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Mirrors > Home > MPE Home > Th. List > assaassr | Structured version Visualization version GIF version |
Description: Right-associative property of an associative algebra. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
isassa.v | ⊢ 𝑉 = (Base‘𝑊) |
isassa.f | ⊢ 𝐹 = (Scalar‘𝑊) |
isassa.b | ⊢ 𝐵 = (Base‘𝐹) |
isassa.s | ⊢ · = ( ·𝑠 ‘𝑊) |
isassa.t | ⊢ × = (.r‘𝑊) |
Ref | Expression |
---|---|
assaassr | ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isassa.v | . . 3 ⊢ 𝑉 = (Base‘𝑊) | |
2 | isassa.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | isassa.b | . . 3 ⊢ 𝐵 = (Base‘𝐹) | |
4 | isassa.s | . . 3 ⊢ · = ( ·𝑠 ‘𝑊) | |
5 | isassa.t | . . 3 ⊢ × = (.r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | assalem 21791 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))) |
7 | 6 | simprd 495 | 1 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 ‘cfv 6548 (class class class)co 7420 Basecbs 17180 .rcmulr 17234 Scalarcsca 17236 ·𝑠 cvsca 17237 AssAlgcasa 21784 |
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-ext 2699 ax-nul 5306 |
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-sb 2061 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rab 3430 df-v 3473 df-sbc 3777 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-iota 6500 df-fv 6556 df-ov 7423 df-assa 21787 |
This theorem is referenced by: assa2ass 21797 issubassa3 21799 sraassab 21801 asclmul2 21820 assamulgscmlem2 21833 mplmon2mul 22013 matinv 22592 cpmadugsumlemC 22790 |
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