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Mirrors > Home > MPE Home > Th. List > ringacl | Structured version Visualization version GIF version |
Description: Closure of the addition operation of a ring. (Contributed by Mario Carneiro, 14-Jan-2014.) |
Ref | Expression |
---|---|
ringacl.b | ⊢ 𝐵 = (Base‘𝑅) |
ringacl.p | ⊢ + = (+g‘𝑅) |
Ref | Expression |
---|---|
ringacl | ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ringgrp 20217 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | ringacl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | ringacl.p | . . 3 ⊢ + = (+g‘𝑅) | |
4 | 2, 3 | grpcl 18931 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1160 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 +gcplusg 17261 Grpcgrp 18923 Ringcrg 20212 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-ext 2697 ax-nul 5303 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-iota 6498 df-fv 6554 df-ov 7419 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-grp 18926 df-ring 20214 |
This theorem is referenced by: ringcomlem 20254 ringcom 20255 ringlghm 20287 ringrghm 20288 imasring 20305 qusring2 20309 cntzsubr 20586 srngadd 20826 issrngd 20830 lmodprop2d 20896 prdslmodd 20942 rhmpreimaidl 21262 ip2subdi 21636 psrlmod 21965 mpfind 22118 coe1add 22251 mat1ghm 22473 scmatghm 22523 mdetrlin2 22597 mdetunilem5 22606 cpmatacl 22706 mdegaddle 26098 deg1addle2 26126 deg1add 26127 ply1divex 26161 frobrhm 33103 deg1addlt 33473 dvhlveclem 40820 baerlem3lem1 41419 mendlmod 42891 cznrng 47674 lmod1lem3 47908 |
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