Theorem nvinvfval 28423

Description: Function for the negative of a vector on a normed complex vector space, in terms of the underlying addition group inverse. (We currently do not have a separate notation for the negative of a vector.) (Contributed by NM, 27-Mar-2008.) (New usage is discouraged.)

Hypotheses

Ref	Expression
nvinvfval.2	⊢ 𝐺 = ( +𝑣 ‘𝑈)
nvinvfval.4	⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
nvinvfval.3	⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V)))

Assertion

Ref	Expression
nvinvfval	⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))

Proof of Theorem nvinvfval

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2798	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		nvinvfval.4	. . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
3	1, 2	nvsf 28402	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4		neg1cn 11739	. . . 4 ⊢ -1 ∈ ℂ
5		nvinvfval.3	. . . . 5 ⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V)))
6	5	curry1f 7784	. . . 4 ⊢ ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
7	3, 4, 6	sylancl 589	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
8	7	ffnd 6488	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
9		nvinvfval.2	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
10	9	nvgrp 28400	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
11	1, 9	bafval 28387	. . . 4 ⊢ (BaseSet‘𝑈) = ran 𝐺
12		eqid 2798	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
13	11, 12	grpoinvf 28315	. . 3 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈))
14		f1ofn 6591	. . 3 ⊢ ((inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈) → (inv‘𝐺) Fn (BaseSet‘𝑈))
15	10, 13, 14	3syl 18	. 2 ⊢ (𝑈 ∈ NrmCVec → (inv‘𝐺) Fn (BaseSet‘𝑈))
16	3	ffnd 6488	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
17	16	adantr 484	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
18	5	curry1val 7783	. . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁‘𝑥) = (-1𝑆𝑥))
19	17, 4, 18	sylancl 589	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = (-1𝑆𝑥))
20	1, 9, 2, 12	nvinv 28422	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥))
21	19, 20	eqtrd 2833	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = ((inv‘𝐺)‘𝑥))
22	8, 15, 21	eqfnfvd 6782	1 ⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  Vcvv 3441  {csn 4525   × cxp 5517  ^◡ccnv 5518   ↾ cres 5521   ∘ ccom 5523   Fn wfn 6319  ⟶wf 6320  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  2^nd c2nd 7670  ℂcc 10524  1c1 10527  -cneg 10860  GrpOpcgr 28272  invcgn 28274  NrmCVeccnv 28367   +_𝑣 cpv 28368  BaseSetcba 28369   ·_𝑠OLD cns 28370

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-ltxr 10669  df-sub 10861  df-neg 10862  df-grpo 28276  df-gid 28277  df-ginv 28278  df-ablo 28328  df-vc 28342  df-nv 28375  df-va 28378  df-ba 28379  df-sm 28380  df-0v 28381  df-nmcv 28383

This theorem is referenced by:  hhssabloilem  29044