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Theorem ofdivdiv2 27522
Description: Function analog of divdiv2 9726. (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 731 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  A  e.  V )
2 simplr 732 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  F : A --> CC )
3 ffn 5591 . . 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 733 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  G : A --> ( CC 
\  { 0 } ) )
6 ffn 5591 . . . 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 734 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC )  /\  ( G : A
--> ( CC  \  {
0 } )  /\  H : A --> ( CC 
\  { 0 } ) ) )  ->  H : A --> ( CC 
\  { 0 } ) )
9 ffn 5591 . . . 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 3550 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6316 . 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 6316 . . 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 6316 . 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 2437 . 2  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
16 eqidd 2437 . 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 5868 . . . . 5  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
182, 17sylan 458 . . . 4  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
19 ffvelrn 5868 . . . . . 6  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  ( G `  x )  e.  ( CC  \  {
0 } ) )
20 eldifsn 3927 . . . . . 6  |-  ( ( G `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( G `  x )  e.  CC  /\  ( G `  x
)  =/=  0 ) )
2119, 20sylib 189 . . . . 5  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  (
( G `  x
)  e.  CC  /\  ( G `  x )  =/=  0 ) )
225, 21sylan 458 . . . 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 5868 . . . . . 6  |-  ( ( H : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  ( H `  x )  e.  ( CC  \  {
0 } ) )
24 eldifsn 3927 . . . . . 6  |-  ( ( H `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( H `  x )  e.  CC  /\  ( H `  x
)  =/=  0 ) )
2523, 24sylib 189 . . . . 5  |-  ( ( H : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  (
( H `  x
)  e.  CC  /\  ( H `  x )  =/=  0 ) )
268, 25sylan 458 . . . 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 9726 . . . 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 1184 . . 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 2437 . . . . 5  |-  ( ( ( ( A  e.  V  /\  F : A
--> CC )  /\  ( G : A --> ( CC 
\  { 0 } )  /\  H : A
--> ( CC  \  {
0 } ) ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
30 eqidd 2437 . . . . 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 6314 . . . 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 6097 . . 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 6314 . . . 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 6314 . . 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 2478 . 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 6325 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 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317   {csn 3814    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    o Fcof 6303   CCcc 8988   0cc0 8990    x. cmul 8995    / cdiv 9677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-po 4503  df-so 4504  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-of 6305  df-riota 6549  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678
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