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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 30327 | . . 3 โข Rel NrmCVec | |
2 | 1st2nd 8019 | . . 3 โข ((Rel NrmCVec โง ๐ โ NrmCVec) โ ๐ = โจ(1st โ๐), (2nd โ๐)โฉ) | |
3 | 1, 2 | mpan 687 | . 2 โข (๐ โ NrmCVec โ ๐ = โจ(1st โ๐), (2nd โ๐)โฉ) |
4 | nvop.6 | . . . . 5 โข ๐ = (normCVโ๐) | |
5 | 4 | nmcvfval 30332 | . . . 4 โข ๐ = (2nd โ๐) |
6 | 5 | opeq2i 4870 | . . 3 โข โจ(1st โ๐), ๐โฉ = โจ(1st โ๐), (2nd โ๐)โฉ |
7 | eqid 2724 | . . . . 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 30334 | . . . 4 โข (๐ โ NrmCVec โ (1st โ๐) = โจ๐บ, ๐โฉ) |
11 | 10 | opeq1d 4872 | . . 3 โข (๐ โ NrmCVec โ โจ(1st โ๐), ๐โฉ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
12 | 6, 11 | eqtr3id 2778 | . 2 โข (๐ โ NrmCVec โ โจ(1st โ๐), (2nd โ๐)โฉ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
13 | 3, 12 | eqtrd 2764 | 1 โข (๐ โ NrmCVec โ ๐ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1533 โ wcel 2098 โจcop 4627 Rel wrel 5672 โcfv 6534 1st c1st 7967 2nd c2nd 7968 NrmCVeccnv 30309 +๐ฃ cpv 30310 ยท๐ OLD cns 30312 normCVcnmcv 30315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 ax-un 7719 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-mpt 5223 df-id 5565 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-iota 6486 df-fun 6536 df-fn 6537 df-f 6538 df-fo 6540 df-fv 6542 df-oprab 7406 df-1st 7969 df-2nd 7970 df-vc 30284 df-nv 30317 df-va 30320 df-sm 30322 df-nmcv 30325 |
This theorem is referenced by: sspval 30448 isph 30547 hilhhi 30889 |
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