Theorem dfgrp2 19073

Description: Alternate definition of a group as semigroup with a left identity and a left inverse for each element. This "definition" is weaker than df-grp 19047, based on the definition of a monoid which provides a left and a right identity. (Contributed by AV, 28-Aug-2021.)

Hypotheses

Ref	Expression
dfgrp2.b	⊢ 𝐵 = (Base‘𝐺)
dfgrp2.p	⊢ + = (+g‘𝐺)

Assertion

Ref	Expression
dfgrp2	⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃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

Proof of Theorem dfgrp2

Dummy variables 𝑎 𝑏 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		grpsgrp 19071	. . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2		grpmnd 19051	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3		dfgrp2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4		eqid 2765	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
5	3, 4	mndidcl 18840	. . . . 5 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
6	2, 5	syl 18	. . . 4 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
7		oveq1 7426	. . . . . . . 8 ⊢ (𝑛 = (0g‘𝐺) → (𝑛 + 𝑥) = ((0g‘𝐺) + 𝑥))
8	7	eqeq1d 2767	. . . . . . 7 ⊢ (𝑛 = (0g‘𝐺) → ((𝑛 + 𝑥) = 𝑥 ↔ ((0g‘𝐺) + 𝑥) = 𝑥))
9		eqeq2 2777	. . . . . . . 8 ⊢ (𝑛 = (0g‘𝐺) → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑥) = (0g‘𝐺)))
10	9	rexbidv 3191	. . . . . . 7 ⊢ (𝑛 = (0g‘𝐺) → (∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺)))
11	8, 10	anbi12d 644	. . . . . 6 ⊢ (𝑛 = (0g‘𝐺) → (((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) ↔ (((0g‘𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺))))
12	11	ralbidv 3190	. . . . 5 ⊢ (𝑛 = (0g‘𝐺) → (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ∀𝑥 ∈ 𝐵 (((0g‘𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺))))
13	12	adantl 487	. . . 4 ⊢ ((𝐺 ∈ Grp ∧ 𝑛 = (0g‘𝐺)) → (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ∀𝑥 ∈ 𝐵 (((0g‘𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺))))
14		dfgrp2.p	. . . . . . . 8 ⊢ + = (+g‘𝐺)
15	3, 14, 4	mndlid 18845	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺) + 𝑥) = 𝑥)
16	2, 15	sylan 592	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺) + 𝑥) = 𝑥)
17	3, 14, 4	grpinvex 19054	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺))
18	16, 17	jca 521	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((0g‘𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺)))
19	18	ralrimiva 3159	. . . 4 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 (((0g‘𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺)))
20	6, 13, 19	rspcedvd 3585	. . 3 ⊢ (𝐺 ∈ Grp → ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛))
21	1, 20	jca 521	. 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)))
22	3	a1i 11	. . . . . 6 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐵 = (Base‘𝐺))
23	14	a1i 11	. . . . . 6 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → + = (+g‘𝐺))
24		sgrpmgm 18814	. . . . . . . 8 ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
25	24	adantl 487	. . . . . . 7 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Mgm)
26	3, 14	mgmcl 18723	. . . . . . 7 ⊢ ((𝐺 ∈ Mgm ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
27	25, 26	syl3an1 1181	. . . . . 6 ⊢ ((((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
28	3, 14	sgrpass 18815	. . . . . . 7 ⊢ ((𝐺 ∈ Smgrp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
29	28	adantll 727	. . . . . 6 ⊢ ((((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
30		simpll 779	. . . . . 6 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝑛 ∈ 𝐵)
31		oveq2 7427	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑛 + 𝑥) = (𝑛 + 𝑎))
32		id 23	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
33	31, 32	eqeq12d 2781	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑛 + 𝑥) = 𝑥 ↔ (𝑛 + 𝑎) = 𝑎))
34		oveq2 7427	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑖 + 𝑥) = (𝑖 + 𝑎))
35	34	eqeq1d 2767	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑎) = 𝑛))
36	35	rexbidv 3191	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛))
37	33, 36	anbi12d 644	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛)))
38	37	rspcv 3579	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛)))
39		simpl 488	. . . . . . . . 9 ⊢ (((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛) → (𝑛 + 𝑎) = 𝑎)
40	38, 39	syl6com 38	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝑎 ∈ 𝐵 → (𝑛 + 𝑎) = 𝑎))
41	40	ad2antlr 740	. . . . . . 7 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎 ∈ 𝐵 → (𝑛 + 𝑎) = 𝑎))
42	41	imp 412	. . . . . 6 ⊢ ((((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎 ∈ 𝐵) → (𝑛 + 𝑎) = 𝑎)
43		oveq1 7426	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑏 → (𝑖 + 𝑎) = (𝑏 + 𝑎))
44	43	eqeq1d 2767	. . . . . . . . . . 11 ⊢ (𝑖 = 𝑏 → ((𝑖 + 𝑎) = 𝑛 ↔ (𝑏 + 𝑎) = 𝑛))
45	44	cbvrexvw 3246	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛 ↔ ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛)
46	45	bilani 510	. . . . . . . . 9 ⊢ (((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛) → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛)
47	38, 46	syl6com 38	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝑎 ∈ 𝐵 → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛))
48	47	ad2antlr 740	. . . . . . 7 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎 ∈ 𝐵 → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛))
49	48	imp 412	. . . . . 6 ⊢ ((((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎 ∈ 𝐵) → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛)
50	22, 23, 27, 29, 30, 42, 49	isgrpde 19068	. . . . 5 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Grp)
51	50	ex 418	. . . 4 ⊢ ((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
52	51	rexlimiva 3160	. . 3 ⊢ (∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
53	52	impcom 413	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) → 𝐺 ∈ Grp)
54	21, 53	impbii 212	1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∀wral 3081  ∃wrex 3091  ‘cfv 6540  (class class class)co 7419  Basecbs 17291  +_gcplusg 17332  0_gc0g 17514  Mgmcmgm 18718  Smgrpcsgrp 18808  Mndcmnd 18824  Grpcgrp 19044

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-iota 6496  df-fun 6542  df-fv 6548  df-riota 7376  df-ov 7422  df-0g 17516  df-mgm 18720  df-sgrp 18809  df-mnd 18825  df-grp 19047

This theorem is used by:  dfgrp2e  19074  dfgrp3  19149