Theorem srgcom 20164

Description: Commutativity of the additive group of a semiring. (Contributed by Thierry Arnoux, 1-Apr-2018.)

Hypotheses

Ref	Expression
srgacl.b	⊢ 𝐵 = (Base‘𝑅)
srgacl.p	⊢ + = (+g‘𝑅)

Assertion

Ref	Expression
srgcom	⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Proof of Theorem srgcom

Step	Hyp	Ref	Expression
1		srgcmn 20147	. 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd)
2		srgacl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		srgacl.p	. . 3 ⊢ + = (+g‘𝑅)
4	2, 3	cmncom 19777	. 2 ⊢ ((𝑅 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
5	1, 4	syl3an1 1163	1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Syntax hints:   → wi 4   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ‘cfv 6530  (class class class)co 7403  Basecbs 17226  +_gcplusg 17269  CMndccmn 19759  SRingcsrg 20144

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-12 2177  ax-ext 2707  ax-nul 5276

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-iota 6483  df-fv 6538  df-ov 7406  df-cmn 19761  df-srg 20145

This theorem is referenced by: (None)