Theorem mnd12g 18963

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)
mnd12g.5	⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Assertion

Ref	Expression
mnd12g	⊢ (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))

Proof of Theorem mnd12g

Step	Hyp	Ref	Expression
1		mnd12g.5	. . 3 ⊢ (𝜑 → (𝑋 + 𝑌) = (𝑌 + 𝑋))
2	1	oveq1d 7455	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑌 + 𝑋) + 𝑍))
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 18958	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
10	3, 4, 5, 6, 9	syl13anc 1400	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = (𝑋 + (𝑌 + 𝑍)))
11	7, 8	mndass 18958	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
12	3, 5, 4, 6, 11	syl13anc 1400	. 2 ⊢ (𝜑 → ((𝑌 + 𝑋) + 𝑍) = (𝑌 + (𝑋 + 𝑍)))
13	2, 10, 12	3eqtr3d 2806	1 ⊢ (𝜑 → (𝑋 + (𝑌 + 𝑍)) = (𝑌 + (𝑋 + 𝑍)))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2147  ‘cfv 6555  (class class class)co 7440  Basecbs 17413  +_gcplusg 17454  Mndcmnd 18949

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-nul 5263

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6509  df-fv 6563  df-ov 7443  df-sgrp 18934  df-mnd 18950

This theorem is used by:  mnd4g  18964  cmn12  20043