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Theorem phop 29809
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.)
Hypotheses
Ref Expression
phop.2 ๐บ = ( +๐‘ฃ โ€˜๐‘ˆ)
phop.4 ๐‘† = ( ยท๐‘ OLD โ€˜๐‘ˆ)
phop.6 ๐‘ = (normCVโ€˜๐‘ˆ)
Assertion
Ref Expression
phop (๐‘ˆ โˆˆ CPreHilOLD โ†’ ๐‘ˆ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)

Proof of Theorem phop
StepHypRef Expression
1 phrel 29806 . . 3 Rel CPreHilOLD
2 1st2nd 7975 . . 3 ((Rel CPreHilOLD โˆง ๐‘ˆ โˆˆ CPreHilOLD) โ†’ ๐‘ˆ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ)
31, 2mpan 689 . 2 (๐‘ˆ โˆˆ CPreHilOLD โ†’ ๐‘ˆ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ)
4 phop.6 . . . . 5 ๐‘ = (normCVโ€˜๐‘ˆ)
54nmcvfval 29598 . . . 4 ๐‘ = (2nd โ€˜๐‘ˆ)
65opeq2i 4838 . . 3 โŸจ(1st โ€˜๐‘ˆ), ๐‘โŸฉ = โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ
7 phnv 29805 . . . . 5 (๐‘ˆ โˆˆ CPreHilOLD โ†’ ๐‘ˆ โˆˆ NrmCVec)
8 eqid 2733 . . . . . 6 (1st โ€˜๐‘ˆ) = (1st โ€˜๐‘ˆ)
98nvvc 29606 . . . . 5 (๐‘ˆ โˆˆ NrmCVec โ†’ (1st โ€˜๐‘ˆ) โˆˆ CVecOLD)
10 vcrel 29551 . . . . . . 7 Rel CVecOLD
11 1st2nd 7975 . . . . . . 7 ((Rel CVecOLD โˆง (1st โ€˜๐‘ˆ) โˆˆ CVecOLD) โ†’ (1st โ€˜๐‘ˆ) = โŸจ(1st โ€˜(1st โ€˜๐‘ˆ)), (2nd โ€˜(1st โ€˜๐‘ˆ))โŸฉ)
1210, 11mpan 689 . . . . . 6 ((1st โ€˜๐‘ˆ) โˆˆ CVecOLD โ†’ (1st โ€˜๐‘ˆ) = โŸจ(1st โ€˜(1st โ€˜๐‘ˆ)), (2nd โ€˜(1st โ€˜๐‘ˆ))โŸฉ)
13 phop.2 . . . . . . . 8 ๐บ = ( +๐‘ฃ โ€˜๐‘ˆ)
1413vafval 29594 . . . . . . 7 ๐บ = (1st โ€˜(1st โ€˜๐‘ˆ))
15 phop.4 . . . . . . . 8 ๐‘† = ( ยท๐‘ OLD โ€˜๐‘ˆ)
1615smfval 29596 . . . . . . 7 ๐‘† = (2nd โ€˜(1st โ€˜๐‘ˆ))
1714, 16opeq12i 4839 . . . . . 6 โŸจ๐บ, ๐‘†โŸฉ = โŸจ(1st โ€˜(1st โ€˜๐‘ˆ)), (2nd โ€˜(1st โ€˜๐‘ˆ))โŸฉ
1812, 17eqtr4di 2791 . . . . 5 ((1st โ€˜๐‘ˆ) โˆˆ CVecOLD โ†’ (1st โ€˜๐‘ˆ) = โŸจ๐บ, ๐‘†โŸฉ)
197, 9, 183syl 18 . . . 4 (๐‘ˆ โˆˆ CPreHilOLD โ†’ (1st โ€˜๐‘ˆ) = โŸจ๐บ, ๐‘†โŸฉ)
2019opeq1d 4840 . . 3 (๐‘ˆ โˆˆ CPreHilOLD โ†’ โŸจ(1st โ€˜๐‘ˆ), ๐‘โŸฉ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
216, 20eqtr3id 2787 . 2 (๐‘ˆ โˆˆ CPreHilOLD โ†’ โŸจ(1st โ€˜๐‘ˆ), (2nd โ€˜๐‘ˆ)โŸฉ = โŸจโŸจ๐บ, ๐‘†โŸฉ, ๐‘โŸฉ)
223, 21eqtrd 2773 1 (๐‘ˆ โˆˆ CPreHilOLD 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 4596  Rel wrel 5642  โ€˜cfv 6500  1st c1st 7923  2nd c2nd 7924  CVecOLDcvc 29549  NrmCVeccnv 29575   +๐‘ฃ cpv 29576   ยท๐‘ OLD cns 29578  normCVcnmcv 29581  CPreHilOLDccphlo 29803
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-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
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-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-1st 7925  df-2nd 7926  df-vc 29550  df-nv 29583  df-va 29586  df-ba 29587  df-sm 29588  df-0v 29589  df-nmcv 29591  df-ph 29804
This theorem is referenced by:  phpar  29815
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