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Theorem vcsm 30366
Description: Functionality of th scalar product of a complex vector space. (Contributed by NM, 3-Nov-2006.) (New usage is discouraged.)
Hypotheses
Ref Expression
vciOLD.1 ๐บ = (1st โ€˜๐‘Š)
vciOLD.2 ๐‘† = (2nd โ€˜๐‘Š)
vciOLD.3 ๐‘‹ = ran ๐บ
Assertion
Ref Expression
vcsm (๐‘Š โˆˆ CVecOLD โ†’ ๐‘†:(โ„‚ ร— ๐‘‹)โŸถ๐‘‹)

Proof of Theorem vcsm
Dummy variables ๐‘ฅ ๐‘ฆ ๐‘ง are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vciOLD.1 . . 3 ๐บ = (1st โ€˜๐‘Š)
2 vciOLD.2 . . 3 ๐‘† = (2nd โ€˜๐‘Š)
3 vciOLD.3 . . 3 ๐‘‹ = ran ๐บ
41, 2, 3vciOLD 30365 . 2 (๐‘Š โˆˆ CVecOLD โ†’ (๐บ โˆˆ AbelOp โˆง ๐‘†:(โ„‚ ร— ๐‘‹)โŸถ๐‘‹ โˆง โˆ€๐‘ฅ โˆˆ ๐‘‹ ((1๐‘†๐‘ฅ) = ๐‘ฅ โˆง โˆ€๐‘ฆ โˆˆ โ„‚ (โˆ€๐‘ง โˆˆ ๐‘‹ (๐‘ฆ๐‘†(๐‘ฅ๐บ๐‘ง)) = ((๐‘ฆ๐‘†๐‘ฅ)๐บ(๐‘ฆ๐‘†๐‘ง)) โˆง โˆ€๐‘ง โˆˆ โ„‚ (((๐‘ฆ + ๐‘ง)๐‘†๐‘ฅ) = ((๐‘ฆ๐‘†๐‘ฅ)๐บ(๐‘ง๐‘†๐‘ฅ)) โˆง ((๐‘ฆ ยท ๐‘ง)๐‘†๐‘ฅ) = (๐‘ฆ๐‘†(๐‘ง๐‘†๐‘ฅ)))))))
54simp2d 1141 1 (๐‘Š โˆˆ CVecOLD โ†’ ๐‘†:(โ„‚ ร— ๐‘‹)โŸถ๐‘‹)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆง wa 395   = wceq 1534   โˆˆ wcel 2099  โˆ€wral 3057   ร— cxp 5671  ran crn 5674  โŸถwf 6539  โ€˜cfv 6543  (class class class)co 7415  1st c1st 7986  2nd c2nd 7987  โ„‚cc 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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