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

Theorem nvop 30480
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 30406 . . 3 Rel NrmCVec
2 1st2nd 8038 . . 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 30411 . . . 4 ๐‘ = (2nd โ€˜๐‘ˆ)
65opeq2i 4874 . . 3 โŸจ(1st โ€˜๐‘ˆ), ๐‘โŸฉ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ
7 eqid 2728 . . . . 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 30413 . . . 4 (๐‘ˆ โˆˆ NrmCVec โ†’ (1st โ€˜๐‘ˆ) = โŸจ๐บ, ๐‘†โŸฉ)
1110opeq1d 4876 . . 3 (๐‘ˆ โˆˆ NrmCVec โ†’ โŸจ(1st โ€˜๐‘ˆ), ๐‘โŸฉ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
126, 11eqtr3id 2782 . 2 (๐‘ˆ โˆˆ NrmCVec โ†’ โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
133, 12eqtrd 2768 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 1534   โˆˆ wcel 2099  โŸจcop 4631  Rel wrel 5678  โ€˜cfv 6543  1st c1st 7986  2nd c2nd 7987  NrmCVeccnv 30388   +๐‘ฃ cpv 30389   ยท๐‘ OLD cns 30391  normCVcnmcv 30394
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5571  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-oprab 7419  df-1st 7988  df-2nd 7989  df-vc 30363  df-nv 30396  df-va 30399  df-sm 30401  df-nmcv 30404
This theorem is referenced by:  sspval  30527  isph  30626  hilhhi  30968
  Copyright terms: Public domain W3C validator