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Theorem iscmnd 18028
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 1260 . . . 4 (𝜑 → ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
43ralrimivv 2953 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5 iscmnd.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
6 iscmnd.p . . . . . . . 8 (𝜑+ = (+g𝐺))
76oveqd 6566 . . . . . . 7 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
86oveqd 6566 . . . . . . 7 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐺)𝑥))
97, 8eqeq12d 2625 . . . . . 6 (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
105, 9raleqbidv 3129 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
115, 10raleqbidv 3129 . . . 4 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
1211anbi2d 736 . . 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 958 . 2 (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
14 eqid 2610 . . 3 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2610 . . 3 (+g𝐺) = (+g𝐺)
1614, 15iscmn 18023 . 2 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
1713, 16sylibr 223 1 (𝜑𝐺 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wral 2896  cfv 5804  (class class class)co 6549  Basecbs 15695  +gcplusg 15768  Mndcmnd 17117  CMndccmn 18016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-cmn 18018
This theorem is referenced by:  isabld  18029  subcmn  18065  prdscmnd  18087  iscrngd  18409  psrcrng  19234  xrsmcmn  19588  2zrngacmnd  41732
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