Theorem vsfval 28410

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 28376	. . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 )
2	1	fveq1i 6671	. . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈)
3		fo1st 7709	. . . . . . . 8 ⊢ 1st :V–onto→V
4		fof 6590	. . . . . . . 8 ⊢ (1st :V–onto→V → 1st :V⟶V)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 1st :V⟶V
6		fco 6531	. . . . . . 7 ⊢ ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V)
7	5, 5, 6	mp2an 690	. . . . . 6 ⊢ (1st ∘ 1st ):V⟶V
8		df-va 28372	. . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st )
9	8	feq1i 6505	. . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V)
10	7, 9	mpbir 233	. . . . 5 ⊢ +𝑣 :V⟶V
11		fvco3 6760	. . . . 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	syl5eq 2868	. . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
14		0ngrp 28288	. . . . . 6 ⊢ ¬ ∅ ∈ GrpOp
15		vex 3497	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
16	15	rnex 7617	. . . . . . . . 9 ⊢ ran 𝑔 ∈ V
17	16, 16	mpoex 7777	. . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18		df-gdiv 28273	. . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
19	17, 18	dmmpti 6492	. . . . . . 7 ⊢ dom /𝑔 = GrpOp
20	19	eleq2i 2904	. . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp)
21	14, 20	mtbir 325	. . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔
22		ndmfv 6700	. . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅)
23	21, 22	mp1i 13	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅)
24		fvprc 6663	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅)
25	24	fveq2d 6674	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅))
26		fvprc 6663	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅)
27	23, 25, 26	3eqtr4rd 2867	. . 3 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
28	13, 27	pm2.61i 184	. 2 ⊢ ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈))
29		vsfval.3	. 2 ⊢ 𝑀 = ( −𝑣 ‘𝑈)
30		vsfval.2	. . 3 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
31	30	fveq2i 6673	. 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈))
32	28, 29, 31	3eqtr4i 2854	1 ⊢ 𝑀 = ( /𝑔 ‘𝐺)

Syntax hints:  ¬ wn 3   = wceq 1537   ∈ wcel 2114  Vcvv 3494  ∅c0 4291  dom cdm 5555  ran crn 5556   ∘ ccom 5559  ⟶wf 6351  –onto→wfo 6353  ‘cfv 6355  (class class class)co 7156   ∈ cmpo 7158  1^st c1st 7687  GrpOpcgr 28266  invcgn 28268   /_𝑔 cgs 28269   +_𝑣 cpv 28362   −_𝑣 cnsb 28366

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-ov 7159  df-oprab 7160  df-mpo 7161  df-1st 7689  df-2nd 7690  df-grpo 28270  df-gdiv 28273  df-va 28372  df-vs 28376

This theorem is referenced by:  nvm  28418  nvmfval  28421  nvnnncan1  28424  nvaddsub  28432