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Theorem cmn4 19730

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 1136	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝐺 ∈ CMnd)
4		cmnmnd 19726	. . 3 ⊢ (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
5	3, 4	syl 17	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝐺 ∈ Mnd)
6		simp2l 1200	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
7		simp2r 1201	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
8		simp3l 1202	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
9		simp3r 1203	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑊 ∈ 𝐵)
10	1, 2	cmncom 19727	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
11	3, 7, 8, 10	syl3anc 1373	. 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 18673	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

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1541 ∈ wcel 2113 ‘cfv 6492 (class class class)co 7358 Basecbs 17136 +_gcplusg 17177 Mndcmnd 18659 CMndccmn 19709

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-12 2184 ax-ext 2708 ax-nul 5251

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-sb 2068 df-clab 2715 df-cleq 2728 df-clel 2811 df-ne 2933 df-ral 3052 df-rex 3061 df-rab 3400 df-v 3442 df-sbc 3741 df-dif 3904 df-un 3906 df-ss 3918 df-nul 4286 df-if 4480 df-sn 4581 df-pr 4583 df-op 4587 df-uni 4864 df-br 5099 df-iota 6448 df-fv 6500 df-ov 7361 df-mgm 18565 df-sgrp 18644 df-mnd 18660 df-cmn 19711

This theorem is referenced by: ablsub4 19739 ghmplusg 19775 lmod4 20863 ip2di 21596 evlslem1 22037 clmsub4 25062 cmn4d 33114 cringm4 33527 lfladdcl 39331

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