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Theorem nvsf 28018
 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 2825 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 28014 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2825 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 28002 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvsf.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 28004 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvsf.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 28003 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vcsm 27961 . 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 1656   ∈ wcel 2164   × cxp 5340  ⟶wf 6119  ‘cfv 6123  1st c1st 7426  ℂcc 10250  CVecOLDcvc 27957  NrmCVeccnv 27983   +𝑣 cpv 27984  BaseSetcba 27985   ·𝑠OLD cns 27986 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-rep 4994  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-iun 4742  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-iota 6086  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-fv 6131  df-ov 6908  df-oprab 6909  df-1st 7428  df-2nd 7429  df-vc 27958  df-nv 27991  df-va 27994  df-ba 27995  df-sm 27996  df-0v 27997  df-nmcv 27999 This theorem is referenced by:  nvinvfval  28039  smcnlem  28096  ssps  28129  hlmulf  28304
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