Theorem archiabl 30134

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 2765	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2765	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
3		eqid 2765	. . . . 5 ⊢ (le‘𝑊) = (le‘𝑊)
4		eqid 2765	. . . . 5 ⊢ (lt‘𝑊) = (lt‘𝑊)
5		eqid 2765	. . . . 5 ⊢ (.g‘𝑊) = (.g‘𝑊)
6		simpll1 1269	. . . . 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 1273	. . . . 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 785	. . . . 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 787	. . . . 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 1167	. . . . . 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 1320	. . . . . 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 1168	. . . . . 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 4813	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((0g‘𝑊)(lt‘𝑊)𝑥 ↔ (0g‘𝑊)(lt‘𝑊)𝑦))
14		breq2 4813	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑣(le‘𝑊)𝑥 ↔ 𝑣(le‘𝑊)𝑦))
15	13, 14	imbi12d 335	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑦 → 𝑣(le‘𝑊)𝑦)))
16	15	rspcv 3457	. . . . . 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 66	. . . . 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 30129	. . . 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 710	. . 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		simpr 477	. . . 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))) → ∃𝑢 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥)))
21		breq2 4813	. . . . . 6 ⊢ (𝑢 = 𝑣 → ((0g‘𝑊)(lt‘𝑊)𝑢 ↔ (0g‘𝑊)(lt‘𝑊)𝑣))
22		breq1 4812	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (𝑢(le‘𝑊)𝑥 ↔ 𝑣(le‘𝑊)𝑥))
23	22	imbi2d 331	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥)))
24	23	ralbidv 3133	. . . . . 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)))
25	21, 24	anbi12d 624	. . . . 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))))
26	25	cbvrexv 3320	. . . 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)))
27	20, 26	sylib 209	. . 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)))
28	19, 27	r19.29a 3225	. 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)
29		simpl1 1242	. . 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)
30		simpl3 1246	. . 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)
31		eqid 2765	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
32		simpl2 1244	. . 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)
33		simpr 477	. . . . . . . . . 10 ⊢ (((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		ralnex 3139	. . . . . . . . . 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)))
35	33, 34	sylibr 225	. . . . . . . . 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)))
36		rexanali 3144	. . . . . . . . . . . 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))
37	36	imbi2i 327	. . . . . . . . . . 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)))
38		imnan 388	. . . . . . . . . . 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)))
39	37, 38	bitri 266	. . . . . . . . . 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)))
40	39	ralbii 3127	. . . . . . . . 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)))
41	35, 40	sylibr 225	. . . . . . . 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)))
42	22	notbid 309	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (¬ 𝑢(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑥))
43	42	anbi2d 622	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → (((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
44	43	rexbidv 3199	. . . . . . . . . 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)))
45	21, 44	imbi12d 335	. . . . . . . . 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))))
46	45	cbvralv 3319	. . . . . . . 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)))
47	41, 46	sylib 209	. . . . . . 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)))
48	47	r19.21bi 3079	. . . . . 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)))
49	14	notbid 309	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ 𝑣(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑦))
50	13, 49	anbi12d 624	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
51	50	cbvrexv 3320	. . . . . 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))
52	48, 51	syl6ib 242	. . . . 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)))
53	52	3impia 1145	. . . 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))
54		simp1l1 1365	. . . . . 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)
55		isogrp 30084	. . . . . . 7 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
56	55	simprbi 490	. . . . . 6 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
57		omndtos 30087	. . . . . 6 ⊢ (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
58	54, 56, 57	3syl 18	. . . . 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)
59		simp2 1167	. . . . 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))
60	1, 3, 4	tltnle 30044	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑦(lt‘𝑊)𝑣 ↔ ¬ 𝑣(le‘𝑊)𝑦))
61	60	bicomd 214	. . . . . . . . 9 ⊢ ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
62	61	3com23 1156	. . . . . . . 8 ⊢ ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
63	62	3expa 1147	. . . . . . 7 ⊢ (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
64	63	anbi2d 622	. . . . . 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)))
65	64	rexbidva 3196	. . . . 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)))
66	58, 59, 65	syl2anc 579	. . . 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)))
67	53, 66	mpbid 223	. . 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))
68	1, 2, 3, 4, 5, 29, 30, 31, 32, 67	archiabllem2 30133	. 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)
69	28, 68	pm2.61dan 847	1 ⊢ ((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652   ∈ wcel 2155  ∀wral 3055  ∃wrex 3056   class class class wbr 4809  ‘cfv 6068  Basecbs 16132  +_gcplusg 16216  lecple 16223  0_gc0g 16368  ltcplt 17209  Tosetctos 17301  Grpcgrp 17691  ._gcmg 17809  opp_gcoppg 18040  Abelcabl 18460  oMndcomnd 30079  oGrpcogrp 30080  Archicarchi 30113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-inf2 8753  ax-cnex 10245  ax-resscn 10246  ax-1cn 10247  ax-icn 10248  ax-addcl 10249  ax-addrcl 10250  ax-mulcl 10251  ax-mulrcl 10252  ax-mulcom 10253  ax-addass 10254  ax-mulass 10255  ax-distr 10256  ax-i2m1 10257  ax-1ne0 10258  ax-1rid 10259  ax-rnegex 10260  ax-rrecex 10261  ax-cnre 10262  ax-pre-lttri 10263  ax-pre-lttrn 10264  ax-pre-ltadd 10265  ax-pre-mulgt0 10266

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2062  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-om 7264  df-1st 7366  df-2nd 7367  df-tpos 7555  df-wrecs 7610  df-recs 7672  df-rdg 7710  df-er 7947  df-en 8161  df-dom 8162  df-sdom 8163  df-pnf 10330  df-mnf 10331  df-xr 10332  df-ltxr 10333  df-le 10334  df-sub 10522  df-neg 10523  df-nn 11275  df-2 11335  df-3 11336  df-4 11337  df-5 11338  df-6 11339  df-7 11340  df-8 11341  df-9 11342  df-n0 11539  df-z 11625  df-dec 11741  df-uz 11887  df-fz 12534  df-seq 13009  df-ndx 16135  df-slot 16136  df-base 16138  df-sets 16139  df-plusg 16229  df-ple 16236  df-0g 16370  df-proset 17196  df-poset 17214  df-plt 17226  df-toset 17302  df-mgm 17510  df-sgrp 17552  df-mnd 17563  df-grp 17694  df-minusg 17695  df-sbg 17696  df-mulg 17810  df-oppg 18041  df-cmn 18461  df-abl 18462  df-omnd 30081  df-ogrp 30082  df-inftm 30114  df-archi 30115

This theorem is referenced by: (None)