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Theorem outsideofeu 35041
Description: Given a nondegenerate ray, there is a unique point congruent to the segment 𝐡𝐢 lying on the ray 𝐴𝑅. Theorem 6.11 of [Schwabhauser] p. 44. (Contributed by Scott Fenton, 23-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
outsideofeu ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ ((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) β†’ βˆƒ!π‘₯ ∈ (π”Όβ€˜π‘)(𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)))
Distinct variable groups:   π‘₯,𝐴   π‘₯,𝐡   π‘₯,𝐢   π‘₯,𝑁   π‘₯,𝑅

Proof of Theorem outsideofeu
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 segcon2 35015 . . . . 5 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ βˆƒπ‘₯ ∈ (π”Όβ€˜π‘)((𝑅 Btwn ⟨𝐴, π‘₯⟩ ∨ π‘₯ Btwn ⟨𝐴, π‘…βŸ©) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩))
21adantr 482 . . . 4 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢)) β†’ βˆƒπ‘₯ ∈ (π”Όβ€˜π‘)((𝑅 Btwn ⟨𝐴, π‘₯⟩ ∨ π‘₯ Btwn ⟨𝐴, π‘…βŸ©) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩))
3 simpl1 1192 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
4 simpl2l 1227 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
5 simpr 486 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ π‘₯ ∈ (π”Όβ€˜π‘))
6 simpl2r 1228 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ 𝑅 ∈ (π”Όβ€˜π‘))
7 broutsideof2 35032 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘))) β†’ (𝐴OutsideOf⟨π‘₯, π‘…βŸ© ↔ (π‘₯ β‰  𝐴 ∧ 𝑅 β‰  𝐴 ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))))
83, 4, 5, 6, 7syl13anc 1373 . . . . . . . . . . 11 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝐴OutsideOf⟨π‘₯, π‘…βŸ© ↔ (π‘₯ β‰  𝐴 ∧ 𝑅 β‰  𝐴 ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))))
98adantr 482 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ ((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)) β†’ (𝐴OutsideOf⟨π‘₯, π‘…βŸ© ↔ (π‘₯ β‰  𝐴 ∧ 𝑅 β‰  𝐴 ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))))
10 simp3 1139 . . . . . . . . . . 11 ((π‘₯ β‰  𝐴 ∧ 𝑅 β‰  𝐴 ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩)) β†’ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))
11 simpllr 775 . . . . . . . . . . . . . . . 16 ((((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩)) β†’ 𝐡 β‰  𝐢)
1211adantl 483 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))) β†’ 𝐡 β‰  𝐢)
13 simprlr 779 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))) β†’ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)
14 simp2l 1200 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
1514anim1i 616 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝐴 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘)))
16 simpl3 1194 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘)))
17 cgrdegen 34914 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ (⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩ β†’ (𝐴 = π‘₯ ↔ 𝐡 = 𝐢)))
183, 15, 16, 17syl3anc 1372 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ (⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩ β†’ (𝐴 = π‘₯ ↔ 𝐡 = 𝐢)))
1918adantr 482 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))) β†’ (⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩ β†’ (𝐴 = π‘₯ ↔ 𝐡 = 𝐢)))
2013, 19mpd 15 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))) β†’ (𝐴 = π‘₯ ↔ 𝐡 = 𝐢))
2120necon3bid 2986 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))) β†’ (𝐴 β‰  π‘₯ ↔ 𝐡 β‰  𝐢))
2212, 21mpbird 257 . . . . . . . . . . . . . 14 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))) β†’ 𝐴 β‰  π‘₯)
2322necomd 2997 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))) β†’ π‘₯ β‰  𝐴)
24 simplll 774 . . . . . . . . . . . . . 14 ((((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩)) β†’ 𝑅 β‰  𝐴)
2524adantl 483 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))) β†’ 𝑅 β‰  𝐴)
26 simprr 772 . . . . . . . . . . . . 13 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))) β†’ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))
