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Theorem nvop 30401
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvop.2 ๐บ = ( +๐‘ฃ โ€˜๐‘ˆ)
nvop.4 ๐‘† = ( ยท๐‘ OLD โ€˜๐‘ˆ)
nvop.6 ๐‘ = (normCVโ€˜๐‘ˆ)
Assertion
Ref Expression
nvop (๐‘ˆ โˆˆ NrmCVec โ†’ ๐‘ˆ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)

Proof of Theorem nvop
StepHypRef Expression
1 nvrel 30327 . . 3 Rel NrmCVec
2 1st2nd 8019 . . 3 ((Rel NrmCVec โˆง ๐‘ˆ โˆˆ NrmCVec) โ†’ ๐‘ˆ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ)
31, 2mpan 687 . 2 (๐‘ˆ โˆˆ NrmCVec โ†’ ๐‘ˆ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ)
4 nvop.6 . . . . 5 ๐‘ = (normCVโ€˜๐‘ˆ)
54nmcvfval 30332 . . . 4 ๐‘ = (2nd โ€˜๐‘ˆ)
65opeq2i 4870 . . 3 โŸจ(1st โ€˜๐‘ˆ), ๐‘โŸฉ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ
7 eqid 2724 . . . . 5 (1st โ€˜๐‘ˆ) = (1st โ€˜๐‘ˆ)
8 nvop.2 . . . . 5 ๐บ = ( +๐‘ฃ โ€˜๐‘ˆ)
9 nvop.4 . . . . 5 ๐‘† = ( ยท๐‘ OLD โ€˜๐‘ˆ)
107, 8, 9nvvop 30334 . . . 4 (๐‘ˆ โˆˆ NrmCVec โ†’ (1st โ€˜๐‘ˆ) = โŸจ๐บ, ๐‘†โŸฉ)
1110opeq1d 4872 . . 3 (๐‘ˆ โˆˆ NrmCVec โ†’ โŸจ(1st โ€˜๐‘ˆ), ๐‘โŸฉ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
126, 11eqtr3id 2778 . 2 (๐‘ˆ โˆˆ NrmCVec โ†’ โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
133, 12eqtrd 2764 1 (๐‘ˆ โˆˆ NrmCVec โ†’ ๐‘ˆ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   = wceq 1533   โˆˆ wcel 2098  โŸจcop 4627  Rel wrel 5672  โ€˜cfv 6534  1st c1st 7967  2nd c2nd 7968  NrmCVeccnv 30309   +๐‘ฃ cpv 30310   ยท๐‘ OLD cns 30312  normCVcnmcv 30315
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540  df-fv 6542  df-oprab 7406  df-1st 7969  df-2nd 7970  df-vc 30284  df-nv 30317  df-va 30320  df-sm 30322  df-nmcv 30325
This theorem is referenced by:  sspval  30448  isph  30547  hilhhi  30889
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