CC  _D  sin )  =  ( CC  _D  ( x  e.  CC  |->  ( sin `  x ) ) ) )
403a1i 11 . . . . . . 7  |-  ( T. 
->  cos : CC --> CC )
4140feqmptd 5934 . . . . . 6  |-  ( T. 
->  cos  =  ( x  e.  CC  |->  ( cos `  x ) ) )
4235, 39, 413eqtr3a 2494 . . . . 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 2429 . . . . . . . 8  |-  ( TopOpen ` fld )  =  ( TopOpen ` fld )
4544cnfldtopon 21714 . . . . . . 7  |-  ( TopOpen ` fld )  e.  (TopOn `  CC )
4645toponunii 19878 . . . . . . . 8  |-  CC  =  U. ( TopOpen ` fld )
4746restid 15291 . . . . . . 7  |-  ( (
TopOpen ` fld )  e.  (TopOn `  CC )  ->  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld ) )
4845, 47ax-mp 5 . . . . . 6  |-  ( (
TopOpen ` fld )t  CC )  =  (
TopOpen ` fld )
4948eqcomi 2442 . . . . 5  |-  ( TopOpen ` fld )  =  ( ( TopOpen ` fld )t  CC )
50 dvtanlem 31694 . . . . . 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 22794 . . . 4  |-  ( T. 
->  ( CC  _D  (
x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( sin `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  ( cos `  x ) ) )
538, 14reccld 10375 . . . . 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 6333 . . . . 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 9972 . . . . 5  |-  ( ( T.  /\  x  e.  ( `' cos " ( CC  \  { 0 } ) ) )  ->  -u ( sin `  x
)  e.  CC )
60 eldifi 3593 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  e.  CC )
61 eldifsni 4129 . . . . . . 7  |-  ( y  e.  ( CC  \  { 0 } )  ->  y  =/=  0
)
6260, 61reccld 10375 . . . . . 6  |-  ( y  e.  ( CC  \  { 0 } )  ->  ( 1  / 
y )  e.  CC )
6362adantl 467 . . . . 5  |-  ( ( T.  /\  y  e.  ( CC  \  {
0 } ) )  ->  ( 1  / 
y )  e.  CC )
64 negex 9872 . . . . . 6  |-  -u (
1  /  ( y ^ 2 ) )  e.  _V
6564a1i 11 . . . . 5  |-  ( ( T.  /\  y  e.  ( CC  \  {
0 } ) )  ->  -u ( 1  / 
( y ^ 2 ) )  e.  _V )
6632negcld 9972 . . . . . 6  |-  ( ( T.  /\  x  e.  CC )  ->  -u ( sin `  x )  e.  CC )
67 dvcos 22812 . . . . . . 7  |-  ( CC 
_D  cos )  =  ( x  e.  CC  |->  -u ( sin `  x ) )
6841oveq2d 6321 . . . . . . 7  |-  ( T. 
->  ( CC  _D  cos )  =  ( CC  _D  ( x  e.  CC  |->  ( cos `  x ) ) ) )
6967, 68syl5reqr 2485 . . . . . 6  |-  ( T. 
->  ( CC  _D  (
x  e.  CC  |->  ( cos `  x ) ) )  =  ( x  e.  CC  |->  -u ( sin `  x ) ) )
7020, 34, 66, 69, 43, 49, 44, 51dvmptres 22794 . . . . 5  |-  ( T. 
->  ( CC  _D  (
x  e.  ( `' cos " ( CC 
\  { 0 } ) )  |->  ( cos `  x ) ) )  =  ( x  e.  ( `' cos " ( CC  \  { 0 } ) )  |->  -u ( sin `  x ) ) )
71 ax-1cn 9596 . . . . . 6  |-  1  e.  CC
72 dvrec 22786 . . . . . 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 6313 . . . . 5  |-  ( y  =  ( cos `  x
)  ->  ( 1  /  y )  =  ( 1  /  ( cos `  x ) ) )
75 oveq1 6312 . . . . . . 7  |-  ( y  =  ( cos `  x
)  ->  ( y ^ 2 )  =  ( ( cos `  x
) ^ 2 ) )
7675oveq2d 6321 . . . . . 6  |-  ( y  =  ( cos `  x
)  ->  ( 1  /  ( y ^
2 ) )  =  ( 1  /  (
( cos `  x
) ^ 2 ) ) )
7776negeqd 9868 . . . . 5  |-  ( y  =  ( cos `  x
)  ->  -u ( 1  /  ( y ^
2 ) )  = 
-u ( 1  / 
( ( cos `  x
) ^ 2 ) ) )
7820, 20, 58, 59, 63, 65, 70, 73, 74, 77dvmptco 22803 . . . 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 22792 . . 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 6333 . . . . 5  |-  ( ( sin `  x )  /  ( cos `  x
) )  e.  _V
8281, 1dmmpti 5725 . . . 4  |-  dom  tan  =  ( `' cos " ( CC  \  {
0 } ) )
8382eqcomi 2442 . . 3  |-  ( `' cos " ( CC 
\  { 0 } ) )  =  dom  tan
848sqcld 12411 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x ) ^ 2 )  e.  CC )
857sqcld 12411 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x ) ^ 2 )  e.  CC )
86 sqne0 12338 . . . . . . . . 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 10420 . . . . . 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 9834 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  +  ( ( sin `  x
) ^ 2 ) )  =  ( ( ( sin `  x
) ^ 2 )  +  ( ( cos `  x ) ^ 2 ) ) )
91 sincossq 14208 . . . . . . . . 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 2470 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  +  ( ( sin `  x
) ^ 2 ) )  =  1 )
9493oveq1d 6320 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( ( cos `  x
) ^ 2 )  +  ( ( sin `  x ) ^ 2 ) )  /  (
( cos `  x
) ^ 2 ) )  =  ( 1  /  ( ( cos `  x ) ^ 2 ) ) )
9589, 94eqtr3d 2472 . . . . 5  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( ( cos `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) )  +  ( ( ( sin `  x
) ^ 2 )  /  ( ( cos `  x ) ^ 2 ) ) )  =  ( 1  /  (
( cos `  x
) ^ 2 ) ) )
968, 14recidd 10377 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x )  x.  (
1  /  ( cos `  x ) ) )  =  1 )
9784, 88dividd 10380 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( cos `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) )  =  1 )
