![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > srgacl | Structured version Visualization version GIF version |
Description: Closure of the addition operation of a semiring. (Contributed by Mario Carneiro, 14-Jan-2014.) (Revised by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
srgacl.b | ⊢ 𝐵 = (Base‘𝑅) |
srgacl.p | ⊢ + = (+g‘𝑅) |
Ref | Expression |
---|---|
srgacl | ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srgmnd 19880 | . 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ Mnd) | |
2 | srgacl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | srgacl.p | . . 3 ⊢ + = (+g‘𝑅) | |
4 | 2, 3 | mndcl 18524 | . 2 ⊢ ((𝑅 ∈ Mnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
5 | 1, 4 | syl3an1 1163 | 1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) ∈ 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ‘cfv 6493 (class class class)co 7351 Basecbs 17043 +gcplusg 17093 Mndcmnd 18516 SRingcsrg 19876 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2708 ax-nul 5261 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-iota 6445 df-fv 6501 df-ov 7354 df-mgm 18457 df-sgrp 18506 df-mnd 18517 df-cmn 19523 df-srg 19877 |
This theorem is referenced by: srglmhm 19906 srgrmhm 19907 sge0tsms 44522 |
Copyright terms: Public domain | W3C validator |