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Theorem iscmnd 13106
Description: Properties that determine a commutative monoid. (Contributed by Mario Carneiro, 7-Jan-2015.)
Hypotheses
Ref Expression
iscmnd.b (𝜑𝐵 = (Base‘𝐺))
iscmnd.p (𝜑+ = (+g𝐺))
iscmnd.g (𝜑𝐺 ∈ Mnd)
iscmnd.c ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
Assertion
Ref Expression
iscmnd (𝜑𝐺 ∈ CMnd)
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦   𝜑,𝑥,𝑦
Allowed substitution hints:   + (𝑥,𝑦)

Proof of Theorem iscmnd
StepHypRef Expression
1 iscmnd.g . . 3 (𝜑𝐺 ∈ Mnd)
2 iscmnd.c . . . . 5 ((𝜑𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥))
323expib 1206 . . . 4 (𝜑 → ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
43ralrimivv 2558 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5 iscmnd.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
6 iscmnd.p . . . . . . . 8 (𝜑+ = (+g𝐺))
76oveqd 5894 . . . . . . 7 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
86oveqd 5894 . . . . . . 7 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐺)𝑥))
97, 8eqeq12d 2192 . . . . . 6 (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
105, 9raleqbidv 2685 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
115, 10raleqbidv 2685 . . . 4 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
1211anbi2d 464 . . 3 (𝜑 → ((𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)) ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥))))
131, 4, 12mpbi2and 943 . 2 (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
14 eqid 2177 . . 3 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2177 . . 3 (+g𝐺) = (+g𝐺)
1614, 15iscmn 13101 . 2 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
1713, 16sylibr 134 1 (𝜑𝐺 ∈ CMnd)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  w3a 978   = wceq 1353  wcel 2148  wral 2455  cfv 5218  (class class class)co 5877  Basecbs 12464  +gcplusg 12538  Mndcmnd 12822  CMndccmn 13093
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 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-iota 5180  df-fv 5226  df-ov 5880  df-cmn 13095
This theorem is referenced by:  isabld  13107  subcmnd  13134  iscrngd  13226
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