Theorem archiabl 32344

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 2733	. . . . 5 ⊢ (Base‘𝑊) = (Base‘𝑊)
2		eqid 2733	. . . . 5 ⊢ (0g‘𝑊) = (0g‘𝑊)
3		eqid 2733	. . . . 5 ⊢ (le‘𝑊) = (le‘𝑊)
4		eqid 2733	. . . . 5 ⊢ (lt‘𝑊) = (lt‘𝑊)
5		eqid 2733	. . . . 5 ⊢ (.g‘𝑊) = (.g‘𝑊)
6		simpll1 1213	. . . . 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 1215	. . . . 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 768	. . . . 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 770	. . . . 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 1138	. . . . . 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 1240	. . . . . 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 1139	. . . . . 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 5153	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ((0g‘𝑊)(lt‘𝑊)𝑥 ↔ (0g‘𝑊)(lt‘𝑊)𝑦))
14		breq2 5153	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑣(le‘𝑊)𝑥 ↔ 𝑣(le‘𝑊)𝑦))
15	13, 14	imbi12d 345	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑦 → 𝑣(le‘𝑊)𝑦)))
16	15	rspcv 3609	. . . . . 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 32339	. . . 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 718	. . 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 486	. . . 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 5153	. . . . . 6 ⊢ (𝑢 = 𝑣 → ((0g‘𝑊)(lt‘𝑊)𝑢 ↔ (0g‘𝑊)(lt‘𝑊)𝑣))
22		breq1 5152	. . . . . . . 8 ⊢ (𝑢 = 𝑣 → (𝑢(le‘𝑊)𝑥 ↔ 𝑣(le‘𝑊)𝑥))
23	22	imbi2d 341	. . . . . . 7 ⊢ (𝑢 = 𝑣 → (((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑢(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑥 → 𝑣(le‘𝑊)𝑥)))
24	23	ralbidv 3178	. . . . . 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 632	. . . . 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 3236	. . . 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 3163	. 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 1192	. . 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 1194	. . 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 2733	. . 3 ⊢ (+g‘𝑊) = (+g‘𝑊)
32		simpl2 1193	. . 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 486	. . . . . . . . . 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 3073	. . . . . . . . . 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 3103	. . . . . . . . . . . 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 336	. . . . . . . . . . 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 401	. . . . . . . . . . 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 275	. . . . . . . . . 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 3094	. . . . . . . . 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 318	. . . . . . . . . . . 12 ⊢ (𝑢 = 𝑣 → (¬ 𝑢(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑥))
43	42	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑢 = 𝑣 → (((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
44	43	rexbidv 3179	. . . . . . . . . 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 345	. . . . . . . . 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 3235	. . . . . . . 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 3249	. . . . . 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 318	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ 𝑣(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑦))
50	13, 49	anbi12d 632	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (((0g‘𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ((0g‘𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
51	50	cbvrexvw 3236	. . . . . 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 1118	. . . 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 1267	. . . . . 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 32220	. . . . . . 7 ⊢ (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
56	55	simprbi 498	. . . . . 6 ⊢ (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
57		omndtos 32223	. . . . . 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 1138	. . . . 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 18375	. . . . . . . . . 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 1127	. . . . . . . 8 ⊢ ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
63	62	3expa 1119	. . . . . . 7 ⊢ (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦 ↔ 𝑦(lt‘𝑊)𝑣))
64	63	anbi2d 630	. . . . . 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 3177	. . . . 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 585	. . . 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 32343	. 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 812	1 ⊢ ((𝑊 ∈ oGrp ∧ (oppg‘𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  ∀wral 3062  ∃wrex 3071   class class class wbr 5149  ‘cfv 6544  Basecbs 17144  +_gcplusg 17197  lecple 17204  0_gc0g 17385  ltcplt 18261  Tosetctos 18369  Grpcgrp 18819  ._gcmg 18950  opp_gcoppg 19209  Abelcabl 19649  oMndcomnd 32215  oGrpcogrp 32216  Archicarchi 32323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725  ax-cnex 11166  ax-resscn 11167  ax-1cn 11168  ax-icn 11169  ax-addcl 11170  ax-addrcl 11171  ax-mulcl 11172  ax-mulrcl 11173  ax-mulcom 11174  ax-addass 11175  ax-mulass 11176  ax-distr 11177  ax-i2m1 11178  ax-1ne0 11179  ax-1rid 11180  ax-rnegex 11181  ax-rrecex 11182  ax-cnre 11183  ax-pre-lttri 11184  ax-pre-lttrn 11185  ax-pre-ltadd 11186  ax-pre-mulgt0 11187

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-riota 7365  df-ov 7412  df-oprab 7413  df-mpo 7414  df-om 7856  df-1st 7975  df-2nd 7976  df-tpos 8211  df-frecs 8266  df-wrecs 8297  df-recs 8371  df-rdg 8410  df-er 8703  df-en 8940  df-dom 8941  df-sdom 8942  df-pnf 11250  df-mnf 11251  df-xr 11252  df-ltxr 11253  df-le 11254  df-sub 11446  df-neg 11447  df-nn 12213  df-2 12275  df-3 12276  df-4 12277  df-5 12278  df-6 12279  df-7 12280  df-8 12281  df-9 12282  df-n0 12473  df-z 12559  df-dec 12678  df-uz 12823  df-fz 13485  df-seq 13967  df-sets 17097  df-slot 17115  df-ndx 17127  df-base 17145  df-plusg 17210  df-ple 17217  df-0g 17387  df-proset 18248  df-poset 18266  df-plt 18283  df-toset 18370  df-mgm 18561  df-sgrp 18610  df-mnd 18626  df-grp 18822  df-minusg 18823  df-sbg 18824  df-mulg 18951  df-oppg 19210  df-cmn 19650  df-abl 19651  df-omnd 32217  df-ogrp 32218  df-inftm 32324  df-archi 32325

This theorem is referenced by: (None)