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Theorem iscmnd 18405
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 1117 . . . 4 (𝜑 → ((𝑥𝐵𝑦𝐵) → (𝑥 + 𝑦) = (𝑦 + 𝑥)))
43ralrimivv 3108 . . 3 (𝜑 → ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥))
5 iscmnd.b . . . . 5 (𝜑𝐵 = (Base‘𝐺))
6 iscmnd.p . . . . . . . 8 (𝜑+ = (+g𝐺))
76oveqd 6830 . . . . . . 7 (𝜑 → (𝑥 + 𝑦) = (𝑥(+g𝐺)𝑦))
86oveqd 6830 . . . . . . 7 (𝜑 → (𝑦 + 𝑥) = (𝑦(+g𝐺)𝑥))
97, 8eqeq12d 2775 . . . . . 6 (𝜑 → ((𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ (𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
105, 9raleqbidv 3291 . . . . 5 (𝜑 → (∀𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
115, 10raleqbidv 3291 . . . 4 (𝜑 → (∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥) ↔ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
1211anbi2d 742 . . 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 994 . 2 (𝜑 → (𝐺 ∈ Mnd ∧ ∀𝑥 ∈ (Base‘𝐺)∀𝑦 ∈ (Base‘𝐺)(𝑥(+g𝐺)𝑦) = (𝑦(+g𝐺)𝑥)))
14 eqid 2760 . . 3 (Base‘𝐺) = (Base‘𝐺)
15 eqid 2760 . . 3 (+g𝐺) = (+g𝐺)
1614, 15iscmn 18400 . 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 383  w3a 1072   = wceq 1632  wcel 2139  wral 3050  cfv 6049  (class class class)co 6813  Basecbs 16059  +gcplusg 16143  Mndcmnd 17495  CMndccmn 18393
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-iota 6012  df-fv 6057  df-ov 6816  df-cmn 18395
This theorem is referenced by:  isabld  18406  subcmn  18442  prdscmnd  18464  iscrngd  18786  psrcrng  19615  xrsmcmn  19971  2zrngacmnd  42452
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