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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 2737 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
| 2 | eqid 2737 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
| 3 | eqid 2737 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
| 4 | eqid 2737 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
| 5 | slmdsrg.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
| 6 | eqid 2737 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
| 7 | eqid 2737 | . . 3 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
| 8 | eqid 2737 | . . 3 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
| 9 | eqid 2737 | . . 3 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
| 10 | eqid 2737 | . . 3 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
| 11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | isslmd 33208 | . 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 1147 | 1 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 .rcmulr 17298 Scalarcsca 17300 ·𝑠 cvsca 17301 0gc0g 17484 CMndccmn 19798 1rcur 20178 SRingcsrg 20183 SLModcslmd 33206 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-nul 5306 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-ov 7434 df-slmd 33207 | 
| This theorem is referenced by: slmdacl 33215 slmdmcl 33216 slmdsn0 33217 slmd0cl 33224 slmd1cl 33225 slmdvs0 33231 gsumvsca2 33233 | 
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