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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 2824 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | assasca.f | . . . 4 ⊢ 𝐹 = (Scalar‘𝑊) | |
3 | eqid 2824 | . . . 4 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
4 | eqid 2824 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | eqid 2824 | . . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | isassa 20091 | . . 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 500 | . 2 ⊢ (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ 𝐹 ∈ CRing)) |
8 | 7 | simp3d 1140 | 1 ⊢ (𝑊 ∈ AssAlg → 𝐹 ∈ CRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∀wral 3141 ‘cfv 6358 (class class class)co 7159 Basecbs 16486 .rcmulr 16569 Scalarcsca 16571 ·𝑠 cvsca 16572 Ringcrg 19300 CRingccrg 19301 LModclmod 19637 AssAlgcasa 20085 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-nul 5213 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-iota 6317 df-fv 6366 df-ov 7162 df-assa 20088 |
This theorem is referenced by: assa2ass 20098 issubassa3 20100 ascldimulOLD 20120 asclrhm 20122 assamulgscmlem2 20132 |
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