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Theorem broutsideof3 35775
Description: Characterization of outsideness in terms of relationship to a fourth point. Theorem 6.3 of [Schwabhauser] p. 43. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
broutsideof3 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))))
Distinct variable groups:   𝑁,𝑐   𝐴,𝑐   𝐡,𝑐   𝑃,𝑐

Proof of Theorem broutsideof3
StepHypRef Expression
1 broutsideof2 35771 . 2 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))))
2 simpl 481 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
3 simpr3 1193 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
4 simpr1 1191 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
5 btwndiff 35676 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ 𝐡 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘)) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐))
62, 3, 4, 5syl3anc 1368 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐))
76adantr 479 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐))
8 df-3an 1086 . . . . . . . . . . . 12 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ↔ ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)))
9 3anass 1092 . . . . . . . . . . . 12 ((((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐) ↔ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ (𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐)))
10 simpr3 1193 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝑃 β‰  𝑐)
1110necomd 2986 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝑐 β‰  𝑃)
12 simp1 1133 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
13 simp23 1205 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
14 simp22 1204 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
15 simp21 1203 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
16 simp3 1135 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝑐 ∈ (π”Όβ€˜π‘))
17 simpr1r 1228 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)
1812, 14, 15, 13, 17btwncomand 35664 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝐴 Btwn ⟨𝐡, π‘ƒβŸ©)
19 simpr2 1192 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝑃 Btwn ⟨𝐡, π‘βŸ©)
2012, 13, 14, 15, 16, 18, 19btwnexch3and 35670 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝑃 Btwn ⟨𝐴, π‘βŸ©)
2111, 20, 193jca 1125 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))
228, 9, 21syl2anbr 597 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩) ∧ (𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐))) β†’ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))
2322expr 455 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ ((𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐) β†’ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
2423an32s 650 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ ((𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐) β†’ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
2524reximdva 3158 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ (βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑃 Btwn ⟨𝐡, π‘βŸ© ∧ 𝑃 β‰  𝑐) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
267, 25mpd 15 . . . . . . 7 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩)) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))
2726expr 455 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
28 simpr2 1192 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
29 btwndiff 35676 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘)) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐))
302, 28, 4, 29syl3anc 1368 . . . . . . . . 9 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐))
3130adantr 479 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐))
32 3anass 1092 . . . . . . . . . . . 12 ((((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐) ↔ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐)))
33 simpr3 1193 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝑃 β‰  𝑐)
3433necomd 2986 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝑐 β‰  𝑃)
35 simpr2 1192 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝑃 Btwn ⟨𝐴, π‘βŸ©)
36 simpr1r 1228 . . . . . . . . . . . . . . 15 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)
3712, 13, 15, 14, 36btwncomand 35664 . . . . . . . . . . . . . 14 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝐡 Btwn ⟨𝐴, π‘ƒβŸ©)
3812, 14, 13, 15, 16, 37, 35btwnexch3and 35670 . . . . . . . . . . . . 13 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ 𝑃 Btwn ⟨𝐡, π‘βŸ©)
3934, 35, 383jca 1125 . . . . . . . . . . . 12 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐)) β†’ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))
408, 32, 39syl2anbr 597 . . . . . . . . . . 11 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ∧ (𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐))) β†’ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))
4140expr 455 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) β†’ ((𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐) β†’ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
4241an32s 650 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ ((𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐) β†’ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
4342reximdva 3158 . . . . . . . 8 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) β†’ (βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 β‰  𝑐) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
4431, 43mpd 15 . . . . . . 7 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))
