MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfgrp3 Structured version   Visualization version   GIF version

Theorem dfgrp3 19057
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 + = (+g𝐺)
Assertion
Ref Expression
dfgrp3 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (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.
StepHypRef Expression
1 grpsgrp 18978 . . 3 (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2 dfgrp3.b . . . 4 𝐵 = (Base‘𝐺)
32grpbn0 18984 . . 3 (𝐺 ∈ Grp → 𝐵 ≠ ∅)
4 simpl 482 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝐺 ∈ Grp)
5 simpr 484 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → 𝑦𝐵)
65adantl 481 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
7 simpl 482 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → 𝑥𝐵)
87adantl 481 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
9 eqid 2737 . . . . . . . 8 (-g𝐺) = (-g𝐺)
102, 9grpsubcl 19038 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝑦(-g𝐺)𝑥) ∈ 𝐵)
114, 6, 8, 10syl3anc 1373 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(-g𝐺)𝑥) ∈ 𝐵)
12 oveq1 7438 . . . . . . . 8 (𝑙 = (𝑦(-g𝐺)𝑥) → (𝑙 + 𝑥) = ((𝑦(-g𝐺)𝑥) + 𝑥))
1312eqeq1d 2739 . . . . . . 7 (𝑙 = (𝑦(-g𝐺)𝑥) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦))
1413adantl 481 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑙 = (𝑦(-g𝐺)𝑥)) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦))
15 dfgrp3.p . . . . . . . 8 + = (+g𝐺)
162, 15, 9grpnpcan 19050 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦)
174, 6, 8, 16syl3anc 1373 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦)
1811, 14, 17rspcedvd 3624 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦)
19 eqid 2737 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
202, 19grpinvcl 19005 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((invg𝐺)‘𝑥) ∈ 𝐵)
2120adantrr 717 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((invg𝐺)‘𝑥) ∈ 𝐵)
222, 15, 4, 21, 6grpcld 18965 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (((invg𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
23 oveq2 7439 . . . . . . . 8 (𝑟 = (((invg𝐺)‘𝑥) + 𝑦) → (𝑥 + 𝑟) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
2423eqeq1d 2739 . . . . . . 7 (𝑟 = (((invg𝐺)‘𝑥) + 𝑦) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦))
2524adantl 481 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑟 = (((invg𝐺)‘𝑥) + 𝑦)) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦))
26 eqid 2737 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
272, 15, 26, 19grprinv 19008 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑥 + ((invg𝐺)‘𝑥)) = (0g𝐺))
2827adantrr 717 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + ((invg𝐺)‘𝑥)) = (0g𝐺))
2928oveq1d 7446 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = ((0g𝐺) + 𝑦))
302, 15grpass 18960 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑥𝐵 ∧ ((invg𝐺)‘𝑥) ∈ 𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
314, 8, 21, 6, 30syl13anc 1374 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
32 grpmnd 18958 . . . . . . . 8 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
332, 15, 26mndlid 18767 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑦𝐵) → ((0g𝐺) + 𝑦) = 𝑦)
3432, 5, 33syl2an 596 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((0g𝐺) + 𝑦) = 𝑦)
3529, 31, 343eqtr3d 2785 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦)
3622, 25, 35rspcedvd 3624 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)
3718, 36jca 511 . . . 4 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦))
3837ralrimivva 3202 . . 3 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦))
391, 3, 383jca 1129 . 2 (𝐺 ∈ Grp → (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)))
40 simp1 1137 . . 3 ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Smgrp)
412, 15dfgrp3lem 19056 . . 3 ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
422, 15dfgrp2 18980 . . 3 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
4340, 41, 42sylanbrc 583 . 2 ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Grp)
4439, 43impbii 209 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  c0 4333  cfv 6561  (class class class)co 7431  Basecbs 17247  +gcplusg 17297  0gc0g 17484  Smgrpcsgrp 18731  Mndcmnd 18747  Grpcgrp 18951  invgcminusg 18952  -gcsg 18953
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-0g 17486  df-mgm 18653  df-sgrp 18732  df-mnd 18748  df-grp 18954  df-minusg 18955  df-sbg 18956
This theorem is referenced by:  dfgrp3e  19058
  Copyright terms: Public domain W3C validator