Theorem mnd4g 18761

Description: Commutative/associative law for commutative 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)
mnd4g.5	⊢ (𝜑 → 𝑊 ∈ 𝐵)
mnd4g.6	⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))

Assertion

Ref	Expression
mnd4g	⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Proof of Theorem mnd4g

Step	Hyp	Ref	Expression
1		mndcl.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
2		mndcl.p	. . . 4 ⊢ + = (+g‘𝐺)
3		mnd4g.1	. . . 4 ⊢ (𝜑 → 𝐺 ∈ Mnd)
4		mnd4g.3	. . . 4 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
5		mnd4g.4	. . . 4 ⊢ (𝜑 → 𝑍 ∈ 𝐵)
6		mnd4g.5	. . . 4 ⊢ (𝜑 → 𝑊 ∈ 𝐵)
7		mnd4g.6	. . . 4 ⊢ (𝜑 → (𝑌 + 𝑍) = (𝑍 + 𝑌))
8	1, 2, 3, 4, 5, 6, 7	mnd12g 18760	. . 3 ⊢ (𝜑 → (𝑌 + (𝑍 + 𝑊)) = (𝑍 + (𝑌 + 𝑊)))
9	8	oveq2d 7447	. 2 ⊢ (𝜑 → (𝑋 + (𝑌 + (𝑍 + 𝑊))) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
10		mnd4g.2	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
11	1, 2	mndcl 18755	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑍 + 𝑊) ∈ 𝐵)
12	3, 5, 6, 11	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑍 + 𝑊) ∈ 𝐵)
13	1, 2	mndass 18756	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ (𝑍 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊))))
14	3, 10, 4, 12, 13	syl13anc 1374	. 2 ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = (𝑋 + (𝑌 + (𝑍 + 𝑊))))
15	1, 2	mndcl 18755	. . . 4 ⊢ ((𝐺 ∈ Mnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵) → (𝑌 + 𝑊) ∈ 𝐵)
16	3, 4, 6, 15	syl3anc 1373	. . 3 ⊢ (𝜑 → (𝑌 + 𝑊) ∈ 𝐵)
17	1, 2	mndass 18756	. . 3 ⊢ ((𝐺 ∈ Mnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵 ∧ (𝑌 + 𝑊) ∈ 𝐵)) → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
18	3, 10, 5, 16, 17	syl13anc 1374	. 2 ⊢ (𝜑 → ((𝑋 + 𝑍) + (𝑌 + 𝑊)) = (𝑋 + (𝑍 + (𝑌 + 𝑊))))
19	9, 14, 18	3eqtr4d 2787	1 ⊢ (𝜑 → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108  ‘cfv 6561  (class class class)co 7431  Basecbs 17247  +_gcplusg 17297  Mndcmnd 18747

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-nul 5306

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-mgm 18653  df-sgrp 18732  df-mnd 18748

This theorem is referenced by:  lsmsubm  19671  pj1ghm  19721  cmn4  19819  gsumzaddlem  19939