Theorem dfgrp2e 17448

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 17447	. 2 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ SGrp ∧ ∃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 6185	. . . . . . . 8 ⊢ (¬ 𝐺 ∈ V → (Base‘𝐺) = ∅)
6	1	eleq2i 2693	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐵 ↔ 𝑛 ∈ (Base‘𝐺))
7		eleq2 2690	. . . . . . . . . 10 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) ↔ 𝑛 ∈ ∅))
8		noel 3919	. . . . . . . . . . 11 ⊢ ¬ 𝑛 ∈ ∅
9	8	pm2.21i 116	. . . . . . . . . 10 ⊢ (𝑛 ∈ ∅ → 𝐺 ∈ V)
10	7, 9	syl6bi 243	. . . . . . . . 9 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ (Base‘𝐺) → 𝐺 ∈ V))
11	6, 10	syl5bi 232	. . . . . . . 8 ⊢ ((Base‘𝐺) = ∅ → (𝑛 ∈ 𝐵 → 𝐺 ∈ V))
12	5, 11	syl 17	. . . . . . 7 ⊢ (¬ 𝐺 ∈ V → (𝑛 ∈ 𝐵 → 𝐺 ∈ V))
13	4, 12	pm2.61i 176	. . . . . 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 3027	. . . 4 ⊢ (∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → 𝐺 ∈ V)
16	1, 2	issgrpv 17286	. . . 4 ⊢ (𝐺 ∈ V → (𝐺 ∈ SGrp ↔ ∀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 ∈ SGrp ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 ((𝑥 + 𝑦) ∈ 𝐵 ∧ ∀𝑧 ∈ 𝐵 ((𝑥 + 𝑦) + 𝑧) = (𝑥 + (𝑦 + 𝑧)))))
18	17	pm5.32ri 670	. 2 ⊢ ((𝐺 ∈ SGrp ∧ ∃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 264	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 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200  ∅c0 3915  ‘cfv 5888  (class class class)co 6650  Basecbs 15857  +_gcplusg 15941  SGrpcsgrp 17283  Grpcgrp 17422

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896  df-riota 6611  df-ov 6653  df-0g 16102  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-grp 17425

This theorem is referenced by: (None)