vsfval - Metamath Proof Explorer

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Theorem vsfval 29886

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 29852	. . . . 5 ⊢ −_𝑣 = ( /_𝑔 ∘ +_𝑣 )
2	1	fveq1i 6893	. . . 4 ⊢ ( −_𝑣 ‘𝑈) = (( /_𝑔 ∘ +_𝑣 )‘𝑈)
3		fo1st 7995	. . . . . . . 8 ⊢ 1^st :V–onto→V
4		fof 6806	. . . . . . . 8 ⊢ (1^st :V–onto→V → 1^st :V⟶V)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 1^st :V⟶V
6		fco 6742	. . . . . . 7 ⊢ ((1^st :V⟶V ∧ 1^st :V⟶V) → (1^st ∘ 1^st ):V⟶V)
7	5, 5, 6	mp2an 691	. . . . . 6 ⊢ (1^st ∘ 1^st ):V⟶V
8		df-va 29848	. . . . . . 7 ⊢ +_𝑣 = (1^st ∘ 1^st )
9	8	feq1i 6709	. . . . . 6 ⊢ ( +_𝑣 :V⟶V ↔ (1^st ∘ 1^st ):V⟶V)
10	7, 9	mpbir 230	. . . . 5 ⊢ +_𝑣 :V⟶V
11		fvco3 6991	. . . . 5 ⊢ (( +_𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /_𝑔 ∘ +_𝑣 )‘𝑈) = ( /_𝑔 ‘( +_𝑣 ‘𝑈)))
12	10, 11	mpan 689	. . . 4 ⊢ (𝑈 ∈ V → (( /_𝑔 ∘ +_𝑣 )‘𝑈) = ( /_𝑔 ‘( +_𝑣 ‘𝑈)))
13	2, 12	eqtrid 2785	. . 3 ⊢ (𝑈 ∈ V → ( −_𝑣 ‘𝑈) = ( /_𝑔 ‘( +_𝑣 ‘𝑈)))
14		0ngrp 29764	. . . . . 6 ⊢ ¬ ∅ ∈ GrpOp
15		vex 3479	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
16	15	rnex 7903	. . . . . . . . 9 ⊢ ran 𝑔 ∈ V
17	16, 16	mpoex 8066	. . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18		df-gdiv 29749	. . . . . . . 8 ⊢ /_𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
19	17, 18	dmmpti 6695	. . . . . . 7 ⊢ dom /_𝑔 = GrpOp
20	19	eleq2i 2826	. . . . . 6 ⊢ (∅ ∈ dom /_𝑔 ↔ ∅ ∈ GrpOp)
21	14, 20	mtbir 323	. . . . 5 ⊢ ¬ ∅ ∈ dom /_𝑔
22		ndmfv 6927	. . . . 5 ⊢ (¬ ∅ ∈ dom /_𝑔 → ( /_𝑔 ‘∅) = ∅)
23	21, 22	mp1i 13	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /_𝑔 ‘∅) = ∅)
24		fvprc 6884	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +_𝑣 ‘𝑈) = ∅)
25	24	fveq2d 6896	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /_𝑔 ‘( +_𝑣 ‘𝑈)) = ( /_𝑔 ‘∅))
26		fvprc 6884	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( −_𝑣 ‘𝑈) = ∅)
27	23, 25, 26	3eqtr4rd 2784	. . 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 6895	. 2 ⊢ ( /_𝑔 ‘𝐺) = ( /_𝑔 ‘( +_𝑣 ‘𝑈))
32	28, 29, 31	3eqtr4i 2771	1 ⊢ 𝑀 = ( /_𝑔 ‘𝐺)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4323 dom cdm 5677 ran crn 5678 ∘ ccom 5681 ⟶wf 6540 –onto→wfo 6542 ‘cfv 6544 (class class class)co 7409 ∈ cmpo 7411 1^st c1st 7973 GrpOpcgr 29742 invcgn 29744 /_𝑔 cgs 29745 +_𝑣 cpv 29838 −_𝑣 cnsb 29842

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-ov 7412 df-oprab 7413 df-mpo 7414 df-1st 7975 df-2nd 7976 df-grpo 29746 df-gdiv 29749 df-va 29848 df-vs 29852

This theorem is referenced by: nvm 29894 nvmfval 29897 nvnnncan1 29900 nvaddsub 29908

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