MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  nvsf Structured version   Visualization version   GIF version

Theorem nvsf 29850
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 2733 . . 3 (1st𝑈) = (1st𝑈)
21nvvc 29846 . 2 (𝑈 ∈ NrmCVec → (1st𝑈) ∈ CVecOLD)
3 eqid 2733 . . . 4 ( +𝑣𝑈) = ( +𝑣𝑈)
43vafval 29834 . . 3 ( +𝑣𝑈) = (1st ‘(1st𝑈))
5 nvsf.4 . . . 4 𝑆 = ( ·𝑠OLD𝑈)
65smfval 29836 . . 3 𝑆 = (2nd ‘(1st𝑈))
7 nvsf.1 . . . 4 𝑋 = (BaseSet‘𝑈)
87, 3bafval 29835 . . 3 𝑋 = ran ( +𝑣𝑈)
94, 6, 8vcsm 29793 . 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 1542  wcel 2107   × cxp 5673  wf 6536  cfv 6540  1st c1st 7968  cc 11104  CVecOLDcvc 29789  NrmCVeccnv 29815   +𝑣 cpv 29816  BaseSetcba 29817   ·𝑠OLD cns 29818
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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7720
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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7407  df-oprab 7408  df-1st 7970  df-2nd 7971  df-vc 29790  df-nv 29823  df-va 29826  df-ba 29827  df-sm 29828  df-0v 29829  df-nmcv 29831
This theorem is referenced by:  nvinvfval  29871  smcnlem  29928  ssps  29961  hlmulf  30135
  Copyright terms: Public domain W3C validator