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Theorem dfgrp3 18851
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 18779 . . 3 (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2 dfgrp3.b . . . 4 𝐵 = (Base‘𝐺)
32grpbn0 18784 . . 3 (𝐺 ∈ Grp → 𝐵 ≠ ∅)
4 simpl 484 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝐺 ∈ Grp)
5 simpr 486 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → 𝑦𝐵)
65adantl 483 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
7 simpl 484 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → 𝑥𝐵)
87adantl 483 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
9 eqid 2733 . . . . . . . 8 (-g𝐺) = (-g𝐺)
102, 9grpsubcl 18832 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝑦(-g𝐺)𝑥) ∈ 𝐵)
114, 6, 8, 10syl3anc 1372 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(-g𝐺)𝑥) ∈ 𝐵)
12 oveq1 7365 . . . . . . . 8 (𝑙 = (𝑦(-g𝐺)𝑥) → (𝑙 + 𝑥) = ((𝑦(-g𝐺)𝑥) + 𝑥))
1312eqeq1d 2735 . . . . . . 7 (𝑙 = (𝑦(-g𝐺)𝑥) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦))
1413adantl 483 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑙 = (𝑦(-g𝐺)𝑥)) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦))
15 dfgrp3.p . . . . . . . 8 + = (+g𝐺)
162, 15, 9grpnpcan 18844 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦)
174, 6, 8, 16syl3anc 1372 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦)
1811, 14, 17rspcedvd 3582 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦)
19 eqid 2733 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
202, 19grpinvcl 18803 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((invg𝐺)‘𝑥) ∈ 𝐵)
2120adantrr 716 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((invg𝐺)‘𝑥) ∈ 𝐵)
222, 15, 4, 21, 6grpcld 18766 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (((invg𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
23 oveq2 7366 . . . . . . . 8 (𝑟 = (((invg𝐺)‘𝑥) + 𝑦) → (𝑥 + 𝑟) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
2423eqeq1d 2735 . . . . . . 7 (𝑟 = (((invg𝐺)‘𝑥) + 𝑦) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦))
2524adantl 483 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑟 = (((invg𝐺)‘𝑥) + 𝑦)) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦))
26 eqid 2733 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
272, 15, 26, 19grprinv 18806 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑥 + ((invg𝐺)‘𝑥)) = (0g𝐺))
2827adantrr 716 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + ((invg𝐺)‘𝑥)) = (0g𝐺))
2928oveq1d 7373 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = ((0g𝐺) + 𝑦))
302, 15grpass 18762 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑥𝐵 ∧ ((invg𝐺)‘𝑥) ∈ 𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
314, 8, 21, 6, 30syl13anc 1373 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
32 grpmnd 18760 . . . . . . . 8 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
332, 15, 26mndlid 18581 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑦𝐵) → ((0g𝐺) + 𝑦) = 𝑦)
3432, 5, 33syl2an 597 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((0g𝐺) + 𝑦) = 𝑦)
3529, 31, 343eqtr3d 2781 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦)
3622, 25, 35rspcedvd 3582 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)
3718, 36jca 513 . . . 4 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦))
3837ralrimivva 3194 . . 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 18850 . . 3 ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
422, 15dfgrp2 18780 . . 3 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
4340, 41, 42sylanbrc 584 . 2 ((𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Grp)
4439, 43impbii 208 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2940  wral 3061  wrex 3070  c0 4283  cfv 6497  (class class class)co 7358  Basecbs 17088  +gcplusg 17138  0gc0g 17326  Smgrpcsgrp 18550  Mndcmnd 18561  Grpcgrp 18753  invgcminusg 18754  -gcsg 18755
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-0g 17328  df-mgm 18502  df-sgrp 18551  df-mnd 18562  df-grp 18756  df-minusg 18757  df-sbg 18758
This theorem is referenced by:  dfgrp3e  18852
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