Theorem sspgval 30758

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 = ( +𝑣 ‘𝑈)
sspg.f	⊢ 𝐹 = ( +𝑣 ‘𝑊)
sspg.h	⊢ 𝐻 = (SubSp‘𝑈)

Assertion

Ref	Expression
sspgval	⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Proof of Theorem sspgval

Step	Hyp	Ref	Expression
1		sspg.y	. . . 4 ⊢ 𝑌 = (BaseSet‘𝑊)
2		sspg.g	. . . 4 ⊢ 𝐺 = ( +𝑣 ‘𝑈)
3		sspg.f	. . . 4 ⊢ 𝐹 = ( +𝑣 ‘𝑊)
4		sspg.h	. . . 4 ⊢ 𝐻 = (SubSp‘𝑈)
5	1, 2, 3, 4	sspg 30757	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
6	5	oveqd 7448	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐴𝐹𝐵) = (𝐴(𝐺 ↾ (𝑌 × 𝑌))𝐵))
7		ovres 7599	. 2 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴(𝐺 ↾ (𝑌 × 𝑌))𝐵) = (𝐴𝐺𝐵))
8	6, 7	sylan9eq 2795	1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1537   ∈ wcel 2106   × cxp 5687   ↾ cres 5691  ‘cfv 6563  (class class class)co 7431  NrmCVeccnv 30613   +_𝑣 cpv 30614  BaseSetcba 30615  SubSpcss 30750

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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-1st 8013  df-2nd 8014  df-grpo 30522  df-ablo 30574  df-vc 30588  df-nv 30621  df-va 30624  df-ba 30625  df-sm 30626  df-0v 30627  df-nmcv 30629  df-ssp 30751

This theorem is referenced by:  sspmval  30762  minvecolem2  30904  hhshsslem2  31297