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Theorem nvop 29660
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 29586 . . 3 Rel NrmCVec
2 1st2nd 7972 . . 3 ((Rel NrmCVec โˆง ๐‘ˆ โˆˆ NrmCVec) โ†’ ๐‘ˆ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ)
31, 2mpan 689 . 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 29591 . . . 4 ๐‘ = (2nd โ€˜๐‘ˆ)
65opeq2i 4835 . . 3 โŸจ(1st โ€˜๐‘ˆ), ๐‘โŸฉ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ
7 eqid 2733 . . . . 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 29593 . . . 4 (๐‘ˆ โˆˆ NrmCVec โ†’ (1st โ€˜๐‘ˆ) = โŸจ๐บ, ๐‘†โŸฉ)
1110opeq1d 4837 . . 3 (๐‘ˆ โˆˆ NrmCVec โ†’ โŸจ(1st โ€˜๐‘ˆ), ๐‘โŸฉ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
126, 11eqtr3id 2787 . 2 (๐‘ˆ โˆˆ NrmCVec โ†’ โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
133, 12eqtrd 2773 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 1542   โˆˆ wcel 2107  โŸจcop 4593  Rel wrel 5639  โ€˜cfv 6497  1st c1st 7920  2nd c2nd 7921  NrmCVeccnv 29568   +๐‘ฃ cpv 29569   ยท๐‘ OLD cns 29571  normCVcnmcv 29574
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 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-oprab 7362  df-1st 7922  df-2nd 7923  df-vc 29543  df-nv 29576  df-va 29579  df-sm 29581  df-nmcv 29584
This theorem is referenced by:  sspval  29707  isph  29806  hilhhi  30148
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