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Theorem vsfval 30141

Description: Value of the function for the vector subtraction operation on a normed complex vector space. (Contributed by NM, 15-Feb-2008.) (Revised by Mario Carneiro, 27-Dec-2014.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
vsfval.2	⊢ 𝐺 = ( +_𝑣 ‘𝑈)
vsfval.3	⊢ 𝑀 = ( −_𝑣 ‘𝑈)

**Assertion**
Ref	Expression
vsfval	⊢ 𝑀 = ( /_𝑔 ‘𝐺)

Proof of Theorem vsfval Dummy variables 𝑥 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-vs 30107	. . . . 5 ⊢ −_𝑣 = ( /_𝑔 ∘ +_𝑣 )
2	1	fveq1i 6892	. . . 4 ⊢ ( −_𝑣 ‘𝑈) = (( /_𝑔 ∘ +_𝑣 )‘𝑈)
3		fo1st 7997	. . . . . . . 8 ⊢ 1^st :V–onto→V
4		fof 6805	. . . . . . . 8 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 1^st :V⟶V
6		fco 6741	. . . . . . 7 ⊢ ((1^st :V⟶V ∧ 1^st :V⟶V) → (1^st ∘ 1^st ):V⟶V)
7	5, 5, 6	mp2an 690	. . . . . 6 ⊢ (1^st ∘ 1^st ):V⟶V
8		df-va 30103	. . . . . . 7 ⊢ +_𝑣 = (1^st ∘ 1^st )
9	8	feq1i 6708	. . . . . 6 ⊢ ( +_𝑣 :V⟶V ↔ (1^st ∘ 1^st ):V⟶V)
10	7, 9	mpbir 230	. . . . 5 ⊢ +_𝑣 :V⟶V
11		fvco3 6990	. . . . 5 ⊢ (( +_𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /_𝑔 ∘ +_𝑣 )‘𝑈) = ( /_𝑔 ‘( +_𝑣 ‘𝑈)))
12	10, 11	mpan 688	. . . 4 ⊢ (𝑈 ∈ V → (( /_𝑔 ∘ +_𝑣 )‘𝑈) = ( /_𝑔 ‘( +_𝑣 ‘𝑈)))
13	2, 12	eqtrid 2784	. . 3 ⊢ (𝑈 ∈ V → ( −_𝑣 ‘𝑈) = ( /_𝑔 ‘( +_𝑣 ‘𝑈)))
14		0ngrp 30019	. . . . . 6 ⊢ ¬ ∅ ∈ GrpOp
15		vex 3478	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
16	15	rnex 7905	. . . . . . . . 9 ⊢ ran 𝑔 ∈ V
17	16, 16	mpoex 8068	. . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18		df-gdiv 30004	. . . . . . . 8 ⊢ /_𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
19	17, 18	dmmpti 6694	. . . . . . 7 ⊢ dom /_𝑔 = GrpOp
20	19	eleq2i 2825	. . . . . 6 ⊢ (∅ ∈ dom /_𝑔 ↔ ∅ ∈ GrpOp)
21	14, 20	mtbir 322	. . . . 5 ⊢ ¬ ∅ ∈ dom /_𝑔
22		ndmfv 6926	. . . . 5 ⊢ (¬ ∅ ∈ dom /_𝑔 → ( /_𝑔 ‘∅) = ∅)
23	21, 22	mp1i 13	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /_𝑔 ‘∅) = ∅)
24		fvprc 6883	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +_𝑣 ‘𝑈) = ∅)
25	24	fveq2d 6895	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /_𝑔 ‘( +_𝑣 ‘𝑈)) = ( /_𝑔 ‘∅))
26		fvprc 6883	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( −_𝑣 ‘𝑈) = ∅)
27	23, 25, 26	3eqtr4rd 2783	. . 3 ⊢ (¬ 𝑈 ∈ V → ( −_𝑣 ‘𝑈) = ( /_𝑔 ‘( +_𝑣 ‘𝑈)))
28	13, 27	pm2.61i 182	. 2 ⊢ ( −_𝑣 ‘𝑈) = ( /_𝑔 ‘( +_𝑣 ‘𝑈))
29		vsfval.3	. 2 ⊢ 𝑀 = ( −_𝑣 ‘𝑈)
30		vsfval.2	. . 3 ⊢ 𝐺 = ( +_𝑣 ‘𝑈)
31	30	fveq2i 6894	. 2 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘( +_𝑣 ‘𝑈))
32	28, 29, 31	3eqtr4i 2770	1 ⊢ 𝑀 = ( /_𝑔 ‘𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ∅c0 4322 dom cdm 5676 ran crn 5677 ∘ ccom 5680 ⟶wf 6539 –onto→wfo 6541 ‘cfv 6543 (class class class)co 7411 ∈ cmpo 7413 1^st c1st 7975 GrpOpcgr 29997 invcgn 29999 /_𝑔 cgs 30000 +_𝑣 cpv 30093 −_𝑣 cnsb 30097

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7414 df-oprab 7415 df-mpo 7416 df-1st 7977 df-2nd 7978 df-grpo 30001 df-gdiv 30004 df-va 30103 df-vs 30107

This theorem is referenced by: nvm 30149 nvmfval 30152 nvnnncan1 30155 nvaddsub 30163

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