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Theorem ofdivrec 27411
Description: Function analog of divrec 9650, a division analog of ofnegsub 9954. (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 957 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  A  e.  V )
2 simp2 958 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  F : A --> CC )
3 ffn 5550 . . 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 9004 . . . 4  |-  1  e.  CC
6 fnconstg 5590 . . . 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 959 . . . 4  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  G : A --> ( CC  \  { 0 } ) )
9 ffn 5550 . . . 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 3510 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6275 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  (
( A  X.  {
1 } )  o F  /  G )  Fn  A )
134, 10, 1, 1, 11offn 6275 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  ( F  o F  /  G
)  Fn  A )
14 eqidd 2405 . 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 2405 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
171, 15, 10, 16ofc1 6286 . 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 5827 . . . . 5  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
192, 18sylan 458 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
20 ffvelrn 5827 . . . . . 6  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  ( G `  x )  e.  ( CC  \  {
0 } ) )
21 eldifsn 3887 . . . . . 6  |-  ( ( G `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( G `  x )  e.  CC  /\  ( G `  x
)  =/=  0 ) )
2220, 21sylib 189 . . . . 5  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  (
( G `  x
)  e.  CC  /\  ( G `  x )  =/=  0 ) )
238, 22sylan 458 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  (
( G `  x
)  e.  CC  /\  ( G `  x )  =/=  0 ) )
24 divrec 9650 . . . . . 6  |-  ( ( ( F `  x
)  e.  CC  /\  ( G `  x )  e.  CC  /\  ( G `  x )  =/=  0 )  ->  (
( F `  x
)  /  ( G `
 x ) )  =  ( ( F `
 x )  x.  ( 1  /  ( G `  x )
) ) )
2524eqcomd 2409 . . . . 5  |-  ( ( ( F `  x
)  e.  CC  /\  ( G `  x )  e.  CC  /\  ( G `  x )  =/=  0 )  ->  (
( F `  x
)  x.  ( 1  /  ( G `  x ) ) )  =  ( ( F `
 x )  / 
( G `  x
) ) )
26253expb 1154 . . . 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 643 . . 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 6273 . . 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 2439 . 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 6284 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 359    /\ w3a 936    = wceq 1649    e. wcel 1721    =/= wne 2567    \ cdif 3277   {csn 3774    X. cxp 4835    Fn wfn 5408   -->wf 5409   ` cfv 5413  (class class class)co 6040    o Fcof 6262   CCcc 8944   0cc0 8946   1c1 8947    x. cmul 8951    / cdiv 9633
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-po 4463  df-so 4464  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-of 6264  df-riota 6508  df-er 6864  df-en 7069  df-dom 7070  df-sdom 7071  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634
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