Theorem dfgrp3 18589

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 18518	. . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2		dfgrp3.b	. . . 4 ⊢ 𝐵 = (Base‘𝐺)
3	2	grpbn0 18523	. . 3 ⊢ (𝐺 ∈ Grp → 𝐵 ≠ ∅)
4		simpl 482	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝐺 ∈ Grp)
5		simpr 484	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐵)
6	5	adantl 481	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑦 ∈ 𝐵)
7		simpl 482	. . . . . . . 8 ⊢ ((𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → 𝑥 ∈ 𝐵)
8	7	adantl 481	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → 𝑥 ∈ 𝐵)
9		eqid 2738	. . . . . . . 8 ⊢ (-g‘𝐺) = (-g‘𝐺)
10	2, 9	grpsubcl 18570	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(-g‘𝐺)𝑥) ∈ 𝐵)
11	4, 6, 8, 10	syl3anc 1369	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(-g‘𝐺)𝑥) ∈ 𝐵)
12		oveq1 7262	. . . . . . . 8 ⊢ (𝑙 = (𝑦(-g‘𝐺)𝑥) → (𝑙 + 𝑥) = ((𝑦(-g‘𝐺)𝑥) + 𝑥))
13	12	eqeq1d 2740	. . . . . . 7 ⊢ (𝑙 = (𝑦(-g‘𝐺)𝑥) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦))
14	13	adantl 481	. . . . . 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 18582	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦)
17	4, 6, 8, 16	syl3anc 1369	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦)
18	11, 14, 17	rspcedvd 3555	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦)
19		eqid 2738	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
20	2, 19	grpinvcl 18542	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
21	20	adantrr 713	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((invg‘𝐺)‘𝑥) ∈ 𝐵)
22	2, 15	grpcl 18500	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ ((invg‘𝐺)‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵) → (((invg‘𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
23	4, 21, 6, 22	syl3anc 1369	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
24		oveq2 7263	. . . . . . . 8 ⊢ (𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦) → (𝑥 + 𝑟) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
25	24	eqeq1d 2740	. . . . . . 7 ⊢ (𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦))
26	25	adantl 481	. . . . . 6 ⊢ (((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) ∧ 𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦)) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦))
27		eqid 2738	. . . . . . . . . 10 ⊢ (0g‘𝐺) = (0g‘𝐺)
28	2, 15, 27, 19	grprinv 18544	. . . . . . . . 9 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (𝑥 + ((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
29	28	adantrr 713	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + ((invg‘𝐺)‘𝑥)) = (0g‘𝐺))
30	29	oveq1d 7270	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = ((0g‘𝐺) + 𝑦))
31	2, 15	grpass 18501	. . . . . . . 8 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ ((invg‘𝐺)‘𝑥) ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
32	4, 8, 21, 6, 31	syl13anc 1370	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
33		grpmnd 18499	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
34	2, 15, 27	mndlid 18320	. . . . . . . 8 ⊢ ((𝐺 ∈ Mnd ∧ 𝑦 ∈ 𝐵) → ((0g‘𝐺) + 𝑦) = 𝑦)
35	33, 5, 34	syl2an 595	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((0g‘𝐺) + 𝑦) = 𝑦)
36	30, 32, 35	3eqtr3d 2786	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦)
37	23, 26, 36	rspcedvd 3555	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)
38	18, 37	jca 511	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
39	38	ralrimivva 3114	. . 3 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
40	1, 3, 39	3jca 1126	. 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))
41		simp1 1134	. . 3 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Smgrp)
42	2, 15	dfgrp3lem 18588	. . 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))
43	2, 15	dfgrp2 18519	. . 3 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑢 ∈ 𝐵 ∀𝑎 ∈ 𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑢)))
44	41, 42, 43	sylanbrc 582	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Grp)
45	40, 44	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 395   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  ∅c0 4253  ‘cfv 6418  (class class class)co 7255  Basecbs 16840  +_gcplusg 16888  0_gc0g 17067  Smgrpcsgrp 18289  Mndcmnd 18300  Grpcgrp 18492  inv_gcminusg 18493  -_gcsg 18494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-0g 17069  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-grp 18495  df-minusg 18496  df-sbg 18497

This theorem is referenced by:  dfgrp3e  18590