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Theorem ofdivdiv2 38527
Description: Function analogue of divdiv2 10737. (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 oF / ( G oF / H ) ) = ( ( F oF x. H ) oF / G ) )
Proof of Theorem ofdivdiv2
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 simpll 790 . 2 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> A e. V )
2 simplr 792 . . 3 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> F : A --> CC )
3 ffn 6045 . . 3 |- ( F : A --> CC -> F Fn A )
42, 3syl 17 . 2 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> F Fn A )
5 simprl 794 . . . 4 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> G : A --> ( CC \ { 0 } ) )
6 ffn 6045 . . . 4 |- ( G : A --> ( CC \ { 0 } ) -> G Fn A )
75, 6syl 17 . . 3 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> G Fn A )
8 simprr 796 . . . 4 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> H : A --> ( CC \ { 0 } ) )
9 ffn 6045 . . . 4 |- ( H : A --> ( CC \ { 0 } ) -> H Fn A )
108, 9syl 17 . . 3 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> H Fn A )
11 inidm 3822 . . 3 |- ( A i^i A ) = A
127, 10, 1, 1, 11offn 6908 . 2 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> ( G oF / H ) Fn A )
134, 10, 1, 1, 11offn 6908 . . 3 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> ( F oF x. H ) Fn A )
1413, 7, 1, 1, 11offn 6908 . 2 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> ( ( F oF x. H ) oF / G ) Fn A )
15 eqidd 2623 . 2 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
16 eqidd 2623 . 2 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) /\ x e. A ) -> ( ( G oF / H ) ` x ) = ( ( G oF / H ) ` x ) )
17 ffvelrn 6357 . . . . 5 |- ( ( F : A --> CC /\ x e. A ) -> ( F ` x ) e. CC )
182, 17sylan 488 . . . 4 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) /\ x e. A ) -> ( F ` x ) e. CC )
19 ffvelrn 6357 . . . . . 6 |- ( ( G : A --> ( CC \ { 0 } ) /\ x e. A ) -> ( G ` x ) e. ( CC \ { 0 } ) )
20 eldifsn 4317 . . . . . 6 |- ( ( G ` x ) e. ( CC \ { 0 } ) <-> ( ( G ` x ) e. CC /\ ( G ` x ) =/= 0 ) )
2119, 20sylib 208 . . . . 5 |- ( ( G : A --> ( CC \ { 0 } ) /\ x e. A ) -> ( ( G ` x ) e. CC /\ ( G ` x ) =/= 0 ) )
225, 21sylan 488 . . . 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 6357 . . . . . 6 |- ( ( H : A --> ( CC \ { 0 } ) /\ x e. A ) -> ( H ` x ) e. ( CC \ { 0 } ) )
24 eldifsn 4317 . . . . . 6 |- ( ( H ` x ) e. ( CC \ { 0 } ) <-> ( ( H ` x ) e. CC /\ ( H ` x ) =/= 0 ) )
2523, 24sylib 208 . . . . 5 |- ( ( H : A --> ( CC \ { 0 } ) /\ x e. A ) -> ( ( H ` x ) e. CC /\ ( H ` x ) =/= 0 ) )
268, 25sylan 488 . . . 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 10737 . . . 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 1326 . . 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 2623 . . . . 5 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) /\ x e. A ) -> ( G ` x ) = ( G ` x ) )
30 eqidd 2623 . . . . 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 6906 . . . 4 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) /\ x e. A ) -> ( ( G oF / H ) ` x ) = ( ( G ` x ) / ( H ` x ) ) )
3231oveq2d 6666 . . 3 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) /\ x e. A ) -> ( ( F ` x ) / ( ( G oF / H ) ` x ) ) = ( ( F ` x ) / ( ( G ` x ) / ( H ` x ) ) ) )
334, 10, 1, 1, 11, 15, 30ofval 6906 . . . 4 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) /\ x e. A ) -> ( ( F oF x. H ) ` x ) = ( ( F ` x ) x. ( H ` x ) ) )
3413, 7, 1, 1, 11, 33, 29ofval 6906 . . 3 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) /\ x e. A ) -> ( ( ( F oF x. H ) oF / G ) ` x ) = ( ( ( F ` x ) x. ( H ` x ) ) / ( G ` x ) ) )
3528, 32, 343eqtr4d 2666 . 2 |- ( ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) /\ x e. A ) -> ( ( F ` x ) / ( ( G oF / H ) ` x ) ) = ( ( ( F oF x. H ) oF / G ) ` x ) )
361, 4, 12, 14, 15, 16, 35offveq 6918 1 |- ( ( ( A e. V /\ F : A --> CC ) /\ ( G : A --> ( CC \ { 0 } ) /\ H : A --> ( CC \ { 0 } ) ) ) -> ( F oF / ( G oF / H ) ) = ( ( F oF x. H ) oF / G ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   \ cdif 3571   {csn 4177   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   0cc0 9936   x. cmul 9941   / cdiv 10684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685
This theorem is referenced by: (None)
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