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Theorem dfgrp3 17711
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 17643 . . 3 (𝐺 ∈ Grp → 𝐺 ∈ SGrp)
2 dfgrp3.b . . . 4 𝐵 = (Base‘𝐺)
32grpbn0 17648 . . 3 (𝐺 ∈ Grp → 𝐵 ≠ ∅)
4 simpl 474 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝐺 ∈ Grp)
5 simpr 479 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → 𝑦𝐵)
65adantl 473 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝑦𝐵)
7 simpl 474 . . . . . . . 8 ((𝑥𝐵𝑦𝐵) → 𝑥𝐵)
87adantl 473 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → 𝑥𝐵)
9 eqid 2756 . . . . . . . 8 (-g𝐺) = (-g𝐺)
102, 9grpsubcl 17692 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → (𝑦(-g𝐺)𝑥) ∈ 𝐵)
114, 6, 8, 10syl3anc 1477 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑦(-g𝐺)𝑥) ∈ 𝐵)
12 oveq1 6816 . . . . . . . 8 (𝑙 = (𝑦(-g𝐺)𝑥) → (𝑙 + 𝑥) = ((𝑦(-g𝐺)𝑥) + 𝑥))
1312eqeq1d 2758 . . . . . . 7 (𝑙 = (𝑦(-g𝐺)𝑥) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦))
1413adantl 473 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑙 = (𝑦(-g𝐺)𝑥)) → ((𝑙 + 𝑥) = 𝑦 ↔ ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦))
15 dfgrp3.p . . . . . . . 8 + = (+g𝐺)
162, 15, 9grpnpcan 17704 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑦𝐵𝑥𝐵) → ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦)
174, 6, 8, 16syl3anc 1477 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑦(-g𝐺)𝑥) + 𝑥) = 𝑦)
1811, 14, 17rspcedvd 3452 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦)
19 eqid 2756 . . . . . . . . 9 (invg𝐺) = (invg𝐺)
202, 19grpinvcl 17664 . . . . . . . 8 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → ((invg𝐺)‘𝑥) ∈ 𝐵)
2120adantrr 755 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((invg𝐺)‘𝑥) ∈ 𝐵)
222, 15grpcl 17627 . . . . . . 7 ((𝐺 ∈ Grp ∧ ((invg𝐺)‘𝑥) ∈ 𝐵𝑦𝐵) → (((invg𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
234, 21, 6, 22syl3anc 1477 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (((invg𝐺)‘𝑥) + 𝑦) ∈ 𝐵)
24 oveq2 6817 . . . . . . . 8 (𝑟 = (((invg𝐺)‘𝑥) + 𝑦) → (𝑥 + 𝑟) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
2524eqeq1d 2758 . . . . . . 7 (𝑟 = (((invg𝐺)‘𝑥) + 𝑦) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦))
2625adantl 473 . . . . . 6 (((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) ∧ 𝑟 = (((invg𝐺)‘𝑥) + 𝑦)) → ((𝑥 + 𝑟) = 𝑦 ↔ (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦))
27 eqid 2756 . . . . . . . . . 10 (0g𝐺) = (0g𝐺)
282, 15, 27, 19grprinv 17666 . . . . . . . . 9 ((𝐺 ∈ Grp ∧ 𝑥𝐵) → (𝑥 + ((invg𝐺)‘𝑥)) = (0g𝐺))
2928adantrr 755 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + ((invg𝐺)‘𝑥)) = (0g𝐺))
3029oveq1d 6824 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = ((0g𝐺) + 𝑦))
312, 15grpass 17628 . . . . . . . 8 ((𝐺 ∈ Grp ∧ (𝑥𝐵 ∧ ((invg𝐺)‘𝑥) ∈ 𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
