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Theorem ofdivrec 27558
Description: Function analog of divrec 9725, a division analog of ofnegsub 10029. (Contributed by Steve Rodriguez, 3-Nov-2015.)
Assertion
Ref Expression
ofdivrec  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  ( F  o F  x.  (
( A  X.  {
1 } )  o F  /  G ) )  =  ( F  o F  /  G
) )

Proof of Theorem ofdivrec
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simp1 958 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  A  e.  V )
2 simp2 959 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  F : A --> CC )
3 ffn 5620 . . 3  |-  ( F : A --> CC  ->  F  Fn  A )
42, 3syl 16 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  F  Fn  A )
5 ax-1cn 9079 . . . 4  |-  1  e.  CC
6 fnconstg 5660 . . . 4  |-  ( 1  e.  CC  ->  ( A  X.  { 1 } )  Fn  A )
75, 6mp1i 12 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  ( A  X.  { 1 } )  Fn  A )
8 simp3 960 . . . 4  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  G : A --> ( CC  \  { 0 } ) )
9 ffn 5620 . . . 4  |-  ( G : A --> ( CC 
\  { 0 } )  ->  G  Fn  A )
108, 9syl 16 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  G  Fn  A )
11 inidm 3535 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6345 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  (
( A  X.  {
1 } )  o F  /  G )  Fn  A )
134, 10, 1, 1, 11offn 6345 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  ( F  o F  /  G
)  Fn  A )
14 eqidd 2443 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
155a1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  1  e.  CC )
16 eqidd 2443 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
171, 15, 10, 16ofc1 6356 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  (
( ( A  X.  { 1 } )  o F  /  G
) `  x )  =  ( 1  / 
( G `  x
) ) )
18 ffvelrn 5897 . . . . 5  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
192, 18sylan 459 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
20 ffvelrn 5897 . . . . . 6  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  ( G `  x )  e.  ( CC  \  {
0 } ) )
21 eldifsn 3951 . . . . . 6  |-  ( ( G `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( G `  x )  e.  CC  /\  ( G `  x
)  =/=  0 ) )
2220, 21sylib 190 . . . . 5  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  (
( G `  x
)  e.  CC  /\  ( G `  x )  =/=  0 ) )
238, 22sylan 459 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  (
( G `  x
)  e.  CC  /\  ( G `  x )  =/=  0 ) )
24 divrec 9725 . . . . . 6  |-  ( ( ( F `  x
)  e.  CC  /\  ( G `  x )  e.  CC  /\  ( G `  x )  =/=  0 )  ->  (
( F `  x
)  /  ( G `
 x ) )  =  ( ( F `
 x )  x.  ( 1  /  ( G `  x )
) ) )
2524eqcomd 2447 . . . . 5  |-  ( ( ( F `  x
)  e.  CC  /\  ( G `  x )  e.  CC  /\  ( G `  x )  =/=  0 )  ->  (
( F `  x
)  x.  ( 1  /  ( G `  x ) ) )  =  ( ( F `
 x )  / 
( G `  x
) ) )
26253expb 1155 . . . 4  |-  ( ( ( F `  x
)  e.  CC  /\  ( ( G `  x )  e.  CC  /\  ( G `  x
)  =/=  0 ) )  ->  ( ( F `  x )  x.  ( 1  /  ( G `  x )
) )  =  ( ( F `  x
)  /  ( G `
 x ) ) )
2719, 23, 26syl2anc 644 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  (
( F `  x
)  x.  ( 1  /  ( G `  x ) ) )  =  ( ( F `
 x )  / 
( G `  x
) ) )
284, 10, 1, 1, 11, 14, 16ofval 6343 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  (
( F  o F  /  G ) `  x )  =  ( ( F `  x
)  /  ( G `
 x ) ) )
2927, 28eqtr4d 2477 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  (
( F `  x
)  x.  ( 1  /  ( G `  x ) ) )  =  ( ( F  o F  /  G
) `  x )
)
301, 4, 12, 13, 14, 17, 29offveq 6354 1  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  ( F  o F  x.  (
( A  X.  {
1 } )  o F  /  G ) )  =  ( F  o F  /  G
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1727    =/= wne 2605    \ cdif 3303   {csn 3838    X. cxp 4905    Fn wfn 5478   -->wf 5479   ` cfv 5483  (class class class)co 6110    o Fcof 6332   CCcc 9019   0cc0 9021   1c1 9022    x. cmul 9026    / cdiv 9708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-rep 4345  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-of 6334  df-riota 6578  df-er 6934  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709
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