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Theorem archiabl 31917
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 2736 . . . . 5 (Base‘𝑊) = (Base‘𝑊)
2 eqid 2736 . . . . 5 (0g𝑊) = (0g𝑊)
3 eqid 2736 . . . . 5 (le‘𝑊) = (le‘𝑊)
4 eqid 2736 . . . . 5 (lt‘𝑊) = (lt‘𝑊)
5 eqid 2736 . . . . 5 (.g𝑊) = (.g𝑊)
6 simpll1 1212 . . . . 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 1214 . . . . 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 767 . . . . 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 769 . . . . 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 1137 . . . . . 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 1239 . . . . . 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 1138 . . . . . 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 5107 . . . . . . . 8 (𝑥 = 𝑦 → ((0g𝑊)(lt‘𝑊)𝑥 ↔ (0g𝑊)(lt‘𝑊)𝑦))
14 breq2 5107 . . . . . . . 8 (𝑥 = 𝑦 → (𝑣(le‘𝑊)𝑥𝑣(le‘𝑊)𝑦))
1513, 14imbi12d 344 . . . . . . 7 (𝑥 = 𝑦 → (((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑦𝑣(le‘𝑊)𝑦)))
1615rspcv 3575 . . . . . 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 31912 . . . 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 717 . . 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 485 . . . 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 5107 . . . . . 6 (𝑢 = 𝑣 → ((0g𝑊)(lt‘𝑊)𝑢 ↔ (0g𝑊)(lt‘𝑊)𝑣))
22 breq1 5106 . . . . . . . 8 (𝑢 = 𝑣 → (𝑢(le‘𝑊)𝑥𝑣(le‘𝑊)𝑥))
2322imbi2d 340 . . . . . . 7 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑥𝑣(le‘𝑊)𝑥)))
2423ralbidv 3172 . . . . . 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 631 . . . . 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))))
2625cbvrexvw 3224 . . . 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 217 . . 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 3157 . 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 1191 . . 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 1193 . . 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 2736 . . 3 (+g𝑊) = (+g𝑊)
32 simpl2 1192 . . 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 485 . . . . . . . . . 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 3073 . . . . . . . . . 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 233 . . . . . . . . 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 3103 . . . . . . . . . . . 12 (∃𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ¬ ∀𝑥 ∈ (Base‘𝑊)((0g𝑊)(lt‘𝑊)𝑥𝑢(le‘𝑊)𝑥))
3736imbi2i 335 . . . . . . . . . . 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 400 . . . . . . . . . . 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 274 . . . . . . . . . 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 3094 . . . . . . . . 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 233 . . . . . . . 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 317 . . . . . . . . . . . 12 (𝑢 = 𝑣 → (¬ 𝑢(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑥))
4342anbi2d 629 . . . . . . . . . . 11 (𝑢 = 𝑣 → (((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑢(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥)))
4443rexbidv 3173 . . . . . . . . . 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 344 . . . . . . . . 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))))
4645cbvralvw 3223 . . . . . . . 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 217 . . . . . . 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 3232 . . . . . 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 317 . . . . . . . 8 (𝑥 = 𝑦 → (¬ 𝑣(le‘𝑊)𝑥 ↔ ¬ 𝑣(le‘𝑊)𝑦))
