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Theorem cmnmnd 13057
Description: A commutative monoid is a monoid. (Contributed by Mario Carneiro, 6-Jan-2015.)
Assertion
Ref Expression
cmnmnd (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)

Proof of Theorem cmnmnd
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2177 . . 3 (Base‘𝐺) = (Base‘𝐺)
2 eqid 2177 . . 3 (+g𝐺) = (+g𝐺)
31, 2iscmn 13049 . 2 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
43simplbi 274 1 (𝐺 ∈ CMnd → 𝐺 ∈ Mnd)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1353  wcel 2148  wral 2455  cfv 5216  (class class class)co 5874  Basecbs 12456  +gcplusg 12530  Mndcmnd 12771  CMndccmn 13041
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 4004  df-iota 5178  df-fv 5224  df-ov 5877  df-cmn 13043
This theorem is referenced by:  cmn32  13060  cmn4  13061  cmn12  13062  cmnmndd  13064  rinvmod  13065  srgmnd  13103
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