Theorem dfgrp2 13600

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 13576, 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 13598	. . 3 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Smgrp)
2		grpmnd 13580	. . . . 5 ⊢ (𝐺 ∈ Grp → 𝐺 ∈ Mnd)
3		dfgrp2.b	. . . . . 6 ⊢ 𝐵 = (Base‘𝐺)
4		eqid 2229	. . . . . 6 ⊢ (0g‘𝐺) = (0g‘𝐺)
5	3, 4	mndidcl 13503	. . . . 5 ⊢ (𝐺 ∈ Mnd → (0g‘𝐺) ∈ 𝐵)
6	2, 5	syl 14	. . . 4 ⊢ (𝐺 ∈ Grp → (0g‘𝐺) ∈ 𝐵)
7		oveq1 6020	. . . . . . . 8 ⊢ (𝑛 = (0g‘𝐺) → (𝑛 + 𝑥) = ((0g‘𝐺) + 𝑥))
8	7	eqeq1d 2238	. . . . . . 7 ⊢ (𝑛 = (0g‘𝐺) → ((𝑛 + 𝑥) = 𝑥 ↔ ((0g‘𝐺) + 𝑥) = 𝑥))
9		eqeq2 2239	. . . . . . . 8 ⊢ (𝑛 = (0g‘𝐺) → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑥) = (0g‘𝐺)))
10	9	rexbidv 2531	. . . . . . 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 2530	. . . . 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 13508	. . . . . . 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 13583	. . . . . 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 2603	. . . 4 ⊢ (𝐺 ∈ Grp → ∀𝑥 ∈ 𝐵 (((0g‘𝐺) + 𝑥) = 𝑥 ∧ ∃𝑖 ∈ 𝐵 (𝑖 + 𝑥) = (0g‘𝐺)))
20	6, 13, 19	rspcedvd 2914	. . 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 13480	. . . . . . . 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 13432	. . . . . . 7 ⊢ ((𝐺 ∈ Mgm ∧ 𝑎 ∈ 𝐵 ∧ 𝑏 ∈ 𝐵) → (𝑎 + 𝑏) ∈ 𝐵)
27	25, 26	syl3an1 1304	. . . . . 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 13481	. . . . . . 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 6021	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → (𝑛 + 𝑥) = (𝑛 + 𝑎))
32		id 19	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → 𝑥 = 𝑎)
33	31, 32	eqeq12d 2244	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑎 → ((𝑛 + 𝑥) = 𝑥 ↔ (𝑛 + 𝑎) = 𝑎))
34		oveq2 6021	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑎 → (𝑖 + 𝑥) = (𝑖 + 𝑎))
35	34	eqeq1d 2238	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑎 → ((𝑖 + 𝑥) = 𝑛 ↔ (𝑖 + 𝑎) = 𝑛))
36	35	rexbidv 2531	. . . . . . . . . . 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 2904	. . . . . . . . 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 6020	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑏 → (𝑖 + 𝑎) = (𝑏 + 𝑎))
44	43	eqeq1d 2238	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑏 → ((𝑖 + 𝑎) = 𝑛 ↔ (𝑏 + 𝑎) = 𝑛))
45	44	cbvrexvw 2770	. . . . . . . . . . 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 13595	. . . . 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 2643	. . 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 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508  ∃wrex 2509  ‘cfv 5324  (class class class)co 6013  Basecbs 13072  +_gcplusg 13150  0_gc0g 13329  Mgmcmgm 13427  Smgrpcsgrp 13474  Mndcmnd 13489  Grpcgrp 13573

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-cnex 8113  ax-resscn 8114  ax-1re 8116  ax-addrcl 8119

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-iota 5284  df-fun 5326  df-fn 5327  df-fv 5332  df-riota 5966  df-ov 6016  df-inn 9134  df-2 9192  df-ndx 13075  df-slot 13076  df-base 13078  df-plusg 13163  df-0g 13331  df-mgm 13429  df-sgrp 13475  df-mnd 13490  df-grp 13576

This theorem is referenced by:  dfgrp2e  13601  dfgrp3m  13672