Theorem abl32 19719

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 19703	. . 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 19716	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
10	3, 4, 5, 6, 9	syl13anc 1374	1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  ‘cfv 6488  (class class class)co 7354  Basecbs 17124  +_gcplusg 17165  CMndccmn 19696  Abelcabl 19697

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 2182  ax-ext 2705  ax-nul 5248

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 2712  df-cleq 2725  df-clel 2808  df-ne 2930  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6444  df-fv 6496  df-ov 7357  df-sgrp 18631  df-mnd 18647  df-cmn 19698  df-abl 19699

This theorem is referenced by:  matunitlindflem1  37679  baerlem5alem1  41830  baerlem5blem1  41831