Theorem abl32 19323

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 19308	. . 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 19320	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))
10	3, 4, 5, 6, 9	syl13anc 1370	1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  CMndccmn 19301  Abelcabl 19302

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-nul 5225

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-iota 6376  df-fv 6426  df-ov 7258  df-sgrp 18290  df-mnd 18301  df-cmn 19303  df-abl 19304

This theorem is referenced by:  matunitlindflem1  35700  baerlem5alem1  39649  baerlem5blem1  39650