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Mirrors > Home > MPE Home > Th. List > assasca | Structured version Visualization version GIF version |
Description: An associative algebra's scalar field is a commutative ring. (Contributed by Mario Carneiro, 7-Jan-2015.) |
Ref | Expression |
---|---|
assasca.f | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
assasca | ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ CRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | assasca.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2733 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | eqid 2733 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | eqid 2733 | . . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | isassa 21091 | . . 3 ⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing) ∧ ∀𝑧 ∈ (Base‘𝐹)∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑧( ·𝑠 ‘𝑊)𝑥)(.r‘𝑊)𝑦) = (𝑧( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦)) ∧ (𝑥(.r‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑦)) = (𝑧( ·𝑠 ‘𝑊)(𝑥(.r‘𝑊)𝑦))))) |
7 | 6 | simplbi 497 | . 2 ⊢ (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing)) |
8 | 7 | simp3d 1142 | 1 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1537 ∈ wcel 2101 ∀wral 3059 ‘cfv 6447 (class class class)co 7295 Basecbs 16940 .rcmulr 16991 Scalarcsca 16993 ·𝑠 cvsca 16994 Ringcrg 19811 CRingccrg 19812 LModclmod 20151 AssAlgcasa 21085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2103 ax-9 2111 ax-ext 2704 ax-nul 5233 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3an 1087 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2063 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2939 df-ral 3060 df-rab 3224 df-v 3436 df-sbc 3719 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4260 df-if 4463 df-sn 4565 df-pr 4567 df-op 4571 df-uni 4842 df-br 5078 df-iota 6399 df-fv 6455 df-ov 7298 df-assa 21088 |
This theorem is referenced by: assa2ass 21098 issubassa3 21100 asclrhm 21122 assamulgscmlem2 21132 asclmulg 31694 |
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