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Theorem nvinvfval 28020
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 2799 . . . . 5 (BaseSet‘𝑈) = (BaseSet‘𝑈)
2 nvinvfval.4 . . . . 5 𝑆 = ( ·𝑠OLD𝑈)
31, 2nvsf 27999 . . . 4 (𝑈 ∈ NrmCVec → 𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈))
4 neg1cn 11434 . . . 4 -1 ∈ ℂ
5 nvinvfval.3 . . . . 5 𝑁 = (𝑆(2nd ↾ ({-1} × V)))
65curry1f 7508 . . . 4 ((𝑆:(ℂ × (BaseSet‘𝑈))⟶(BaseSet‘𝑈) ∧ -1 ∈ ℂ) → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
73, 4, 6sylancl 581 . . 3 (𝑈 ∈ NrmCVec → 𝑁:(BaseSet‘𝑈)⟶(BaseSet‘𝑈))
87ffnd 6257 . 2 (𝑈 ∈ NrmCVec → 𝑁 Fn (BaseSet‘𝑈))
9 nvinvfval.2 . . . 4 𝐺 = ( +𝑣𝑈)
109nvgrp 27997 . . 3 (𝑈 ∈ NrmCVec → 𝐺 ∈ GrpOp)
111, 9bafval 27984 . . . 4 (BaseSet‘𝑈) = ran 𝐺
12 eqid 2799 . . . 4 (inv‘𝐺) = (inv‘𝐺)
1311, 12grpoinvf 27912 . . 3 (𝐺 ∈ GrpOp → (inv‘𝐺):(BaseSet‘𝑈)–1-1-onto→(BaseSet‘𝑈))
14 f1ofn 6357 . . 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 6257 . . . . 5 (𝑈 ∈ NrmCVec → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
1716adantr 473 . . . 4 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → 𝑆 Fn (ℂ × (BaseSet‘𝑈)))
185curry1val 7507 . . . 4 ((𝑆 Fn (ℂ × (BaseSet‘𝑈)) ∧ -1 ∈ ℂ) → (𝑁𝑥) = (-1𝑆𝑥))
1917, 4, 18sylancl 581 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁𝑥) = (-1𝑆𝑥))
201, 9, 2, 12nvinv 28019 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (-1𝑆𝑥) = ((inv‘𝐺)‘𝑥))
2119, 20eqtrd 2833 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑥 ∈ (BaseSet‘𝑈)) → (𝑁𝑥) = ((inv‘𝐺)‘𝑥))
228, 15, 21eqfnfvd 6540 1 (𝑈 ∈ NrmCVec → 𝑁 = (inv‘𝐺))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  Vcvv 3385  {csn 4368   × cxp 5310  ccnv 5311  cres 5314  ccom 5316   Fn wfn 6096  wf 6097  1-1-ontowf1o 6100  cfv 6101  (class class class)co 6878  2nd c2nd 7400  cc 10222  1c1 10225  -cneg 10557  GrpOpcgr 27869  invcgn 27871  NrmCVeccnv 27964   +𝑣 cpv 27965  BaseSetcba 27966   ·𝑠OLD cns 27967
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183  ax-resscn 10281  ax-1cn 10282  ax-icn 10283  ax-addcl 10284  ax-addrcl 10285  ax-mulcl 10286  ax-mulrcl 10287  ax-mulcom 10288  ax-addass 10289  ax-mulass 10290  ax-distr 10291  ax-i2m1 10292  ax-1ne0 10293  ax-1rid 10294  ax-rnegex 10295  ax-rrecex 10296  ax-cnre 10297  ax-pre-lttri 10298  ax-pre-lttrn 10299  ax-pre-ltadd 10300
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-nel 3075  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-po 5233  df-so 5234  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-riota 6839  df-ov 6881  df-oprab 6882  df-mpt2 6883  df-1st 7401  df-2nd 7402  df-er 7982  df-en 8196  df-dom 8197  df-sdom 8198  df-pnf 10365  df-mnf 10366  df-ltxr 10368  df-sub 10558  df-neg 10559  df-grpo 27873  df-gid 27874  df-ginv 27875  df-ablo 27925  df-vc 27939  df-nv 27972  df-va 27975  df-ba 27976  df-sm 27977  df-0v 27978  df-nmcv 27980
This theorem is referenced by:  hhssabloilem  28643
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