Theorem dfgrp3m 13431

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 13357	. . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2		dfgrp3.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐺)
3		eqid 2205	. . . . 5 ⊢ (0g‘𝐺) = (0g‘𝐺)
4	2, 3	grpidcl 13361	. . . 4 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
5		elex2 2788	. . . 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 2205	. . . . . . . 8 ⊢ (-g‘𝐺) = (-g‘𝐺)
13	2, 12	grpsubcl 13412	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → (𝑦(-g‘𝐺)𝑥) ∈ 𝐵)
14	7, 9, 11, 13	syl3anc 1250	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑦(-g‘𝐺)𝑥) ∈ 𝐵)
15		oveq1 5951	. . . . . . . 8 ⊢ (𝑙 = (𝑦(-g‘𝐺)𝑥) → (𝑙 + 𝑥) = ((𝑦(-g‘𝐺)𝑥) + 𝑥))
16	15	eqeq1d 2214	. . . . . . 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 13424	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ 𝑦 ∈ 𝐵 ∧ 𝑥 ∈ 𝐵) → ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦)
20	7, 9, 11, 19	syl3anc 1250	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑦(-g‘𝐺)𝑥) + 𝑥) = 𝑦)
21	14, 17, 20	rspcedvd 2883	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦)
22		eqid 2205	. . . . . . . . 9 ⊢ (invg‘𝐺) = (invg‘𝐺)
23	2, 22	grpinvcl 13380	. . . . . . . 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 13346	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (((invg‘𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
26		oveq2 5952	. . . . . . . 8 ⊢ (𝑟 = (((invg‘𝐺)‘𝑥) + 𝑦) → (𝑥 + 𝑟) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
27	26	eqeq1d 2214	. . . . . . 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 13383	. . . . . . . . 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 5959	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = ((0g‘𝐺) + 𝑦))
32	2, 18	grpass 13341	. . . . . . . 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 1252	. . . . . . 7 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑥 + ((invg‘𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)))
34		grpmnd 13339	. . . . . . . 8 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
35	2, 18, 3	mndlid 13267	. . . . . . . 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 2246	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ (𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑥 + (((invg‘𝐺)‘𝑥) + 𝑦)) = 𝑦)
38	25, 28, 37	rspcedvd 2883	. . . . 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 2588	. . 3 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦))
41	1, 6, 40	3jca 1180	. 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Smgrp ∧ ∃𝑤 𝑤 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (∃𝑙 ∈ 𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟 ∈ 𝐵 (𝑥 + 𝑟) = 𝑦)))
42		simp1 1000	. . 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 13430	. . 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 13359	. . 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 981   = wceq 1373  ∃wex 1515   ∈ wcel 2176  ∀wral 2484  ∃wrex 2485  ‘cfv 5271  (class class class)co 5944  Basecbs 12832  +_gcplusg 12909  0_gc0g 13088  Smgrpcsgrp 13233  Mndcmnd 13248  Grpcgrp 13332  inv_gcminusg 13333  -_gcsg 13334

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-cnex 8016  ax-resscn 8017  ax-1re 8019  ax-addrcl 8022

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-inn 9037  df-2 9095  df-ndx 12835  df-slot 12836  df-base 12838  df-plusg 12922  df-0g 13090  df-mgm 13188  df-sgrp 13234  df-mnd 13249  df-grp 13335  df-minusg 13336  df-sbg 13337

This theorem is referenced by:  dfgrp3me  13432