Theorem ofnegsub 11018

Description: Function analogue of negsub 10329. (Contributed by Mario Carneiro, 24-Jul-2014.)

Assertion

Ref	Expression
ofnegsub	|- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF + ( ( A X. { -u 1 } ) oF x. G ) ) = ( F oF - G ) )

Proof of Theorem ofnegsub

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 ) -> A e. V )
2		simp2 1062	. . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F : A --> CC )
3		ffn 6045	. . 3 |- ( F : A --> CC -> F Fn A )
4	2, 3	syl 17	. 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F Fn A )
5		ax-1cn 9994	. . . . 5 |- 1 e. CC
6	5	negcli 10349	. . . 4 |- -u 1 e. CC
7		fnconstg 6093	. . . 4 |- ( -u 1 e. CC -> ( A X. { -u 1 } ) Fn A )
8	6, 7	mp1i 13	. . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( A X. { -u 1 } ) Fn A )
9		simp3 1063	. . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G : A --> CC )
10		ffn 6045	. . . 4 |- ( G : A --> CC -> G Fn A )
11	9, 10	syl 17	. . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G Fn A )
12		inidm 3822	. . 3 |- ( A i^i A ) = A
13	8, 11, 1, 1, 12	offn 6908	. 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( ( A X. { -u 1 } ) oF x. G ) Fn A )
14	4, 11, 1, 1, 12	offn 6908	. 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF - G ) Fn A )
15		eqidd 2623	. 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
16	6	a1i 11	. . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> -u 1 e. CC )
17		eqidd 2623	. . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) = ( G ` x ) )
18	1, 16, 11, 17	ofc1 6920	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( ( A X. { -u 1 } ) oF x. G ) ` x ) = ( -u 1 x. ( G ` x ) ) )
19	9	ffvelrnda 6359	. . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) e. CC )
20	19	mulm1d 10482	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( -u 1 x. ( G ` x ) ) = -u ( G ` x ) )
21	18, 20	eqtrd 2656	. 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( ( A X. { -u 1 } ) oF x. G ) ` x ) = -u ( G ` x ) )
22	2	ffvelrnda 6359	. . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) e. CC )
23	22, 19	negsubd 10398	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) + -u ( G ` x ) ) = ( ( F ` x ) - ( G ` x ) ) )
24	4, 11, 1, 1, 12, 15, 17	ofval 6906	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F oF - G ) ` x ) = ( ( F ` x ) - ( G ` x ) ) )
25	23, 24	eqtr4d 2659	. 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) + -u ( G ` x ) ) = ( ( F oF - G ) ` x ) )
26	1, 4, 13, 14, 15, 21, 25	offveq 6918	1 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF + ( ( A X. { -u 1 } ) oF x. G ) ) = ( F oF - G ) )

Syntax hints:   -> wi 4   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   {csn 4177   X. cxp 5112   Fn wfn 5883   -->wf 5884   ` cfv 5888  (class class class)co 6650   oFcof 6895   CCcc 9934   1c1 9937   + caddc 9939   x. cmul 9941   - cmin 10266   -ucneg 10267

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

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-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-ltxr 10079  df-sub 10268  df-neg 10269

This theorem is referenced by:  i1fsub  23475  itg1sub  23476  plysub  23975  coesub  24013  dgrsub  24028  basellem9  24815  expgrowth  38534