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Theorem sspsval 29089
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 29088 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))
65oveqd 7288 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐴𝑅𝐵) = (𝐴(𝑆 ↾ (ℂ × 𝑌))𝐵))
7 ovres 7432 . 2 ((𝐴 ∈ ℂ ∧ 𝐵𝑌) → (𝐴(𝑆 ↾ (ℂ × 𝑌))𝐵) = (𝐴𝑆𝐵))
86, 7sylan9eq 2800 1 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1542  wcel 2110   × cxp 5588  cres 5592  cfv 6432  (class class class)co 7271  cc 10870  NrmCVeccnv 28942  BaseSetcba 28944   ·𝑠OLD cns 28945  SubSpcss 29079
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-id 5490  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-oprab 7275  df-1st 7824  df-2nd 7825  df-vc 28917  df-nv 28950  df-va 28953  df-ba 28954  df-sm 28955  df-0v 28956  df-nmcv 28958  df-ssp 29080
This theorem is referenced by:  sspmval  29091  minvecolem2  29233  hhshsslem2  29626
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