Theorem ofdivcan4 38526

Description: Function analogue of divcan4 10712. (Contributed by Steve Rodriguez, 4-Nov-2015.)

Assertion

Ref	Expression
ofdivcan4	|- ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) -> ( ( F oF x. G ) oF / G ) = F )

Proof of Theorem ofdivcan4

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1 1061	. 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) -> A e. V )
2		simp2 1062	. . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) -> F : A --> CC )
3		ffn 6045	. . . 4 |- ( F : A --> CC -> F Fn A )
4	2, 3	syl 17	. . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) -> F Fn A )
5		simp3 1063	. . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) -> G : A --> ( CC \ { 0 } ) )
6		ffn 6045	. . . 4 |- ( G : A --> ( CC \ { 0 } ) -> G Fn A )
7	5, 6	syl 17	. . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) -> G Fn A )
8		inidm 3822	. . 3 |- ( A i^i A ) = A
9	4, 7, 1, 1, 8	offn 6908	. 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) -> ( F oF x. G ) Fn A )
10		eqidd 2623	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
11		eqidd 2623	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) /\ x e. A ) -> ( G ` x ) = ( G ` x ) )
12	4, 7, 1, 1, 8, 10, 11	ofval 6906	. 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) /\ x e. A ) -> ( ( F oF x. G ) ` x ) = ( ( F ` x ) x. ( G ` x ) ) )
13		ffvelrn 6357	. . . 4 |- ( ( F : A --> CC /\ x e. A ) -> ( F ` x ) e. CC )
14	2, 13	sylan 488	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) /\ x e. A ) -> ( F ` x ) e. CC )
15		ffvelrn 6357	. . . . 5 |- ( ( G : A --> ( CC \ { 0 } ) /\ x e. A ) -> ( G ` x ) e. ( CC \ { 0 } ) )
16		eldifsn 4317	. . . . 5 |- ( ( G ` x ) e. ( CC \ { 0 } ) <-> ( ( G ` x ) e. CC /\ ( G ` x ) =/= 0 ) )
17	15, 16	sylib 208	. . . 4 |- ( ( G : A --> ( CC \ { 0 } ) /\ x e. A ) -> ( ( G ` x ) e. CC /\ ( G ` x ) =/= 0 ) )
18	5, 17	sylan 488	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) /\ x e. A ) -> ( ( G ` x ) e. CC /\ ( G ` x ) =/= 0 ) )
19		divcan4 10712	. . . 4 |- ( ( ( F ` x ) e. CC /\ ( G ` x ) e. CC /\ ( G ` x ) =/= 0 ) -> ( ( ( F ` x ) x. ( G ` x ) ) / ( G ` x ) ) = ( F ` x ) )
20	19	3expb 1266	. . 3 |- ( ( ( F ` x ) e. CC /\ ( ( G ` x ) e. CC /\ ( G ` x ) =/= 0 ) ) -> ( ( ( F ` x ) x. ( G ` x ) ) / ( G ` x ) ) = ( F ` x ) )
21	14, 18, 20	syl2anc 693	. 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) /\ x e. A ) -> ( ( ( F ` x ) x. ( G ` x ) ) / ( G ` x ) ) = ( F ` x ) )
22	1, 9, 7, 4, 12, 11, 21	offveq 6918	1 |- ( ( A e. V /\ F : A --> CC /\ G : A --> ( CC \ { 0 } ) ) -> ( ( F oF x. G ) oF / G ) = F )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = 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:  expgrowth  38534  binomcxplemnotnn0  38555