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Theorem nvsf 30648
Description: Mapping for the scalar multiplication operation. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
nvsf.1 𝑋 = (BaseSet‘𝑈)
nvsf.4 𝑆 = ( ·𝑠OLD𝑈)
Assertion
Ref Expression
nvsf (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋)

Proof of Theorem nvsf
StepHypRef Expression
1 eqid 2735 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 30644 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2735 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 30632 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvsf.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 30634 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvsf.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 30633 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vcsm 30591 . 2 ((1st𝑈) ∈ CVecOLD𝑆:(ℂ × 𝑋)⟶𝑋)
102, 9syl 17 1 (𝑈 ∈ NrmCVec → 𝑆:(ℂ × 𝑋)⟶𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106   × cxp 5687  wf 6559  cfv 6563  1st c1st 8011  cc 11151  CVecOLDcvc 30587  NrmCVeccnv 30613   +𝑣 cpv 30614  BaseSetcba 30615   ·𝑠OLD cns 30616
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-1st 8013  df-2nd 8014  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629
This theorem is referenced by:  nvinvfval  30669  smcnlem  30726  ssps  30759  hlmulf  30933
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