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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdsrg | Structured version Visualization version GIF version |
Description: The scalar component of a semimodule is a semiring. (Contributed by NM, 8-Dec-2013.) (Revised by Mario Carneiro, 19-Jun-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdsrg.1 | ⊢ 𝐹 = (Scalar‘𝑊) |
Ref | Expression |
---|---|
slmdsrg | ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2738 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | eqid 2738 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | eqid 2738 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | slmdsrg.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | eqid 2738 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
7 | eqid 2738 | . . 3 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
8 | eqid 2738 | . . 3 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
9 | eqid 2738 | . . 3 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
10 | eqid 2738 | . . 3 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | isslmd 31455 | . 2 ⊢ (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ 𝐹 ∈ SRing ∧ ∀𝑤 ∈ (Base‘𝐹)∀𝑧 ∈ (Base‘𝐹)∀𝑥 ∈ (Base‘𝑊)∀𝑦 ∈ (Base‘𝑊)(((𝑧( ·𝑠 ‘𝑊)𝑦) ∈ (Base‘𝑊) ∧ (𝑧( ·𝑠 ‘𝑊)(𝑦(+g‘𝑊)𝑥)) = ((𝑧( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑥)) ∧ ((𝑤(+g‘𝐹)𝑧)( ·𝑠 ‘𝑊)𝑦) = ((𝑤( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑦))) ∧ (((𝑤(.r‘𝐹)𝑧)( ·𝑠 ‘𝑊)𝑦) = (𝑤( ·𝑠 ‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑦)) ∧ ((1r‘𝐹)( ·𝑠 ‘𝑊)𝑦) = 𝑦 ∧ ((0g‘𝐹)( ·𝑠 ‘𝑊)𝑦) = (0g‘𝑊))))) |
12 | 11 | simp2bi 1145 | 1 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ‘cfv 6433 (class class class)co 7275 Basecbs 16912 +gcplusg 16962 .rcmulr 16963 Scalarcsca 16965 ·𝑠 cvsca 16966 0gc0g 17150 CMndccmn 19386 1rcur 19737 SRingcsrg 19741 SLModcslmd 31453 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-iota 6391 df-fv 6441 df-ov 7278 df-slmd 31454 |
This theorem is referenced by: slmdacl 31462 slmdmcl 31463 slmdsn0 31464 slmd0cl 31471 slmd1cl 31472 slmdvs0 31478 gsumvsca2 31480 |
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