Theorem dfgrp2 13102

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 13078, 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 13100	. . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2		grpmnd 13082	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3		dfgrp2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4		eqid 2193	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
5	3, 4	mndidcl 13014	. . . . 5 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
6	2, 5	syl 14	. . . 4 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
7		oveq1 5926	. . . . . . . 8 ⊢ (𝑛 = (0g‘𝐺) → (𝑛 + 𝑥) = ((0g‘𝐺) + 𝑥))
8	7	eqeq1d 2202	. . . . . . 7 ⊢ (𝑛 = (0g‘𝐺) → ((𝑛 + 𝑥) = 𝑥 ↔ ((0g‘𝐺) + 𝑥) = 𝑥))
9		eqeq2 2203	. . . . . . . 8 ⊢ (𝑛 = (0g‘𝐺) → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑥) = (0g‘𝐺)))
10	9	rexbidv 2495	. . . . . . 7 ⊢ (𝑛 = (0g‘𝐺) → (∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺)))
11	8, 10	anbi12d 473	. . . . . 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 2494	. . . . 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 277	. . . 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 13019	. . . . . . 7 ⊢ ((𝐺 ∈ Mnd ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺) + 𝑥) = 𝑥)
16	2, 15	sylan 283	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ((0g‘𝐺) + 𝑥) = 𝑥)
17	3, 14, 4	grpinvex 13085	. . . . . 6 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺))
18	16, 17	jca 306	. . . . 5 ⊢ ((𝐺 ∈ Grp ∧ 𝑥 ∈ 𝐵) → (((0g‘𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺)))
19	18	ralrimiva 2567	. . . 4 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 (((0g‘𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺)))
20	6, 13, 19	rspcedvd 2871	. . 3 ⊢ (𝐺 ∈ Grp → ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛))
21	1, 20	jca 306	. 2 ⊢ (𝐺 ∈ Grp → (𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)))
22	3	a1i 9	. . . . . 6 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐵 = (Base‘𝐺))
23	14	a1i 9	. . . . . 6 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → + = (+g‘𝐺))
24		sgrpmgm 12993	. . . . . . . 8 ⊢ (𝐺 ∈ Smgrp → 𝐺 ∈ Mgm)
25	24	adantl 277	. . . . . . 7 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Mgm)
26	3, 14	mgmcl 12945	. . . . . . 7 ⊢ ((𝐺 ∈ Mgm ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
27	25, 26	syl3an1 1282	. . . . . 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 12994	. . . . . . 7 ⊢ ((𝐺 ∈ Smgrp ∧ (𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵 ∧ 𝑐 ∈ 𝐵)) → ((𝑎 + 𝑏) + 𝑐) = (𝑎 + (𝑏 + 𝑐)))
29	28	adantll 476	. . . . . 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 527	. . . . . 6 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝑛 ∈ 𝐵)
31		oveq2 5927	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑛 + 𝑥) = (𝑛 + 𝑎))
32		id 19	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
33	31, 32	eqeq12d 2208	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑛 + 𝑥) = 𝑥 ↔ (𝑛 + 𝑎) = 𝑎))
34		oveq2 5927	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑖 + 𝑥) = (𝑖 + 𝑎))
35	34	eqeq1d 2202	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑎) = 𝑛))
36	35	rexbidv 2495	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → (∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛 ↔ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛))
37	33, 36	anbi12d 473	. . . . . . . . . 10 ⊢ (𝑥 = 𝑎 → (((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) ↔ ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛)))
38	37	rspcv 2861	. . . . . . . . 9 ⊢ (𝑎 ∈ 𝐵 → (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → ((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛)))
39		simpl 109	. . . . . . . . 9 ⊢ (((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛) → (𝑛 + 𝑎) = 𝑎)
40	38, 39	syl6com 35	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝑎 ∈ 𝐵 → (𝑛 + 𝑎) = 𝑎))
41	40	ad2antlr 489	. . . . . . 7 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → (𝑎 ∈ 𝐵 → (𝑛 + 𝑎) = 𝑎))
42	41	imp 124	. . . . . 6 ⊢ ((((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) ∧ 𝑎 ∈ 𝐵) → (𝑛 + 𝑎) = 𝑎)
43		oveq1 5926	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑏 → (𝑖 + 𝑎) = (𝑏 + 𝑎))
44	43	eqeq1d 2202	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑏 → ((𝑖 + 𝑎) = 𝑛 ↔ (𝑏 + 𝑎) = 𝑛))
45	44	cbvrexvw 2731	. . . . . . . . . . 11 ⊢ (∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛 ↔ ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛)
46	45	biimpi 120	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛 → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛)
47	46	adantl 277	. . . . . . . . 9 ⊢ (((𝑛 + 𝑎) = 𝑎 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑎) = 𝑛) → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛)
48	38, 47	syl6com 35	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝑎 ∈ 𝐵 → ∃𝑏 ∈ 𝐵 (𝑏 + 𝑎) = 𝑛))
49	48	ad2antlr 489	. . . . . . 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 124	. . . . . 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 13097	. . . . 5 ⊢ (((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) ∧ 𝐺 ∈ Smgrp) → 𝐺 ∈ Grp)
52	51	ex 115	. . . 4 ⊢ ((𝑛 ∈ 𝐵 ∧ ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
53	52	rexlimiva 2606	. . 3 ⊢ (∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛) → (𝐺 ∈ Smgrp → 𝐺 ∈ Grp))
54	53	impcom 125	. 2 ⊢ ((𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)) → 𝐺 ∈ Grp)
55	21, 54	impbii 126	1 ⊢ (𝐺 ∈ Grp ↔ (𝐺 ∈ Smgrp ∧ ∃𝑛 ∈ 𝐵 ∀𝑥 ∈ 𝐵 ((𝑛 + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = 𝑛)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  ∀wral 2472  ∃wrex 2473  ‘cfv 5255  (class class class)co 5919  Basecbs 12621  +_gcplusg 12698  0_gc0g 12870  Mgmcmgm 12940  Smgrpcsgrp 12987  Mndcmnd 13000  Grpcgrp 13075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-cnex 7965  ax-resscn 7966  ax-1re 7968  ax-addrcl 7971

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-iota 5216  df-fun 5257  df-fn 5258  df-fv 5263  df-riota 5874  df-ov 5922  df-inn 8985  df-2 9043  df-ndx 12624  df-slot 12625  df-base 12627  df-plusg 12711  df-0g 12872  df-mgm 12942  df-sgrp 12988  df-mnd 13001  df-grp 13078

This theorem is referenced by:  dfgrp2e  13103  dfgrp3m  13174