Theorem archiabl 33063

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 2726	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2726	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
3		eqid 2726	. . . . 5 ⊢ (le‘𝑊) = (le‘𝑊)
4		eqid 2726	. . . . 5 ⊢ (lt‘𝑊) = (lt‘𝑊)
5		eqid 2726	. . . . 5 ⊢ (.g‘𝑊) = (.g‘𝑊)
6		simpll1 1209	. . . . 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 1211	. . . . 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 767	. . . . 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 769	. . . . 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 1134	. . . . . 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 1236	. . . . . 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 1135	. . . . . 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 5157	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((0g‘𝑊)(lt‘𝑊)𝑥 ↔ (0g‘𝑊)(lt‘𝑊)𝑦))
14		breq2 5157	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑣(le‘𝑊)𝑥 ↔ 𝑣(le‘𝑊)𝑦))
15	13, 14	imbi12d 343	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑦 → 𝑣(le‘𝑊)𝑦)))
16	15	rspcv 3604	. . . . . 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 33058	. . . 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 717	. . 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 483	. . . 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 5157	. . . . . 6 ⊢ (𝑢 = 𝑣 → ((0g‘𝑊)(lt‘𝑊)𝑢 ↔ (0g‘𝑊)(lt‘𝑊)𝑣))
22		breq1 5156	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (𝑢(le‘𝑊)𝑥 ↔ 𝑣(le‘𝑊)𝑥))
23	22	imbi2d 339	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥)))
24	23	ralbidv 3168	. . . . . 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 630	. . . . 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	cbvrexvw 3226	. . . 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 217	. . 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 3152	. 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 1188	. . 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 1190	. . 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 2726	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
32		simpl2 1189	. . 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 483	. . . . . . . . . 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 3062	. . . . . . . . . 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 233	. . . . . . . . 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 3092	. . . . . . . . . . . 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 335	. . . . . . . . . . 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 398	. . . . . . . . . . 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 274	. . . . . . . . . 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 3083	. . . . . . . . 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 233	. . . . . . . 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 317	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (¬ 𝑢(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑥))
43	42	anbi2d 628	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → (((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
44	43	rexbidv 3169	. . . . . . . . . 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 343	. . . . . . . . 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	cbvralvw 3225	. . . . . . . 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 217	. . . . . . 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 3239	. . . . . 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 317	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ 𝑣(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑦))
50	13, 49	anbi12d 630	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
51	50	cbvrexvw 3226	. . . . . 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	imbitrdi 250	. . . . 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 1114	. . . 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 1263	. . . . . 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 32937	. . . . . . 7 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
56	55	simprbi 495	. . . . . 6 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
57		omndtos 32940	. . . . . 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 1134	. . . . 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 18447	. . . . . . . . . 10 ⊢ ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑦(lt‘𝑊)𝑣 ↔ ¬ 𝑣(le‘𝑊)𝑦))
61	60	bicomd 222	. . . . . . . . 9 ⊢ ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
62	61	3com23 1123	. . . . . . . 8 ⊢ ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
63	62	3expa 1115	. . . . . . 7 ⊢ (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
64	63	anbi2d 628	. . . . . 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 3167	. . . . 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 582	. . . 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 231	. . 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 33062	. 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 811	1 ⊢ ((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∧ w3a 1084   = wceq 1534   ∈ wcel 2099  ∀wral 3051  ∃wrex 3060   class class class wbr 5153  ‘cfv 6554  Basecbs 17213  +_gcplusg 17266  lecple 17273  0_gc0g 17454  ltcplt 18333  Tosetctos 18441  Grpcgrp 18928  ._gcmg 19061  opp_gcoppg 19339  Abelcabl 19779  oMndcomnd 32932  oGrpcogrp 32933  Archicarchi 33042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3967  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-tr 5271  df-id 5580  df-eprel 5586  df-po 5594  df-so 5595  df-fr 5637  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-ord 6379  df-on 6380  df-lim 6381  df-suc 6382  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-om 7877  df-1st 8003  df-2nd 8004  df-tpos 8241  df-frecs 8296  df-wrecs 8327  df-recs 8401  df-rdg 8440  df-er 8734  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12611  df-dec 12730  df-uz 12875  df-fz 13539  df-seq 14022  df-sets 17166  df-slot 17184  df-ndx 17196  df-base 17214  df-plusg 17279  df-ple 17286  df-0g 17456  df-proset 18320  df-poset 18338  df-plt 18355  df-toset 18442  df-mgm 18633  df-sgrp 18712  df-mnd 18728  df-grp 18931  df-minusg 18932  df-sbg 18933  df-mulg 19062  df-oppg 19340  df-cmn 19780  df-abl 19781  df-omnd 32934  df-ogrp 32935  df-inftm 33043  df-archi 33044

This theorem is referenced by: (None)