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Theorem grpprop 18644

Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by NM, 11-Oct-2013.)

**Hypotheses**
Ref	Expression
grpprop.b	⊢ (Base‘𝐾) = (Base‘𝐿)
grpprop.p	⊢ (+_g‘𝐾) = (+_g‘𝐿)

**Assertion**
Ref	Expression
grpprop	⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Proof of Theorem grpprop Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqidd 2737	. . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐾))
2		grpprop.b	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐿)
3	2	a1i 11	. . 3 ⊢ (⊤ → (Base‘𝐾) = (Base‘𝐿))
4		grpprop.p	. . . . 5 ⊢ (+_g‘𝐾) = (+_g‘𝐿)
5	4	oveqi 7320	. . . 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	grppropd 18643	. 2 ⊢ (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
8	7	mptru 1546	1 ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Colors of variables: wff setvar class

Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1539 ⊤wtru 1540 ∈ wcel 2104 ‘cfv 6458 (class class class)co 7307 Basecbs 16961 +_gcplusg 17011 Grpcgrp 18626

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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pr 5361

This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3306 df-v 3439 df-sbc 3722 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-mpt 5165 df-id 5500 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-iota 6410 df-fun 6460 df-fv 6466 df-ov 7310 df-0g 17201 df-mgm 18375 df-sgrp 18424 df-mnd 18435 df-grp 18629

This theorem is referenced by: grppropstr 18645 grpss 18646 opprring 19922 opprsubg 19927 rmodislmod 20240 rmodislmodOLD 20241 lmod1 46077

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