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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 18906 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | ringacl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | ringacl.p | . . 3 ⊢ + = (+g‘𝑅) | |
4 | 2, 3 | grpcl 17784 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1208 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1113 = wceq 1658 ∈ wcel 2166 ‘cfv 6123 (class class class)co 6905 Basecbs 16222 +gcplusg 16305 Grpcgrp 17776 Ringcrg 18901 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-nul 5013 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-iota 6086 df-fv 6131 df-ov 6908 df-mgm 17595 df-sgrp 17637 df-mnd 17648 df-grp 17779 df-ring 18903 |
This theorem is referenced by: ringcom 18933 ringlghm 18958 ringrghm 18959 imasring 18973 qusring2 18974 cntzsubr 19168 srngadd 19213 issrngd 19217 lmodprop2d 19281 prdslmodd 19328 psrlmod 19762 mpfind 19896 coe1add 19994 ip2subdi 20351 mat1ghm 20657 scmatghm 20707 mdetrlin2 20781 mdetunilem5 20790 cpmatacl 20891 mdegaddle 24233 deg1addle2 24261 deg1add 24262 ply1divex 24295 dvhlveclem 37183 baerlem3lem1 37782 mendlmod 38606 cznrng 42802 lmod1lem3 43125 |
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