Theorem grppropd 19081

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 18870	. . 3 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
5	1, 2, 3	grpidpropd 18761	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
6	5	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿))
7	3, 6	eqeq12d 2778	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
8	7	anass1rs 668	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
9	8	rexbidva 3186	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
10	9	ralbidva 3185	. . . 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 3322	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
12	1, 11	raleqbidv 3336	. . . 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 3322	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿) ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
14	2, 13	raleqbidv 3336	. . . 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 312	. . 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 644	. 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 2762	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2762	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
19		eqid 2762	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
20	17, 18, 19	isgrp 19069	. 2 ⊢ (𝐾 ∈ Grp ↔ (𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
21		eqid 2762	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
22		eqid 2762	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
23		eqid 2762	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
24	21, 22, 23	isgrp 19069	. 2 ⊢ (𝐿 ∈ Grp ↔ (𝐿 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
25	16, 20, 24	3bitr4g 317	1 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088  ‘cfv 6537  (class class class)co 7417  Basecbs 17307  +_gcplusg 17348  0_gc0g 17530  Mndcmnd 18842  Grpcgrp 19063

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-iota 6493  df-fun 6539  df-fv 6545  df-ov 7420  df-0g 17532  df-mgm 18736  df-sgrp 18827  df-mnd 18843  df-grp 19066

This theorem is used by:  grpprop  19082  ghmpropd  19389  oppggrpb  19491  ablpropd  19925  ringpropd  20436  lmodprop2d  21114  sralmod  21377  psrgrp  22177  nmpropd2  24827  ngppropd  24869  tngngp2  24884  tnggrpr  24887  zhmnrg  34483