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Theorem archiabl 29880
Description: Archimedean left- and right- ordered groups are Abelian. (Contributed by Thierry Arnoux, 1-May-2018.)
Assertion
Ref Expression
archiabl ((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)

Proof of Theorem archiabl
Dummy variables 𝑣 𝑢 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2651 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2651 . . . . 5 (0g𝑊) = (0g𝑊)
3 eqid 2651 . . . . 5 (le‘𝑊) = (le‘𝑊)
4 eqid 2651 . . . . 5 (lt‘𝑊) = (lt‘𝑊)
5 eqid 2651 . . . . 5 (.g𝑊) = (.g𝑊)
6 simpll1 1120 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ oGrp)
7 simpll3 1122 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Archi)
8 simplr 807 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑣 ∈ (Base‘𝑊))
9 simprl 809 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → (0g𝑊)(lt‘𝑊)𝑣)
10 simp2 1082 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → 𝑦 ∈ (Base‘𝑊))
11 simp1rr 1147 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))
12 simp3 1083 . . . . . 6 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → (0g𝑊)(lt‘𝑊)𝑦)
13 breq2 4689 . . . . . . . 8 (𝑥 = 𝑦 → ((0g𝑊)(lt‘𝑊)𝑥 ↔ (0g𝑊)(lt‘𝑊)𝑦))
14 breq2 4689 . . . . . . . 8 (𝑥 = 𝑦 → (𝑣(le‘𝑊)𝑥𝑣(le‘𝑊)𝑦))
1513, 14imbi12d 333 . . . . . . 7 (𝑥 = 𝑦 → (((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑦𝑣(le‘𝑊)𝑦)))
1615rspcv 3336 . . . . . 6 (𝑦 ∈ (Base‘𝑊) → (∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥) → ((0g𝑊)(lt‘𝑊)𝑦𝑣(le‘𝑊)𝑦)))
1710, 11, 12, 16syl3c 66 . . . . 5 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) ∧ 𝑦 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑦) → 𝑣(le‘𝑊)𝑦)
181, 2, 3, 4, 5, 6, 7, 8, 9, 17archiabllem1 29875 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
1918adantllr 755 . . 3 (((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) ∧ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
20 simpr 476 . . . 4 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
21 breq2 4689 . . . . . 6 (𝑢 = 𝑣 → ((0g𝑊)(lt‘𝑊)𝑢 ↔ (0g𝑊)(lt‘𝑊)𝑣))
22 breq1 4688 . . . . . . . 8 (𝑢 = 𝑣 → (𝑢(le‘𝑊)𝑥𝑣(le‘𝑊)𝑥))
2322imbi2d 329 . . . . . . 7 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2423ralbidv 3015 . . . . . 6 (𝑢 = 𝑣 → (∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥) ↔ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2521, 24anbi12d 747 . . . . 5 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥))))
2625cbvrexv 3202 . . . 4 (∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ∃𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2720, 26sylib 208 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∃𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2819, 27r19.29a 3107 . 2 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
29 simpl1 1084 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ oGrp)
30 simpl3 1086 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Archi)
31 eqid 2651 . . 3 (+g𝑊) = (+g𝑊)
32 simpl2 1085 . . 3 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → (oppg𝑊) ∈ oGrp)
33 simpr 476 . . . . . . . . . 10 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
34 ralnex 3021 . . . . . . . . . 10 (∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
3533, 34sylibr 224 . . . . . . . . 9 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
36 rexanali 3027 . . . . . . . . . . . 12 (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))
3736imbi2i 325 . . . . . . . . . . 11 (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑢 → ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
38 imnan 437 . . . . . . . . . . 11 (((0g𝑊)(lt‘𝑊)𝑢 → ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)) ↔ ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
3937, 38bitri 264 . . . . . . . . . 10 (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
4039ralbii 3009 . . . . . . . . 9 (∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ∀𝑢 ∈ (Base‘𝑊) ¬ ((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥)))
4135, 40sylibr 224 . . . . . . . 8 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)))
4222notbid 307 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (¬ 𝑢(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑥))
4342anbi2d 740 . . . . . . . . . . 11 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4443rexbidv 3081 . . . . . . . . . 10 (𝑢 = 𝑣 → (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4521, 44imbi12d 333 . . . . . . . . 9 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥))))
