Theorem vsfval 29283

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 29249	. . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 )
2	1	fveq1i 6826	. . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈)
3		fo1st 7919	. . . . . . . 8 ⊢ 1st :V–onto→V
4		fof 6739	. . . . . . . 8 ⊢ (1st :V–onto→V → 1st :V⟶V)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 1st :V⟶V
6		fco 6675	. . . . . . 7 ⊢ ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V)
7	5, 5, 6	mp2an 689	. . . . . 6 ⊢ (1st ∘ 1st ):V⟶V
8		df-va 29245	. . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st )
9	8	feq1i 6642	. . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V)
10	7, 9	mpbir 230	. . . . 5 ⊢ +𝑣 :V⟶V
11		fvco3 6923	. . . . 5 ⊢ (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
12	10, 11	mpan 687	. . . 4 ⊢ (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
13	2, 12	eqtrid 2788	. . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
14		0ngrp 29161	. . . . . 6 ⊢ ¬ ∅ ∈ GrpOp
15		vex 3445	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
16	15	rnex 7827	. . . . . . . . 9 ⊢ ran 𝑔 ∈ V
17	16, 16	mpoex 7988	. . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18		df-gdiv 29146	. . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
19	17, 18	dmmpti 6628	. . . . . . 7 ⊢ dom /𝑔 = GrpOp
20	19	eleq2i 2828	. . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp)
21	14, 20	mtbir 322	. . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔
22		ndmfv 6860	. . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅)
23	21, 22	mp1i 13	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅)
24		fvprc 6817	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅)
25	24	fveq2d 6829	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅))
26		fvprc 6817	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅)
27	23, 25, 26	3eqtr4rd 2787	. . 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 6828	. 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈))
32	28, 29, 31	3eqtr4i 2774	1 ⊢ 𝑀 = ( /𝑔 ‘𝐺)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2105  Vcvv 3441  ∅c0 4269  dom cdm 5620  ran crn 5621   ∘ ccom 5624  ⟶wf 6475  –onto→wfo 6477  ‘cfv 6479  (class class class)co 7337   ∈ cmpo 7339  1^st c1st 7897  GrpOpcgr 29139  invcgn 29141   /_𝑔 cgs 29142   +_𝑣 cpv 29235   −_𝑣 cnsb 29239

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-oprab 7341  df-mpo 7342  df-1st 7899  df-2nd 7900  df-grpo 29143  df-gdiv 29146  df-va 29245  df-vs 29249

This theorem is referenced by:  nvm  29291  nvmfval  29294  nvnnncan1  29297  nvaddsub  29305