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Theorem ablpropd 19811
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 18970 . . 3 (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
51, 2, 3cmnpropd 19810 . . 3 (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
64, 5anbi12d 632 . 2 (𝜑 → ((𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd) ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd)))
7 isabl 19803 . 2 (𝐾 ∈ Abel ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd))
8 isabl 19803 . 2 (𝐿 ∈ Abel ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd))
96, 7, 83bitr4g 314 1 (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1539  wcel 2107  cfv 6560  (class class class)co 7432  Basecbs 17248  +gcplusg 17298  Grpcgrp 18952  CMndccmn 19799  Abelcabl 19800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-mpt 5225  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-iota 6513  df-fun 6562  df-fv 6568  df-ov 7435  df-0g 17487  df-mgm 18654  df-sgrp 18733  df-mnd 18749  df-grp 18955  df-cmn 19801  df-abl 19802
This theorem is referenced by:  ablprop  19812  rngpropd  20172
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