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Theorem iscmnd 18126
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 1265 . . . 4 (𝜑 → ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
43ralrimivv 2964 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5 iscmnd.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
6 iscmnd.p . . . . . . . 8 (𝜑+ = (+g𝐺))
76oveqd 6621 . . . . . . 7 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
86oveqd 6621 . . . . . . 7 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐺)𝑥))
97, 8eqeq12d 2636 . . . . . 6 (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
105, 9raleqbidv 3141 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
115, 10raleqbidv 3141 . . . 4 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
1211anbi2d 739 . . 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 955 . 2 (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
14 eqid 2621 . . 3 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2621 . . 3 (+g𝐺) = (+g𝐺)
1614, 15iscmn 18121 . 2 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
1713, 16sylibr 224 1 (𝜑𝐺 ∈ CMnd)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  cfv 5847  (class class class)co 6604  Basecbs 15781  +gcplusg 15862  Mndcmnd 17215  CMndccmn 18114
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 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-iota 5810  df-fv 5855  df-ov 6607  df-cmn 18116
This theorem is referenced by:  isabld  18127  subcmn  18163  prdscmnd  18185  iscrngd  18507  psrcrng  19332  xrsmcmn  19688  2zrngacmnd  41230
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