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Theorem nvinvfval 30573
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 2726 . . . . 5 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 nvinvfval.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
31, 2nvsf 30552 . . . 4 (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4 neg1cn 12378 . . . 4 -1 ∈ ℂ
5 nvinvfval.3 . . . . 5 𝑁 = (𝑆(2nd ↾ ({-1} × V)))
65curry1f 8120 . . . 4 ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
73, 4, 6sylancl 584 . . 3 (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
87ffnd 6729 . 2 (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
9 nvinvfval.2 . . . 4 𝐺 = ( +𝑣𝑈)
109nvgrp 30550 . . 3 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
111, 9bafval 30537 . . . 4 (BaseSet‘𝑈) = ran 𝐺
12 eqid 2726 . . . 4 (inv‘𝐺) = (inv‘𝐺)
1311, 12grpoinvf 30465 . . 3 (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈))
14 f1ofn 6844 . . 3 ((inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈) → (inv‘𝐺) Fn (BaseSet‘𝑈))
1510, 13, 143syl 18 . 2 (𝑈 ∈ NrmCVec → (inv‘𝐺) Fn (BaseSet‘𝑈))
163ffnd 6729 . . . . 5 (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
1716adantr 479 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
185curry1val 8119 . . . 4 ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁𝑥) = (-1𝑆𝑥))
1917, 4, 18sylancl 584 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁𝑥) = (-1𝑆𝑥))
201, 9, 2, 12nvinv 30572 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥))
2119, 20eqtrd 2766 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁𝑥) = ((inv‘𝐺)‘𝑥))
228, 15, 21eqfnfvd 7047 1 (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1534  wcel 2099  Vcvv 3462  {csn 4633   × cxp 5680  ccnv 5681  cres 5684  ccom 5686   Fn wfn 6549  wf 6550  1-1-ontowf1o 6553  cfv 6554  (class class class)co 7424  2nd c2nd 8002  cc 11156  1c1 11159  -cneg 11495  GrpOpcgr 30422  invcgn 30424  NrmCVeccnv 30517   +𝑣 cpv 30518  BaseSetcba 30519   ·𝑠OLD cns 30520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-ltxr 11303  df-sub 11496  df-neg 11497  df-grpo 30426  df-gid 30427  df-ginv 30428  df-ablo 30478  df-vc 30492  df-nv 30525  df-va 30528  df-ba 30529  df-sm 30530  df-0v 30531  df-nmcv 30533
This theorem is referenced by:  hhssabloilem  31194
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