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Theorem nvsid 28509
 Description: Identity element for the scalar product of a normed complex vector space. (Contributed by NM, 4-Dec-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvscl.1 𝑋 = (BaseSet‘𝑈)
nvscl.4 𝑆 = ( ·𝑠OLD𝑈)
Assertion
Ref Expression
nvsid ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1𝑆𝐴) = 𝐴)

Proof of Theorem nvsid
StepHypRef Expression
1 eqid 2758 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 28497 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2758 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 28485 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvscl.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 28487 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvscl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 28486 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vcidOLD 28446 . 2 (((1st𝑈) ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
102, 9sylan 583 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1𝑆𝐴) = 𝐴)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ‘cfv 6335  (class class class)co 7150  1st c1st 7691  1c1 10576  CVecOLDcvc 28440  NrmCVeccnv 28466   +𝑣 cpv 28467  BaseSetcba 28468   ·𝑠OLD cns 28469 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5156  ax-sep 5169  ax-nul 5176  ax-pr 5298  ax-un 7459 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-iun 4885  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-ov 7153  df-oprab 7154  df-1st 7693  df-2nd 7694  df-vc 28441  df-nv 28474  df-va 28477  df-ba 28478  df-sm 28479  df-0v 28480  df-nmcv 28482 This theorem is referenced by:  nvmul0or  28532  nvpi  28549  nvge0  28555  ipval2lem3  28587  ipval2  28589  ipidsq  28592  lnoadd  28640  ip1ilem  28708  ip2i  28710  ipdirilem  28711  ipasslem1  28713  ipasslem4  28716  ipasslem10  28721  minvecolem2  28757  hlmulid  28787
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