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Theorem cmn4 19714

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 1135	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝐺 ∈ CMnd)
4		cmnmnd 19710	. . 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 1198	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑋 ∈ 𝐵)
7		simp2r 1199	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑌 ∈ 𝐵)
8		simp3l 1200	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑍 ∈ 𝐵)
9		simp3r 1201	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑍 ∈ 𝐵 ∧ 𝑊 ∈ 𝐵)) → 𝑊 ∈ 𝐵)
10	1, 2	cmncom 19711	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 + 𝑍) = (𝑍 + 𝑌))
11	3, 7, 8, 10	syl3anc 1370	. 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 18676	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 1540 ∈ wcel 2105 ‘cfv 6543 (class class class)co 7412 Basecbs 17151 +_gcplusg 17204 Mndcmnd 18662 CMndccmn 19693

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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-12 2170 ax-ext 2702 ax-nul 5306

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-sb 2067 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3778 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-iota 6495 df-fv 6551 df-ov 7415 df-mgm 18568 df-sgrp 18647 df-mnd 18663 df-cmn 19695

This theorem is referenced by: ablsub4 19723 ghmplusg 19759 lmod4 20670 ip2di 21417 evlslem1 21869 clmsub4 24866 cringm4 32854 lfladdcl 38257

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