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Theorem cmn4 19934

Description: Commutative/associative law for commutative monoids. (Contributed by NM, 4-Feb-2014.) (Revised by Mario Carneiro, 21-Apr-2016.)

**Hypotheses**
Ref	Expression
ablcom.b	⊢ 𝐵 = (Base‘𝐺)
ablcom.p	⊢ + = (+_g‘𝐺)

**Assertion**
Ref	Expression
cmn4	⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

**Proof of Theorem cmn4**
Step	Hyp	Ref	Expression
1		ablcom.b	. 2 ⊢ 𝐵 = (Base‘𝐺)
2		ablcom.p	. 2 ⊢ + = (+_g‘𝐺)
3		simp1 1154	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝐺 ∈ CMnd)
4		cmnmnd 19930	. . 3 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
5	3, 4	syl 18	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝐺 ∈ Mnd)
6		simp2l 1218	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
7		simp2r 1219	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
8		simp3l 1220	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
9		simp3r 1221	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑊 ∈ 𝐵)
10	1, 2	cmncom 19931	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
11	3, 7, 8, 10	syl3anc 1398	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
12	1, 2, 5, 6, 7, 8, 9, 11	mnd4g 18857	1 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → ((𝑋 + 𝑌) + (𝑍 + 𝑊)) = ((𝑋 + 𝑍) + (𝑌 + 𝑊)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ‘cfv 6537 (class class class)co 7417 Basecbs 17307 +_gcplusg 17348 Mndcmnd 18842 CMndccmn 19913

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 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-12 2215 ax-ext 2734 ax-nul 5267

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 2100 df-clab 2741 df-cleq 2754 df-clel 2837 df-ne 2958 df-ral 3079 df-rex 3089 df-rab 3415 df-v 3455 df-sbc 3743 df-dif 3905 df-un 3907 df-ss 3919 df-nul 4283 df-if 4486 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-br 5108 df-iota 6493 df-fv 6545 df-ov 7420 df-mgm 18736 df-sgrp 18827 df-mnd 18843 df-cmn 19915

This theorem is used by: ablsub4 19943 ghmplusg 19979 lmod4 21102 cringm4 21540 ip2di 21860 evlslem1 22304 clmsub4 25340 cmn4d 33480 lfladdcl 39952

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