Theorem archiabl 33826

Description: Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)

Assertion

Ref	Expression
archiabl	⊢ ((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)

Proof of Theorem archiabl

Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2763	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2763	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
3		eqid 2763	. . . . 5 ⊢ (le‘𝑊) = (le‘𝑊)
4		eqid 2763	. . . . 5 ⊢ (lt‘𝑊) = (lt‘𝑊)
5		eqid 2763	. . . . 5 ⊢ (.g‘𝑊) = (.g‘𝑊)
6		simpll1 1231	. . . . 5 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ oGrp)
7		simpll3 1233	. . . . 5 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Archi)
8		simplr 781	. . . . 5 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))) → 𝑣 ∈ (Base‘𝑊))
9		simprl 783	. . . . 5 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))) → (0g‘𝑊)(lt‘𝑊)𝑣)
10		simp2 1155	. . . . . 6 ⊢ (((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g‘𝑊)(lt‘𝑊)𝑦) → 𝑦 ∈ (Base‘𝑊))
11		simp1rr 1258	. . . . . 6 ⊢ (((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g‘𝑊)(lt‘𝑊)𝑦) → ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))
12		simp3 1156	. . . . . 6 ⊢ (((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g‘𝑊)(lt‘𝑊)𝑦) → (0g‘𝑊)(lt‘𝑊)𝑦)
13		breq2 5110	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((0g‘𝑊)(lt‘𝑊)𝑥 ↔ (0g‘𝑊)(lt‘𝑊)𝑦))
14		breq2 5110	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑣(le‘𝑊)𝑥 ↔ 𝑣(le‘𝑊)𝑦))
15	13, 14	imbi12d 347	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑦 → 𝑣(le‘𝑊)𝑦)))
16	15	rspcv 3576	. . . . . 6 ⊢ (𝑦 ∈ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥) → ((0g‘𝑊)(lt‘𝑊)𝑦 → 𝑣(le‘𝑊)𝑦)))
17	10, 11, 12, 16	syl3c 67	. . . . 5 ⊢ (((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g‘𝑊)(lt‘𝑊)𝑦) → 𝑣(le‘𝑊)𝑦)
18	1, 2, 3, 4, 5, 6, 7, 8, 9, 17	archiabllem1 33821	. . . 4 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
19	18	adantllr 732	. . 3 ⊢ (((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
20		breq2 5110	. . . . . 6 ⊢ (𝑢 = 𝑣 → ((0g‘𝑊)(lt‘𝑊)𝑢 ↔ (0g‘𝑊)(lt‘𝑊)𝑣))
21		breq1 5109	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (𝑢(le‘𝑊)𝑥 ↔ 𝑣(le‘𝑊)𝑥))
22	21	imbi2d 343	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥)))
23	22	ralbidv 3189	. . . . . 6 ⊢ (𝑢 = 𝑣 → (∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥) ↔ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥)))
24	20, 23	anbi12d 644	. . . . 5 ⊢ (𝑢 = 𝑣 → (((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)) ↔ ((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥))))
25	24	cbvrexvw 3245	. . . 4 ⊢ (∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)) ↔ ∃𝑣 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥)))
26	25	bilani 510	. . 3 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) → ∃𝑣 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥)))
27	19, 26	r19.29a 3174	. 2 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
28		simpl1 1210	. . 3 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ oGrp)
29		simpl3 1212	. . 3 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Archi)
30		eqid 2763	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
31		simpl2 1211	. . 3 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) → (oppg‘𝑊) ∈ oGrp)
32		ralnex 3092	. . . . . . . . . 10 ⊢ (∀𝑢 ∈ (Base‘𝑊) ¬ ((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)) ↔ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)))
33	32	bilanri 512	. . . . . . . . 9 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) → ∀𝑢 ∈ (Base‘𝑊) ¬ ((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)))
34		rexanali 3120	. . . . . . . . . . . 12 ⊢ (∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ¬ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))
35	34	imbi2i 339	. . . . . . . . . . 11 ⊢ (((0g‘𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ((0g‘𝑊)(lt‘𝑊)𝑢 → ¬ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)))
36		imnan 405	. . . . . . . . . . 11 ⊢ (((0g‘𝑊)(lt‘𝑊)𝑢 → ¬ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)) ↔ ¬ ((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)))
37	35, 36	bitri 278	. . . . . . . . . 10 ⊢ (((0g‘𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ¬ ((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)))
38	37	ralbii 3112	. . . . . . . . 9 ⊢ (∀𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ∀𝑢 ∈ (Base‘𝑊) ¬ ((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)))
39	33, 38	sylibr 237	. . . . . . . 8 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) → ∀𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)))
