Theorem nvinvfval 30728

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 2737	. . . . 5 ⊢ (BaseSet‘𝑈) = (BaseSet‘𝑈)
2		nvinvfval.4	. . . . 5 ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈)
3	1, 2	nvsf 30707	. . . 4 ⊢ (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4		neg1cn 12142	. . . 4 ⊢ -1 ∈ ℂ
5		nvinvfval.3	. . . . 5 ⊢ 𝑁 = (𝑆 ∘ ◡(2nd ↾ ({-1} × V)))
6	5	curry1f 8058	. . . 4 ⊢ ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
7	3, 4, 6	sylancl 587	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
8	7	ffnd 6671	. 2 ⊢ (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
9		nvinvfval.2	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
10	9	nvgrp 30705	. . 3 ⊢ (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
11	1, 9	bafval 30692	. . . 4 ⊢ (BaseSet‘𝑈) = ran 𝐺
12		eqid 2737	. . . 4 ⊢ (inv‘𝐺) = (inv‘𝐺)
13	11, 12	grpoinvf 30620	. . 3 ⊢ (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈))
14		f1ofn 6783	. . 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 6671	. . . . 5 ⊢ (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
17	16	adantr 480	. . . 4 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
18	5	curry1val 8057	. . . 4 ⊢ ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁‘𝑥) = (-1𝑆𝑥))
19	17, 4, 18	sylancl 587	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = (-1𝑆𝑥))
20	1, 9, 2, 12	nvinv 30727	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥))
21	19, 20	eqtrd 2772	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁‘𝑥) = ((inv‘𝐺)‘𝑥))
22	8, 15, 21	eqfnfvd 6988	1 ⊢ (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {csn 4582   × cxp 5630  ^◡ccnv 5631   ↾ cres 5634   ∘ ccom 5636   Fn wfn 6495  ⟶wf 6496  –1-1-onto→wf1o 6499  ‘cfv 6500  (class class class)co 7368  2^nd c2nd 7942  ℂcc 11036  1c1 11039  -cneg 11377  GrpOpcgr 30577  invcgn 30579  NrmCVeccnv 30672   +_𝑣 cpv 30673  BaseSetcba 30674   ·_𝑠OLD cns 30675

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-reu 3353  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-po 5540  df-so 5541  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-riota 7325  df-ov 7371  df-oprab 7372  df-mpo 7373  df-1st 7943  df-2nd 7944  df-er 8645  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11180  df-mnf 11181  df-ltxr 11183  df-sub 11378  df-neg 11379  df-grpo 30581  df-gid 30582  df-ginv 30583  df-ablo 30633  df-vc 30647  df-nv 30680  df-va 30683  df-ba 30684  df-sm 30685  df-0v 30686  df-nmcv 30688

This theorem is referenced by:  hhssabloilem  31349