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Theorem ofdivrec 27543
Description: Function analog of divrec 9440, a division analog of ofnegsub 9744. (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 955 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  A  e.  V )
2 simp2 956 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  F : A --> CC )
3 ffn 5389 . . 3  |-  ( F : A --> CC  ->  F  Fn  A )
42, 3syl 15 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  F  Fn  A )
5 ax-1cn 8795 . . . 4  |-  1  e.  CC
6 fnconstg 5429 . . . 4  |-  ( 1  e.  CC  ->  ( A  X.  { 1 } )  Fn  A )
75, 6mp1i 11 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  ( A  X.  { 1 } )  Fn  A )
8 simp3 957 . . . 4  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  G : A --> ( CC  \  { 0 } ) )
9 ffn 5389 . . . 4  |-  ( G : A --> ( CC 
\  { 0 } )  ->  G  Fn  A )
108, 9syl 15 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  G  Fn  A )
11 inidm 3378 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6089 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  (
( A  X.  {
1 } )  o F  /  G )  Fn  A )
134, 10, 1, 1, 11offn 6089 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  ( F  o F  /  G
)  Fn  A )
14 eqidd 2284 . 2  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  ( F `  x )  =  ( F `  x ) )
155a1i 10 . . 3  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  1  e.  CC )
16 eqidd 2284 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
171, 15, 10, 16ofc1 6100 . 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 5663 . . . . 5  |-  ( ( F : A --> CC  /\  x  e.  A )  ->  ( F `  x
)  e.  CC )
192, 18sylan 457 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  ( F `  x )  e.  CC )
20 ffvelrn 5663 . . . . . 6  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  ( G `  x )  e.  ( CC  \  {
0 } ) )
21 eldifsn 3749 . . . . . 6  |-  ( ( G `  x )  e.  ( CC  \  { 0 } )  <-> 
( ( G `  x )  e.  CC  /\  ( G `  x
)  =/=  0 ) )
2220, 21sylib 188 . . . . 5  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  (
( G `  x
)  e.  CC  /\  ( G `  x )  =/=  0 ) )
238, 22sylan 457 . . . 4  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  (
( G `  x
)  e.  CC  /\  ( G `  x )  =/=  0 ) )
24 divrec 9440 . . . . . 6  |-  ( ( ( F `  x
)  e.  CC  /\  ( G `  x )  e.  CC  /\  ( G `  x )  =/=  0 )  ->  (
( F `  x
)  /  ( G `
 x ) )  =  ( ( F `
 x )  x.  ( 1  /  ( G `  x )
) ) )
2524eqcomd 2288 . . . . 5  |-  ( ( ( F `  x
)  e.  CC  /\  ( G `  x )  e.  CC  /\  ( G `  x )  =/=  0 )  ->  (
( F `  x
)  x.  ( 1  /  ( G `  x ) ) )  =  ( ( F `
 x )  / 
( G `  x
) ) )
26253expb 1152 . . . 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 642 . . 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 6087 . . 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 2318 . 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 6098 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 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446    \ cdif 3149   {csn 3640    X. cxp 4687    Fn wfn 5250   -->wf 5251   ` cfv 5255  (class class class)co 5858    o Fcof 6076   CCcc 8735   0cc0 8737   1c1 8738    x. cmul 8742    / cdiv 9423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-po 4314  df-so 4315  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-of 6078  df-riota 6304  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424
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