Theorem vsfval 30605

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 30571	. . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 )
2	1	fveq1i 6818	. . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈)
3		fo1st 7936	. . . . . . . 8 ⊢ 1st :V–onto→V
4		fof 6730	. . . . . . . 8 ⊢ (1st :V–onto→V → 1st :V⟶V)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 1st :V⟶V
6		fco 6670	. . . . . . 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 30567	. . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st )
9	8	feq1i 6637	. . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V)
10	7, 9	mpbir 231	. . . . 5 ⊢ +𝑣 :V⟶V
11		fvco3 6916	. . . . 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 2778	. . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
14		0ngrp 30483	. . . . . 6 ⊢ ¬ ∅ ∈ GrpOp
15		vex 3440	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
16	15	rnex 7835	. . . . . . . . 9 ⊢ ran 𝑔 ∈ V
17	16, 16	mpoex 8006	. . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18		df-gdiv 30468	. . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
19	17, 18	dmmpti 6620	. . . . . . 7 ⊢ dom /𝑔 = GrpOp
20	19	eleq2i 2823	. . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp)
21	14, 20	mtbir 323	. . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔
22		ndmfv 6849	. . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅)
23	21, 22	mp1i 13	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅)
24		fvprc 6809	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅)
25	24	fveq2d 6821	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅))
26		fvprc 6809	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅)
27	23, 25, 26	3eqtr4rd 2777	. . 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 6820	. 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈))
32	28, 29, 31	3eqtr4i 2764	1 ⊢ 𝑀 = ( /𝑔 ‘𝐺)

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ∅c0 4278  dom cdm 5611  ran crn 5612   ∘ ccom 5615  ⟶wf 6472  –onto→wfo 6474  ‘cfv 6476  (class class class)co 7341   ∈ cmpo 7343  1^st c1st 7914  GrpOpcgr 30461  invcgn 30463   /_𝑔 cgs 30464   +_𝑣 cpv 30557   −_𝑣 cnsb 30561

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5212  ax-sep 5229  ax-nul 5239  ax-pow 5298  ax-pr 5365  ax-un 7663

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4279  df-if 4471  df-pw 4547  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4855  df-iun 4938  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5506  df-xp 5617  df-rel 5618  df-cnv 5619  df-co 5620  df-dm 5621  df-rn 5622  df-res 5623  df-ima 5624  df-iota 6432  df-fun 6478  df-fn 6479  df-f 6480  df-f1 6481  df-fo 6482  df-f1o 6483  df-fv 6484  df-ov 7344  df-oprab 7345  df-mpo 7346  df-1st 7916  df-2nd 7917  df-grpo 30465  df-gdiv 30468  df-va 30567  df-vs 30571

This theorem is referenced by:  nvm  30613  nvmfval  30616  nvnnncan1  30619  nvaddsub  30627