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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 2818 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2818 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2818 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
4 | eqid 2818 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | eqid 2818 | . . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | isassa 20016 | . . 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 498 | . 2 ⊢ (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ (Scalar‘𝑊) ∈ CRing)) |
8 | 7 | simp2d 1135 | 1 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ Ring) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ‘cfv 6348 (class class class)co 7145 Basecbs 16471 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 Ringcrg 19226 CRingccrg 19227 LModclmod 19563 AssAlgcasa 20010 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-nul 5201 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-iota 6307 df-fv 6356 df-ov 7148 df-assa 20013 |
This theorem is referenced by: issubassa 20026 assapropd 20029 aspval 20030 asclmul1 20042 asclmul2 20043 ascldimul 20044 ascldimulOLD 20045 asclrhm 20047 rnascl 20048 aspval2 20055 assamulgscmlem1 20056 assamulgscmlem2 20057 mplind 20210 evlseu 20224 pf1subrg 20439 zlmassa 20599 matinv 21214 selvval2lem4 39014 assaascl0 44361 assaascl1 44362 |
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