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Theorem ofdivdiv2 29607
Description: Function analog of divdiv2 10048. (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  oF  /  ( G  oF  /  H )
)  =  ( ( F  oF  x.  H )  oF  /  G ) )

Proof of Theorem ofdivdiv2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simpll 753 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  A  e.  V )
2 simplr 754 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  F : A --> CC )
3 ffn 5564 . . 3  |-  ( F : A --> CC  ->  F  Fn  A )
42, 3syl 16 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  F  Fn  A )
5 simprl 755 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  G : A --> ( CC 
\  { 0 } ) )
6 ffn 5564 . . . 4  |-  ( G : A --> ( CC 
\  { 0 } )  ->  G  Fn  A )
75, 6syl 16 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  G  Fn  A )
8 simprr 756 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  H : A --> ( CC 
\  { 0 } ) )
9 ffn 5564 . . . 4  |-  ( H : A --> ( CC 
\  { 0 } )  ->  H  Fn  A )
108, 9syl 16 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  H  Fn  A )
11 inidm 3564 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6336 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  -> 
( G  oF  /  H )  Fn  A )
134, 10, 1, 1, 11offn 6336 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  -> 
( F  oF  x.  H )  Fn  A )
1413, 7, 1, 1, 11offn 6336 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  -> 
( ( F  oF  x.  H )  oF  /  G
)  Fn  A )
15 eqidd 2444 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
16 eqidd 2444 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( G  oF  /  H ) `  x )  =  ( ( G  oF  /  H ) `  x ) )
17 ffvelrn 5846 . . . . 5  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
182, 17sylan 471 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
19 ffvelrn 5846 . . . . . 6  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  ( G `  x )  e.  ( CC  \  {
0 } ) )
20 eldifsn 4005 . . . . . 6  |-  ( ( G `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( G `  x )  e.  CC  /\  ( G `  x
)  =/=  0 ) )
2119, 20sylib 196 . . . . 5  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  (
( G `  x
)  e.  CC  /\  ( G `  x )  =/=  0 ) )
225, 21sylan 471 . . . 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 5846 . . . . . 6  |-  ( ( H : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  ( H `  x )  e.  ( CC  \  {
0 } ) )
24 eldifsn 4005 . . . . . 6  |-  ( ( H `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( H `  x )  e.  CC  /\  ( H `  x
)  =/=  0 ) )
2523, 24sylib 196 . . . . 5  |-  ( ( H : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  (
( H `  x
)  e.  CC  /\  ( H `  x )  =/=  0 ) )
268, 25sylan 471 . . . 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 10048 . . . 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 1218 . . 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 2444 . . . . 5  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
30 eqidd 2444 . . . . 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 6334 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( G  oF  /  H ) `  x )  =  ( ( G `  x
)  /  ( H `
 x ) ) )
3231oveq2d 6112 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( F `  x
)  /  ( ( G  oF  /  H ) `  x
) )  =  ( ( F `  x
)  /  ( ( G `  x )  /  ( H `  x ) ) ) )
334, 10, 1, 1, 11, 15, 30ofval 6334 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( F  oF  x.  H ) `  x )  =  ( ( F `  x
)  x.  ( H `
 x ) ) )
3413, 7, 1, 1, 11, 33, 29ofval 6334 . . 3  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( ( F  oF  x.  H )  oF  /  G
) `  x )  =  ( ( ( F `  x )  x.  ( H `  x ) )  / 
( G `  x
) ) )
3528, 32, 343eqtr4d 2485 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  (
( F `  x
)  /  ( ( G  oF  /  H ) `  x
) )  =  ( ( ( F  oF  x.  H )  oF  /  G
) `  x )
)
361, 4, 12, 14, 15, 16, 35offveq 6346 1  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  -> 
( F  oF  /  ( G  oF  /  H )
)  =  ( ( F  oF  x.  H )  oF  /  G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2611    \ cdif 3330   {csn 3882    Fn wfn 5418   -->wf 5419   ` cfv 5423  (class class class)co 6096    oFcof 6323   CCcc 9285   0cc0 9287    x. cmul 9292    / cdiv 9998
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363  ax-pre-mulgt0 9364
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-reu 2727  df-rmo 2728  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-riota 6057  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-of 6325  df-er 7106  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-sub 9602  df-neg 9603  df-div 9999
This theorem is referenced by: (None)
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