mnd32g - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > mnd32g		Structured version Visualization version GIF version

Theorem mnd32g 18185

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

**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 7229	. 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 18182	. . 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 18182	. . 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 2787	1 ⊢ (𝜑 → ((𝑋 + 𝑌) + 𝑍) = ((𝑋 + 𝑍) + 𝑌))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2110 ‘cfv 6380 (class class class)co 7213 Basecbs 16760 +_gcplusg 16802 Mndcmnd 18173

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-nul 5199

This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-br 5054 df-iota 6338 df-fv 6388 df-ov 7216 df-sgrp 18163 df-mnd 18174

This theorem is referenced by: cmn32 19189