Theorem grppropd 13092

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

Dummy variables 𝑧 𝑤 are mutually distinct and distinct from all other variables.

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 13024	. . 3 ⊢ (𝜑 → (𝐾 ∈ Mnd ↔ 𝐿 ∈ Mnd))
5	1	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐾))
6	2	adantr 276	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐵 = (Base‘𝐿))
7		simprl 529	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
8	5, 7	basmexd 12681	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ V)
9	6, 7	basmexd 12681	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ V)
10	3	ralrimivva 2576	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
11		oveq1 5926	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑥(+g‘𝐾)𝑦) = (𝑧(+g‘𝐾)𝑦))
12		oveq1 5926	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑥(+g‘𝐿)𝑦) = (𝑧(+g‘𝐿)𝑦))
13	11, 12	eqeq12d 2208	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦) ↔ (𝑧(+g‘𝐾)𝑦) = (𝑧(+g‘𝐿)𝑦)))
14		oveq2 5927	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → (𝑧(+g‘𝐾)𝑦) = (𝑧(+g‘𝐾)𝑤))
15		oveq2 5927	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → (𝑧(+g‘𝐿)𝑦) = (𝑧(+g‘𝐿)𝑤))
16	14, 15	eqeq12d 2208	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ((𝑧(+g‘𝐾)𝑦) = (𝑧(+g‘𝐿)𝑦) ↔ (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤)))
17	13, 16	cbvral2v 2739	. . . . . . . . . . . . . 14 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦) ↔ ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
18	10, 17	sylib 122	. . . . . . . . . . . . 13 ⊢ (𝜑 → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
19	18	adantr 276	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∀𝑧 ∈ 𝐵 ∀𝑤 ∈ 𝐵 (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
20	19	r19.21bi 2582	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → ∀𝑤 ∈ 𝐵 (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
21	20	r19.21bi 2582	. . . . . . . . . 10 ⊢ ((((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) ∧ 𝑤 ∈ 𝐵) → (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
22	21	anasss 399	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ (𝑧 ∈ 𝐵 ∧ 𝑤 ∈ 𝐵)) → (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
23	5, 6, 8, 9, 22	grpidpropdg 12960	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿))
24	3, 23	eqeq12d 2208	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
25	24	anass1rs 571	. . . . . 6 ⊢ (((𝜑 ∧ 𝑦 ∈ 𝐵) ∧ 𝑥 ∈ 𝐵) → ((𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
26	25	rexbidva 2491	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
27	26	ralbidva 2490	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
28	1	rexeqdv 2697	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
29	1, 28	raleqbidv 2706	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
30	2	rexeqdv 2697	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿) ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
31	2, 30	raleqbidv 2706	. . . 4 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿) ↔ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
32	27, 29, 31	3bitr3d 218	. . 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)))
33	4, 32	anbi12d 473	. 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))))
34		eqid 2193	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
35		eqid 2193	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
36		eqid 2193	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
37	34, 35, 36	isgrp 13081	. 2 ⊢ (𝐾 ∈ Grp ↔ (𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
38		eqid 2193	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
39		eqid 2193	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
40		eqid 2193	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
41	38, 39, 40	isgrp 13081	. 2 ⊢ (𝐿 ∈ Grp ↔ (𝐿 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐿)∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
42	33, 37, 41	3bitr4g 223	1 ⊢ (𝜑 → (𝐾 ∈ Grp ↔ 𝐿 ∈ Grp))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  Vcvv 2760  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  +_gcplusg 12698  0_gc0g 12870  Mndcmnd 13000  Grpcgrp 13075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078

This theorem is referenced by:  grpprop  13093  grppropstrg  13094  ghmpropd  13356  ablpropd  13369  ringpropd  13537  opprring  13578  opprsubgg  13583  lmodprop2d  13847  sralmod  13949