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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 27585 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝑅 = (𝑆 ↾ (ℂ × 𝑌))) |
6 | 5 | oveqd 6667 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐴𝑅𝐵) = (𝐴(𝑆 ↾ (ℂ × 𝑌))𝐵)) |
7 | ovres 6800 | . 2 ⊢ ((𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌) → (𝐴(𝑆 ↾ (ℂ × 𝑌))𝐵) = (𝐴𝑆𝐵)) | |
8 | 6, 7 | sylan9eq 2676 | 1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ ℂ ∧ 𝐵 ∈ 𝑌)) → (𝐴𝑅𝐵) = (𝐴𝑆𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 × cxp 5112 ↾ cres 5116 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 NrmCVeccnv 27439 BaseSetcba 27441 ·𝑠OLD cns 27442 SubSpcss 27576 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-1st 7168 df-2nd 7169 df-vc 27414 df-nv 27447 df-va 27450 df-ba 27451 df-sm 27452 df-0v 27453 df-nmcv 27455 df-ssp 27577 |
This theorem is referenced by: sspmval 27588 minvecolem2 27731 hhshsslem2 28125 |
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