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Theorem dfgrp3 17283
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 ∈ SGrp ∧ 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 17215 . . 3 (𝐺 ∈ Grp → 𝐺 ∈ SGrp)
2 dfgrp3.b . . . 4 𝐵 = (Base‘𝐺)
32grpbn0 17220 . . 3 (𝐺 ∈ Grp → 𝐵 ≠ ∅)
4 simpl 471 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝐺 ∈ Grp)
5 simpr 475 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → 𝑦𝐵)
65adantl 480 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
7 simpl 471 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → 𝑥𝐵)
87adantl 480 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
9 eqid 2609 . . . . . . . 8 (-g𝐺) = (-g𝐺)
102, 9grpsubcl 17264 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝑦(-g𝐺)𝑥) ∈ 𝐵)
114, 6, 8, 10syl3anc 1317 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(-g𝐺)𝑥) ∈ 𝐵)
12 oveq1 6534 . . . . . . . 8 (𝑙 = (𝑦(-g𝐺)𝑥) → (𝑙 + 𝑥) = ((𝑦(-g𝐺)𝑥) + 𝑥))
1312eqeq1d 2611 . . . . . . 7 (𝑙 = (𝑦(-g𝐺)𝑥) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦))
1413adantl 480 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑙 = (𝑦(-g𝐺)𝑥)) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦))
15 dfgrp3.p . . . . . . . 8 + = (+g𝐺)
162, 15, 9grpnpcan 17276 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦)
174, 6, 8, 16syl3anc 1317 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦)
1811, 14, 17rspcedvd 3288 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦)
19 eqid 2609 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
202, 19grpinvcl 17236 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((invg𝐺)‘𝑥) ∈ 𝐵)
2120adantrr 748 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((invg𝐺)‘𝑥) ∈ 𝐵)
222, 15grpcl 17199 . . . . . . 7 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑥) ∈ 𝐵𝑦𝐵) → (((invg𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
234, 21, 6, 22syl3anc 1317 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (((invg𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
24 oveq2 6535 . . . . . . . 8 (𝑟 = (((invg𝐺)‘𝑥) + 𝑦) → (𝑥 + 𝑟) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
2524eqeq1d 2611 . . . . . . 7 (𝑟 = (((invg𝐺)‘𝑥) + 𝑦) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦))
2625adantl 480 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑟 = (((invg𝐺)‘𝑥) + 𝑦)) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦))
27 eqid 2609 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
282, 15, 27, 19grprinv 17238 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑥 + ((invg𝐺)‘𝑥)) = (0g𝐺))
2928adantrr 748 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + ((invg𝐺)‘𝑥)) = (0g𝐺))
3029oveq1d 6542 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = ((0g𝐺) + 𝑦))
312, 15grpass 17200 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑥𝐵 ∧ ((invg𝐺)‘𝑥) ∈ 𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
324, 8, 21, 6, 31syl13anc 1319 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
33 grpmnd 17198 . . . . . . . 8 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
342, 15, 27mndlid 17080 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑦𝐵) → ((0g𝐺) + 𝑦) = 𝑦)
3533, 5, 34syl2an 492 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((0g𝐺) + 𝑦) = 𝑦)
3630, 32, 353eqtr3d 2651 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦)
3723, 26, 36rspcedvd 3288 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)
3818, 37jca 552 . . . 4 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦))
3938ralrimivva 2953 . . 3 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦))
401, 3, 393jca 1234 . 2 (𝐺 ∈ Grp → (𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)))
41 simp1 1053 . . 3 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ SGrp)
422, 15dfgrp3lem 17282 . . 3 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
432, 15dfgrp2 17216 . . 3 (𝐺 ∈ Grp ↔ (𝐺 ∈ SGrp ∧ ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
4441, 42, 43sylanbrc 694 . 2 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Grp)
4540, 44impbii 197 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 194  wa 382  w3a 1030   = wceq 1474  wcel 1976  wne 2779  wral 2895  wrex 2896  c0 3873  cfv 5790  (class class class)co 6527  Basecbs 15641  +gcplusg 15714  0gc0g 15869  SGrpcsgrp 17052  Mndcmnd 17063  Grpcgrp 17191  invgcminusg 17192  -gcsg 17193
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-rep 4693  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6824
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-reu 2902  df-rmo 2903  df-rab 2904  df-v 3174  df-sbc 3402  df-csb 3499  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-pw 4109  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-iun 4451  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-f1 5795  df-fo 5796  df-f1o 5797  df-fv 5798  df-riota 6489  df-ov 6530  df-oprab 6531  df-mpt2 6532  df-1st 7036  df-2nd 7037  df-0g 15871  df-mgm 17011  df-sgrp 17053  df-mnd 17064  df-grp 17194  df-minusg 17195  df-sbg 17196
This theorem is referenced by:  dfgrp3e  17284
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