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Theorem ablpropd 18895
Description: If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 6-Dec-2014.)
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
ablpropd (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem ablpropd
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, 3grppropd 18096 . . 3 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
51, 2, 3cmnpropd 18894 . . 3 (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
64, 5anbi12d 633 . 2 (𝜑 → ((𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd) ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd)))
7 isabl 18888 . 2 (𝐾 ∈ Abel ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd))
8 isabl 18888 . 2 (𝐿 ∈ Abel ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd))
96, 7, 83bitr4g 317 1 (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2115  cfv 6328  (class class class)co 7130  Basecbs 16461  +gcplusg 16543  Grpcgrp 18081  CMndccmn 18884  Abelcabl 18885
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 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303
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 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-iota 6287  df-fun 6330  df-fv 6336  df-ov 7133  df-0g 16693  df-mgm 17830  df-sgrp 17879  df-mnd 17890  df-grp 18084  df-cmn 18886  df-abl 18887
This theorem is referenced by:  ablprop  18896
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