Theorem ablprop 19398

Description: If two structures have the same group components (properties), one is an Abelian group iff the other one is. (Contributed by NM, 11-Oct-2013.)

Hypotheses

Ref	Expression
ablprop.b	⊢ (Base‘𝐾) = (Base‘𝐿)
ablprop.p	⊢ (+g‘𝐾) = (+g‘𝐿)

Assertion

Ref	Expression
ablprop	⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Proof of Theorem ablprop

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2739	. . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐾))
2		ablprop.b	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐿)
3	2	a1i 11	. . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐿))
4		ablprop.p	. . . . 5 ⊢ (+g‘𝐾) = (+g‘𝐿)
5	4	oveqi 7288	. . . 4 ⊢ (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦)
6	5	a1i 11	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑦 ∈ (Base‘𝐾))) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
7	1, 3, 6	ablpropd 19397	. 2 ⊢ (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
8	7	mptru 1546	1 ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1539  ⊤wtru 1540   ∈ wcel 2106  ‘cfv 6433  (class class class)co 7275  Basecbs 16912  +_gcplusg 16962  Abelcabl 19387

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fv 6441  df-ov 7278  df-0g 17152  df-mgm 18326  df-sgrp 18375  df-mnd 18386  df-grp 18580  df-cmn 19388  df-abl 19389

This theorem is referenced by:  zlmlmod  20728  dvaabl  39038  cznabel  45512