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Theorem iscmnd 18848
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 1114 . . . 4 (𝜑 → ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
43ralrimivv 3187 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5 iscmnd.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
6 iscmnd.p . . . . . . . 8 (𝜑+ = (+g𝐺))
76oveqd 7162 . . . . . . 7 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
86oveqd 7162 . . . . . . 7 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐺)𝑥))
97, 8eqeq12d 2834 . . . . . 6 (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
105, 9raleqbidv 3399 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
115, 10raleqbidv 3399 . . . 4 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
1211anbi2d 628 . . 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 708 . 2 (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
14 eqid 2818 . . 3 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2818 . . 3 (+g𝐺) = (+g𝐺)
1614, 15iscmn 18843 . 2 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
1713, 16sylibr 235 1 (𝜑𝐺 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  cfv 6348  (class class class)co 7145  Basecbs 16471  +gcplusg 16553  Mndcmnd 17899  CMndccmn 18835
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-iota 6307  df-fv 6356  df-ov 7148  df-cmn 18837
This theorem is referenced by:  isabld  18849  subcmn  18886  cntrcmnd  18891  prdscmnd  18910  iscrngd  19265  psrcrng  20121  xrsmcmn  20496  2zrngacmnd  44141
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