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Theorem nvinvfval 27344
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 = ( +𝑣𝑈)
nvinvfval.4 𝑆 = ( ·𝑠OLD𝑈)
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.
StepHypRef Expression
1 eqid 2621 . . . . 5 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 nvinvfval.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
31, 2nvsf 27323 . . . 4 (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4 neg1cn 11068 . . . 4 -1 ∈ ℂ
5 nvinvfval.3 . . . . 5 𝑁 = (𝑆(2nd ↾ ({-1} × V)))
65curry1f 7216 . . . 4 ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
73, 4, 6sylancl 693 . . 3 (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
8 ffn 6002 . . 3 (𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈) → 𝑁 Fn (BaseSet‘𝑈))
97, 8syl 17 . 2 (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
10 nvinvfval.2 . . . 4 𝐺 = ( +𝑣𝑈)
1110nvgrp 27321 . . 3 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
121, 10bafval 27308 . . . 4 (BaseSet‘𝑈) = ran 𝐺
13 eqid 2621 . . . 4 (inv‘𝐺) = (inv‘𝐺)
1412, 13grpoinvf 27235 . . 3 (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈))
15 f1ofn 6095 . . 3 ((inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈) → (inv‘𝐺) Fn (BaseSet‘𝑈))
1611, 14, 153syl 18 . 2 (𝑈 ∈ NrmCVec → (inv‘𝐺) Fn (BaseSet‘𝑈))
17 ffn 6002 . . . . . 6 (𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
183, 17syl 17 . . . . 5 (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
1918adantr 481 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
205curry1val 7215 . . . 4 ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁𝑥) = (-1𝑆𝑥))
2119, 4, 20sylancl 693 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁𝑥) = (-1𝑆𝑥))
221, 10, 2, 13nvinv 27343 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥))
2321, 22eqtrd 2655 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁𝑥) = ((inv‘𝐺)‘𝑥))
249, 16, 23eqfnfvd 6270 1 (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  {csn 4148   × cxp 5072  ccnv 5073  cres 5076  ccom 5078   Fn wfn 5842  wf 5843  1-1-ontowf1o 5846  cfv 5847  (class class class)co 6604  2nd c2nd 7112  cc 9878  1c1 9881  -cneg 10211  GrpOpcgr 27192  invcgn 27194  NrmCVeccnv 27288   +𝑣 cpv 27289  BaseSetcba 27290   ·𝑠OLD cns 27291
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-po 4995  df-so 4996  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-ltxr 10023  df-sub 10212  df-neg 10213  df-grpo 27196  df-gid 27197  df-ginv 27198  df-ablo 27248  df-vc 27263  df-nv 27296  df-va 27299  df-ba 27300  df-sm 27301  df-0v 27302  df-nmcv 27304
This theorem is referenced by:  hhssabloilem  27967
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