![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 19822 | . 2 ⊢ ((𝑊 ∈ AssAlg ∧ (𝐴 ∈ 𝐵 ∧ 𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → (((𝐴 · 𝑋) × 𝑌) = (𝐴 · (𝑋 × 𝑌)) ∧ (𝑋 × (𝐴 · 𝑌)) = (𝐴 · (𝑋 × 𝑌)))) |
7 | 6 | simprd 488 | 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 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 ‘cfv 6185 (class class class)co 6974 Basecbs 16337 .rcmulr 16420 Scalarcsca 16422 ·𝑠 cvsca 16423 AssAlgcasa 19815 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2744 ax-nul 5063 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-br 4926 df-iota 6149 df-fv 6193 df-ov 6977 df-assa 19818 |
This theorem is referenced by: assa2ass 19828 issubassa 19830 asclmul2 19846 asclrhm 19848 assamulgscmlem2 19855 mplmon2mul 20006 matinv 21002 cpmadugsumlemC 21199 |
Copyright terms: Public domain | W3C validator |