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Theorem broutsideof3 35645
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 35641 . 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 482 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑁 ∈ β„•)
3 simpr3 1194 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐡 ∈ (π”Όβ€˜π‘))
4 simpr1 1192 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝑃 ∈ (π”Όβ€˜π‘))
5 btwndiff 35546 . . . . . . . . . 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 1369 . . . . . . . . 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 480 . . . . . . . 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 1087 . . . . . . . . . . . 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 1093 . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . 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 2991 . . . . . . . . . . . . 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 1134 . . . . . . . . . . . . . 14 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
13 simp23 1206 . . . . . . . . . . . . . 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 1205 . . . . . . . . . . . . . 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 1204 . . . . . . . . . . . . . 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 1136 . . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . 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 35534 . . . . . . . . . . . . . 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 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 Btwn ⟨𝐡, π‘βŸ©)
2012, 13, 14, 15, 16, 18, 19btwnexch3and 35540 . . . . . . . . . . . . 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 1126 . . . . . . . . . . . 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 598 . . . . . . . . . . 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 456 . . . . . . . . . 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 651 . . . . . . . . 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 3163 . . . . . . . 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 456 . . . . . 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 1193 . . . . . . . . . 10 ((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) β†’ 𝐴 ∈ (π”Όβ€˜π‘))
29 btwndiff 35546 . . . . . . . . . 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 1369 . . . . . . . . 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 480 . . . . . . . 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 1093 . . . . . . . . . . . 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 1194 . . . . . . . . . . . . . 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 2991 . . . . . . . . . . . . 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 1193 . . . . . . . . . . . . 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 1229 . . . . . . . . . . . . . . 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 35534 . . . . . . . . . . . . . 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 35540 . . . . . . . . . . . . 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 1126 . . . . . . . . . . . 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 598 . . . . . . . . . . 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 456 . . . . . . . . . 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 651 . . . . . . . . 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 3163 . . . . . . . 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 456 . . . . . 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 858 . . . . 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 1219 . . . . . . . . 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 766 . . . . . . . . . 10 (((𝑁 ∈ β„• ∧ (𝑃 ∈ (π”Όβ€˜π‘) ∧ 𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘))) ∧ 𝑐 ∈ (π”Όβ€˜π‘)) β†’ 𝑁 ∈ β„•)
49 simplr1 1213 . . . . . . . . . 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 1214 . . . . . . . . . 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 484 . . . . . . . . . 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 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 ⟨𝐴, π‘βŸ©)
5348, 49, 50, 51, 52btwncomand 35534 . . . . . . . . 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 1215 . . . . . . . . . 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 1221 . . . . . . . . . 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 35534 . . . . . . . . 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 35621 . . . . . . . . . . 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 1377 . . . . . . . . . 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 480 . . . . . . . . 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 1461 . . . . . . . 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 456 . . . . . . 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 651 . . . . . 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 3150 . . . . 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 578 . . 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 1087 . . 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 1087 . . 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 314 . 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 279 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 395   ∨ wo 846   ∧ w3a 1085   ∈ wcel 2099   β‰  wne 2935  βˆƒwrex 3065  βŸ¨cop 4630   class class class wbr 5142  β€˜cfv 6542  β„•cn 12228  π”Όcee 28673   Btwn cbtwn 28674  OutsideOfcoutsideof 35638
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7732  ax-inf2 9650  ax-cnex 11180  ax-resscn 11181  ax-1cn 11182  ax-icn 11183  ax-addcl 11184  ax-addrcl 11185  ax-mulcl 11186  ax-mulrcl 11187  ax-mulcom 11188  ax-addass 11189  ax-mulass 11190  ax-distr 11191  ax-i2m1 11192  ax-1ne0 11193  ax-1rid 11194  ax-rnegex 11195  ax-rrecex 11196  ax-cnre 11197  ax-pre-lttri 11198  ax-pre-lttrn 11199  ax-pre-ltadd 11200  ax-pre-mulgt0 11201  ax-pre-sup 11202
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-nel 3042  df-ral 3057  df-rex 3066  df-rmo 3371  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-int 4945  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-tr 5260  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-se 5628  df-we 5629  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7370  df-ov 7417  df-oprab 7418  df-mpo 7419  df-om 7863  df-1st 7985  df-2nd 7986  df-frecs 8278  df-wrecs 8309  df-recs 8383  df-rdg 8422  df-1o 8478  df-er 8716  df-map 8836  df-en 8954  df-dom 8955  df-sdom 8956  df-fin 8957  df-sup 9451  df-oi 9519  df-card 9948  df-pnf 11266  df-mnf 11267  df-xr 11268  df-ltxr 11269  df-le 11270  df-sub 11462  df-neg 11463  df-div 11888  df-nn 12229  df-2 12291  df-3 12292  df-n0 12489  df-z 12575  df-uz 12839  df-rp 12993  df-ico 13348  df-icc 13349  df-fz 13503  df-fzo 13646  df-seq 13985  df-exp 14045  df-hash 14308  df-cj 15064  df-re 15065  df-im 15066  df-sqrt 15200  df-abs 15201  df-clim 15450  df-sum 15651  df-ee 28676  df-btwn 28677  df-cgr 28678  df-ofs 35502  df-colinear 35558  df-ifs 35559  df-cgr3 35560  df-fs 35561  df-outsideof 35639
This theorem is referenced by: (None)
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