Theorem dfgrp3 18918

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 18842	. . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2		dfgrp3.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	2	grpbn0 18847	. . 3 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
4		simpl 483	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
5		simpr 485	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
6	5	adantl 482	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
7		simpl 483	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	adantl 482	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
9		eqid 2732	. . . . . . . 8 ⊢ (-g‘𝐺) = (-g‘𝐺)
10	2, 9	grpsubcl 18899	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(-g‘𝐺)𝑥) ∈ 𝐵)
11	4, 6, 8, 10	syl3anc 1371	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(-g‘𝐺)𝑥) ∈ 𝐵)
12		oveq1 7412	. . . . . . . 8 ⊢ (𝑙 = (𝑦(-g‘𝐺)𝑥) → (𝑙 + 𝑥) = ((𝑦(-g‘𝐺)𝑥) + 𝑥))
13	12	eqeq1d 2734	. . . . . . 7 ⊢ (𝑙 = (𝑦(-g‘𝐺)𝑥) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦))
14	13	adantl 482	. . . . . 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 18911	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦)
17	4, 6, 8, 16	syl3anc 1371	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦)
18	11, 14, 17	rspcedvd 3614	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦)
19		eqid 2732	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
20	2, 19	grpinvcl 18868	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
21	20	adantrr 715	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
22	2, 15, 4, 21, 6	grpcld 18829	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
23		oveq2 7413	. . . . . . . 8 ⊢ (𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦) → (𝑥 + 𝑟) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
24	23	eqeq1d 2734	. . . . . . 7 ⊢ (𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦))
25	24	adantl 482	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦)) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦))
26		eqid 2732	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
27	2, 15, 26, 19	grprinv 18871	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥 + ((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
28	27	adantrr 715	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + ((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
29	28	oveq1d 7420	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = ((0g‘𝐺) + 𝑦))
30	2, 15	grpass 18824	. . . . . . . 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 1372	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
32		grpmnd 18822	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
33	2, 15, 26	mndlid 18641	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → ((0g‘𝐺) + 𝑦) = 𝑦)
34	32, 5, 33	syl2an 596	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((0g‘𝐺) + 𝑦) = 𝑦)
35	29, 31, 34	3eqtr3d 2780	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦)
36	22, 25, 35	rspcedvd 3614	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)
37	18, 36	jca 512	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
38	37	ralrimivva 3200	. . 3 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
39	1, 3, 38	3jca 1128	. 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))
40		simp1 1136	. . 3 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Smgrp)
41	2, 15	dfgrp3lem 18917	. . 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 18843	. . 3 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑢 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑢)))
43	40, 41, 42	sylanbrc 583	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Grp)
44	39, 43	impbii 208	1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))

Syntax hints:   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  ∅c0 4321  ‘cfv 6540  (class class class)co 7405  Basecbs 17140  +_gcplusg 17193  0_gc0g 17381  Smgrpcsgrp 18605  Mndcmnd 18621  Grpcgrp 18815  inv_gcminusg 18816  -_gcsg 18817

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3376  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7971  df-2nd 7972  df-0g 17383  df-mgm 18557  df-sgrp 18606  df-mnd 18622  df-grp 18818  df-minusg 18819  df-sbg 18820

This theorem is referenced by:  dfgrp3e  18919