Theorem dfgrp3m 13402

Description: Alternate definition of a group as semigroup (with at least one element) which is also a quasigroup, i.e. a magma in which solutions 𝑥 and 𝑦 of the equations (𝑎 + 𝑥) = 𝑏 and (𝑥 + 𝑎) = 𝑏 exist. Theorem 3.2 of [Bruck] p. 28. (Contributed by AV, 28-Aug-2021.)

Hypotheses

Ref	Expression
dfgrp3.b	⊢ 𝐵 = (Base‘𝐺)
dfgrp3.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
dfgrp3m	⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))

Distinct variable groups:   𝐵,𝑙,𝑟,𝑤,𝑥,𝑦   𝐺,𝑙,𝑟,𝑤,𝑥,𝑦   + ,𝑙,𝑟,𝑤,𝑥,𝑦

Proof of Theorem dfgrp3m

Dummy variables 𝑎 𝑖 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpsgrp 13328	. . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2		dfgrp3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2204	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	2, 3	grpidcl 13332	. . . 4 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
5		elex2 2787	. . . 4 ⊢ ((0g‘𝐺) ∈ 𝐵 → ∃𝑤 𝑤 ∈ 𝐵)
6	4, 5	syl 14	. . 3 ⊢ (𝐺 ∈ Grp → ∃𝑤 𝑤 ∈ 𝐵)
7		simpl 109	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
8		simpr 110	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
9	8	adantl 277	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
10		simpl 109	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
11	10	adantl 277	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
12		eqid 2204	. . . . . . . 8 ⊢ (-g‘𝐺) = (-g‘𝐺)
13	2, 12	grpsubcl 13383	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(-g‘𝐺)𝑥) ∈ 𝐵)
14	7, 9, 11, 13	syl3anc 1249	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(-g‘𝐺)𝑥) ∈ 𝐵)
15		oveq1 5950	. . . . . . . 8 ⊢ (𝑙 = (𝑦(-g‘𝐺)𝑥) → (𝑙 + 𝑥) = ((𝑦(-g‘𝐺)𝑥) + 𝑥))
16	15	eqeq1d 2213	. . . . . . 7 ⊢ (𝑙 = (𝑦(-g‘𝐺)𝑥) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦))
17	16	adantl 277	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑙 = (𝑦(-g‘𝐺)𝑥)) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦))
18		dfgrp3.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
19	2, 18, 12	grpnpcan 13395	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦)
20	7, 9, 11, 19	syl3anc 1249	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦)
21	14, 17, 20	rspcedvd 2882	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦)
22		eqid 2204	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
23	2, 22	grpinvcl 13351	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
24	23	adantrr 479	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
25	2, 18, 7, 24, 9	grpcld 13317	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
26		oveq2 5951	. . . . . . . 8 ⊢ (𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦) → (𝑥 + 𝑟) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
27	26	eqeq1d 2213	. . . . . . 7 ⊢ (𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦))
28	27	adantl 277	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦)) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦))
29	2, 18, 3, 22	grprinv 13354	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥 + ((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
30	29	adantrr 479	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + ((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
31	30	oveq1d 5958	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = ((0g‘𝐺) + 𝑦))
32	2, 18	grpass 13312	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
33	7, 11, 24, 9, 32	syl13anc 1251	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
34		grpmnd 13310	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
35	2, 18, 3	mndlid 13238	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → ((0g‘𝐺) + 𝑦) = 𝑦)
36	34, 8, 35	syl2an 289	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((0g‘𝐺) + 𝑦) = 𝑦)
37	31, 33, 36	3eqtr3d 2245	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦)
38	25, 28, 37	rspcedvd 2882	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)
39	21, 38	jca 306	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
40	39	ralrimivva 2587	. . 3 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
41	1, 6, 40	3jca 1179	. 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))
42		simp1 999	. . 3 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Smgrp)
43	2, 18	dfgrp3mlem 13401	. . 3 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑢))
44	2, 18	dfgrp2 13330	. . 3 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑢 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑢)))
45	42, 43, 44	sylanbrc 417	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Grp)
46	41, 45	impbii 126	1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))

Syntax hints:   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1372  ∃wex 1514   ∈ wcel 2175  ∀wral 2483  ∃wrex 2484  ‘cfv 5270  (class class class)co 5943  Basecbs 12803  +_gcplusg 12880  0_gc0g 13059  Smgrpcsgrp 13204  Mndcmnd 13219  Grpcgrp 13303  inv_gcminusg 13304  -_gcsg 13305

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-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-coll 4158  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1re 8018  ax-addrcl 8021

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rmo 2491  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-iun 3928  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-ima 4687  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-f1 5275  df-fo 5276  df-f1o 5277  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-inn 9036  df-2 9094  df-ndx 12806  df-slot 12807  df-base 12809  df-plusg 12893  df-0g 13061  df-mgm 13159  df-sgrp 13205  df-mnd 13220  df-grp 13306  df-minusg 13307  df-sbg 13308

This theorem is referenced by:  dfgrp3me  13403