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Theorem ofdivdiv2 27648
Description: Function analog of divdiv2 9488. (Contributed by Steve Rodriguez, 23-Nov-2015.)
Assertion
Ref Expression
ofdivdiv2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  -> 
( F  o F  /  ( G  o F  /  H ) )  =  ( ( F  o F  x.  H
)  o F  /  G ) )

Proof of Theorem ofdivdiv2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 730 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  A  e.  V )
2 simplr 731 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  F : A --> CC )
3 ffn 5405 . . 3  |-  ( F : A --> CC  ->  F  Fn  A )
42, 3syl 15 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  F  Fn  A )
5 simprl 732 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  G : A --> ( CC 
\  { 0 } ) )
6 ffn 5405 . . . 4  |-  ( G : A --> ( CC 
\  { 0 } )  ->  G  Fn  A )
75, 6syl 15 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  G  Fn  A )
8 simprr 733 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  H : A --> ( CC 
\  { 0 } ) )
9 ffn 5405 . . . 4  |-  ( H : A --> ( CC 
\  { 0 } )  ->  H  Fn  A )
108, 9syl 15 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  H  Fn  A )
11 inidm 3391 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6105 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  -> 
( G  o F  /  H )  Fn  A )
134, 10, 1, 1, 11offn 6105 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  -> 
( F  o F  x.  H )  Fn  A )
1413, 7, 1, 1, 11offn 6105 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  -> 
( ( F  o F  x.  H )  o F  /  G
)  Fn  A )
15 eqidd 2297 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
16 eqidd 2297 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( G  o F  /  H ) `  x )  =  ( ( G  o F  /  H ) `  x ) )
17 ffvelrn 5679 . . . . 5  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
182, 17sylan 457 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
19 ffvelrn 5679 . . . . . 6  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  ( G `  x )  e.  ( CC  \  {
0 } ) )
20 eldifsn 3762 . . . . . 6  |-  ( ( G `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( G `  x )  e.  CC  /\  ( G `  x
)  =/=  0 ) )
2119, 20sylib 188 . . . . 5  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  (
( G `  x
)  e.  CC  /\  ( G `  x )  =/=  0 ) )
225, 21sylan 457 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( G `  x
)  e.  CC  /\  ( G `  x )  =/=  0 ) )
23 ffvelrn 5679 . . . . . 6  |-  ( ( H : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  ( H `  x )  e.  ( CC  \  {
0 } ) )
24 eldifsn 3762 . . . . . 6  |-  ( ( H `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( H `  x )  e.  CC  /\  ( H `  x
)  =/=  0 ) )
2523, 24sylib 188 . . . . 5  |-  ( ( H : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  (
( H `  x
)  e.  CC  /\  ( H `  x )  =/=  0 ) )
268, 25sylan 457 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( H `  x
)  e.  CC  /\  ( H `  x )  =/=  0 ) )
27 divdiv2 9488 . . . 4  |-  ( ( ( F `  x
)  e.  CC  /\  ( ( G `  x )  e.  CC  /\  ( G `  x
)  =/=  0 )  /\  ( ( H `
 x )  e.  CC  /\  ( H `
 x )  =/=  0 ) )  -> 
( ( F `  x )  /  (
( G `  x
)  /  ( H `
 x ) ) )  =  ( ( ( F `  x
)  x.  ( H `
 x ) )  /  ( G `  x ) ) )
2818, 22, 26, 27syl3anc 1182 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( F `  x
)  /  ( ( G `  x )  /  ( H `  x ) ) )  =  ( ( ( F `  x )  x.  ( H `  x ) )  / 
( G `  x
) ) )
29 eqidd 2297 . . . . 5  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
30 eqidd 2297 . . . . 5  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  ( H `  x )  =  ( H `  x ) )
317, 10, 1, 1, 11, 29, 30ofval 6103 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( G  o F  /  H ) `  x )  =  ( ( G `  x
)  /  ( H `
 x ) ) )
3231oveq2d 5890 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( F `  x
)  /  ( ( G  o F  /  H ) `  x
) )  =  ( ( F `  x
)  /  ( ( G `  x )  /  ( H `  x ) ) ) )
334, 10, 1, 1, 11, 15, 30ofval 6103 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( F  o F  x.  H ) `  x )  =  ( ( F `  x
)  x.  ( H `
 x ) ) )
3413, 7, 1, 1, 11, 33, 29ofval 6103 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( ( F  o F  x.  H )  o F  /  G
) `  x )  =  ( ( ( F `  x )  x.  ( H `  x ) )  / 
( G `  x
) ) )
3528, 32, 343eqtr4d 2338 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( F `  x
)  /  ( ( G  o F  /  H ) `  x
) )  =  ( ( ( F  o F  x.  H )  o F  /  G
) `  x )
)
361, 4, 12, 14, 15, 16, 35offveq 6114 1  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  -> 
( F  o F  /  ( G  o F  /  H ) )  =  ( ( F  o F  x.  H
)  o F  /  G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459    \ cdif 3162   {csn 3653    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    o Fcof 6092   CCcc 8751   0cc0 8753    x. cmul 8758    / cdiv 9439
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-po 4330  df-so 4331  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-of 6094  df-riota 6320  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440
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