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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 2733 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2733 | . . 3 ⊢ (mulGrp‘𝑅) = (mulGrp‘𝑅) | |
3 | eqid 2733 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2733 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | eqid 2733 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | 1, 2, 3, 4, 5 | issrg 19993 | . 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 1146 | 1 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ‘cfv 6535 (class class class)co 7396 Basecbs 17131 +gcplusg 17184 .rcmulr 17185 0gc0g 17372 Mndcmnd 18612 CMndccmn 19632 mulGrpcmgp 19970 SRingcsrg 19991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5302 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rab 3434 df-v 3477 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4321 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-br 5145 df-iota 6487 df-fv 6543 df-ov 7399 df-srg 19992 |
This theorem is referenced by: srgmnd 19995 srgcom 20011 srgsummulcr 20028 sgsummulcl 20029 srgbinomlem3 20033 srgbinomlem4 20034 srgbinomlem 20035 gsumvsca2 32343 |
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