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Theorem dfgrp2e 18890
Description: Alternate definition of a group as a set with a closed, associative operation, a left identity and a left inverse for each element. Alternate definition in [Lang] p. 7. (Contributed by NM, 10-Oct-2006.) (Revised by AV, 28-Aug-2021.)
Hypotheses
Ref Expression
dfgrp2.b 𝐵 = (Base‘𝐺)
dfgrp2.p + = (+g𝐺)
Assertion
Ref Expression
dfgrp2e (𝐺 ∈ Grp ↔ (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
Distinct variable groups:   𝐵,𝑖,𝑛,𝑥   𝑖,𝐺,𝑛,𝑥   + ,𝑖,𝑛,𝑥   𝑦,𝐵,𝑧,𝑥   𝑦,𝐺,𝑧   𝑦, + ,𝑧

Proof of Theorem dfgrp2e
StepHypRef Expression
1 dfgrp2.b . . 3 𝐵 = (Base‘𝐺)
2 dfgrp2.p . . 3 + = (+g𝐺)
31, 2dfgrp2 18889 . 2 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
4 ax-1 6 . . . . . . 7 (𝐺 ∈ V → (𝑛𝐵𝐺 ∈ V))
5 fvprc 6876 . . . . . . . 8 𝐺 ∈ V → (Base‘𝐺) = ∅)
61eleq2i 2819 . . . . . . . . 9 (𝑛𝐵𝑛 ∈ (Base‘𝐺))
7 eleq2 2816 . . . . . . . . . 10 ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) ↔ 𝑛 ∈ ∅))
8 noel 4325 . . . . . . . . . . 11 ¬ 𝑛 ∈ ∅
98pm2.21i 119 . . . . . . . . . 10 (𝑛 ∈ ∅ → 𝐺 ∈ V)
107, 9biimtrdi 252 . . . . . . . . 9 ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) → 𝐺 ∈ V))
116, 10biimtrid 241 . . . . . . . 8 ((Base‘𝐺) = ∅ → (𝑛𝐵𝐺 ∈ V))
125, 11syl 17 . . . . . . 7 𝐺 ∈ V → (𝑛𝐵𝐺 ∈ V))
134, 12pm2.61i 182 . . . . . 6 (𝑛𝐵𝐺 ∈ V)
1413a1d 25 . . . . 5 (𝑛𝐵 → (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → 𝐺 ∈ V))
1514rexlimiv 3142 . . . 4 (∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → 𝐺 ∈ V)
161, 2issgrpv 18651 . . . 4 (𝐺 ∈ V → (𝐺 ∈ Smgrp ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))))
1715, 16syl 17 . . 3 (∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → (𝐺 ∈ Smgrp ↔ ∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))))
1817pm5.32ri 575 . 2 ((𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)) ↔ (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
193, 18bitri 275 1 (𝐺 ∈ Grp ↔ (∀𝑥𝐵𝑦𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  Vcvv 3468  c0 4317  cfv 6536  (class class class)co 7404  Basecbs 17150  +gcplusg 17203  Smgrpcsgrp 18648  Grpcgrp 18860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6488  df-fun 6538  df-fv 6544  df-riota 7360  df-ov 7407  df-0g 17393  df-mgm 18570  df-sgrp 18649  df-mnd 18665  df-grp 18863
This theorem is referenced by: (None)
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