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| Mirrors > Home > MPE Home > Th. List > assasca | Structured version Visualization version GIF version | ||
| Description: The scalars of an associative algebra form a ring. (Contributed by Mario Carneiro, 7-Jan-2015.) (Revised by SN, 2-Mar-2025.) |
| Ref | Expression |
|---|---|
| assasca.f | ⊢ 𝐹 = (Scalar‘𝑊) |
| Ref | Expression |
|---|---|
| assasca | ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | assalmod 21880 | . 2 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod) | |
| 2 | assasca.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 3 | 2 | lmodring 20866 | . 2 ⊢ (𝑊 ∈ LMod → 𝐹 ∈ Ring) |
| 4 | 1, 3 | syl 17 | 1 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ Ring) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Scalarcsca 17300 Ringcrg 20230 LModclmod 20858 AssAlgcasa 21870 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-lmod 20860 df-assa 21873 |
| This theorem is referenced by: assa2ass 21883 assa2ass2 21884 asclrhm 21910 assamulgscmlem2 21920 asclmulg 21922 |
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