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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 2738 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | srgmgp.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | eqid 2738 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2738 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | eqid 2738 | . . 3 ⊢ (0g‘𝑅) = (0g‘𝑅) | |
6 | 1, 2, 3, 4, 5 | issrg 19732 | . 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 1145 | 1 ⊢ (𝑅 ∈ SRing → 𝐺 ∈ Mnd) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∀wral 3064 ‘cfv 6428 (class class class)co 7269 Basecbs 16901 +gcplusg 16951 .rcmulr 16952 0gc0g 17139 Mndcmnd 18374 CMndccmn 19375 mulGrpcmgp 19709 SRingcsrg 19730 |
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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-nul 5230 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3433 df-sbc 3718 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5076 df-iota 6386 df-fv 6436 df-ov 7272 df-srg 19731 |
This theorem is referenced by: srgcl 19737 srgass 19738 srgideu 19739 srgidcl 19743 srgidmlem 19745 srg1zr 19754 srgpcomp 19757 srgpcompp 19758 srgpcomppsc 19759 srg1expzeq1 19764 srgbinomlem1 19765 srgbinomlem4 19768 srgbinomlem 19769 |
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