5013, 49anbi12d 631 . . . . . . 7 (𝑥 = 𝑦 → (((0g𝑊)(lt‘𝑊)𝑥 ∧ ¬ 𝑣(le‘𝑊)𝑥) ↔ ((0g𝑊)(lt‘𝑊)𝑦 ∧ ¬ 𝑣(le‘𝑊)𝑦)))
5150cbvrexvw 3224 . . . . . 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 250 . . . . 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 1117 . . . 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 1266 . . . . . 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 31793 . . . . . . 7 (𝑊 ∈ oGrp ↔ (𝑊 ∈ Grp ∧ 𝑊 ∈ oMnd))
5655simprbi 497 . . . . . 6 (𝑊 ∈ oGrp → 𝑊 ∈ oMnd)
57 omndtos 31796 . . . . . 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 1137 . . . . 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 18303 . . . . . . . . . 10 ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (𝑦(lt‘𝑊)𝑣 ↔ ¬ 𝑣(le‘𝑊)𝑦))
6160bicomd 222 . . . . . . . . 9 ((𝑊 ∈ Toset ∧ 𝑦 ∈ (Base‘𝑊) ∧ 𝑣 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
62613com23 1126 . . . . . . . 8 ((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
63623expa 1118 . . . . . . 7 (((𝑊 ∈ Toset ∧ 𝑣 ∈ (Base‘𝑊)) ∧ 𝑦 ∈ (Base‘𝑊)) → (¬ 𝑣(le‘𝑊)𝑦𝑦(lt‘𝑊)𝑣))
6463anbi2d 629 . . . . . 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 3171 . . . . 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 584 . . . 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 231 . . 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 31916 . 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 811 1 ((𝑊 ∈ oGrp ∧ (oppg𝑊) ∈ oGrp ∧ 𝑊 ∈ Archi) → 𝑊 ∈ Abel)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wcel 2106  wral 3062  wrex 3071   class class class wbr 5103  cfv 6493  Basecbs 17075  +gcplusg 17125  lecple 17132  0gc0g 17313  ltcplt 18189  Tosetctos 18297  Grpcgrp 18740  .gcmg 18863  oppgcoppg 19114  Abelcabl 19554  oMndcomnd 31788  oGrpcogrp 31789  Archicarchi 31896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pow 5318  ax-pr 5382  ax-un 7668  ax-cnex 11103  ax-resscn 11104  ax-1cn 11105  ax-icn 11106  ax-addcl 11107  ax-addrcl 11108  ax-mulcl 11109  ax-mulrcl 11110  ax-mulcom 11111  ax-addass 11112  ax-mulass 11113  ax-distr 11114  ax-i2m1 11115  ax-1ne0 11116  ax-1rid 11117  ax-rnegex 11118  ax-rrecex 11119  ax-cnre 11120  ax-pre-lttri 11121  ax-pre-lttrn 11122  ax-pre-ltadd 11123  ax-pre-mulgt0 11124
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-nel 3048  df-ral 3063  df-rex 3072  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4281  df-if 4485  df-pw 4560  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-iun 4954  df-br 5104  df-opab 5166  df-mpt 5187  df-tr 5221  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6251  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-riota 7309  df-ov 7356  df-oprab 7357  df-mpo 7358  df-om 7799  df-1st 7917  df-2nd 7918  df-tpos 8153  df-frecs 8208  df-wrecs 8239  df-recs 8313  df-rdg 8352  df-er 8644  df-en 8880  df-dom 8881  df-sdom 8882  df-pnf 11187  df-mnf 11188  df-xr 11189  df-ltxr 11190  df-le 11191  df-sub 11383  df-neg 11384  df-nn 12150  df-2 12212  df-3 12213  df-4 12214  df-5 12215  df-6 12216  df-7 12217  df-8 12218  df-9 12219  df-n0 12410  df-z 12496  df-dec 12615  df-uz 12760  df-fz 13417  df-seq 13899  df-sets 17028  df-slot 17046  df-ndx 17058  df-base 17076  df-plusg 17138  df-ple 17145  df-0g 17315  df-proset 18176  df-poset 18194  df-plt 18211  df-toset 18298  df-mgm 18489  df-sgrp 18538  df-mnd 18549  df-grp 18743  df-minusg 18744  df-sbg 18745  df-mulg 18864  df-oppg 19115  df-cmn 19555  df-abl 19556  df-omnd 31790  df-ogrp 31791  df-inftm 31897  df-archi 31898
This theorem is referenced by: (None)
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