Theorem nvinvfval 29002

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 2738	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		nvinvfval.4	. . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
3	1, 2	nvsf 28981	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4		neg1cn 12087	. . . 4 ⊢ -1 ∈ ℂ
5		nvinvfval.3	. . . . 5 ⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V)))
6	5	curry1f 7946	. . . 4 ⊢ ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
7	3, 4, 6	sylancl 586	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
8	7	ffnd 6601	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
9		nvinvfval.2	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
10	9	nvgrp 28979	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
11	1, 9	bafval 28966	. . . 4 ⊢ (BaseSet‘𝑈) = ran 𝐺
12		eqid 2738	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
13	11, 12	grpoinvf 28894	. . 3 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈))
14		f1ofn 6717	. . 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 6601	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
17	16	adantr 481	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
18	5	curry1val 7945	. . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁‘𝑥) = (-1𝑆𝑥))
19	17, 4, 18	sylancl 586	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = (-1𝑆𝑥))
20	1, 9, 2, 12	nvinv 29001	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥))
21	19, 20	eqtrd 2778	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = ((inv‘𝐺)‘𝑥))
22	8, 15, 21	eqfnfvd 6912	1 ⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106  Vcvv 3432  {csn 4561   × cxp 5587  ^◡ccnv 5588   ↾ cres 5591   ∘ ccom 5593   Fn wfn 6428  ⟶wf 6429  –1-1-onto→wf1o 6432  ‘cfv 6433  (class class class)co 7275  2^nd c2nd 7830  ℂcc 10869  1c1 10872  -cneg 11206  GrpOpcgr 28851  invcgn 28853  NrmCVeccnv 28946   +_𝑣 cpv 28947  BaseSetcba 28948   ·_𝑠OLD cns 28949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-ltxr 11014  df-sub 11207  df-neg 11208  df-grpo 28855  df-gid 28856  df-ginv 28857  df-ablo 28907  df-vc 28921  df-nv 28954  df-va 28957  df-ba 28958  df-sm 28959  df-0v 28960  df-nmcv 28962

This theorem is referenced by:  hhssabloilem  29623