Theorem ablcomd 33517

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 19952	. 2 ⊢ ((𝐺 ∈ Abel ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
7	1, 2, 3, 6	syl3anc 1398	1 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  ‘cfv 6535  (class class class)co 7416  Basecbs 17326  +_gcplusg 17367  Abelcabl 19934

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 2147  ax-9 2155  ax-12 2213  ax-ext 2732

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 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6491  df-fv 6543  df-ov 7419  df-cmn 19935  df-abl 19936

This theorem is used by:  dflring2  33936  vietalem  34122