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Mirrors > Home > ILE Home > Th. List > ringacl | 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 13115 | . 2 ⊢ (𝑅 ∈ Ring → 𝑅 ∈ Grp) | |
2 | ringacl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | ringacl.p | . . 3 ⊢ + = (+g‘𝑅) | |
4 | 2, 3 | grpcl 12817 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1271 | 1 ⊢ ((𝑅 ∈ Ring ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ w3a 978 = wceq 1353 ∈ wcel 2148 ‘cfv 5215 (class class class)co 5872 Basecbs 12454 +gcplusg 12528 Grpcgrp 12809 Ringcrg 13110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-cnex 7899 ax-resscn 7900 ax-1re 7902 ax-addrcl 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-mpt 4065 df-id 4292 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-iota 5177 df-fun 5217 df-fn 5218 df-fv 5223 df-ov 5875 df-inn 8916 df-2 8974 df-3 8975 df-ndx 12457 df-slot 12458 df-base 12460 df-plusg 12541 df-mulr 12542 df-mgm 12707 df-sgrp 12740 df-mnd 12750 df-grp 12812 df-ring 13112 |
This theorem is referenced by: ringcom 13145 opprring 13180 mulgass3 13185 |
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