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Mirrors > Home > MPE Home > Th. List > srgmgp | Structured version Visualization version GIF version |
Description: A semiring is a monoid under multiplication. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
Ref | Expression |
---|---|
srgmgp.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
srgmgp | ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | srgmgp.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | eqid 2798 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2798 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | eqid 2798 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | 1, 2, 3, 4, 5 | issrg 19250 | . 2 ⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ 𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝑅)(∀𝑦 ∈ (Base‘𝑅)∀𝑧 ∈ (Base‘𝑅)((𝑥(.r‘𝑅)(𝑦(+g‘𝑅)𝑧)) = ((𝑥(.r‘𝑅)𝑦)(+g‘𝑅)(𝑥(.r‘𝑅)𝑧)) ∧ ((𝑥(+g‘𝑅)𝑦)(.r‘𝑅)𝑧) = ((𝑥(.r‘𝑅)𝑧)(+g‘𝑅)(𝑦(.r‘𝑅)𝑧))) ∧ (((0g‘𝑅)(.r‘𝑅)𝑥) = (0g‘𝑅) ∧ (𝑥(.r‘𝑅)(0g‘𝑅)) = (0g‘𝑅))))) |
7 | 6 | simp2bi 1143 | 1 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 ‘cfv 6324 (class class class)co 7135 Basecbs 16475 +gcplusg 16557 .rcmulr 16558 0gc0g 16705 Mndcmnd 17903 CMndccmn 18898 mulGrpcmgp 19232 SRingcsrg 19248 |
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-srg 19249 |
This theorem is referenced by: srgcl 19255 srgass 19256 srgideu 19257 srgidcl 19261 srgidmlem 19263 srg1zr 19272 srgpcomp 19275 srgpcompp 19276 srgpcomppsc 19277 srg1expzeq1 19282 srgbinomlem1 19283 srgbinomlem4 19286 srgbinomlem 19287 |
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