4544expr 455 . . . . . 6 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ (𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩ β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
4627, 45jaod 857 . . . . 5 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) β†’ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
47 simprr1 1218 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))) β†’ 𝑐 β‰  𝑃)
48 simpll 765 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
49 simplr1 1212 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
50 simplr2 1213 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
51 simpr 483 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝑐 ∈ (π”Όβ€˜π‘))
52 simprr2 1219 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))) β†’ 𝑃 Btwn ⟨𝐴, π‘βŸ©)
5348, 49, 50, 51, 52btwncomand 35664 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))) β†’ 𝑃 Btwn βŸ¨π‘, 𝐴⟩)
54 simplr3 1214 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
55 simprr3 1220 . . . . . . . . . 10 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))) β†’ 𝑃 Btwn ⟨𝐡, π‘βŸ©)
5648, 49, 54, 51, 55btwncomand 35664 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))) β†’ 𝑃 Btwn βŸ¨π‘, 𝐡⟩)
57 btwnconn2 35751 . . . . . . . . . . 11 ((𝑁 ∈ β„• ∧ (𝑐 ∈ (π”Όβ€˜π‘) ∧ 𝑃 ∈ (π”Όβ€˜π‘)) ∧ (𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝑐 β‰  𝑃 ∧ 𝑃 Btwn βŸ¨π‘, 𝐴⟩ ∧ 𝑃 Btwn βŸ¨π‘, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
5848, 51, 49, 50, 54, 57syl122anc 1376 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ ((𝑐 β‰  𝑃 ∧ 𝑃 Btwn βŸ¨π‘, 𝐴⟩ ∧ 𝑃 Btwn βŸ¨π‘, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
5958adantr 479 . . . . . . . . 9 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))) β†’ ((𝑐 β‰  𝑃 ∧ 𝑃 Btwn βŸ¨π‘, 𝐴⟩ ∧ 𝑃 Btwn βŸ¨π‘, 𝐡⟩) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
6047, 53, 56, 59mp3and 1460 . . . . . . . 8 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩))
6160expr 455 . . . . . . 7 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ ((𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
6261an32s 650 . . . . . 6 ((((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ ((𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
6362rexlimdva 3145 . . . . 5 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ (βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©) β†’ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
6446, 63impbid 211 . . . 4 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃)) β†’ ((𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩) ↔ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
6564pm5.32da 577 . . 3 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) ↔ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))))
66 df-3an 1086 . . 3 ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) ↔ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)))
67 df-3an 1086 . . 3 ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)) ↔ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃) ∧ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©)))
6865, 66, 673bitr4g 313 . 2 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ ((𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ (𝐴 Btwn βŸ¨π‘ƒ, 𝐡⟩ ∨ 𝐡 Btwn βŸ¨π‘ƒ, 𝐴⟩)) ↔ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))))
691, 68bitrd 278 1 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ (𝑃OutsideOf⟨𝐴, 𝐡⟩ ↔ (𝐴 β‰  𝑃 ∧ 𝐡 β‰  𝑃 ∧ βˆƒπ‘ ∈ (π”Όβ€˜π‘)(𝑐 β‰  𝑃 ∧ 𝑃 Btwn ⟨𝐴, π‘βŸ© ∧ 𝑃 Btwn ⟨𝐡, π‘βŸ©))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 845   ∧ w3a 1084   ∈ wcel 2098   β‰  wne 2930  βˆƒwrex 3060  βŸ¨cop 4631   class class class wbr 5144  β€˜cfv 6543  β„•cn 12237  π”Όcee 28738   Btwn cbtwn 28739  OutsideOfcoutsideof 35768
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5360  ax-pr 5424  ax-un 7735  ax-inf2 9659  ax-cnex 11189  ax-resscn 11190  ax-1cn 11191  ax-icn 11192  ax-addcl 11193  ax-addrcl 11194  ax-mulcl 11195  ax-mulrcl 11196  ax-mulcom 11197  ax-addass 11198  ax-mulass 11199  ax-distr 11200  ax-i2m1 11201  ax-1ne0 11202  ax-1rid 11203  ax-rnegex 11204  ax-rrecex 11205  ax-cnre 11206  ax-pre-lttri 11207  ax-pre-lttrn 11208  ax-pre-ltadd 11209  ax-pre-mulgt0 11210  ax-pre-sup 11211
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3465  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-pss 3961  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-int 4946  df-iun 4994  df-br 5145  df-opab 5207  df-mpt 5228  df-tr 5262  df-id 5571  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-se 5629  df-we 5630  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552  df-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7866  df-1st 7987  df-2nd 7988  df-frecs 8280  df-wrecs 8311  df-recs 8385  df-rdg 8424  df-1o 8480  df-er 8718  df-map 8840  df-en 8958  df-dom 8959  df-sdom 8960  df-fin 8961  df-sup 9460  df-oi 9528  df-card 9957  df-pnf 11275  df-mnf 11276  df-xr 11277  df-ltxr 11278  df-le 11279  df-sub 11471  df-neg 11472  df-div 11897  df-nn 12238  df-2 12300  df-3 12301  df-n0 12498  df-z 12584  df-uz 12848  df-rp 13002  df-ico 13357  df-icc 13358  df-fz 13512  df-fzo 13655  df-seq 13994  df-exp 14054  df-hash 14317  df-cj 15073  df-re 15074  df-im 15075  df-sqrt 15209  df-abs 15210  df-clim 15459  df-sum 15660  df-ee 28741  df-btwn 28742  df-cgr 28743  df-ofs 35632  df-colinear 35688  df-ifs 35689  df-cgr3 35690  df-fs 35691  df-outsideof 35769
This theorem is referenced by: (None)
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