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Theorem ablprop 19698

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 2731	. . 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 7354	. . . 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 19697	. 2 ⊢ (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
8	7	mptru 1548	1 ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1541 ⊤wtru 1542 ∈ wcel 2110 ‘cfv 6477 (class class class)co 7341 Basecbs 17112 +_gcplusg 17153 Abelcabl 19686

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-sep 5232 ax-nul 5242 ax-pr 5368

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 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 3394 df-v 3436 df-sbc 3740 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-iota 6433 df-fun 6479 df-fv 6485 df-ov 7344 df-0g 17337 df-mgm 18540 df-sgrp 18619 df-mnd 18635 df-grp 18841 df-cmn 19687 df-abl 19688

This theorem is referenced by: opprrng 20256 zlmlmod 21452 dvaabl 41042 opprablb 42525 cznabel 48270

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