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Mirrors > Home > MPE Home > Th. List > srgcmn | Structured version Visualization version GIF version |
Description: A semiring is a commutative monoid. (Contributed by Thierry Arnoux, 21-Mar-2018.) |
Ref | Expression |
---|---|
srgcmn | ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2821 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2821 | . . 3 ⊢ (mulGrp‘𝑅) = (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 19251 | . 2 ⊢ (𝑅 ∈ SRing ↔ (𝑅 ∈ CMnd ∧ (mulGrp‘𝑅) ∈ 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 | simp1bi 1141 | 1 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ‘cfv 6349 (class class class)co 7150 Basecbs 16477 +gcplusg 16559 .rcmulr 16560 0gc0g 16707 Mndcmnd 17905 CMndccmn 18900 mulGrpcmgp 19233 SRingcsrg 19249 |
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 5202 |
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 4561 df-pr 4563 df-op 4567 df-uni 4832 df-br 5059 df-iota 6308 df-fv 6357 df-ov 7153 df-srg 19250 |
This theorem is referenced by: srgmnd 19253 srgcom 19269 srgsummulcr 19281 sgsummulcl 19282 srgbinomlem3 19286 srgbinomlem4 19287 srgbinomlem 19288 gsumvsca2 30850 |
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