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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 29842 | . . 3 โข Rel NrmCVec | |
2 | 1st2nd 8021 | . . 3 โข ((Rel NrmCVec โง ๐ โ NrmCVec) โ ๐ = โจ(1st โ๐), (2nd โ๐)โฉ) | |
3 | 1, 2 | mpan 688 | . 2 โข (๐ โ NrmCVec โ ๐ = โจ(1st โ๐), (2nd โ๐)โฉ) |
4 | nvop.6 | . . . . 5 โข ๐ = (normCVโ๐) | |
5 | 4 | nmcvfval 29847 | . . . 4 โข ๐ = (2nd โ๐) |
6 | 5 | opeq2i 4876 | . . 3 โข โจ(1st โ๐), ๐โฉ = โจ(1st โ๐), (2nd โ๐)โฉ |
7 | eqid 2732 | . . . . 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 29849 | . . . 4 โข (๐ โ NrmCVec โ (1st โ๐) = โจ๐บ, ๐โฉ) |
11 | 10 | opeq1d 4878 | . . 3 โข (๐ โ NrmCVec โ โจ(1st โ๐), ๐โฉ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
12 | 6, 11 | eqtr3id 2786 | . 2 โข (๐ โ NrmCVec โ โจ(1st โ๐), (2nd โ๐)โฉ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
13 | 3, 12 | eqtrd 2772 | 1 โข (๐ โ NrmCVec โ ๐ = โจโจ๐บ, ๐โฉ, ๐โฉ) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 = wceq 1541 โ wcel 2106 โจcop 4633 Rel wrel 5680 โcfv 6540 1st c1st 7969 2nd c2nd 7970 NrmCVeccnv 29824 +๐ฃ cpv 29825 ยท๐ OLD cns 29827 normCVcnmcv 29830 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-fo 6546 df-fv 6548 df-oprab 7409 df-1st 7971 df-2nd 7972 df-vc 29799 df-nv 29832 df-va 29835 df-sm 29837 df-nmcv 29840 |
This theorem is referenced by: sspval 29963 isph 30062 hilhhi 30404 |
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