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Mirrors > Home > MPE Home > Th. List > srgcom | Structured version Visualization version GIF version |
Description: Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.) |
Ref | Expression |
---|---|
srgacl.b | ⊢ 𝐵 = (Base‘𝑅) |
srgacl.p | ⊢ + = (+g‘𝑅) |
Ref | Expression |
---|---|
srgcom | ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | srgcmn 20136 | . 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd) | |
2 | srgacl.b | . . 3 ⊢ 𝐵 = (Base‘𝑅) | |
3 | srgacl.p | . . 3 ⊢ + = (+g‘𝑅) | |
4 | 2, 3 | cmncom 19760 | . 2 ⊢ ((𝑅 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
5 | 1, 4 | syl3an1 1160 | 1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ‘cfv 6553 (class class class)co 7426 Basecbs 17187 +gcplusg 17240 CMndccmn 19742 SRingcsrg 20133 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2166 ax-ext 2699 ax-nul 5310 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-iota 6505 df-fv 6561 df-ov 7429 df-cmn 19744 df-srg 20134 |
This theorem is referenced by: (None) |
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