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Mirrors > Home > ILE Home > Th. List > ringgrp | GIF version |
Description: A ring is a group. (Contributed by NM, 15-Sep-2011.) |
Ref | Expression |
---|---|
ringgrp | ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2177 | . . 3 ⊢ (Base‘𝑅) = (Base‘𝑅) | |
2 | eqid 2177 | . . 3 ⊢ (mulGrp‘𝑅) = (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 ∧ (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))))) |
6 | 5 | simp1bi 1012 | 1 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) |
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: ringgrpd 13141 ringmnd 13142 ring0cl 13157 ringacl 13166 ringcom 13167 ringabl 13168 ringlz 13175 ringrz 13176 ringnegl 13181 rngnegr 13182 ringmneg1 13183 ringmneg2 13184 ringm2neg 13185 ringsubdi 13186 rngsubdir 13187 mulgass2 13188 ringressid 13191 opprring 13202 dvdsrneg 13225 unitnegcl 13252 dvrdir 13265 subrgsubg 13308 cnfld0 13356 cnfldneg 13358 cnfldsub 13360 cnsubglem 13364 zringgrp 13376 |
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