Theorem abl32 19896

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 19880	. . 3 ⊢ (𝐺 ∈ Abel → 𝐺 ∈ CMnd)
3	1, 2	syl 18	. 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 19893	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
10	3, 4, 5, 6, 9	syl13anc 1399	1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ‘cfv 6540  (class class class)co 7416  Basecbs 17286  +_gcplusg 17327  CMndccmn 19873  Abelcabl 19874

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 2148  ax-9 2156  ax-12 2216  ax-ext 2737  ax-nul 5271

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 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-iota 6496  df-fv 6548  df-ov 7419  df-sgrp 18798  df-mnd 18814  df-cmn 19875  df-abl 19876

This theorem is used by:  matunitlindflem1  38300  baerlem5alem1  42515  baerlem5blem1  42516