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Mirrors > Home > MPE Home > Th. List > phop | Structured version Visualization version GIF version |
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 2-Apr-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
phop.2 | โข ๐บ = ( +๐ฃ โ๐) |
phop.4 | โข ๐ = ( ยท๐ OLD โ๐) |
phop.6 | โข ๐ = (normCVโ๐) |
Ref | Expression |
---|---|
phop | โข (๐ โ CPreHilOLD โ ๐ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | phrel 29806 | . . 3 โข Rel CPreHilOLD | |
2 | 1st2nd 7975 | . . 3 โข ((Rel CPreHilOLD โง ๐ โ CPreHilOLD) โ ๐ = โจ(1st โ๐), (2nd โ๐)โฉ) | |
3 | 1, 2 | mpan 689 | . 2 โข (๐ โ CPreHilOLD โ ๐ = โจ(1st โ๐), (2nd โ๐)โฉ) |
4 | phop.6 | . . . . 5 โข ๐ = (normCVโ๐) | |
5 | 4 | nmcvfval 29598 | . . . 4 โข ๐ = (2nd โ๐) |
6 | 5 | opeq2i 4838 | . . 3 โข โจ(1st โ๐), ๐โฉ = โจ(1st โ๐), (2nd โ๐)โฉ |
7 | phnv 29805 | . . . . 5 โข (๐ โ CPreHilOLD โ ๐ โ NrmCVec) | |
8 | eqid 2733 | . . . . . 6 โข (1st โ๐) = (1st โ๐) | |
9 | 8 | nvvc 29606 | . . . . 5 โข (๐ โ NrmCVec โ (1st โ๐) โ CVecOLD) |
10 | vcrel 29551 | . . . . . . 7 โข Rel CVecOLD | |
11 | 1st2nd 7975 | . . . . . . 7 โข ((Rel CVecOLD โง (1st โ๐) โ CVecOLD) โ (1st โ๐) = โจ(1st โ(1st โ๐)), (2nd โ(1st โ๐))โฉ) | |
12 | 10, 11 | mpan 689 | . . . . . 6 โข ((1st โ๐) โ CVecOLD โ (1st โ๐) = โจ(1st โ(1st โ๐)), (2nd โ(1st โ๐))โฉ) |
13 | phop.2 | . . . . . . . 8 โข ๐บ = ( +๐ฃ โ๐) | |
14 | 13 | vafval 29594 | . . . . . . 7 โข ๐บ = (1st โ(1st โ๐)) |
15 | phop.4 | . . . . . . . 8 โข ๐ = ( ยท๐ OLD โ๐) | |
16 | 15 | smfval 29596 | . . . . . . 7 โข ๐ = (2nd โ(1st โ๐)) |
17 | 14, 16 | opeq12i 4839 | . . . . . 6 โข โจ๐บ, ๐โฉ = โจ(1st โ(1st โ๐)), (2nd โ(1st โ๐))โฉ |
18 | 12, 17 | eqtr4di 2791 | . . . . 5 โข ((1st โ๐) โ CVecOLD โ (1st โ๐) = โจ๐บ, ๐โฉ) |
19 | 7, 9, 18 | 3syl 18 | . . . 4 โข (๐ โ CPreHilOLD โ (1st โ๐) = โจ๐บ, ๐โฉ) |
20 | 19 | opeq1d 4840 | . . 3 โข (๐ โ CPreHilOLD โ โจ(1st โ๐), ๐โฉ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
21 | 6, 20 | eqtr3id 2787 | . 2 โข (๐ โ CPreHilOLD โ โจ(1st โ๐), (2nd โ๐)โฉ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
22 | 3, 21 | eqtrd 2773 | 1 โข (๐ โ CPreHilOLD โ ๐ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โจcop 4596 Rel wrel 5642 โcfv 6500 1st c1st 7923 2nd c2nd 7924 CVecOLDcvc 29549 NrmCVeccnv 29575 +๐ฃ 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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