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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 2818 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2818 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | eqid 2818 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | eqid 2818 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | slmdsrg.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | eqid 2818 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
7 | eqid 2818 | . . 3 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
8 | eqid 2818 | . . 3 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
9 | eqid 2818 | . . 3 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
10 | eqid 2818 | . . 3 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | isslmd 30757 | . 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 1138 | 1 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
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 +gcplusg 16553 .rcmulr 16554 Scalarcsca 16556 ·𝑠 cvsca 16557 0gc0g 16701 CMndccmn 18835 1rcur 19180 SRingcsrg 19184 SLModcslmd 30755 |
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-slmd 30756 |
This theorem is referenced by: slmdacl 30764 slmdmcl 30765 slmdsn0 30766 slmd0cl 30773 slmd1cl 30774 slmdvs0 30780 gsumvsca2 30782 |
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