2723, 25, 263jca 1129 . . . . . . . . . . . 12 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))) β†’ (π‘₯ β‰  𝐴 ∧ 𝑅 β‰  𝐴 ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩)))
2827expr 458 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ ((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)) β†’ ((π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩) β†’ (π‘₯ β‰  𝐴 ∧ 𝑅 β‰  𝐴 ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩))))
2910, 28impbid2 225 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ ((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)) β†’ ((π‘₯ β‰  𝐴 ∧ 𝑅 β‰  𝐴 ∧ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩)) ↔ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩)))
309, 29bitrd 279 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ ((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)) β†’ (𝐴OutsideOf⟨π‘₯, π‘…βŸ© ↔ (π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩)))
31 orcom 869 . . . . . . . . 9 ((π‘₯ Btwn ⟨𝐴, π‘…βŸ© ∨ 𝑅 Btwn ⟨𝐴, π‘₯⟩) ↔ (𝑅 Btwn ⟨𝐴, π‘₯⟩ ∨ π‘₯ Btwn ⟨𝐴, π‘…βŸ©))
3230, 31bitrdi 287 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ ((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)) β†’ (𝐴OutsideOf⟨π‘₯, π‘…βŸ© ↔ (𝑅 Btwn ⟨𝐴, π‘₯⟩ ∨ π‘₯ Btwn ⟨𝐴, π‘…βŸ©)))
3332expr 458 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢)) β†’ (⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩ β†’ (𝐴OutsideOf⟨π‘₯, π‘…βŸ© ↔ (𝑅 Btwn ⟨𝐴, π‘₯⟩ ∨ π‘₯ Btwn ⟨𝐴, π‘…βŸ©))))
3433pm5.32rd 579 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) ∧ (𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢)) β†’ ((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ↔ ((𝑅 Btwn ⟨𝐴, π‘₯⟩ ∨ π‘₯ Btwn ⟨𝐴, π‘…βŸ©) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)))
3534an32s 651 . . . . 5 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢)) ∧ π‘₯ ∈ (π”Όβ€˜π‘)) β†’ ((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ↔ ((𝑅 Btwn ⟨𝐴, π‘₯⟩ ∨ π‘₯ Btwn ⟨𝐴, π‘…βŸ©) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)))
3635rexbidva 3177 . . . 4 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢)) β†’ (βˆƒπ‘₯ ∈ (π”Όβ€˜π‘)(𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ↔ βˆƒπ‘₯ ∈ (π”Όβ€˜π‘)((𝑅 Btwn ⟨𝐴, π‘₯⟩ ∨ π‘₯ Btwn ⟨𝐴, π‘…βŸ©) ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)))
372, 36mpbird 257 . . 3 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢)) β†’ βˆƒπ‘₯ ∈ (π”Όβ€˜π‘)(𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩))
38 simpl1 1192 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
39 simpl2l 1227 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
40 simpl2r 1228 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ 𝑅 ∈ (π”Όβ€˜π‘))
41 simpl3l 1229 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
4239, 40, 413jca 1129 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)))
43 simpl3r 1230 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ 𝐢 ∈ (π”Όβ€˜π‘))
44 simprl 770 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ π‘₯ ∈ (π”Όβ€˜π‘))
45 simprr 772 . . . . . . . . 9 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ 𝑦 ∈ (π”Όβ€˜π‘))
4643, 44, 453jca 1129 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ (𝐢 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘)))
4738, 42, 463jca 1129 . . . . . . 7 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ (𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))))
48 simpr 486 . . . . . . 7 (((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ© ∧ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩))) β†’ ((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ© ∧ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩)))
49 outsideofeq 35040 . . . . . . . 8 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) β†’ (((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ© ∧ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩)) β†’ π‘₯ = 𝑦))
5049imp 408 . . . . . . 7 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ© ∧ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩))) β†’ π‘₯ = 𝑦)
5147, 48, 50syl2an 597 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘))) ∧ ((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) ∧ ((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ© ∧ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩)))) β†’ π‘₯ = 𝑦)
