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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 29545 | . 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 1144 | 1 โข (๐ โ CVecOLD โ ๐:(โ ร ๐)โถ๐) |
Colors of variables: wff setvar class |
Syntax hints: โ wi 4 โง wa 397 = wceq 1542 โ wcel 2107 โwral 3061 ร cxp 5632 ran crn 5635 โถwf 6493 โcfv 6497 (class class class)co 7358 1st c1st 7920 2nd c2nd 7921 โcc 11054 1c1 11057 + caddc 11059 ยท cmul 11061 AbelOpcablo 29528 CVecOLDcvc 29542 |
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-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-fv 6505 df-ov 7361 df-1st 7922 df-2nd 7923 df-vc 29543 |
This theorem is referenced by: vccl 29547 nvsf 29603 |
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