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Mirrors > Home > MPE Home > Th. List > assaring | Structured version Visualization version GIF version |
Description: An associative algebra is a ring. (Contributed by Mario Carneiro, 5-Dec-2014.) |
Ref | Expression |
---|---|
assaring | ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2778 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2778 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2778 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
4 | eqid 2778 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | eqid 2778 | . . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | isassa 19716 | . . 3 ⊢ (𝑊 ∈ AssAlg ↔ ((𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ (Scalar‘𝑊) ∈ CRing) ∧ ∀𝑧 ∈ (Base‘(Scalar‘𝑊))∀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 493 | . 2 ⊢ (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ (Scalar‘𝑊) ∈ CRing)) |
8 | 7 | simp2d 1134 | 1 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 ∧ w3a 1071 = wceq 1601 ∈ wcel 2107 ∀wral 3090 ‘cfv 6137 (class class class)co 6924 Basecbs 16259 .rcmulr 16343 Scalarcsca 16345 ·𝑠 cvsca 16346 Ringcrg 18938 CRingccrg 18939 LModclmod 19259 AssAlgcasa 19710 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-nul 5027 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4674 df-br 4889 df-iota 6101 df-fv 6145 df-ov 6927 df-assa 19713 |
This theorem is referenced by: issubassa 19725 assapropd 19728 aspval 19729 asclmul1 19740 asclmul2 19741 asclrhm 19743 rnascl 19744 aspval2 19748 assamulgscmlem1 19749 assamulgscmlem2 19750 mplind 19902 evlseu 19916 pf1subrg 20112 zlmassa 20272 matinv 20893 assaascl0 43192 assaascl1 43193 |
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