Theorem grpprop 18928

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 7380	. . . 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 18927	. 2 ⊢ (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
8	7	mptru 1549	1 ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1542  ⊤wtru 1543   ∈ wcel 2114  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  +_gcplusg 17220  Grpcgrp 18909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-iota 6454  df-fun 6500  df-fv 6506  df-ov 7370  df-0g 17404  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-grp 18912

This theorem is referenced by:  grppropstr  18929  grpss  18930  opprsubg  20332  rmodislmod  20925  opprgrpb  42957  lmod1  48968