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Mirrors > Home > MPE Home > Th. List > sspgval | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
sspg.y | ⊢ 𝑌 = (BaseSet‘𝑊) |
sspg.g | ⊢ 𝐺 = ( +𝑣 ‘𝑈) |
sspg.f | ⊢ 𝐹 = ( +𝑣 ‘𝑊) |
sspg.h | ⊢ 𝐻 = (SubSp‘𝑈) |
Ref | Expression |
---|---|
sspgval | ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) |
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 30537 | . . 3 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → 𝐹 = (𝐺 ↾ (𝑌 × 𝑌))) |
6 | 5 | oveqd 7437 | . 2 ⊢ ((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) → (𝐴𝐹𝐵) = (𝐴(𝐺 ↾ (𝑌 × 𝑌))𝐵)) |
7 | ovres 7587 | . 2 ⊢ ((𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌) → (𝐴(𝐺 ↾ (𝑌 × 𝑌))𝐵) = (𝐴𝐺𝐵)) | |
8 | 6, 7 | sylan9eq 2788 | 1 ⊢ (((𝑈 ∈ NrmCVec ∧ 𝑊 ∈ 𝐻) ∧ (𝐴 ∈ 𝑌 ∧ 𝐵 ∈ 𝑌)) → (𝐴𝐹𝐵) = (𝐴𝐺𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 × cxp 5676 ↾ cres 5680 ‘cfv 6548 (class class class)co 7420 NrmCVeccnv 30393 +𝑣 cpv 30394 BaseSetcba 30395 SubSpcss 30530 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-ov 7423 df-oprab 7424 df-1st 7993 df-2nd 7994 df-grpo 30302 df-ablo 30354 df-vc 30368 df-nv 30401 df-va 30404 df-ba 30405 df-sm 30406 df-0v 30407 df-nmcv 30409 df-ssp 30531 |
This theorem is referenced by: sspmval 30542 minvecolem2 30684 hhshsslem2 31077 |
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