Theorem grpprop 18977

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 2762	. . 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 7405	. . . 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 18976	. 2 ⊢ (⊤ → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))
8	7	mptru 1566	1 ⊢ (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp)

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559  ⊤wtru 1560   ∈ wcel 2141  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  Grpcgrp 18958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5245  ax-nul 5255  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3745  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  df-id 5540  df-xp 5651  df-rel 5652  df-cnv 5653  df-co 5654  df-dm 5655  df-iota 6473  df-fun 6519  df-fv 6525  df-ov 7395  df-0g 17453  df-mgm 18657  df-sgrp 18736  df-mnd 18752  df-grp 18961

This theorem is referenced by:  grppropstr  18978  grpss  18979  opprsubg  20380  rmodislmod  20977  opprgrpb  43098  lmod1  49078