Theorem cmncom 19817

Description: A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)

Hypotheses

Ref	Expression
ablcom.b	⊢ 𝐵 = (Base‘𝐺)
ablcom.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
cmncom	⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Proof of Theorem cmncom

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ablcom.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
2		ablcom.p	. . . . . 6 ⊢ + = (+g‘𝐺)
3	1, 2	iscmn 19808	. . . . 5 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
4	3	simprbi 496	. . . 4 ⊢ (𝐺 ∈ CMnd → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5		rsp2 3276	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
6	5	imp 406	. . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
7	4, 6	sylan 580	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
8	7	caovcomg 7629	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
9	8	3impb 1114	1 ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1539   ∈ wcel 2107  ∀wral 3060  ‘cfv 6560  (class class class)co 7432  Basecbs 17248  +_gcplusg 17298  Mndcmnd 18748  CMndccmn 19799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-iota 6513  df-fv 6568  df-ov 7435  df-cmn 19801

This theorem is referenced by:  ablcom  19818  cmn32  19819  cmn4  19820  cmn12  19821  cmnbascntr  19824  rinvmod  19825  mulgnn0di  19844  ghmcmn  19850  subcmn  19856  cntzcmn  19859  prdscmnd  19880  srgcom  20204  srgbinomlem4  20227  csrgbinom  20230  crngcom  20249  ip2di  21660  lply1binom  22315  chfacfscmulgsum  22867  chfacfpmmulgsum  22871  cpmadugsumlemF  22883  cmn246135  33039  cmn145236  33040  gsumwun  33069  omndadd2d  33086  rlocaddval  33273  ofldchr  33345  elrspunsn  33458  evl1deg2  33603  evl1deg3  33604  primrootsunit1  42099  aks6d1c5lem3  42139