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Theorem cmnpropd 18248
Description: If two structures have the same group components (properties), one is a commutative monoid iff the other one is. (Contributed by Mario Carneiro, 6-Jan-2015.)
Hypotheses
Ref Expression
ablpropd.1 (𝜑𝐵 = (Base‘𝐾))
ablpropd.2 (𝜑𝐵 = (Base‘𝐿))
ablpropd.3 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
Assertion
Ref Expression
cmnpropd (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem cmnpropd
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ablpropd.1 . . . 4 (𝜑𝐵 = (Base‘𝐾))
2 ablpropd.2 . . . 4 (𝜑𝐵 = (Base‘𝐿))
3 ablpropd.3 . . . 4 ((𝜑 ∧ (𝑥𝐵𝑦𝐵)) → (𝑥(+g𝐾)𝑦) = (𝑥(+g𝐿)𝑦))
41, 2, 3mndpropd 17363 . . 3 (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
53oveqrspc2v 6713 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
63oveqrspc2v 6713 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → (𝑣(+g𝐾)𝑢) = (𝑣(+g𝐿)𝑢))
76ancom2s 861 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑣(+g𝐾)𝑢) = (𝑣(+g𝐿)𝑢))
85, 7eqeq12d 2666 . . . . 5 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → ((𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
982ralbidva 3017 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
101raleqdv 3174 . . . . 5 (𝜑 → (∀𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
111, 10raleqbidv 3182 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
122raleqdv 3174 . . . . 5 (𝜑 → (∀𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
132, 12raleqbidv 3182 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
149, 11, 133bitr3d 298 . . 3 (𝜑 → (∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
154, 14anbi12d 747 . 2 (𝜑 → ((𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)) ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢))))
16 eqid 2651 . . 3 (Base‘𝐾) = (Base‘𝐾)
17 eqid 2651 . . 3 (+g𝐾) = (+g𝐾)
1816, 17iscmn 18246 . 2 (𝐾 ∈ CMnd ↔ (𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
19 eqid 2651 . . 3 (Base‘𝐿) = (Base‘𝐿)
20 eqid 2651 . . 3 (+g𝐿) = (+g𝐿)
2119, 20iscmn 18246 . 2 (𝐿 ∈ CMnd ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
2215, 18, 213bitr4g 303 1 (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wcel 2030  wral 2941  cfv 5926  (class class class)co 6690  Basecbs 15904  +gcplusg 15988  Mndcmnd 17341  CMndccmn 18239
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-nul 4822  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-iota 5889  df-fv 5934  df-ov 6693  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-cmn 18241
This theorem is referenced by:  ablpropd  18249  crngpropd  18629  prdscrngd  18659  resvcmn  29966
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