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Theorem cmncom 13036
Description: A commutative monoid is commutative. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ablcom.b 𝐵 = (Base‘𝐺)
ablcom.p + = (+g𝐺)
Assertion
Ref Expression
cmncom ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))

Proof of Theorem cmncom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablcom.b . . . . . 6 𝐵 = (Base‘𝐺)
2 ablcom.p . . . . . 6 + = (+g𝐺)
31, 2iscmn 13027 . . . . 5 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
43simprbi 275 . . . 4 (𝐺 ∈ CMnd → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5 rsp2 2527 . . . . 5 (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) → ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
65imp 124 . . . 4 ((∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
74, 6sylan 283 . . 3 ((𝐺 ∈ CMnd ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
87caovcomg 6027 . 2 ((𝐺 ∈ CMnd ∧ (𝑋𝐵𝑌𝐵)) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
983impb 1199 1 ((𝐺 ∈ CMnd ∧ 𝑋𝐵𝑌𝐵) → (𝑋 + 𝑌) = (𝑌 + 𝑋))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wral 2455  cfv 5215  (class class class)co 5872  Basecbs 12454  +gcplusg 12528  Mndcmnd 12749  CMndccmn 13019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-un 3133  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4003  df-iota 5177  df-fv 5223  df-ov 5875  df-cmn 13021
This theorem is referenced by:  ablcom  13037  cmn32  13038  cmn4  13039  cmn12  13040  rinvmod  13043  subcmnd  13060  srgcom  13097  crngcom  13128
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