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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 19137 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))) |
7 | 6 | simprd 478 | 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 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ‘cfv 5804 (class class class)co 6549 Basecbs 15695 .rcmulr 15769 Scalarcsca 15771 ·𝑠 cvsca 15772 AssAlgcasa 19130 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-nul 4717 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-iota 5768 df-fv 5812 df-ov 6552 df-assa 19133 |
This theorem is referenced by: assa2ass 19143 issubassa 19145 asclmul2 19161 asclrhm 19163 assamulgscmlem2 19170 mplmon2mul 19322 matinv 20302 cpmadugsumlemC 20499 |
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