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Mirrors > Home > MPE Home > Th. List > Mathboxes > rngogrpo | Structured version Visualization version GIF version |
Description: A ring's addition operation is a group operation. (Contributed by Steve Rodriguez, 9-Sep-2007.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ringgrp.1 | ⊢ 𝐺 = (1st ‘𝑅) |
Ref | Expression |
---|---|
rngogrpo | ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringgrp.1 | . . 3 ⊢ 𝐺 = (1st ‘𝑅) | |
2 | 1 | rngoablo 34037 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp) |
3 | ablogrpo 27741 | . 2 ⊢ (𝐺 ∈ AbelOp → 𝐺 ∈ GrpOp) | |
4 | 2, 3 | syl 17 | 1 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ GrpOp) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1631 ∈ wcel 2145 ‘cfv 6030 1st c1st 7317 GrpOpcgr 27683 AbelOpcablo 27738 RingOpscrngo 34023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pow 4975 ax-pr 5035 ax-un 7100 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-nul 4064 df-if 4227 df-sn 4318 df-pr 4320 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-mpt 4865 df-id 5158 df-xp 5256 df-rel 5257 df-cnv 5258 df-co 5259 df-dm 5260 df-rn 5261 df-iota 5993 df-fun 6032 df-fn 6033 df-f 6034 df-fv 6038 df-ov 6799 df-1st 7319 df-2nd 7320 df-ablo 27739 df-rngo 34024 |
This theorem is referenced by: rngone0 34040 rngogcl 34041 rngoaass 34043 rngorcan 34046 rngolcan 34047 rngo0cl 34048 rngo0rid 34049 rngo0lid 34050 rngolz 34051 rngorz 34052 rngosn3 34053 rngonegcl 34056 rngoaddneg1 34057 rngoaddneg2 34058 rngosub 34059 rngodm1dm2 34061 rngorn1 34062 rngonegmn1l 34070 rngonegmn1r 34071 rngogrphom 34100 rngohom0 34101 rngohomsub 34102 rngokerinj 34104 keridl 34161 dmncan1 34205 |
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