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Theorem ablpropd 19815

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 18976	. . 3 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
5	1, 2, 3	cmnpropd 19814	. . 3 ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
6	4, 5	anbi12d 641	. 2 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd) ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd)))
7		isabl 19807	. 2 ⊢ (𝐾 ∈ Abel ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd))
8		isabl 19807	. 2 ⊢ (𝐿 ∈ Abel ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd))
9	6, 7, 8	3bitr4g 316	1 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ‘cfv 6517 (class class class)co 7392 Basecbs 17228 +_gcplusg 17269 Grpcgrp 18958 CMndccmn 19803 Abelcabl 19804

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-nul 5255 ax-pr 5389

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3745 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-iota 6473 df-fun 6519 df-fv 6525 df-ov 7395 df-0g 17453 df-mgm 18657 df-sgrp 18736 df-mnd 18752 df-grp 18961 df-cmn 19805 df-abl 19806

This theorem is referenced by: ablprop 19816 rngpropd 20203

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