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Theorem cmnpropd 18912
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 17932 . . 3 (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
53oveqrspc2v 7172 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
63oveqrspc2v 7172 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → (𝑣(+g𝐾)𝑢) = (𝑣(+g𝐿)𝑢))
76ancom2s 649 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑣(+g𝐾)𝑢) = (𝑣(+g𝐿)𝑢))
85, 7eqeq12d 2840 . . . . 5 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → ((𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
982ralbidva 3193 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
101raleqdv 3403 . . . . 5 (𝜑 → (∀𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
111, 10raleqbidv 3393 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
122raleqdv 3403 . . . . 5 (𝜑 → (∀𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
132, 12raleqbidv 3393 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
149, 11, 133bitr3d 312 . . 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 633 . 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 2824 . . 3 (Base‘𝐾) = (Base‘𝐾)
17 eqid 2824 . . 3 (+g𝐾) = (+g𝐾)
1816, 17iscmn 18910 . 2 (𝐾 ∈ CMnd ↔ (𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
19 eqid 2824 . . 3 (Base‘𝐿) = (Base‘𝐿)
20 eqid 2824 . . 3 (+g𝐿) = (+g𝐿)
2119, 20iscmn 18910 . 2 (𝐿 ∈ CMnd ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
2215, 18, 213bitr4g 317 1 (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  wral 3133  cfv 6343  (class class class)co 7145  Basecbs 16479  +gcplusg 16561  Mndcmnd 17907  CMndccmn 18902
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-nul 5196  ax-pow 5253
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-iota 6302  df-fv 6351  df-ov 7148  df-mgm 17848  df-sgrp 17897  df-mnd 17908  df-cmn 18904
This theorem is referenced by:  ablpropd  18913  crngpropd  19329  prdscrngd  19359  resvcmn  30936
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