Theorem dfgrp2e 18895

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	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
dfgrp2e	⊢ (𝐺 ∈ Grp ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (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

Step	Hyp	Ref	Expression
1		dfgrp2.b	. . 3 ⊢ 𝐵 = (Base‘𝐺)
2		dfgrp2.p	. . 3 ⊢ + = (+g‘𝐺)
3	1, 2	dfgrp2 18894	. 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)))
4		ax-1 6	. . . . . . 7 ⊢ (𝐺 ∈ V → (𝑛 ∈ 𝐵 → 𝐺 ∈ V))
5		fvprc 6850	. . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
6	1	eleq2i 2820	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐵 ↔ 𝑛 ∈ (Base‘𝐺))
7		eleq2 2817	. . . . . . . . . 10 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) ↔ 𝑛 ∈ ∅))
8		noel 4301	. . . . . . . . . . 11 ⊢ ¬ 𝑛 ∈ ∅
9	8	pm2.21i 119	. . . . . . . . . 10 ⊢ (𝑛 ∈ ∅ → 𝐺 ∈ V)
10	7, 9	biimtrdi 253	. . . . . . . . 9 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) → 𝐺 ∈ V))
11	6, 10	biimtrid 242	. . . . . . . 8 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ 𝐵 → 𝐺 ∈ V))
12	5, 11	syl 17	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (𝑛 ∈ 𝐵 → 𝐺 ∈ V))
13	4, 12	pm2.61i 182	. . . . . 6 ⊢ (𝑛 ∈ 𝐵 → 𝐺 ∈ V)
14	13	a1d 25	. . . . 5 ⊢ (𝑛 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → 𝐺 ∈ V))
15	14	rexlimiv 3127	. . . 4 ⊢ (∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → 𝐺 ∈ V)
16	1, 2	issgrpv 18648	. . . 4 ⊢ (𝐺 ∈ V → (𝐺 ∈ Smgrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))))
17	15, 16	syl 17	. . 3 ⊢ (∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝐺 ∈ Smgrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))))
18	17	pm5.32ri 575	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)))
19	3, 18	bitri 275	1 ⊢ (𝐺 ∈ Grp ↔ (∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧))) ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053  Vcvv 3447  ∅c0 4296  ‘cfv 6511  (class class class)co 7387  Basecbs 17179  +_gcplusg 17220  Smgrpcsgrp 18645  Grpcgrp 18865

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-iota 6464  df-fun 6513  df-fv 6519  df-riota 7344  df-ov 7390  df-0g 17404  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-grp 18868

This theorem is referenced by: (None)