Theorem ablcomd 33226

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 19840	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
7	1, 2, 3, 6	syl3anc 1391	1 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Syntax hints:   → wi 4   = wceq 1561   ∈ wcel 2143  ‘cfv 6522  (class class class)co 7397  Basecbs 17246  +_gcplusg 17287  Abelcabl 19822

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-iota 6478  df-fv 6530  df-ov 7400  df-cmn 19823  df-abl 19824

This theorem is referenced by:  dflring2  33690  vietalem  33877