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Mathbox for Thierry Arnoux |
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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 2731 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2731 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | eqid 2731 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | eqid 2731 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | slmdsrg.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | eqid 2731 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
7 | eqid 2731 | . . 3 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
8 | eqid 2731 | . . 3 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
9 | eqid 2731 | . . 3 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
10 | eqid 2731 | . . 3 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | isslmd 32218 | . 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 1146 | 1 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3060 ‘cfv 6532 (class class class)co 7393 Basecbs 17126 +gcplusg 17179 .rcmulr 17180 Scalarcsca 17182 ·𝑠 cvsca 17183 0gc0g 17367 CMndccmn 19612 1rcur 19963 SRingcsrg 19967 SLModcslmd 32216 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-nul 5299 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rab 3432 df-v 3475 df-sbc 3774 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-iota 6484 df-fv 6540 df-ov 7396 df-slmd 32217 |
This theorem is referenced by: slmdacl 32225 slmdmcl 32226 slmdsn0 32227 slmd0cl 32234 slmd1cl 32235 slmdvs0 32241 gsumvsca2 32243 |
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