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Theorem ablpropd 19729

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	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
ablpropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
ablpropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)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**
Step	Hyp	Ref	Expression
1		ablpropd.1	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		ablpropd.2	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		ablpropd.3	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+_g‘𝐾)𝑦) = (𝑥(+_g‘𝐿)𝑦))
4	1, 2, 3	grppropd 18890	. . 3 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
5	1, 2, 3	cmnpropd 19728	. . 3 ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
6	4, 5	anbi12d 632	. 2 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd) ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd)))
7		isabl 19721	. 2 ⊢ (𝐾 ∈ Abel ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd))
8		isabl 19721	. 2 ⊢ (𝐿 ∈ Abel ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd))
9	6, 7, 8	3bitr4g 314	1 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2109 ‘cfv 6514 (class class class)co 7390 Basecbs 17186 +_gcplusg 17227 Grpcgrp 18872 CMndccmn 19717 Abelcabl 19718

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-iota 6467 df-fun 6516 df-fv 6522 df-ov 7393 df-0g 17411 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-grp 18875 df-cmn 19719 df-abl 19720

This theorem is referenced by: ablprop 19730 rngpropd 20090

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