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Theorem iscmn 19807
Description: The predicate "is a commutative monoid". (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
iscmn.b 𝐵 = (Base‘𝐺)
iscmn.p + = (+g𝐺)
Assertion
Ref Expression
iscmn (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐺,𝑦
Allowed substitution hints:   + (𝑥,𝑦)

Proof of Theorem iscmn
Dummy variable 𝑔 is distinct from all other variables.
StepHypRef Expression
1 fveq2 6906 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 iscmn.b . . . . 5 𝐵 = (Base‘𝐺)
31, 2eqtr4di 2795 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 raleq 3323 . . . . 5 ((Base‘𝑔) = 𝐵 → (∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑦𝐵 (𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥)))
54raleqbi1dv 3338 . . . 4 ((Base‘𝑔) = 𝐵 → (∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥)))
63, 5syl 17 . . 3 (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥)))
7 fveq2 6906 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
8 iscmn.p . . . . . . 7 + = (+g𝐺)
97, 8eqtr4di 2795 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
109oveqd 7448 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
119oveqd 7448 . . . . 5 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
1210, 11eqeq12d 2753 . . . 4 (𝑔 = 𝐺 → ((𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13122ralbidv 3221 . . 3 (𝑔 = 𝐺 → (∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
146, 13bitrd 279 . 2 (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
15 df-cmn 19800 . 2 CMnd = {𝑔 ∈ Mnd ∣ ∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥)}
1614, 15elrab2 3695 1 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  Mndcmnd 18747  CMndccmn 19798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-iota 6514  df-fv 6569  df-ov 7434  df-cmn 19800
This theorem is referenced by:  isabl2  19808  cmnpropd  19809  iscmnd  19812  cmnmnd  19815  cmncom  19816  ghmcmn  19849  submcmn2  19857  cycsubmcmn  19907  iscrng2  20249  xrs1cmn  21424  abliso  33041  gicabl  43111  pgrpgt2nabl  48282
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