Theorem abl32 19744

Description: Commutative/associative law for Abelian groups. (Contributed by Stefan O'Rear, 10-Apr-2015.) (Revised by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
ablcom.b	⊢ 𝐵 = (Base‘𝐺)
ablcom.p	⊢ + = (+g‘𝐺)
abl32.g	⊢ (𝜑 → 𝐺 ∈ Abel)
abl32.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
abl32.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
abl32.z	⊢ (𝜑 → 𝑍 ∈ 𝐵)

Assertion

Ref	Expression
abl32	⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Proof of Theorem abl32

Step	Hyp	Ref	Expression
1		abl32.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ Abel)
2		ablcmn 19728	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
3	1, 2	syl 17	. 2 ⊢ (𝜑 → 𝐺 ∈ CMnd)
4		abl32.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		abl32.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		abl32.z	. 2 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
7		ablcom.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
8		ablcom.p	. . 3 ⊢ + = (+g‘𝐺)
9	7, 8	cmn32 19741	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
10	3, 4, 5, 6, 9	syl13anc 1375	1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ‘cfv 6500  (class class class)co 7368  Basecbs 17148  +_gcplusg 17189  CMndccmn 19721  Abelcabl 19722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-12 2185  ax-ext 2709  ax-nul 5253

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-iota 6456  df-fv 6508  df-ov 7371  df-sgrp 18656  df-mnd 18672  df-cmn 19723  df-abl 19724

This theorem is referenced by:  matunitlindflem1  37864  baerlem5alem1  42081  baerlem5blem1  42082