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Theorem cmnpropd 19652
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 18646 . . 3 (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
53oveqrspc2v 7431 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑢(+g𝐾)𝑣) = (𝑢(+g𝐿)𝑣))
63oveqrspc2v 7431 . . . . . . 7 ((𝜑 ∧ (𝑣𝐵𝑢𝐵)) → (𝑣(+g𝐾)𝑢) = (𝑣(+g𝐿)𝑢))
76ancom2s 649 . . . . . 6 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → (𝑣(+g𝐾)𝑢) = (𝑣(+g𝐿)𝑢))
85, 7eqeq12d 2749 . . . . 5 ((𝜑 ∧ (𝑢𝐵𝑣𝐵)) → ((𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
982ralbidva 3217 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
101raleqdv 3326 . . . . 5 (𝜑 → (∀𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
111, 10raleqbidv 3343 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
122raleqdv 3326 . . . . 5 (𝜑 → (∀𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢) ↔ ∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
132, 12raleqbidv 3343 . . . 4 (𝜑 → (∀𝑢𝐵𝑣𝐵 (𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢) ↔ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
149, 11, 133bitr3d 309 . . 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 632 . 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 2733 . . 3 (Base‘𝐾) = (Base‘𝐾)
17 eqid 2733 . . 3 (+g𝐾) = (+g𝐾)
1816, 17iscmn 19650 . 2 (𝐾 ∈ CMnd ↔ (𝐾 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐾)∀𝑣 ∈ (Base‘𝐾)(𝑢(+g𝐾)𝑣) = (𝑣(+g𝐾)𝑢)))
19 eqid 2733 . . 3 (Base‘𝐿) = (Base‘𝐿)
20 eqid 2733 . . 3 (+g𝐿) = (+g𝐿)
2119, 20iscmn 19650 . 2 (𝐿 ∈ CMnd ↔ (𝐿 ∈ Mnd ∧ ∀𝑢 ∈ (Base‘𝐿)∀𝑣 ∈ (Base‘𝐿)(𝑢(+g𝐿)𝑣) = (𝑣(+g𝐿)𝑢)))
2215, 18, 213bitr4g 314 1 (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  cfv 6540  (class class class)co 7404  Basecbs 17140  +gcplusg 17193  Mndcmnd 18621  CMndccmn 19641
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-iota 6492  df-fv 6548  df-ov 7407  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-cmn 19643
This theorem is referenced by:  ablpropd  19653  crngpropd  20093  prdscrngd  20125  resvcmn  32426
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