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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 21407 | . 2 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod) | |
2 | assasca.f | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | 2 | lmodring 20472 | . 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 1542 ∈ wcel 2107 ‘cfv 6541 Scalarcsca 17197 Ringcrg 20050 LModclmod 20464 AssAlgcasa 21397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5306 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rab 3434 df-v 3477 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6493 df-fv 6549 df-ov 7409 df-lmod 20466 df-assa 21400 |
This theorem is referenced by: assa2ass 21410 asclrhm 21436 assamulgscmlem2 21446 asclmulg 32624 |
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