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Theorem dfgrp2e 18061
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 18060 . 2 (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛)))
4 ax-1 6 . . . . . . 7 (𝐺 ∈ V → (𝑛𝐵𝐺 ∈ V))
5 fvprc 6659 . . . . . . . 8 𝐺 ∈ V → (Base‘𝐺) = ∅)
61eleq2i 2908 . . . . . . . . 9 (𝑛𝐵𝑛 ∈ (Base‘𝐺))
7 eleq2 2905 . . . . . . . . . 10 ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) ↔ 𝑛 ∈ ∅))
8 noel 4299 . . . . . . . . . . 11 ¬ 𝑛 ∈ ∅
98pm2.21i 119 . . . . . . . . . 10 (𝑛 ∈ ∅ → 𝐺 ∈ V)
107, 9syl6bi 254 . . . . . . . . 9 ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) → 𝐺 ∈ V))
116, 10syl5bi 243 . . . . . . . 8 ((Base‘𝐺) = ∅ → (𝑛𝐵𝐺 ∈ V))
125, 11syl 17 . . . . . . 7 𝐺 ∈ V → (𝑛𝐵𝐺 ∈ V))
134, 12pm2.61i 183 . . . . . 6 (𝑛𝐵𝐺 ∈ V)
1413a1d 25 . . . . 5 (𝑛𝐵 → (∀𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → 𝐺 ∈ V))
1514rexlimiv 3284 . . . 4 (∃𝑛𝐵𝑥𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖𝐵 (𝑖 + 𝑥) = 𝑛) → 𝐺 ∈ V)
161, 2issgrpv 17894 . . . 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 576 . 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 276 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 207  wa 396   = wceq 1530  wcel 2107  wral 3142  wrex 3143  Vcvv 3499  c0 4294  cfv 6351  (class class class)co 7151  Basecbs 16475  +gcplusg 16557  Smgrpcsgrp 17891  Grpcgrp 18035
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rmo 3150  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359  df-riota 7109  df-ov 7154  df-0g 16707  df-mgm 17844  df-sgrp 17892  df-mnd 17903  df-grp 18038
This theorem is referenced by: (None)
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