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Theorem nvinvfval 28430
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 2824 . . . . 5 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 nvinvfval.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
31, 2nvsf 28409 . . . 4 (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4 neg1cn 11748 . . . 4 -1 ∈ ℂ
5 nvinvfval.3 . . . . 5 𝑁 = (𝑆(2nd ↾ ({-1} × V)))
65curry1f 7797 . . . 4 ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
73, 4, 6sylancl 589 . . 3 (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
87ffnd 6504 . 2 (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
9 nvinvfval.2 . . . 4 𝐺 = ( +𝑣𝑈)
109nvgrp 28407 . . 3 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
111, 9bafval 28394 . . . 4 (BaseSet‘𝑈) = ran 𝐺
12 eqid 2824 . . . 4 (inv‘𝐺) = (inv‘𝐺)
1311, 12grpoinvf 28322 . . 3 (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈))
14 f1ofn 6607 . . 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 6504 . . . . 5 (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
1716adantr 484 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
185curry1val 7796 . . . 4 ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁𝑥) = (-1𝑆𝑥))
1917, 4, 18sylancl 589 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁𝑥) = (-1𝑆𝑥))
201, 9, 2, 12nvinv 28429 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥))
2119, 20eqtrd 2859 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁𝑥) = ((inv‘𝐺)‘𝑥))
228, 15, 21eqfnfvd 6796 1 (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2115  Vcvv 3480  {csn 4550   × cxp 5540  ccnv 5541  cres 5544  ccom 5546   Fn wfn 6338  wf 6339  1-1-ontowf1o 6342  cfv 6343  (class class class)co 7149  2nd c2nd 7683  cc 10533  1c1 10536  -cneg 10869  GrpOpcgr 28279  invcgn 28281  NrmCVeccnv 28374   +𝑣 cpv 28375  BaseSetcba 28376   ·𝑠OLD cns 28377
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611
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 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-id 5447  df-po 5461  df-so 5462  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-1st 7684  df-2nd 7685  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-ltxr 10678  df-sub 10870  df-neg 10871  df-grpo 28283  df-gid 28284  df-ginv 28285  df-ablo 28335  df-vc 28349  df-nv 28382  df-va 28385  df-ba 28386  df-sm 28387  df-0v 28388  df-nmcv 28390
This theorem is referenced by:  hhssabloilem  29051
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