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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 37915 | . 2 ⊢ (𝑅 ∈ RingOps → 𝐺 ∈ AbelOp) |
| 3 | ablogrpo 30566 | . 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 1540 ∈ wcel 2108 ‘cfv 6561 1st c1st 8012 GrpOpcgr 30508 AbelOpcablo 30563 RingOpscrngo 37901 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-fv 6569 df-ov 7434 df-1st 8014 df-2nd 8015 df-ablo 30564 df-rngo 37902 |
| This theorem is referenced by: rngone0 37918 rngogcl 37919 rngoaass 37921 rngorcan 37924 rngolcan 37925 rngo0cl 37926 rngo0rid 37927 rngo0lid 37928 rngolz 37929 rngorz 37930 rngosn3 37931 rngonegcl 37934 rngoaddneg1 37935 rngoaddneg2 37936 rngosub 37937 rngodm1dm2 37939 rngorn1 37940 rngonegmn1l 37948 rngonegmn1r 37949 rngogrphom 37978 rngohom0 37979 rngohomsub 37980 rngokerinj 37982 keridl 38039 dmncan1 38083 |
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