40	21	notbid 321	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (¬ 𝑢(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑥))
41	40	anbi2d 642	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → (((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
42	41	rexbidv 3190	. . . . . . . . . 10 ⊢ (𝑢 = 𝑣 → (∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
43	20, 42	imbi12d 347	. . . . . . . . 9 ⊢ (𝑢 = 𝑣 → (((0g‘𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ((0g‘𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥))))
44	43	cbvralvw 3244	. . . . . . . 8 ⊢ (∀𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ∀𝑣 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
45	39, 44	sylib 221	. . . . . . 7 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) → ∀𝑣 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
46	45	r19.21bi 3258	. . . . . 6 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) → ((0g‘𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
47	14	notbid 321	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ 𝑣(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑦))
48	13, 47	anbi12d 644	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
49	48	cbvrexvw 3245	. . . . . 6 ⊢ (∃𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦))
50	46, 49	imbitrdi 254	. . . . 5 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) → ((0g‘𝑊)(lt‘𝑊)𝑣 → ∃𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
51	50	3impia 1135	. . . 4 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g‘𝑊)(lt‘𝑊)𝑣) → ∃𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦))
52		simp1l1 1285	. . . . . 6 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g‘𝑊)(lt‘𝑊)𝑣) → 𝑊 ∈ oGrp)
53		isogrp 20365	. . . . . . 7 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
54	53	simprbi 503	. . . . . 6 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
55		omndtos 20368	. . . . . 6 ⊢ (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
56	52, 54, 55	3syl 19	. . . . 5 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g‘𝑊)(lt‘𝑊)𝑣) → 𝑊 ∈ Toset)
57		simp2 1155	. . . . 5 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g‘𝑊)(lt‘𝑊)𝑣) → 𝑣 ∈ (Base‘𝑊))
58	1, 3, 4	tltnle 18620	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑦(lt‘𝑊)𝑣 ↔ ¬ 𝑣(le‘𝑊)𝑦))
59	58	bicomd 226	. . . . . . . . 9 ⊢ ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
60	59	3com23 1144	. . . . . . . 8 ⊢ ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
61	60	3expa 1136	. . . . . . 7 ⊢ (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
62	61	anbi2d 642	. . . . . 6 ⊢ (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (((0g‘𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ((0g‘𝑊)(lt‘𝑊)𝑦 ∧ 𝑦(lt‘𝑊)𝑣)))
63	62	rexbidva 3188	. . . . 5 ⊢ ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) → (∃𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑦 ∧ 𝑦(lt‘𝑊)𝑣)))
64	56, 57, 63	syl2anc 596	. . . 4 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g‘𝑊)(lt‘𝑊)𝑣) → (∃𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑦 ∧ 𝑦(lt‘𝑊)𝑣)))
65	51, 64	mpbid 235	. . 3 ⊢ ((((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g‘𝑊)(lt‘𝑊)𝑣) → ∃𝑦 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑦 ∧ 𝑦(lt‘𝑊)𝑣))
66	1, 2, 3, 4, 5, 28, 29, 30, 31, 65	archiabllem2 33825	. 2 ⊢ (((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
67	27, 66	pm2.61dan 825	1 ⊢ ((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   class class class wbr 5106  ‘cfv 6555  Basecbs 17413  +_gcplusg 17454  lecple 17461  0_gc0g 17636  ltcplt 18508  Tosetctos 18614  Grpcgrp 19170  ._gcmg 19303  opp_gcoppg 19586  Abelcabl 20022  oMndcomnd 20360  oGrpcogrp 20361  Archicarchi 33805

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-tpos 8268  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-en 8999  df-dom 9000  df-sdom 9001  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-nn 12360  df-2 12429  df-3 12430  df-4 12431  df-5 12432  df-6 12433  df-7 12434  df-8 12435  df-9 12436  df-n0 12631  df-z 12718  df-dec 12839  df-uz 12990  df-fz 13664  df-seq 14169  df-sets 17368  df-slot 17386  df-ndx 17398  df-base 17414  df-plusg 17467  df-ple 17474  df-0g 17638  df-proset 18494  df-poset 18513  df-plt 18528  df-toset 18615  df-mgm 18842  df-sgrp 18934  df-mnd 18950  df-grp 19173  df-minusg 19174  df-sbg 19175  df-mulg 19304  df-oppg 19587  df-cmn 20023  df-abl 20024  df-omnd 20362  df-ogrp 20363  df-inftm 33806  df-archi 33807

This theorem is used by: (None)