MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvop Structured version   Visualization version   GIF version

Theorem nvop 29916
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 29842 . . 3 Rel NrmCVec
2 1st2nd 8021 . . 3 ((Rel NrmCVec โˆง ๐‘ˆ โˆˆ NrmCVec) โ†’ ๐‘ˆ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ)
31, 2mpan 688 . 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 29847 . . . 4 ๐‘ = (2nd โ€˜๐‘ˆ)
65opeq2i 4876 . . 3 โŸจ(1st โ€˜๐‘ˆ), ๐‘โŸฉ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ
7 eqid 2732 . . . . 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 29849 . . . 4 (๐‘ˆ โˆˆ NrmCVec โ†’ (1st โ€˜๐‘ˆ) = โŸจ๐บ, ๐‘†โŸฉ)
1110opeq1d 4878 . . 3 (๐‘ˆ โˆˆ NrmCVec โ†’ โŸจ(1st โ€˜๐‘ˆ), ๐‘โŸฉ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
126, 11eqtr3id 2786 . 2 (๐‘ˆ โˆˆ NrmCVec โ†’ โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
133, 12eqtrd 2772 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 1541   โˆˆ wcel 2106  โŸจcop 4633  Rel wrel 5680  โ€˜cfv 6540  1st c1st 7969  2nd c2nd 7970  NrmCVeccnv 29824   +๐‘ฃ cpv 29825   ยท๐‘ OLD cns 29827  normCVcnmcv 29830
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-oprab 7409  df-1st 7971  df-2nd 7972  df-vc 29799  df-nv 29832  df-va 29835  df-sm 29837  df-nmcv 29840
This theorem is referenced by:  sspval  29963  isph  30062  hilhhi  30404
  Copyright terms: Public domain W3C validator