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Mirrors > Home > MPE Home > Th. List > vcsm | Structured version Visualization version GIF version |
Description: Functionality of th scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.) |
Ref | Expression |
---|---|
vciOLD.1 | โข ๐บ = (1st โ๐) |
vciOLD.2 | โข ๐ = (2nd โ๐) |
vciOLD.3 | โข ๐ = ran ๐บ |
Ref | Expression |
---|---|
vcsm | โข (๐ โ CVecOLD โ ๐:(โ ร ๐)โถ๐) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vciOLD.1 | . . 3 โข ๐บ = (1st โ๐) | |
2 | vciOLD.2 | . . 3 โข ๐ = (2nd โ๐) | |
3 | vciOLD.3 | . . 3 โข ๐ = ran ๐บ | |
4 | 1, 2, 3 | vciOLD 30365 | . 2 โข (๐ โ CVecOLD โ (๐บ โ AbelOp โง ๐:(โ ร ๐)โถ๐ โง โ๐ฅ โ ๐ ((1๐๐ฅ) = ๐ฅ โง โ๐ฆ โ โ (โ๐ง โ ๐ (๐ฆ๐(๐ฅ๐บ๐ง)) = ((๐ฆ๐๐ฅ)๐บ(๐ฆ๐๐ง)) โง โ๐ง โ โ (((๐ฆ + ๐ง)๐๐ฅ) = ((๐ฆ๐๐ฅ)๐บ(๐ง๐๐ฅ)) โง ((๐ฆ ยท ๐ง)๐๐ฅ) = (๐ฆ๐(๐ง๐๐ฅ))))))) |
5 | 4 | simp2d 1141 | 1 โข (๐ โ CVecOLD โ ๐:(โ ร ๐)โถ๐) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 395 = wceq 1534 โ wcel 2099 โwral 3057 ร cxp 5671 ran crn 5674 โถwf 6539 โcfv 6543 (class class class)co 7415 1st c1st 7986 2nd c2nd 7987 โcc 11131 1c1 11134 + caddc 11136 ยท cmul 11138 AbelOpcablo 30348 CVecOLDcvc 30362 |
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-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fv 6551 df-ov 7418 df-1st 7988 df-2nd 7989 df-vc 30363 |
This theorem is referenced by: vccl 30367 nvsf 30423 |
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