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Theorem sspsval 28435
Description: Scalar multiplication on a subspace in terms of scalar multiplication on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
ssps.y 𝑌 = (BaseSet‘𝑊)
ssps.s 𝑆 = ( ·𝑠OLD𝑈)
ssps.r 𝑅 = ( ·𝑠OLD𝑊)
ssps.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
sspsval (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵))

Proof of Theorem sspsval
StepHypRef Expression
1 ssps.y . . . 4 𝑌 = (BaseSet‘𝑊)
2 ssps.s . . . 4 𝑆 = ( ·𝑠OLD𝑈)
3 ssps.r . . . 4 𝑅 = ( ·𝑠OLD𝑊)
4 ssps.h . . . 4 𝐻 = (SubSp‘𝑈)
51, 2, 3, 4ssps 28434 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))
65oveqd 7162 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐴𝑅𝐵) = (𝐴(𝑆 ↾ (ℂ × 𝑌))𝐵))
7 ovres 7303 . 2 ((𝐴 ∈ ℂ ∧ 𝐵𝑌) → (𝐴(𝑆 ↾ (ℂ × 𝑌))𝐵) = (𝐴𝑆𝐵))
86, 7sylan9eq 2873 1 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wcel 2105   × cxp 5546  cres 5550  cfv 6348  (class class class)co 7145  cc 10523  NrmCVeccnv 28288  BaseSetcba 28290   ·𝑠OLD cns 28291  SubSpcss 28425
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-1st 7678  df-2nd 7679  df-vc 28263  df-nv 28296  df-va 28299  df-ba 28300  df-sm 28301  df-0v 28302  df-nmcv 28304  df-ssp 28426
This theorem is referenced by:  sspmval  28437  minvecolem2  28579  hhshsslem2  28972
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