Theorem ablcomd 33308

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 19871	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
7	1, 2, 3, 6	syl3anc 1396	1 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  ‘cfv 6539  (class class class)co 7413  Basecbs 17271  +_gcplusg 17312  Abelcabl 19853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-iota 6495  df-fv 6547  df-ov 7416  df-cmn 19854  df-abl 19855

This theorem is referenced by:  dflring2  33730  vietalem  33916