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Theorem vcsm 29546
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 29545 . 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 1144 1 (๐‘Š โˆˆ CVecOLD โ†’ ๐‘†:(โ„‚ ร— ๐‘‹)โŸถ๐‘‹)
Colors of variables: wff setvar class
Syntax hints:   โ†’ wi 4   โˆง wa 397   = wceq 1542   โˆˆ wcel 2107  โˆ€wral 3061   ร— cxp 5632  ran crn 5635  โŸถwf 6493  โ€˜cfv 6497  (class class class)co 7358  1st c1st 7920  2nd c2nd 7921  โ„‚cc 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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