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Theorem ablprop 19766

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 2741	. . 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 7376	. . . 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 19765	. 2 ⊢ (⊤ → (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel))
8	7	mptru 1554	1 ⊢ (𝐾 ∈ Abel ↔ 𝐿 ∈ Abel)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 207 ∧ wa 396 = wceq 1547 ⊤wtru 1548 ∈ wcel 2119 ‘cfv 6492 (class class class)co 7363 Basecbs 17177 +_gcplusg 17218 Abelcabl 19754

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2712 ax-sep 5225 ax-nul 5235 ax-pr 5369

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2719 df-cleq 2732 df-clel 2815 df-nfc 2889 df-ne 2936 df-ral 3055 df-rex 3065 df-rab 3393 df-v 3434 df-sbc 3731 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4269 df-if 4462 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4846 df-br 5080 df-opab 5142 df-mpt 5161 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-iota 6448 df-fun 6494 df-fv 6500 df-ov 7366 df-0g 17402 df-mgm 18606 df-sgrp 18685 df-mnd 18701 df-grp 18910 df-cmn 19755 df-abl 19756

This theorem is referenced by: opprrng 20323 zlmlmod 21504 dvaabl 41523 opprablb 43010 cznabel 48758

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