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Mirrors > Home > MPE Home > Th. List > nvop | Structured version Visualization version GIF version |
Description: A complex inner product space in terms of ordered pair components. (Contributed by NM, 11-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
nvop.2 | โข ๐บ = ( +๐ฃ โ๐) |
nvop.4 | โข ๐ = ( ยท๐ OLD โ๐) |
nvop.6 | โข ๐ = (normCVโ๐) |
Ref | Expression |
---|---|
nvop | โข (๐ โ NrmCVec โ ๐ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nvrel 29586 | . . 3 โข Rel NrmCVec | |
2 | 1st2nd 7972 | . . 3 โข ((Rel NrmCVec โง ๐ โ NrmCVec) โ ๐ = โจ(1st โ๐), (2nd โ๐)โฉ) | |
3 | 1, 2 | mpan 689 | . 2 โข (๐ โ NrmCVec โ ๐ = โจ(1st โ๐), (2nd โ๐)โฉ) |
4 | nvop.6 | . . . . 5 โข ๐ = (normCVโ๐) | |
5 | 4 | nmcvfval 29591 | . . . 4 โข ๐ = (2nd โ๐) |
6 | 5 | opeq2i 4835 | . . 3 โข โจ(1st โ๐), ๐โฉ = โจ(1st โ๐), (2nd โ๐)โฉ |
7 | eqid 2733 | . . . . 5 โข (1st โ๐) = (1st โ๐) | |
8 | nvop.2 | . . . . 5 โข ๐บ = ( +๐ฃ โ๐) | |
9 | nvop.4 | . . . . 5 โข ๐ = ( ยท๐ OLD โ๐) | |
10 | 7, 8, 9 | nvvop 29593 | . . . 4 โข (๐ โ NrmCVec โ (1st โ๐) = โจ๐บ, ๐โฉ) |
11 | 10 | opeq1d 4837 | . . 3 โข (๐ โ NrmCVec โ โจ(1st โ๐), ๐โฉ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
12 | 6, 11 | eqtr3id 2787 | . 2 โข (๐ โ NrmCVec โ โจ(1st โ๐), (2nd โ๐)โฉ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
13 | 3, 12 | eqtrd 2773 | 1 โข (๐ โ NrmCVec โ ๐ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1542 โ wcel 2107 โจcop 4593 Rel wrel 5639 โcfv 6497 1st c1st 7920 2nd c2nd 7921 NrmCVeccnv 29568 +๐ฃ 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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