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Theorem sspgval 30538
Description: Vector addition on a subspace in terms of vector addition on the parent space. (Contributed by NM, 28-Jan-2008.) (New usage is discouraged.)
Hypotheses
Ref Expression
sspg.y 𝑌 = (BaseSet‘𝑊)
sspg.g 𝐺 = ( +𝑣𝑈)
sspg.f 𝐹 = ( +𝑣𝑊)
sspg.h 𝐻 = (SubSp‘𝑈)
Assertion
Ref Expression
sspgval (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝐴𝑌𝐵𝑌)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Proof of Theorem sspgval
StepHypRef Expression
1 sspg.y . . . 4 𝑌 = (BaseSet‘𝑊)
2 sspg.g . . . 4 𝐺 = ( +𝑣𝑈)
3 sspg.f . . . 4 𝐹 = ( +𝑣𝑊)
4 sspg.h . . . 4 𝐻 = (SubSp‘𝑈)
51, 2, 3, 4sspg 30537 . . 3 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
65oveqd 7437 . 2 ((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) → (𝐴𝐹𝐵) = (𝐴(𝐺 ↾ (𝑌 × 𝑌))𝐵))
7 ovres 7587 . 2 ((𝐴𝑌𝐵𝑌) → (𝐴(𝐺 ↾ (𝑌 × 𝑌))𝐵) = (𝐴𝐺𝐵))
86, 7sylan9eq 2788 1 (((𝑈 ∈ NrmCVec ∧ 𝑊𝐻) ∧ (𝐴𝑌𝐵𝑌)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099   × cxp 5676  cres 5680  cfv 6548  (class class class)co 7420  NrmCVeccnv 30393   +𝑣 cpv 30394  BaseSetcba 30395  SubSpcss 30530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3374  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-ov 7423  df-oprab 7424  df-1st 7993  df-2nd 7994  df-grpo 30302  df-ablo 30354  df-vc 30368  df-nv 30401  df-va 30404  df-ba 30405  df-sm 30406  df-0v 30407  df-nmcv 30409  df-ssp 30531
This theorem is referenced by:  sspmval  30542  minvecolem2  30684  hhshsslem2  31077
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