Theorem grppropd 18982

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 18785	. . 3 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
5	1, 2, 3	grpidpropd 18688	. . . . . . . . 9 ⊢ (𝜑 → (0g‘𝐾) = (0g‘𝐿))
6	5	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿))
7	3, 6	eqeq12d 2751	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
8	7	anass1rs 655	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
9	8	rexbidva 3175	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
10	9	ralbidva 3174	. . . 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 3325	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
12	1, 11	raleqbidv 3344	. . . 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 3325	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿) ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
14	2, 13	raleqbidv 3344	. . . 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 309	. . 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 632	. 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 2735	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
18		eqid 2735	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
19		eqid 2735	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
20	17, 18, 19	isgrp 18970	. 2 ⊢ (𝐾 ∈ Grp ↔ (𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
21		eqid 2735	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
22		eqid 2735	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
23		eqid 2735	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
24	21, 22, 23	isgrp 18970	. 2 ⊢ (𝐿 ∈ Grp ↔ (𝐿 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
25	16, 20, 24	3bitr4g 314	1 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059  ∃wrex 3068  ‘cfv 6563  (class class class)co 7431  Basecbs 17245  +_gcplusg 17298  0_gc0g 17486  Mndcmnd 18760  Grpcgrp 18964

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-0g 17488  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-grp 18967

This theorem is referenced by:  grpprop  18983  ghmpropd  19287  oppggrpb  19392  ablpropd  19825  ringpropd  20302  lmodprop2d  20939  sralmod  21212  psrgrp  21994  nmpropd2  24624  ngppropd  24666  tngngp2  24689  tnggrpr  24692  zhmnrg  33928