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Theorem sspgval 28485

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 28484	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌)))
6	5	oveqd 7154	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐴𝐹𝐵) = (𝐴(𝐺 ↾ (𝑌 × 𝑌))𝐵))
7		ovres 7295	. 2 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴(𝐺 ↾ (𝑌 × 𝑌))𝐵) = (𝐴𝐺𝐵))
8	6, 7	sylan9eq 2875	1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 398 = wceq 1537 ∈ wcel 2114 × cxp 5534 ↾ cres 5538 ‘cfv 6336 (class class class)co 7137 NrmCVeccnv 28340 +_𝑣 cpv 28341 BaseSetcba 28342 SubSpcss 28477

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2792 ax-rep 5171 ax-sep 5184 ax-nul 5191 ax-pow 5247 ax-pr 5311 ax-un 7442

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2891 df-nfc 2959 df-ne 3012 df-ral 3138 df-rex 3139 df-reu 3140 df-rab 3142 df-v 3483 df-sbc 3759 df-csb 3867 df-dif 3922 df-un 3924 df-in 3926 df-ss 3935 df-nul 4275 df-if 4449 df-pw 4522 df-sn 4549 df-pr 4551 df-op 4555 df-uni 4820 df-iun 4902 df-br 5048 df-opab 5110 df-mpt 5128 df-id 5441 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-iota 6295 df-fun 6338 df-fn 6339 df-f 6340 df-f1 6341 df-fo 6342 df-f1o 6343 df-fv 6344 df-ov 7140 df-oprab 7141 df-1st 7670 df-2nd 7671 df-grpo 28249 df-ablo 28301 df-vc 28315 df-nv 28348 df-va 28351 df-ba 28352 df-sm 28353 df-0v 28354 df-nmcv 28356 df-ssp 28478

This theorem is referenced by: sspmval 28489 minvecolem2 28631 hhshsslem2 29024

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