9896, 97eqtr4d 2473 . . . . . 6  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( cos `  x )  x.  (
1  /  ( cos `  x ) ) )  =  ( ( ( cos `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) ) )
997, 7, 84, 88div23d 10419 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x )  x.  ( sin `  x
) )  /  (
( cos `  x
) ^ 2 ) )  =  ( ( ( sin `  x
)  /  ( ( cos `  x ) ^ 2 ) )  x.  ( sin `  x
) ) )
1007sqvald 12410 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x ) ^ 2 )  =  ( ( sin `  x )  x.  ( sin `  x
) ) )
101100oveq1d 6320 . . . . . . 7  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( ( sin `  x ) ^ 2 )  / 
( ( cos `  x
) ^ 2 ) )  =  ( ( ( sin `  x
)  x.  ( sin `  x ) )  / 
( ( cos `  x
) ^ 2 ) ) )
10284, 88reccld 10375 . . . . . . . . . 10  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( 1  / 
( ( cos `  x
) ^ 2 ) )  e.  CC )
103102, 7mul2negd 10072 . . . . . . . . 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 10386 . . . . . . . . 9  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( ( sin `  x )  /  (
( cos `  x
) ^ 2 ) )  =  ( ( 1  /  ( ( cos `  x ) ^ 2 ) )  x.  ( sin `  x
) ) )
105103, 104eqtr4d 2473 . . . . . . . 8  |-  ( x  e.  ( `' cos " ( CC  \  {
0 } ) )  ->  ( -u (
1  /  ( ( cos `  x ) ^ 2 ) )  x.  -u ( sin `  x
) )  =  ( ( sin `  x
)  /  ( ( cos `  x ) ^ 2 ) ) )
106105oveq1d 6320 . . . . . . 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 2481 . . . . . 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 6323 . . . . 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 10886 . . . . . 6  |-  2  e.  NN0
110 expneg 12277 . . . . . 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 2480 . . . 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 2792 . . 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 4505 . . 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 2462 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 1870    =/= wne 2625   A.wral 2782   _Vcvv 3087    \ cdif 3439    C_ wss 3442   {csn 4002   {cpr 4004    |-> cmpt 4484   `'ccnv 4853   dom cdm 4854   "cima 4857    Fn wfn 5596   -->wf 5597   ` cfv 5601  (class class class)co 6305   CCcc 9536   RRcr 9537   0cc0 9538   1c1 9539    + caddc 9541    x. cmul 9543   -ucneg 9860    / cdiv 10268   2c2 10659   NN0cn0 10869   ^cexp 12269   sincsin 14094   cosccos 14095   tanctan 14096   ↾t crest 15278   TopOpenctopn 15279  ℂfldccnfld 18905  TopOnctopon 19849    _D cdv 22695
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-inf2 8146  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616  ax-addf 9617  ax-mulf 9618
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 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-iin 4305  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-se 4814  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-isom 5610  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-supp 6926  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-map 7482  df-pm 7483  df-ixp 7531  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-fsupp 7890  df-fi 7931  df-sup 7962  df-inf 7963  df-oi 8025  df-card 8372  df-cda 8596  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-q 11265  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ico 11641  df-icc 11642  df-fz 11783  df-fzo 11914  df-fl 12025  df-seq 12211  df-exp 12270  df-fac 12457  df-bc 12485  df-hash 12513  df-shft 13109  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-limsup 13504  df-clim 13530  df-rlim 13531  df-sum 13731  df-ef 14099  df-sin 14101  df-cos 14102  df-tan 14103  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-sets 15090  df-ress 15091  df-plusg 15165  df-mulr 15166  df-starv 15167  df-sca 15168  df-vsca 15169  df-ip 15170  df-tset 15171  df-ple 15172  df-ds 15174  df-unif 15175  df-hom 15176  df-cco 15177  df-rest 15280  df-topn 15281  df-0g 15299  df-gsum 15300  df-topgen 15301  df-pt 15302  df-prds 15305  df-xrs 15359  df-qtop 15364  df-imas 15365  df-xps 15367  df-mre 15443  df-mrc 15444  df-acs 15446  df-mgm 16439  df-sgrp 16478  df-mnd 16488  df-submnd 16534  df-mulg 16627  df-cntz 16922  df-cmn 17367  df-psmet 18897  df-xmet 18898  df-met 18899  df-bl 18900  df-mopn 18901  df-fbas 18902  df-fg 18903  df-cnfld 18906  df-top 19852  df-bases 19853  df-topon 19854  df-topsp 19855  df-cld 19965  df-ntr 19966  df-cls 19967  df-nei 20045  df-lp 20083  df-perf 20084  df-cn 20174  df-cnp 20175  df-t1 20261  df-haus 20262  df-tx 20508  df-hmeo 20701  df-fil 20792  df-fm 20884  df-flim 20885  df-flf 20886  df-xms 21266  df-ms 21267  df-tms 21268  df-cncf 21806  df-limc 22698  df-dv 22699
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator