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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 18552 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | ringacl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | ringacl.p | . . 3 ⊢ + = (+g‘𝑅) | |
4 | 2, 3 | grpcl 17430 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1359 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ‘cfv 5888 (class class class)co 6650 Basecbs 15857 +gcplusg 15941 Grpcgrp 17422 Ringcrg 18547 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-iota 5851 df-fv 5896 df-ov 6653 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-grp 17425 df-ring 18549 |
This theorem is referenced by: ringcom 18579 ringlghm 18604 ringrghm 18605 imasring 18619 qusring2 18620 cntzsubr 18812 srngadd 18857 issrngd 18861 lmodprop2d 18925 prdslmodd 18969 psrlmod 19401 mpfind 19536 coe1add 19634 ip2subdi 19989 mat1ghm 20289 scmatghm 20339 mdetrlin2 20413 mdetunilem5 20422 cpmatacl 20521 mdegaddle 23834 deg1addle2 23862 deg1add 23863 ply1divex 23896 dvhlveclem 36397 baerlem3lem1 36996 mendlmod 37763 cznrng 41955 lmod1lem3 42278 |
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