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Theorem iscmn 18132
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 6153 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 iscmn.b . . . . 5 𝐵 = (Base‘𝐺)
31, 2syl6eqr 2673 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 raleq 3130 . . . . 5 ((Base‘𝑔) = 𝐵 → (∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑦𝐵 (𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥)))
54raleqbi1dv 3138 . . . 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 6153 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
8 iscmn.p . . . . . . 7 + = (+g𝐺)
97, 8syl6eqr 2673 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
109oveqd 6627 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
119oveqd 6627 . . . . 5 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
1210, 11eqeq12d 2636 . . . 4 (𝑔 = 𝐺 → ((𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13122ralbidv 2984 . . 3 (𝑔 = 𝐺 → (∀𝑥𝐵𝑦𝐵 (𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
146, 13bitrd 268 . 2 (𝑔 = 𝐺 → (∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
15 df-cmn 18127 . 2 CMnd = {𝑔 ∈ Mnd ∣ ∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥)}
1614, 15elrab2 3352 1 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  cfv 5852  (class class class)co 6610  Basecbs 15792  +gcplusg 15873  Mndcmnd 17226  CMndccmn 18125
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-iota 5815  df-fv 5860  df-ov 6613  df-cmn 18127
This theorem is referenced by:  isabl2  18133  cmnpropd  18134  iscmnd  18137  cmnmnd  18140  cmncom  18141  ghmcmn  18169  submcmn2  18176  iscrng2  18495  xrs1cmn  19718  abliso  29505  gicabl  37184  pgrpgt2nabl  41461
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