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Mirrors > Home > MPE Home > Th. List > assalmod | Structured version Visualization version GIF version |
Description: An associative algebra is a left module. (Contributed by Mario Carneiro, 5-Dec-2014.) |
Ref | Expression |
---|---|
assalmod | ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . . 4 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2821 | . . . 4 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
3 | eqid 2821 | . . . 4 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
4 | eqid 2821 | . . . 4 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
5 | eqid 2821 | . . . 4 ⊢ (.r‘𝑊) = (.r‘𝑊) | |
6 | 1, 2, 3, 4, 5 | isassa 20088 | . . 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 500 | . 2 ⊢ (𝑊 ∈ AssAlg → (𝑊 ∈ LMod ∧ 𝑊 ∈ Ring ∧ (Scalar‘𝑊) ∈ CRing)) |
8 | 7 | simp1d 1138 | 1 ⊢ (𝑊 ∈ AssAlg → 𝑊 ∈ LMod) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ‘cfv 6355 (class class class)co 7156 Basecbs 16483 .rcmulr 16566 Scalarcsca 16568 ·𝑠 cvsca 16569 Ringcrg 19297 CRingccrg 19298 LModclmod 19634 AssAlgcasa 20082 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-iota 6314 df-fv 6363 df-ov 7159 df-assa 20085 |
This theorem is referenced by: assa2ass 20095 issubassa3 20097 issubassa 20098 assapropd 20101 aspval 20102 asplss 20103 ascldimul 20116 ascldimulOLD 20117 asclrhm 20119 rnascl 20120 issubassa2 20121 aspval2 20127 assamulgscmlem1 20128 assamulgscmlem2 20129 mplmon2mul 20281 mplind 20282 matinv 21286 selvval2lem4 39156 assaascl0 44453 assaascl1 44454 |
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