sspsval - Metamath Proof Explorer

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Theorem sspsval 30817

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 = ( ·_𝑠OLD ‘𝑈)
ssps.r	⊢ 𝑅 = ( ·_𝑠OLD ‘𝑊)
ssps.h	⊢ 𝐻 = (SubSp‘𝑈)

**Assertion**
Ref	Expression
sspsval	⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵))

**Proof of Theorem sspsval**
Step	Hyp	Ref	Expression
1		ssps.y	. . . 4 ⊢ 𝑌 = (BaseSet‘𝑊)
2		ssps.s	. . . 4 ⊢ 𝑆 = ( ·_𝑠OLD ‘𝑈)
3		ssps.r	. . . 4 ⊢ 𝑅 = ( ·_𝑠OLD ‘𝑊)
4		ssps.h	. . . 4 ⊢ 𝐻 = (SubSp‘𝑈)
5	1, 2, 3, 4	ssps 30816	. . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌)))
6	5	oveqd 7377	. 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐴𝑅𝐵) = (𝐴(𝑆 ↾ (ℂ × 𝑌))𝐵))
7		ovres 7526	. 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌) → (𝐴(𝑆 ↾ (ℂ × 𝑌))𝐵) = (𝐴𝑆𝐵))
8	6, 7	sylan9eq 2792	1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1542 ∈ wcel 2114 × cxp 5622 ↾ cres 5626 ‘cfv 6492 (class class class)co 7360 ℂcc 11027 NrmCVeccnv 30670 BaseSetcba 30672 ·_𝑠OLD cns 30673 SubSpcss 30807

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5212 ax-sep 5231 ax-nul 5241 ax-pow 5302 ax-pr 5370 ax-un 7682

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5519 df-xp 5630 df-rel 5631 df-cnv 5632 df-co 5633 df-dm 5634 df-rn 5635 df-res 5636 df-ima 5637 df-iota 6448 df-fun 6494 df-fn 6495 df-f 6496 df-f1 6497 df-fo 6498 df-f1o 6499 df-fv 6500 df-ov 7363 df-oprab 7364 df-1st 7935 df-2nd 7936 df-vc 30645 df-nv 30678 df-va 30681 df-ba 30682 df-sm 30683 df-0v 30684 df-nmcv 30686 df-ssp 30808

This theorem is referenced by: sspmval 30819 minvecolem2 30961 hhshsslem2 31354

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