Theorem grppropd 13558

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 13481	. . 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 13101	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐾 ∈ V)
9	6, 7	basmexd 13101	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐿 ∈ V)
10	3	ralrimivva 2612	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦))
11		oveq1 6014	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑥(+g‘𝐾)𝑦) = (𝑧(+g‘𝐾)𝑦))
12		oveq1 6014	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑧 → (𝑥(+g‘𝐿)𝑦) = (𝑧(+g‘𝐿)𝑦))
13	11, 12	eqeq12d 2244	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑧 → ((𝑥(+g‘𝐾)𝑦) = (𝑥(+g‘𝐿)𝑦) ↔ (𝑧(+g‘𝐾)𝑦) = (𝑧(+g‘𝐿)𝑦)))
14		oveq2 6015	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → (𝑧(+g‘𝐾)𝑦) = (𝑧(+g‘𝐾)𝑤))
15		oveq2 6015	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑤 → (𝑧(+g‘𝐿)𝑦) = (𝑧(+g‘𝐿)𝑤))
16	14, 15	eqeq12d 2244	. . . . . . . . . . . . . . 15 ⊢ (𝑦 = 𝑤 → ((𝑧(+g‘𝐾)𝑦) = (𝑧(+g‘𝐿)𝑦) ↔ (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤)))
17	13, 16	cbvral2v 2778	. . . . . . . . . . . . . 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 2618	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑧 ∈ 𝐵) → ∀𝑤 ∈ 𝐵 (𝑧(+g‘𝐾)𝑤) = (𝑧(+g‘𝐿)𝑤))
21	20	r19.21bi 2618	. . . . . . . . . 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 13415	. . . . . . . 8 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (0g‘𝐾) = (0g‘𝐿))
24	3, 23	eqeq12d 2244	. . . . . . 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 2527	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐵) → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
27	26	ralbidva 2526	. . . 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 2735	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐾)𝑦) = (0g‘𝐾) ↔ ∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
29	1, 28	raleqbidv 2744	. . . 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 2735	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ 𝐵 (𝑥(+g‘𝐿)𝑦) = (0g‘𝐿) ↔ ∃𝑥 ∈ (Base‘𝐿)(𝑥(+g‘𝐿)𝑦) = (0g‘𝐿)))
31	2, 30	raleqbidv 2744	. . . 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 2229	. . 3 ⊢ (Base‘𝐾) = (Base‘𝐾)
35		eqid 2229	. . 3 ⊢ (+g‘𝐾) = (+g‘𝐾)
36		eqid 2229	. . 3 ⊢ (0g‘𝐾) = (0g‘𝐾)
37	34, 35, 36	isgrp 13547	. 2 ⊢ (𝐾 ∈ Grp ↔ (𝐾 ∈ Mnd ∧ ∀𝑦 ∈ (Base‘𝐾)∃𝑥 ∈ (Base‘𝐾)(𝑥(+g‘𝐾)𝑦) = (0g‘𝐾)))
38		eqid 2229	. . 3 ⊢ (Base‘𝐿) = (Base‘𝐿)
39		eqid 2229	. . 3 ⊢ (+g‘𝐿) = (+g‘𝐿)
40		eqid 2229	. . 3 ⊢ (0g‘𝐿) = (0g‘𝐿)
41	38, 39, 40	isgrp 13547	. 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 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  Vcvv 2799  ‘cfv 5318  (class class class)co 6007  Basecbs 13040  +_gcplusg 13118  0_gc0g 13297  Mndcmnd 13457  Grpcgrp 13541

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-cnex 8098  ax-resscn 8099  ax-1re 8101  ax-addrcl 8104

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-fv 5326  df-riota 5960  df-ov 6010  df-inn 9119  df-2 9177  df-ndx 13043  df-slot 13044  df-base 13046  df-plusg 13131  df-0g 13299  df-mgm 13397  df-sgrp 13443  df-mnd 13458  df-grp 13544

This theorem is referenced by:  grpprop  13559  grppropstrg  13560  ghmpropd  13828  ablpropd  13841  ringpropd  14009  opprring  14050  opprsubgg  14055  lmodprop2d  14320  sralmod  14422  psrgrp  14657