324, 8, 21, 6, 31syl13anc 1479 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((𝑥 + ((invg𝐺)‘𝑥)) + 𝑦) = (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)))
33 grpmnd 17626 . . . . . . . 8 (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
342, 15, 27mndlid 17508 . . . . . . . 8 ((𝐺 ∈ Mnd ∧ 𝑦𝐵) → ((0g𝐺) + 𝑦) = 𝑦)
3533, 5, 34syl2an 495 . . . . . . 7 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ((0g𝐺) + 𝑦) = 𝑦)
3630, 32, 353eqtr3d 2798 . . . . . 6 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (𝑥 + (((invg𝐺)‘𝑥) + 𝑦)) = 𝑦)
3723, 26, 36rspcedvd 3452 . . . . 5 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)
3818, 37jca 555 . . . 4 ((𝐺 ∈ Grp ∧ (𝑥𝐵𝑦𝐵)) → (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦))
3938ralrimivva 3105 . . 3 (𝐺 ∈ Grp → ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦))
401, 3, 393jca 1123 . 2 (𝐺 ∈ Grp → (𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)))
41 simp1 1131 . . 3 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ SGrp)
422, 15dfgrp3lem 17710 . . 3 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢))
432, 15dfgrp2 17644 . . 3 (𝐺 ∈ Grp ↔ (𝐺 ∈ SGrp ∧ ∃𝑢𝐵𝑎𝐵 ((𝑢 + 𝑎) = 𝑎 ∧ ∃𝑖𝐵 (𝑖 + 𝑎) = 𝑢)))
4441, 42, 43sylanbrc 701 . 2 ((𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)) → 𝐺 ∈ Grp)
4540, 44impbii 199 1 (𝐺 ∈ Grp ↔ (𝐺 ∈ SGrp ∧ 𝐵 ≠ ∅ ∧ ∀𝑥𝐵𝑦𝐵 (∃𝑙𝐵 (𝑙 + 𝑥) = 𝑦 ∧ ∃𝑟𝐵 (𝑥 + 𝑟) = 𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  w3a 1072   = wceq 1628  wcel 2135  wne 2928  wral 3046  wrex 3047  c0 4054  cfv 6045  (class class class)co 6809  Basecbs 16055  +gcplusg 16139  0gc0g 16298  SGrpcsgrp 17480  Mndcmnd 17491  Grpcgrp 17619  invgcminusg 17620  -gcsg 17621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1867  ax-4 1882  ax-5 1984  ax-6 2050  ax-7 2086  ax-8 2137  ax-9 2144  ax-10 2164  ax-11 2179  ax-12 2192  ax-13 2387  ax-ext 2736  ax-rep 4919  ax-sep 4929  ax-nul 4937  ax-pow 4988  ax-pr 5051  ax-un 7110
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1631  df-ex 1850  df-nf 1855  df-sb 2043  df-eu 2607  df-mo 2608  df-clab 2743  df-cleq 2749  df-clel 2752  df-nfc 2887  df-ne 2929  df-ral 3051  df-rex 3052  df-reu 3053  df-rmo 3054  df-rab 3055  df-v 3338  df-sbc 3573  df-csb 3671  df-dif 3714  df-un 3716  df-in 3718  df-ss 3725  df-nul 4055  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4585  df-iun 4670  df-br 4801  df-opab 4861  df-mpt 4878  df-id 5170  df-xp 5268  df-rel 5269  df-cnv 5270  df-co 5271  df-dm 5272  df-rn 5273  df-res 5274  df-ima 5275  df-iota 6008  df-fun 6047  df-fn 6048  df-f 6049  df-f1 6050  df-fo 6051  df-f1o 6052  df-fv 6053  df-riota 6770  df-ov 6812  df-oprab 6813  df-mpt2 6814  df-1st 7329  df-2nd 7330  df-0g 16300  df-mgm 17439  df-sgrp 17481  df-mnd 17492  df-grp 17622  df-minusg 17623  df-sbg 17624
This theorem is referenced by:  dfgrp3e  17712
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