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Theorem ofdivrec 26866
Description: Function analog of divrec 9530, a division analog of ofnegsub 9834. (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 5472 . . 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 8885 . . . 4  |-  1  e.  CC
6 fnconstg 5512 . . . 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 5472 . . . 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 3454 . . 3  |-  ( A  i^i  A )  =  A
127, 10, 1, 1, 11offn 6176 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  (
( A  X.  {
1 } )  o F  /  G )  Fn  A )
134, 10, 1, 1, 11offn 6176 . 2  |-  ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  ->  ( F  o F  /  G
)  Fn  A )
14 eqidd 2359 . 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 2359 . . 3  |-  ( ( ( A  e.  V  /\  F : A --> CC  /\  G : A --> ( CC 
\  { 0 } ) )  /\  x  e.  A )  ->  ( G `  x )  =  ( G `  x ) )
171, 15, 10, 16ofc1 6187 . 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 5746 . . . . 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 5746 . . . . . 6  |-  ( ( G : A --> ( CC 
\  { 0 } )  /\  x  e.  A )  ->  ( G `  x )  e.  ( CC  \  {
0 } ) )
21 eldifsn 3825 . . . . . 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 9530 . . . . . 6  |-  ( ( ( F `  x
)  e.  CC  /\  ( G `  x )  e.  CC  /\  ( G `  x )  =/=  0 )  ->  (
( F `  x
)  /  ( G `
 x ) )  =  ( ( F `
 x )  x.  ( 1  /  ( G `  x )
) ) )
2524eqcomd 2363 . . . . 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 6174 . . 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 2393 . 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 6185 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 1642    e. wcel 1710    =/= wne 2521    \ cdif 3225   {csn 3716    X. cxp 4769    Fn wfn 5332   -->wf 5333   ` cfv 5337  (class class class)co 5945    o Fcof 6163   CCcc 8825   0cc0 8827   1c1 8828    x. cmul 8832    / cdiv 9513
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-rep 4212  ax-sep 4222  ax-nul 4230  ax-pow 4269  ax-pr 4295  ax-un 4594  ax-resscn 8884  ax-1cn 8885  ax-icn 8886  ax-addcl 8887  ax-addrcl 8888  ax-mulcl 8889  ax-mulrcl 8890  ax-mulcom 8891  ax-addass 8892  ax-mulass 8893  ax-distr 8894  ax-i2m1 8895  ax-1ne0 8896  ax-1rid 8897  ax-rnegex 8898  ax-rrecex 8899  ax-cnre 8900  ax-pre-lttri 8901  ax-pre-lttrn 8902  ax-pre-ltadd 8903  ax-pre-mulgt0 8904
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-nel 2524  df-ral 2624  df-rex 2625  df-reu 2626  df-rmo 2627  df-rab 2628  df-v 2866  df-sbc 3068  df-csb 3158  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-iun 3988  df-br 4105  df-opab 4159  df-mpt 4160  df-id 4391  df-po 4396  df-so 4397  df-xp 4777  df-rel 4778  df-cnv 4779  df-co 4780  df-dm 4781  df-rn 4782  df-res 4783  df-ima 4784  df-iota 5301  df-fun 5339  df-fn 5340  df-f 5341  df-f1 5342  df-fo 5343  df-f1o 5344  df-fv 5345  df-ov 5948  df-oprab 5949  df-mpt2 5950  df-of 6165  df-riota 6391  df-er 6747  df-en 6952  df-dom 6953  df-sdom 6954  df-pnf 8959  df-mnf 8960  df-xr 8961  df-ltxr 8962  df-le 8963  df-sub 9129  df-neg 9130  df-div 9514
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