Theorem vsfval 31281

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 31247	. . . . 5 ⊢ −𝑣 = ( /𝑔 ∘ +𝑣 )
2	1	fveq1i 6915	. . . 4 ⊢ ( −𝑣 ‘𝑈) = (( /𝑔 ∘ +𝑣 )‘𝑈)
3		fo1st 8053	. . . . . . . 8 ⊢ 1st :V⟶ontoV
4		fof 6824	. . . . . . . 8 ⊢ (1st :V⟶ontoV → 1st :V⟶V)
5	3, 4	ax-mp 5	. . . . . . 7 ⊢ 1st :V⟶V
6		fco 6762	. . . . . . 7 ⊢ ((1st :V⟶V ∧ 1st :V⟶V) → (1st ∘ 1st ):V⟶V)
7	5, 5, 6	mp2an 705	. . . . . 6 ⊢ (1st ∘ 1st ):V⟶V
8		df-va 31243	. . . . . . 7 ⊢ +𝑣 = (1st ∘ 1st )
9	8	feq1i 6728	. . . . . 6 ⊢ ( +𝑣 :V⟶V ↔ (1st ∘ 1st ):V⟶V)
10	7, 9	mpbir 234	. . . . 5 ⊢ +𝑣 :V⟶V
11		fvco3 7014	. . . . 5 ⊢ (( +𝑣 :V⟶V ∧ 𝑈 ∈ V) → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
12	10, 11	mpan 703	. . . 4 ⊢ (𝑈 ∈ V → (( /𝑔 ∘ +𝑣 )‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
13	2, 12	eqtrid 2810	. . 3 ⊢ (𝑈 ∈ V → ( −𝑣 ‘𝑈) = ( /𝑔 ‘( +𝑣 ‘𝑈)))
14		0ngrp 31159	. . . . . 6 ⊢ ¬ ∅ ∈ GrpOp
15		vex 3458	. . . . . . . . . 10 ⊢ 𝑔 ∈ V
16	15	rnex 7954	. . . . . . . . 9 ⊢ ran 𝑔 ∈ V
17	16, 16	mpoex 8125	. . . . . . . 8 ⊢ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))) ∈ V
18		df-gdiv 31144	. . . . . . . 8 ⊢ /𝑔 = (𝑔 ∈ GrpOp ↦ (𝑥 ∈ ran 𝑔, 𝑦 ∈ ran 𝑔 ↦ (𝑥𝑔((inv‘𝑔)‘𝑦))))
19	17, 18	dmmpti 6711	. . . . . . 7 ⊢ dom /𝑔 = GrpOp
20	19	eleq2i 2856	. . . . . 6 ⊢ (∅ ∈ dom /𝑔 ↔ ∅ ∈ GrpOp)
21	14, 20	mtbir 326	. . . . 5 ⊢ ¬ ∅ ∈ dom /𝑔
22		ndmfv 6946	. . . . 5 ⊢ (¬ ∅ ∈ dom /𝑔 → ( /𝑔 ‘∅) = ∅)
23	21, 22	mp1i 14	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘∅) = ∅)
24		fvprc 6906	. . . . 5 ⊢ (¬ 𝑈 ∈ V → ( +𝑣 ‘𝑈) = ∅)
25	24	fveq2d 6918	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( /𝑔 ‘( +𝑣 ‘𝑈)) = ( /𝑔 ‘∅))
26		fvprc 6906	. . . 4 ⊢ (¬ 𝑈 ∈ V → ( −𝑣 ‘𝑈) = ∅)
27	23, 25, 26	3eqtr4rd 2809	. . 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 6917	. 2 ⊢ ( /𝑔 ‘𝐺) = ( /𝑔 ‘( +𝑣 ‘𝑈))
32	28, 29, 31	3eqtr4i 2796	1 ⊢ 𝑀 = ( /𝑔 ‘𝐺)

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2147  Vcvv 3454  ∅c0 4281  dom cdm 5656  ran crn 5657   ∘ ccom 5660  ⟶wf 6560  ⟶ontowfo 6562  ‘cfv 6564  (class class class)co 7449   ∈ cmpo 7451  1^st c1st 8031  GrpOpcgr 31137  invcgn 31139   /_𝑔 cgs 31140   +_𝑣 cpv 31233   −_𝑣 cnsb 31237

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-ov 7452  df-oprab 7453  df-mpo 7454  df-1st 8033  df-2nd 8034  df-grpo 31141  df-gdiv 31144  df-va 31243  df-vs 31247

This theorem is used by:  nvm  31289  nvmfval  31292  nvnnncan1  31295  nvaddsub  31303