Theorem srgcom 20319

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 20302	. 2 ⊢ (𝑅 ∈ SRing → 𝑅 ∈ CMnd)
2		srgacl.b	. . 3 ⊢ 𝐵 = (Base‘𝑅)
3		srgacl.p	. . 3 ⊢ + = (+g‘𝑅)
4	2, 3	cmncom 19899	. 2 ⊢ ((𝑅 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
5	1, 4	syl3an1 1181	1 ⊢ ((𝑅 ∈ SRing ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

This proof depends on syntax axioms:   → wi 4   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ‘cfv 6543  (class class class)co 7423  Basecbs 17294  +_gcplusg 17335  CMndccmn 19881  SRingcsrg 20299

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-12 2216  ax-ext 2738  ax-nul 5274

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ne 2962  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6499  df-fv 6551  df-ov 7426  df-cmn 19883  df-srg 20300

This theorem is used by: (None)