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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 2622 | . . 3 ⊢ (Base‘𝑊) = (Base‘𝑊) | |
2 | eqid 2622 | . . 3 ⊢ (+g‘𝑊) = (+g‘𝑊) | |
3 | eqid 2622 | . . 3 ⊢ ( ·𝑠 ‘𝑊) = ( ·𝑠 ‘𝑊) | |
4 | eqid 2622 | . . 3 ⊢ (0g‘𝑊) = (0g‘𝑊) | |
5 | slmdsrg.1 | . . 3 ⊢ 𝐹 = (Scalar‘𝑊) | |
6 | eqid 2622 | . . 3 ⊢ (Base‘𝐹) = (Base‘𝐹) | |
7 | eqid 2622 | . . 3 ⊢ (+g‘𝐹) = (+g‘𝐹) | |
8 | eqid 2622 | . . 3 ⊢ (.r‘𝐹) = (.r‘𝐹) | |
9 | eqid 2622 | . . 3 ⊢ (1r‘𝐹) = (1r‘𝐹) | |
10 | eqid 2622 | . . 3 ⊢ (0g‘𝐹) = (0g‘𝐹) | |
11 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 | isslmd 29755 | . 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 1077 | 1 ⊢ (𝑊 ∈ SLMod → 𝐹 ∈ SRing) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 .rcmulr 15942 Scalarcsca 15944 ·𝑠 cvsca 15945 0gc0g 16100 CMndccmn 18193 1rcur 18501 SRingcsrg 18505 SLModcslmd 29753 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-slmd 29754 |
This theorem is referenced by: slmdacl 29762 slmdmcl 29763 slmdsn0 29764 slmd0cl 29771 slmd1cl 29772 slmdvs0 29778 gsumvsca2 29783 |
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