4645cbvralv 3201 . . . . . . . 8 (∀𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥)) ↔ ∀𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4741, 46sylib 208 . . . . . . 7 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → ∀𝑣 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4847r19.21bi 2961 . . . . . 6 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) → ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4914notbid 307 . . . . . . . 8 (𝑥 = 𝑦 → (¬ 𝑣(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑦))
5013, 49anbi12d 747 . . . . . . 7 (𝑥 = 𝑦 → (((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
5150cbvrexv 3202 . . . . . 6 (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦))
5248, 51syl6ib 241 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊)) → ((0g𝑊)(lt‘𝑊)𝑣 → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
53523impia 1280 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦))
54 simp1l1 1174 . . . . . 6 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑊 ∈ oGrp)
55 isogrp 29830 . . . . . . 7 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
5655simprbi 479 . . . . . 6 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
57 omndtos 29833 . . . . . 6 (𝑊 ∈ oMnd → 𝑊 ∈ Toset)
5854, 56, 573syl 18 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑊 ∈ Toset)
59 simp2 1082 . . . . 5 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → 𝑣 ∈ (Base‘𝑊))
601, 3, 4tltnle 29790 . . . . . . . . . 10 ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑦(lt‘𝑊)𝑣 ↔ ¬ 𝑣(le‘𝑊)𝑦))
6160bicomd 213 . . . . . . . . 9 ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
62613com23 1291 . . . . . . . 8 ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
63623expa 1284 . . . . . . 7 (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
6463anbi2d 740 . . . . . 6 (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6564rexbidva 3078 . . . . 5 ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) → (∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6658, 59, 65syl2anc 694 . . . 4 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → (∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦) ↔ ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣)))
6753, 66mpbid 222 . . 3 ((((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) ∧ 𝑣 ∈ (Base‘𝑊) ∧ (0g𝑊)(lt‘𝑊)𝑣) → ∃𝑦 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
681, 2, 3, 4, 5, 29, 30, 31, 32, 67archiabllem2 29879 . 2 (((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) ∧ ¬ ∃𝑢 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑢 ∧ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))) → 𝑊 ∈ Abel)
6928, 68pm2.61dan 849 1 ((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942   class class class wbr 4685  cfv 5926  Basecbs 15904  +gcplusg 15988  lecple 15995  0gc0g 16147  ltcplt 16988  Tosetctos 17080  Grpcgrp 17469  .gcmg 17587  oppgcoppg 17821  Abelcabl 18240  oMndcomnd 29825  oGrpcogrp 29826  Archicarchi 29859
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-inf2 8576  ax-cnex 10030  ax-resscn 10031  ax-1cn 10032  ax-icn 10033  ax-addcl 10034  ax-addrcl 10035  ax-mulcl 10036  ax-mulrcl 10037  ax-mulcom 10038  ax-addass 10039  ax-mulass 10040  ax-distr 10041  ax-i2m1 10042  ax-1ne0 10043  ax-1rid 10044  ax-rnegex 10045  ax-rrecex 10046  ax-cnre 10047  ax-pre-lttri 10048  ax-pre-lttrn 10049  ax-pre-ltadd 10050  ax-pre-mulgt0 10051
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-1st 7210  df-2nd 7211  df-tpos 7397  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-er 7787  df-en 7998  df-dom 7999  df-sdom 8000  df-pnf 10114  df-mnf 10115  df-xr 10116  df-ltxr 10117  df-le 10118  df-sub 10306  df-neg 10307  df-nn 11059  df-2 11117  df-3 11118  df-4 11119  df-5 11120  df-6 11121  df-7 11122  df-8 11123  df-9 11124  df-n0 11331  df-z 11416  df-dec 11532  df-uz 11726  df-fz 12365  df-seq 12842  df-ndx 15907  df-slot 15908  df-base 15910  df-sets 15911  df-plusg 16001  df-ple 16008  df-0g 16149  df-preset 16975  df-poset 16993  df-plt 17005  df-toset 17081  df-mgm 17289  df-sgrp 17331  df-mnd 17342  df-grp 17472  df-minusg 17473  df-sbg 17474  df-mulg 17588  df-oppg 17822  df-cmn 18241  df-abl 18242  df-omnd 29827  df-ogrp 29828  df-inftm 29860  df-archi 29861
This theorem is referenced by: (None)
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