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Theorem iscmn 18420
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 6353 . . . . 5 (𝑔 = 𝐺 → (Base‘𝑔) = (Base‘𝐺))
2 iscmn.b . . . . 5 𝐵 = (Base‘𝐺)
31, 2syl6eqr 2812 . . . 4 (𝑔 = 𝐺 → (Base‘𝑔) = 𝐵)
4 raleq 3277 . . . . 5 ((Base‘𝑔) = 𝐵 → (∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ ∀𝑦𝐵 (𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥)))
54raleqbi1dv 3285 . . . 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 6353 . . . . . . 7 (𝑔 = 𝐺 → (+g𝑔) = (+g𝐺))
8 iscmn.p . . . . . . 7 + = (+g𝐺)
97, 8syl6eqr 2812 . . . . . 6 (𝑔 = 𝐺 → (+g𝑔) = + )
109oveqd 6831 . . . . 5 (𝑔 = 𝐺 → (𝑥(+g𝑔)𝑦) = (𝑥 + 𝑦))
119oveqd 6831 . . . . 5 (𝑔 = 𝐺 → (𝑦(+g𝑔)𝑥) = (𝑦 + 𝑥))
1210, 11eqeq12d 2775 . . . 4 (𝑔 = 𝐺 → ((𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥) ↔ (𝑥 + 𝑦) = (𝑦 + 𝑥)))
13122ralbidv 3127 . . 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 18415 . 2 CMnd = {𝑔 ∈ Mnd ∣ ∀𝑥 ∈ (Base‘𝑔)∀𝑦 ∈ (Base‘𝑔)(𝑥(+g𝑔)𝑦) = (𝑦(+g𝑔)𝑥)}
1614, 15elrab2 3507 1 (𝐺 ∈ CMnd ↔ (𝐺 ∈ Mnd ∧ ∀𝑥𝐵𝑦𝐵 (𝑥 + 𝑦) = (𝑦 + 𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  cfv 6049  (class class class)co 6814  Basecbs 16079  +gcplusg 16163  Mndcmnd 17515  CMndccmn 18413
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 6817  df-cmn 18415
This theorem is referenced by:  isabl2  18421  cmnpropd  18422  iscmnd  18425  cmnmnd  18428  cmncom  18429  ghmcmn  18457  submcmn2  18464  iscrng2  18783  xrs1cmn  20008  abliso  30026  gicabl  38189  pgrpgt2nabl  42675
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