Theorem cmncom 19771

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 19762	. . . . 5 ⊢ (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
4	3	simprbi 498	. . . 4 ⊢ (𝐺 ∈ CMnd → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5		rsp2 3257	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) → ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
6	5	imp 407	. . . 4 ⊢ ((∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
7	4, 6	sylan 586	. . 3 ⊢ ((𝐺 ∈ CMnd ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
8	7	caovcomg 7558	. 2 ⊢ ((𝐺 ∈ CMnd ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
9	8	3impb 1120	1 ⊢ ((𝐺 ∈ CMnd ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∀wral 3054  ‘cfv 6492  (class class class)co 7363  Basecbs 17177  +_gcplusg 17218  Mndcmnd 18700  CMndccmn 19753

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-12 2189  ax-ext 2712

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-iota 6448  df-fv 6500  df-ov 7366  df-cmn 19755

This theorem is referenced by:  ablcom  19772  cmn32  19773  cmn4  19774  cmn12  19775  cmnbascntr  19778  rinvmod  19779  mulgnn0di  19798  ghmcmn  19804  subcmn  19810  cntzcmn  19813  prdscmnd  19834  omndadd2d  20103  srgcom  20185  srgbinomlem4  20208  csrgbinom  20211  crngcom  20230  ofldchr  21558  ip2di  21623  lply1binom  22303  chfacfscmulgsum  22850  chfacfpmmulgsum  22854  cpmadugsumlemF  22866  cmn246135  33119  cmn145236  33120  gsumwun  33164  rlocaddval  33356  elrspunsn  33519  evl1deg2  33667  evl1deg3  33668  primrootsunit1  42589  aks6d1c5lem3  42629