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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 2739 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | assasca.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2739 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | eqid 2739 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | eqid 2739 | . . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | isassa 20685 | . . 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 501 | . 2 ⊢ (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing)) |
8 | 7 | simp3d 1145 | 1 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ‘cfv 6350 (class class class)co 7183 Basecbs 16599 .rcmulr 16682 Scalarcsca 16684 ·𝑠 cvsca 16685 Ringcrg 19429 CRingccrg 19430 LModclmod 19766 AssAlgcasa 20679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-nul 5184 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-ral 3059 df-rex 3060 df-rab 3063 df-v 3402 df-sbc 3686 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-br 5041 df-iota 6308 df-fv 6358 df-ov 7186 df-assa 20682 |
This theorem is referenced by: assa2ass 20692 issubassa3 20694 ascldimulOLD 20715 asclrhm 20717 assamulgscmlem2 20727 asclmulg 31251 |
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