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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 2821 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | srgmgp.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | eqid 2821 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2821 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | eqid 2821 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | 1, 2, 3, 4, 5 | issrg 19256 | . 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 1142 | 1 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ‘cfv 6354 (class class class)co 7155 Basecbs 16482 +gcplusg 16564 .rcmulr 16565 0gc0g 16712 Mndcmnd 17910 CMndccmn 18905 mulGrpcmgp 19238 SRingcsrg 19254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-nul 5209 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-br 5066 df-iota 6313 df-fv 6362 df-ov 7158 df-srg 19255 |
This theorem is referenced by: srgcl 19261 srgass 19262 srgideu 19263 srgidcl 19267 srgidmlem 19269 srg1zr 19278 srgpcomp 19281 srgpcompp 19282 srgpcomppsc 19283 srg1expzeq1 19288 srgbinomlem1 19289 srgbinomlem4 19292 srgbinomlem 19293 |
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