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| Description: Closure of the addition operation of a non-unital ring. (Contributed by AV, 16-Feb-2025.) | 
| Ref | Expression | 
|---|---|
| rngacl.b | ⊢ 𝐵 = (Base‘𝑅) | 
| rngacl.p | ⊢ + = (+g‘𝑅) | 
| Ref | Expression | 
|---|---|
| rngacl | ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | rnggrp 20155 | . 2 ⊢ (𝑅 ∈ Rng → 𝑅 ∈ Grp) | |
| 2 | rngacl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
| 3 | rngacl.p | . . 3 ⊢ + = (+g‘𝑅) | |
| 4 | 2, 3 | grpcl 18959 | . 2 ⊢ ((𝑅 ∈ Grp ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) | 
| 5 | 1, 4 | syl3an1 1164 | 1 ⊢ ((𝑅 ∈ Rng ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 (class class class)co 7431 Basecbs 17247 +gcplusg 17297 Grpcgrp 18951 Rngcrng 20149 | 
| 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-ext 2708 ax-nul 5306 | 
| 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-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 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-iota 6514 df-fv 6569 df-ov 7434 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-grp 18954 df-abl 19801 df-rng 20150 | 
| This theorem is referenced by: imasrng 20174 qusrng 20177 cntzsubrng 20567 rngqiprngghmlem2 21298 rngqiprngghm 21309 | 
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