Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  iscmn Structured version   Visualization version   GIF version

Theorem iscmn 18986
 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 6662 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 iscmn.b . . . . 5 𝐵 = (Base‘𝐺)
31, 2eqtr4di 2811 . . . 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 3321 . . . 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 6662 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
8 iscmn.p . . . . . . 7 + = (+g𝐺)
97, 8eqtr4di 2811 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
109oveqd 7172 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
119oveqd 7172 . . . . 5 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
1210, 11eqeq12d 2774 . . . 4 (𝑔 = 𝐺 → ((𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13122ralbidv 3128 . . 3 (𝑔 = 𝐺 → (∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
146, 13bitrd 282 . 2 (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
15 df-cmn 18980 . 2 CMnd = {𝑔 ∈ Mnd ∣ ∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥)}
1614, 15elrab2 3607 1 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∀wral 3070  ‘cfv 6339  (class class class)co 7155  Basecbs 16546  +gcplusg 16628  Mndcmnd 17982  CMndccmn 18978 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rab 3079  df-v 3411  df-un 3865  df-in 3867  df-ss 3877  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-br 5036  df-iota 6298  df-fv 6347  df-ov 7158  df-cmn 18980 This theorem is referenced by:  isabl2  18987  cmnpropd  18988  iscmnd  18991  cmnmnd  18994  cmncom  18995  ghmcmn  19025  submcmn2  19032  cycsubmcmn  19081  iscrng2  19389  xrs1cmn  20211  abliso  30835  gicabl  40444  pgrpgt2nabl  45163
 Copyright terms: Public domain W3C validator