Theorem dfgrp3 19151

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
dfgrp3	⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))

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

Proof of Theorem dfgrp3

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

Step	Hyp	Ref	Expression
1		grpsgrp 19073	. . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2		dfgrp3.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	2	grpbn0 19079	. . 3 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
4		simpl 488	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
5		simpr 490	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
6	5	adantl 487	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
7		simpl 488	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	adantl 487	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
9		eqid 2765	. . . . . . . 8 ⊢ (-g‘𝐺) = (-g‘𝐺)
10	2, 9	grpsubcl 19132	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(-g‘𝐺)𝑥) ∈ 𝐵)
11	4, 6, 8, 10	syl3anc 1398	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(-g‘𝐺)𝑥) ∈ 𝐵)
12		oveq1 7426	. . . . . . . 8 ⊢ (𝑙 = (𝑦(-g‘𝐺)𝑥) → (𝑙 + 𝑥) = ((𝑦(-g‘𝐺)𝑥) + 𝑥))
13	12	eqeq1d 2767	. . . . . . 7 ⊢ (𝑙 = (𝑦(-g‘𝐺)𝑥) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦))
14	13	adantl 487	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑙 = (𝑦(-g‘𝐺)𝑥)) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦))
15		dfgrp3.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
16	2, 15, 9	grpnpcan 19144	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦)
17	4, 6, 8, 16	syl3anc 1398	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦)
18	11, 14, 17	rspcedvd 3585	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦)
19		eqid 2765	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
20	2, 19	grpinvcl 19100	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
21	20	adantrr 730	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
22	2, 15, 4, 21, 6	grpcld 19060	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
23		oveq2 7427	. . . . . . . 8 ⊢ (𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦) → (𝑥 + 𝑟) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
24	23	eqeq1d 2767	. . . . . . 7 ⊢ (𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦))
25	24	adantl 487	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦)) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦))
26		eqid 2765	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
27	2, 15, 26, 19	grprinv 19103	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥 + ((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
28	27	adantrr 730	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + ((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
29	28	oveq1d 7434	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = ((0g‘𝐺) + 𝑦))
30	2, 15	grpass 19055	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
31	4, 8, 21, 6, 30	syl13anc 1399	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
32		grpmnd 19053	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
33	2, 15, 26	mndlid 18847	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → ((0g‘𝐺) + 𝑦) = 𝑦)
34	32, 5, 33	syl2an 608	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((0g‘𝐺) + 𝑦) = 𝑦)
35	29, 31, 34	3eqtr3d 2808	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦)
36	22, 25, 35	rspcedvd 3585	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)
37	18, 36	jca 521	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
38	37	ralrimivva 3210	. . 3 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
39	1, 3, 38	3jca 1146	. 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))
40		simp1 1154	. . 3 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Smgrp)
41	2, 15	dfgrp3lem 19150	. . 3 ⊢ ((𝐺 ∈ Smgrp ∧ 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))
42	2, 15	dfgrp2 19075	. . 3 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑢 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑢)))
43	40, 41, 42	sylanbrc 595	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Grp)
44	39, 43	impbii 212	1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  ∅c0 4286  ‘cfv 6540  (class class class)co 7419  Basecbs 17293  +_gcplusg 17334  0_gc0g 17516  Smgrpcsgrp 18810  Mndcmnd 18826  Grpcgrp 19046  inv_gcminusg 19047  -_gcsg 19048

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-1st 7992  df-2nd 7993  df-0g 17518  df-mgm 18722  df-sgrp 18811  df-mnd 18827  df-grp 19049  df-minusg 19050  df-sbg 19051

This theorem is used by:  dfgrp3e  19152