Theorem vsfval 30595

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 30561	. . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 )
2	1	fveq1i 6827	. . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈)
3		fo1st 7951	. . . . . . . 8 ⊢ 1st :V–onto→V
4		fof 6740	. . . . . . . 8 ⊢ (1st :V–onto→V → 1st :V⟶V)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 1st :V⟶V
6		fco 6680	. . . . . . 7 ⊢ ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V)
7	5, 5, 6	mp2an 692	. . . . . 6 ⊢ (1st ∘ 1st ):V⟶V
8		df-va 30557	. . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st )
9	8	feq1i 6647	. . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V)
10	7, 9	mpbir 231	. . . . 5 ⊢ +𝑣 :V⟶V
11		fvco3 6926	. . . . 5 ⊢ (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
12	10, 11	mpan 690	. . . 4 ⊢ (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
13	2, 12	eqtrid 2776	. . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
14		0ngrp 30473	. . . . . 6 ⊢ ¬ ∅ ∈ GrpOp
15		vex 3442	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
16	15	rnex 7850	. . . . . . . . 9 ⊢ ran 𝑔 ∈ V
17	16, 16	mpoex 8021	. . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18		df-gdiv 30458	. . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
19	17, 18	dmmpti 6630	. . . . . . 7 ⊢ dom /𝑔 = GrpOp
20	19	eleq2i 2820	. . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp)
21	14, 20	mtbir 323	. . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔
22		ndmfv 6859	. . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅)
23	21, 22	mp1i 13	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅)
24		fvprc 6818	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅)
25	24	fveq2d 6830	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅))
26		fvprc 6818	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅)
27	23, 25, 26	3eqtr4rd 2775	. . 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 6829	. 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈))
32	28, 29, 31	3eqtr4i 2762	1 ⊢ 𝑀 = ( /𝑔 ‘𝐺)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  Vcvv 3438  ∅c0 4286  dom cdm 5623  ran crn 5624   ∘ ccom 5627  ⟶wf 6482  –onto→wfo 6484  ‘cfv 6486  (class class class)co 7353   ∈ cmpo 7355  1^st c1st 7929  GrpOpcgr 30451  invcgn 30453   /_𝑔 cgs 30454   +_𝑣 cpv 30547   −_𝑣 cnsb 30551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-ov 7356  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932  df-grpo 30455  df-gdiv 30458  df-va 30557  df-vs 30561

This theorem is referenced by:  nvm  30603  nvmfval  30606  nvnnncan1  30609  nvaddsub  30617