Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sspsval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ssps.y | ⊢ 𝑌 = (BaseSet‘𝑊) |
ssps.s | ⊢ 𝑆 = ( ·𝑠OLD ‘𝑈) |
ssps.r | ⊢ 𝑅 = ( ·𝑠OLD ‘𝑊) |
ssps.h | ⊢ 𝐻 = (SubSp‘𝑈) |
Ref | Expression |
---|---|
sspsval | ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵)) |
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 28434 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌))) |
6 | 5 | oveqd 7162 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐴𝑅𝐵) = (𝐴(𝑆 ↾ (ℂ × 𝑌))𝐵)) |
7 | ovres 7303 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌) → (𝐴(𝑆 ↾ (ℂ × 𝑌))𝐵) = (𝐴𝑆𝐵)) | |
8 | 6, 7 | sylan9eq 2873 | 1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 × cxp 5546 ↾ cres 5550 ‘cfv 6348 (class class class)co 7145 ℂcc 10523 NrmCVeccnv 28288 BaseSetcba 28290 ·𝑠OLD cns 28291 SubSpcss 28425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-1st 7678 df-2nd 7679 df-vc 28263 df-nv 28296 df-va 28299 df-ba 28300 df-sm 28301 df-0v 28302 df-nmcv 28304 df-ssp 28426 |
This theorem is referenced by: sspmval 28437 minvecolem2 28579 hhshsslem2 28972 |
Copyright terms: Public domain | W3C validator |