Theorem grppropd 18976

Description: If two structures have the same group components (properties), one is a group iff the other one is. (Contributed by Stefan O'Rear, 27-Nov-2014.) (Revised by Mario Carneiro, 2-Oct-2015.)

Hypotheses

Ref	Expression
grppropd.1	⊢ (𝜑 → 𝐵 = (Base‘𝐾))
grppropd.2	⊢ (𝜑 → 𝐵 = (Base‘𝐿))
grppropd.3	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))

Assertion

Ref	Expression
grppropd	⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦   𝑥,𝐿,𝑦   𝜑,𝑥,𝑦

Proof of Theorem grppropd

Step	Hyp	Ref	Expression
1		grppropd.1	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐾))
2		grppropd.2	. . . 4 ⊢ (𝜑 → 𝐵 = (Base‘𝐿))
3		grppropd.3	. . . 4 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
4	1, 2, 3	mndpropd 18776	. . 3 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
5	1, 2, 3	grpidpropd 18679	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
6	5	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿))
7	3, 6	eqeq12d 2777	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
8	7	anass1rs 665	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
9	8	rexbidva 3183	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
10	9	ralbidva 3182	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
11	1	rexeqdv 3320	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
12	1, 11	raleqbidv 3335	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
13	2	rexeqdv 3320	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿) ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
14	2, 13	raleqbidv 3335	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿) ↔ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
15	10, 12, 14	3bitr3d 311	. . 3 ⊢ (𝜑 → (∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
16	4, 15	anbi12d 641	. 2 ⊢ (𝜑 → ((𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)) ↔ (𝐿 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿))))
17		eqid 2761	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2761	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
19		eqid 2761	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
20	17, 18, 19	isgrp 18964	. 2 ⊢ (𝐾 ∈ Grp ↔ (𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
21		eqid 2761	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
22		eqid 2761	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
23		eqid 2761	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
24	21, 22, 23	isgrp 18964	. 2 ⊢ (𝐿 ∈ Grp ↔ (𝐿 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
25	16, 20, 24	3bitr4g 316	1 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∀wral 3075  ∃wrex 3085  ‘cfv 6517  (class class class)co 7392  Basecbs 17228  +_gcplusg 17269  0_gc0g 17451  Mndcmnd 18751  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:  grpprop  18977  ghmpropd  19279  oppggrpb  19381  ablpropd  19815  ringpropd  20317  lmodprop2d  20971  sralmod  21234  psrgrp  21988  nmpropd2  24635  ngppropd  24677  tngngp2  24692  tnggrpr  24695  zhmnrg  34223