Theorem mnd32g 18673

Description: Commutative/associative law for monoids, with an explicit commutativity hypothesis. (Contributed by Mario Carneiro, 21-Apr-2016.)

Hypotheses

Ref	Expression
mndcl.b	⊢ 𝐵 = (Base‘𝐺)
mndcl.p	⊢ + = (+g‘𝐺)
mnd4g.1	⊢ (𝜑 → 𝐺 ∈ Mnd)
mnd4g.2	⊢ (𝜑 → 𝑋 ∈ 𝐵)
mnd4g.3	⊢ (𝜑 → 𝑌 ∈ 𝐵)
mnd4g.4	⊢ (𝜑 → 𝑍 ∈ 𝐵)
mnd32g.5	⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))

Assertion

Ref	Expression
mnd32g	⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Proof of Theorem mnd32g

Step	Hyp	Ref	Expression
1		mnd32g.5	. . 3 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
2	1	oveq2d 7403	. 2 ⊢ (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑋 + (𝑍 + 𝑌)))
3		mnd4g.1	. . 3 ⊢ (𝜑 → 𝐺 ∈ Mnd)
4		mnd4g.2	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
5		mnd4g.3	. . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
6		mnd4g.4	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
7		mndcl.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
8		mndcl.p	. . . 4 ⊢ + = (+g‘𝐺)
9	7, 8	mndass 18670	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
10	3, 4, 5, 6, 9	syl13anc 1374	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
11	7, 8	mndass 18670	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → ((𝑋 + 𝑍) + 𝑌) = (𝑋 + (𝑍 + 𝑌)))
12	3, 4, 6, 5, 11	syl13anc 1374	. 2 ⊢ (𝜑 → ((𝑋 + 𝑍) + 𝑌) = (𝑋 + (𝑍 + 𝑌)))
13	2, 10, 12	3eqtr4d 2774	1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  Mndcmnd 18661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-nul 5261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-iota 6464  df-fv 6519  df-ov 7390  df-sgrp 18646  df-mnd 18662

This theorem is referenced by:  cmn32  19730  gsumwun  33005