5251an4s 659 . . . . 5 ((((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢)) ∧ ((π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) ∧ ((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ© ∧ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩)))) β†’ π‘₯ = 𝑦)
5352exp32 422 . . . 4 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢)) β†’ ((π‘₯ ∈ (π”Όβ€˜π‘) ∧ 𝑦 ∈ (π”Όβ€˜π‘)) β†’ (((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ© ∧ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩)) β†’ π‘₯ = 𝑦)))
5453ralrimivv 3199 . . 3 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢)) β†’ βˆ€π‘₯ ∈ (π”Όβ€˜π‘)βˆ€π‘¦ ∈ (π”Όβ€˜π‘)(((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ© ∧ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩)) β†’ π‘₯ = 𝑦))
55 opeq1 4872 . . . . . 6 (π‘₯ = 𝑦 β†’ ⟨π‘₯, π‘…βŸ© = βŸ¨π‘¦, π‘…βŸ©)
5655breq2d 5159 . . . . 5 (π‘₯ = 𝑦 β†’ (𝐴OutsideOf⟨π‘₯, π‘…βŸ© ↔ 𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ©))
57 opeq2 4873 . . . . . 6 (π‘₯ = 𝑦 β†’ ⟨𝐴, π‘₯⟩ = ⟨𝐴, π‘¦βŸ©)
5857breq1d 5157 . . . . 5 (π‘₯ = 𝑦 β†’ (⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩ ↔ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩))
5956, 58anbi12d 632 . . . 4 (π‘₯ = 𝑦 β†’ ((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ↔ (𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ© ∧ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩)))
6059reu4 3726 . . 3 (βˆƒ!π‘₯ ∈ (π”Όβ€˜π‘)(𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ↔ (βˆƒπ‘₯ ∈ (π”Όβ€˜π‘)(𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ βˆ€π‘₯ ∈ (π”Όβ€˜π‘)βˆ€π‘¦ ∈ (π”Όβ€˜π‘)(((𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩) ∧ (𝐴OutsideOfβŸ¨π‘¦, π‘…βŸ© ∧ ⟨𝐴, π‘¦βŸ©Cgr⟨𝐡, 𝐢⟩)) β†’ π‘₯ = 𝑦)))
6137, 54, 60sylanbrc 584 . 2 (((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) ∧ (𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢)) β†’ βˆƒ!π‘₯ ∈ (π”Όβ€˜π‘)(𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩))
6261ex 414 1 ((𝑁 ∈ β„• ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑅 ∈ (π”Όβ€˜π‘)) ∧ (𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝐢 ∈ (π”Όβ€˜π‘))) β†’ ((𝑅 β‰  𝐴 ∧ 𝐡 β‰  𝐢) β†’ βˆƒ!π‘₯ ∈ (π”Όβ€˜π‘)(𝐴OutsideOf⟨π‘₯, π‘…βŸ© ∧ ⟨𝐴, π‘₯⟩Cgr⟨𝐡, 𝐢⟩)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2941  βˆ€wral 3062  βˆƒwrex 3071  βˆƒ!wreu 3375  βŸ¨cop 4633   class class class wbr 5147  β€˜cfv 6540  β„•cn 12208  π”Όcee 28126   Btwn cbtwn 28127  Cgrccgr 28128  OutsideOfcoutsideof 35029
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-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720  ax-inf2 9632  ax-cnex 11162  ax-resscn 11163  ax-1cn 11164  ax-icn 11165  ax-addcl 11166  ax-addrcl 11167  ax-mulcl 11168  ax-mulrcl 11169  ax-mulcom 11170  ax-addass 11171  ax-mulass 11172  ax-distr 11173  ax-i2m1 11174  ax-1ne0 11175  ax-1rid 11176  ax-rnegex 11177  ax-rrecex 11178  ax-cnre 11179  ax-pre-lttri 11180  ax-pre-lttrn 11181  ax-pre-ltadd 11182  ax-pre-mulgt0 11183  ax-pre-sup 11184
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 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6297  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-isom 6549  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7851  df-1st 7970  df-2nd 7971  df-frecs 8261  df-wrecs 8292  df-recs 8366  df-rdg 8405  df-1o 8461  df-er 8699  df-map 8818  df-en 8936  df-dom 8937  df-sdom 8938  df-fin 8939  df-sup 9433  df-oi 9501  df-card 9930  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-nn 12209  df-2 12271  df-3 12272  df-n0 12469  df-z 12555  df-uz 12819  df-rp 12971  df-ico 13326  df-icc 13327  df-fz 13481  df-fzo 13624  df-seq 13963  df-exp 14024  df-hash 14287  df-cj 15042  df-re 15043  df-im 15044  df-sqrt 15178  df-abs 15179  df-clim 15428  df-sum 15629  df-ee 28129  df-btwn 28130  df-cgr 28131  df-ofs 34893  df-colinear 34949  df-ifs 34950  df-cgr3 34951  df-fs 34952  df-outsideof 35030
This theorem is referenced by: (None)
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