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Mirrors > Home > MPE Home > Th. List > Mathboxes > slmdcmn | Structured version Visualization version GIF version |
Description: A semimodule is a commutative monoid. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
slmdcmn | ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2798 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | eqid 2798 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | eqid 2798 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | eqid 2798 | . . 3 ⊢ (Scalar‘𝑊) = (Scalar‘𝑊) | |
6 | eqid 2798 | . . 3 ⊢ (Base‘(Scalar‘𝑊)) = (Base‘(Scalar‘𝑊)) | |
7 | eqid 2798 | . . 3 ⊢ (+g‘(Scalar‘𝑊)) = (+g‘(Scalar‘𝑊)) | |
8 | eqid 2798 | . . 3 ⊢ (.r‘(Scalar‘𝑊)) = (.r‘(Scalar‘𝑊)) | |
9 | eqid 2798 | . . 3 ⊢ (1r‘(Scalar‘𝑊)) = (1r‘(Scalar‘𝑊)) | |
10 | eqid 2798 | . . 3 ⊢ (0g‘(Scalar‘𝑊)) = (0g‘(Scalar‘𝑊)) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | isslmd 30880 | . 2 ⊢ (𝑊 ∈ SLMod ↔ (𝑊 ∈ CMnd ∧ (Scalar‘𝑊) ∈ SRing ∧ ∀𝑤 ∈ (Base‘(Scalar‘𝑊))∀𝑧 ∈ (Base‘(Scalar‘𝑊))∀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‘(Scalar‘𝑊))𝑧)( ·𝑠 ‘𝑊)𝑦) = ((𝑤( ·𝑠 ‘𝑊)𝑦)(+g‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑦))) ∧ (((𝑤(.r‘(Scalar‘𝑊))𝑧)( ·𝑠 ‘𝑊)𝑦) = (𝑤( ·𝑠 ‘𝑊)(𝑧( ·𝑠 ‘𝑊)𝑦)) ∧ ((1r‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑦) = 𝑦 ∧ ((0g‘(Scalar‘𝑊))( ·𝑠 ‘𝑊)𝑦) = (0g‘𝑊))))) |
12 | 11 | simp1bi 1142 | 1 ⊢ (𝑊 ∈ SLMod → 𝑊 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 Scalarcsca 16560 ·𝑠 cvsca 16561 0gc0g 16705 CMndccmn 18898 1rcur 19244 SRingcsrg 19248 SLModcslmd 30878 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-nul 5174 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-ov 7138 df-slmd 30879 |
This theorem is referenced by: slmdmnd 30884 gsumvsca1 30904 gsumvsca2 30905 |
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