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Mirrors > Home > ILE Home > Th. List > ringmgp | GIF version |
Description: A ring is a monoid under multiplication. (Contributed by Mario Carneiro, 6-Jan-2015.) |
Ref | Expression |
---|---|
ringmgp.g | ⊢ 𝐺 = (mulGrp‘𝑅) |
Ref | Expression |
---|---|
ringmgp | ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | ringmgp.g | . . 3 ⊢ 𝐺 = (mulGrp‘𝑅) | |
3 | eqid 2177 | . . 3 ⊢ (+g‘𝑅) = (+g‘𝑅) | |
4 | eqid 2177 | . . 3 ⊢ (.r‘𝑅) = (.r‘𝑅) | |
5 | 1, 2, 3, 4 | isring 13136 | . 2 ⊢ (𝑅 ∈ Ring ↔ (𝑅 ∈ Grp ∧ 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))))) |
6 | 5 | simp2bi 1013 | 1 ⊢ (𝑅 ∈ Ring → 𝐺 ∈ Mnd) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ∀wral 2455 ‘cfv 5216 (class class class)co 5874 Basecbs 12456 +gcplusg 12530 .rcmulr 12531 Mndcmnd 12771 Grpcgrp 12831 mulGrpcmgp 13083 Ringcrg 13132 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-cnex 7901 ax-resscn 7902 ax-1re 7904 ax-addrcl 7907 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4004 df-opab 4065 df-mpt 4066 df-id 4293 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-iota 5178 df-fun 5218 df-fn 5219 df-fv 5224 df-ov 5877 df-inn 8918 df-2 8976 df-3 8977 df-ndx 12459 df-slot 12460 df-base 12462 df-plusg 12543 df-mulr 12544 df-ring 13134 |
This theorem is referenced by: mgpf 13147 ringcl 13149 iscrng2 13151 ringass 13152 ringideu 13153 ringidcl 13156 ringidmlem 13158 ringsrg 13177 unitsubm 13241 invrpropdg 13271 subrgcrng 13306 subrgsubm 13315 subrgugrp 13321 issubrg3 13328 cnfldexp 13362 |
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