Theorem srgcom 19010

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 18993	. 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd)
2		srgacl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		srgacl.p	. . 3 ⊢ + = (+g‘𝑅)
4	2, 3	cmncom 18694	. 2 ⊢ ((𝑅 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
5	1, 4	syl3an1 1144	1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Syntax hints:   → wi 4   ∧ w3a 1069   = wceq 1508   ∈ wcel 2051  ‘cfv 6185  (class class class)co 6974  Basecbs 16337  +_gcplusg 16419  CMndccmn 18678  SRingcsrg 18990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743  ax-nul 5063

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ral 3086  df-rex 3087  df-rab 3090  df-v 3410  df-sbc 3675  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-nul 4173  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-iota 6149  df-fv 6193  df-ov 6977  df-cmn 18680  df-srg 18991

This theorem is referenced by: (None)