Theorem grpprop 18891

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 2731	. . 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 7403	. . . 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 18890	. 2 ⊢ (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
8	7	mptru 1547	1 ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ⊤wtru 1541   ∈ wcel 2109  ‘cfv 6514  (class class class)co 7390  Basecbs 17186  +_gcplusg 17227  Grpcgrp 18872

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  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 3409  df-v 3452  df-sbc 3757  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-0g 17411  df-mgm 18574  df-sgrp 18653  df-mnd 18669  df-grp 18875

This theorem is referenced by:  grppropstr  18892  grpss  18893  opprsubg  20268  rmodislmod  20843  opprgrpb  42507  lmod1  48485