Theorem ablcomd 33130

Description: An abelian group operation is commutative, deduction version. (Contributed by Thierry Arnoux, 15-Feb-2026.)

Hypotheses

Ref	Expression
ablcomd.1	⊢ 𝐵 = (Base‘𝐺)
ablcomd.2	⊢ + = (+g‘𝐺)
ablcomd.3	⊢ (𝜑 → 𝐺 ∈ Abel)
ablcomd.4	⊢ (𝜑 → 𝑋 ∈ 𝐵)
ablcomd.5	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
ablcomd	⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Proof of Theorem ablcomd

Step	Hyp	Ref	Expression
1		ablcomd.3	. 2 ⊢ (𝜑 → 𝐺 ∈ Abel)
2		ablcomd.4	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		ablcomd.5	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
4		ablcomd.1	. . 3 ⊢ 𝐵 = (Base‘𝐺)
5		ablcomd.2	. . 3 ⊢ + = (+g‘𝐺)
6	4, 5	ablcom 19730	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
7	1, 2, 3, 6	syl3anc 1373	1 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6492  (class class class)co 7358  Basecbs 17138  +_gcplusg 17179  Abelcabl 19712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-ov 7361  df-cmn 19713  df-abl 19714

This theorem is referenced by:  vietalem  33737