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Theorem ablpropd 19786

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 18941	. . 3 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
5	1, 2, 3	cmnpropd 19785	. . 3 ⊢ (𝜑 → (𝐾 ∈ CMnd ↔ 𝐿 ∈ CMnd))
6	4, 5	anbi12d 630	. 2 ⊢ (𝜑 → ((𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd) ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd)))
7		isabl 19778	. 2 ⊢ (𝐾 ∈ Abel ↔ (𝐾 ∈ Grp ∧ 𝐾 ∈ CMnd))
8		isabl 19778	. 2 ⊢ (𝐿 ∈ Abel ↔ (𝐿 ∈ Grp ∧ 𝐿 ∈ CMnd))
9	6, 7, 8	3bitr4g 313	1 ⊢ (𝜑 → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 = wceq 1534 ∈ wcel 2099 ‘cfv 6546 (class class class)co 7416 Basecbs 17208 +_gcplusg 17261 Grpcgrp 18923 CMndccmn 19774 Abelcabl 19775

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fv 6554 df-ov 7419 df-0g 17451 df-mgm 18628 df-sgrp 18707 df-mnd 18723 df-grp 18926 df-cmn 19776 df-abl 19777

This theorem is referenced by: ablprop 19787 rngpropd 20153

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