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Theorem nvsid 28321
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 2825 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 28309 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2825 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 28297 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvscl.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 28299 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvscl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 28298 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vcidOLD 28258 . 2 (((1st𝑈) ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
102, 9sylan 580 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1𝑆𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1530  wcel 2107  cfv 6351  (class class class)co 7151  1st c1st 7681  1c1 10530  CVecOLDcvc 28252  NrmCVeccnv 28278   +𝑣 cpv 28279  BaseSetcba 28280   ·𝑠OLD cns 28281
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-oprab 7155  df-1st 7683  df-2nd 7684  df-vc 28253  df-nv 28286  df-va 28289  df-ba 28290  df-sm 28291  df-0v 28292  df-nmcv 28294
This theorem is referenced by:  nvmul0or  28344  nvpi  28361  nvge0  28367  ipval2lem3  28399  ipval2  28401  ipidsq  28404  lnoadd  28452  ip1ilem  28520  ip2i  28522  ipdirilem  28523  ipasslem1  28525  ipasslem4  28528  ipasslem10  28533  minvecolem2  28569  hlmulid  28599
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