Theorem ofnegsub 9293

Description: Function analogue of negsub 8574. (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 variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		addcl 8304	. . 3 |- ( ( x e. CC /\ y e. CC ) -> ( x + y ) e. CC )
2	1	adantl 277	. 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ ( x e. CC /\ y e. CC ) ) -> ( x + y ) e. CC )
3		simp2 1029	. 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F : A --> CC )
4		mulcl 8306	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x x. y ) e. CC )
5	4	adantl 277	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ ( x e. CC /\ y e. CC ) ) -> ( x x. y ) e. CC )
6		ax-1cn 8272	. . . . . 6 |- 1 e. CC
7	6	negcli 8594	. . . . 5 |- -u 1 e. CC
8	7	fconst6 5592	. . . 4 |- ( A X. { -u 1 } ) : A --> CC
9	8	a1i 9	. . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( A X. { -u 1 } ) : A --> CC )
10		simp3 1030	. . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G : A --> CC )
11		simp1 1028	. . 3 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> A e. V )
12		inidm 3440	. . 3 |- ( A i^i A ) = A
13	5, 9, 10, 11, 11, 12	off 6315	. 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( ( A X. { -u 1 } ) oF x. G ) : A --> CC )
14		subcl 8525	. . . 4 |- ( ( x e. CC /\ y e. CC ) -> ( x - y ) e. CC )
15	14	adantl 277	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ ( x e. CC /\ y e. CC ) ) -> ( x - y ) e. CC )
16	15, 3, 10, 11, 11, 12	off 6315	. 2 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF - G ) : A --> CC )
17		eqidd 2239	. 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) = ( F ` x ) )
18	7	a1i 9	. . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> -u 1 e. CC )
19	10	ffnd 5534	. . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> G Fn A )
20		eqidd 2239	. . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) = ( G ` x ) )
21	7	a1i 9	. . . . 5 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> -u 1 e. CC )
22	10	ffvelcdmda 5843	. . . . 5 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( G ` x ) e. CC )
23	21, 22	mulcld 8346	. . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( -u 1 x. ( G ` x ) ) e. CC )
24	11, 18, 19, 20, 23	ofc1g 6324	. . 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 ) ) )
25	22	mulm1d 8737	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( -u 1 x. ( G ` x ) ) = -u ( G ` x ) )
26	24, 25	eqtrd 2271	. 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( ( A X. { -u 1 } ) oF x. G ) ` x ) = -u ( G ` x ) )
27	3	ffvelcdmda 5843	. . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( F ` x ) e. CC )
28	27, 22	negsubd 8643	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) + -u ( G ` x ) ) = ( ( F ` x ) - ( G ` x ) ) )
29	3	ffnd 5534	. . . 4 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> F Fn A )
30	27, 22	subcld 8637	. . . 4 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) - ( G ` x ) ) e. CC )
31	29, 19, 11, 11, 12, 17, 20, 30	ofvalg 6312	. . 3 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F oF - G ) ` x ) = ( ( F ` x ) - ( G ` x ) ) )
32	28, 31	eqtr4d 2274	. 2 |- ( ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) /\ x e. A ) -> ( ( F ` x ) + -u ( G ` x ) ) = ( ( F oF - G ) ` x ) )
33	2, 3, 13, 11, 11, 12, 16, 17, 26, 32	offeq 6316	1 |- ( ( A e. V /\ F : A --> CC /\ G : A --> CC ) -> ( F oF + ( ( A X. { -u 1 } ) oF x. G ) ) = ( F oF - G ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209   {csn 3709   X. cxp 4772   -->wf 5373   ` cfv 5377  (class class class)co 6085   oFcof 6300   CCcc 8177   1c1 8180   + caddc 8182   x. cmul 8184   - cmin 8497   -ucneg 8498

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-setind 4684  ax-resscn 8271  ax-1cn 8272  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-sub 8499  df-neg 8500

This theorem is used by:  plysub  15856