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Theorem nvsid 29374
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 2738 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 29362 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2738 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 29350 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvscl.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 29352 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvscl.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 29351 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vcidOLD 29311 . 2 (((1st𝑈) ∈ CVecOLD𝐴𝑋) → (1𝑆𝐴) = 𝐴)
102, 9sylan 581 1 ((𝑈 ∈ NrmCVec ∧ 𝐴𝑋) → (1𝑆𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  cfv 6492  (class class class)co 7350  1st c1st 7910  1c1 10986  CVecOLDcvc 29305  NrmCVeccnv 29331   +𝑣 cpv 29332  BaseSetcba 29333   ·𝑠OLD cns 29334
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2709  ax-rep 5241  ax-sep 5255  ax-nul 5262  ax-pr 5383  ax-un 7663
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7353  df-oprab 7354  df-1st 7912  df-2nd 7913  df-vc 29306  df-nv 29339  df-va 29342  df-ba 29343  df-sm 29344  df-0v 29345  df-nmcv 29347
This theorem is referenced by:  nvmul0or  29397  nvpi  29414  nvge0  29420  ipval2lem3  29452  ipval2  29454  ipidsq  29457  lnoadd  29505  ip1ilem  29573  ip2i  29575  ipdirilem  29576  ipasslem1  29578  ipasslem4  29581  ipasslem10  29586  minvecolem2  29622  hlmulid  29652
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