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Theorem nvsf 28395
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 2821 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 28391 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2821 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 28379 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvsf.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 28381 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvsf.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 28380 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vcsm 28338 . 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 1533  wcel 2110   × cxp 5552  wf 6350  cfv 6354  1st c1st 7686  cc 10534  CVecOLDcvc 28334  NrmCVeccnv 28360   +𝑣 cpv 28361  BaseSetcba 28362   ·𝑠OLD cns 28363
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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-rep 5189  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7158  df-oprab 7159  df-1st 7688  df-2nd 7689  df-vc 28335  df-nv 28368  df-va 28371  df-ba 28372  df-sm 28373  df-0v 28374  df-nmcv 28376
This theorem is referenced by:  nvinvfval  28416  smcnlem  28473  ssps  28506  hlmulf  28680
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