Theorem dfgrp2 18120

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 18098, 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 18119	. . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2		grpmnd 18102	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3		dfgrp2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4		eqid 2798	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
5	3, 4	mndidcl 17918	. . . . 5 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
6	2, 5	syl 17	. . . 4 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
7		oveq1 7142	. . . . . . . 8 ⊢ (𝑛 = (0g‘𝐺) → (𝑛 + 𝑥) = ((0g‘𝐺) + 𝑥))
8	7	eqeq1d 2800	. . . . . . 7 ⊢ (𝑛 = (0g‘𝐺) → ((𝑛 + 𝑥) = 𝑥 ↔ ((0g‘𝐺) + 𝑥) = 𝑥))
9		eqeq2 2810	. . . . . . . 8 ⊢ (𝑛 = (0g‘𝐺) → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑥) = (0g‘𝐺)))
10	9	rexbidv 3256	. . . . . . 7 ⊢ (𝑛 = (0g‘𝐺) → (∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺)))
11	8, 10	anbi12d 633	. . . . . 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 3162	. . . . 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 485	. . . 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 17923	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺) + 𝑥) = 𝑥)
16	2, 15	sylan 583	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺) + 𝑥) = 𝑥)
17	3, 14, 4	grpinvex 18105	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺))
18	16, 17	jca 515	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((0g‘𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺)))
19	18	ralrimiva 3149	. . . 4 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 (((0g‘𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺)))
20	6, 13, 19	rspcedvd 3574	. . 3 ⊢ (𝐺 ∈ Grp → ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛))
21	1, 20	jca 515	. 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 17898	. . . . . . . 8 ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
25	24	adantl 485	. . . . . . 7 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Mgm)
26	3, 14	mgmcl 17847	. . . . . . 7 ⊢ ((𝐺 ∈ Mgm ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
27	25, 26	syl3an1 1160	. . . . . 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 17899	. . . . . . 7 ⊢ ((𝐺 ∈ Smgrp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
29	28	adantll 713	. . . . . 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 766	. . . . . 6 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝑛 ∈ 𝐵)
31		oveq2 7143	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑛 + 𝑥) = (𝑛 + 𝑎))
32		id 22	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
33	31, 32	eqeq12d 2814	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑛 + 𝑥) = 𝑥 ↔ (𝑛 + 𝑎) = 𝑎))
34		oveq2 7143	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑖 + 𝑥) = (𝑖 + 𝑎))
35	34	eqeq1d 2800	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑎) = 𝑛))
36	35	rexbidv 3256	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛))
37	33, 36	anbi12d 633	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛)))
38	37	rspcv 3566	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛)))
39		simpl 486	. . . . . . . . 9 ⊢ (((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛) → (𝑛 + 𝑎) = 𝑎)
40	38, 39	syl6com 37	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝑎 ∈ 𝐵 → (𝑛 + 𝑎) = 𝑎))
41	40	ad2antlr 726	. . . . . . 7 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎 ∈ 𝐵 → (𝑛 + 𝑎) = 𝑎))
42	41	imp 410	. . . . . 6 ⊢ ((((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎 ∈ 𝐵) → (𝑛 + 𝑎) = 𝑎)
43		oveq1 7142	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑏 → (𝑖 + 𝑎) = (𝑏 + 𝑎))
44	43	eqeq1d 2800	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑏 → ((𝑖 + 𝑎) = 𝑛 ↔ (𝑏 + 𝑎) = 𝑛))
45	44	cbvrexvw 3397	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛 ↔ ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛)
46	45	biimpi 219	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛 → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛)
47	46	adantl 485	. . . . . . . . 9 ⊢ (((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛) → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛)
48	38, 47	syl6com 37	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝑎 ∈ 𝐵 → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛))
49	48	ad2antlr 726	. . . . . . 7 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎 ∈ 𝐵 → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛))
50	49	imp 410	. . . . . 6 ⊢ ((((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎 ∈ 𝐵) → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛)
51	22, 23, 27, 29, 30, 42, 50	isgrpde 18116	. . . . 5 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Grp)
52	51	ex 416	. . . 4 ⊢ ((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
53	52	rexlimiva 3240	. . 3 ⊢ (∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
54	53	impcom 411	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) → 𝐺 ∈ Grp)
55	21, 54	impbii 212	1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  ‘cfv 6324  (class class class)co 7135  Basecbs 16475  +_gcplusg 16557  0_gc0g 16705  Mgmcmgm 17842  Smgrpcsgrp 17892  Mndcmnd 17903  Grpcgrp 18095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-riota 7093  df-ov 7138  df-0g 16707  df-mgm 17844  df-sgrp 17893  df-mnd 17904  df-grp 18098

This theorem is referenced by:  dfgrp2e  18121  dfgrp3  18190