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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 30406 | . . 3 โข Rel NrmCVec | |
2 | 1st2nd 8038 | . . 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 30411 | . . . 4 โข ๐ = (2nd โ๐) |
6 | 5 | opeq2i 4874 | . . 3 โข โจ(1st โ๐), ๐โฉ = โจ(1st โ๐), (2nd โ๐)โฉ |
7 | eqid 2728 | . . . . 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 30413 | . . . 4 โข (๐ โ NrmCVec โ (1st โ๐) = โจ๐บ, ๐โฉ) |
11 | 10 | opeq1d 4876 | . . 3 โข (๐ โ NrmCVec โ โจ(1st โ๐), ๐โฉ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
12 | 6, 11 | eqtr3id 2782 | . 2 โข (๐ โ NrmCVec โ โจ(1st โ๐), (2nd โ๐)โฉ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
13 | 3, 12 | eqtrd 2768 | 1 โข (๐ โ NrmCVec โ ๐ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1534 โ wcel 2099 โจcop 4631 Rel wrel 5678 โcfv 6543 1st c1st 7986 2nd c2nd 7987 NrmCVeccnv 30388 +๐ฃ 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 |
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