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Theorem fvtransport 35537
Description: Calculate the value of the TransportTo function. This function takes four points, 𝐴 through 𝐷, where 𝐢 and 𝐷 are distinct. It then returns the point that extends 𝐢𝐷 by the length of 𝐴𝐡. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)
Assertion
Ref Expression
fvtransport ((𝑁 ∈ β„• ∧ ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷)) β†’ (⟨𝐴, 𝐡⟩TransportTo⟨𝐢, 𝐷⟩) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
Distinct variable groups:   𝑁,π‘Ÿ   𝐴,π‘Ÿ   𝐡,π‘Ÿ   𝐢,π‘Ÿ   𝐷,π‘Ÿ

Proof of Theorem fvtransport
Dummy variables 𝑛 𝑝 π‘ž π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ov 7408 . 2 (⟨𝐴, 𝐡⟩TransportTo⟨𝐢, 𝐷⟩) = (TransportToβ€˜βŸ¨βŸ¨π΄, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩)
2 opelxpi 5706 . . . . . . 7 ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) β†’ ⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)))
323ad2ant1 1130 . . . . . 6 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ ⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)))
4 opelxpi 5706 . . . . . . 7 ((𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) β†’ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)))
543ad2ant2 1131 . . . . . 6 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)))
6 simp3 1135 . . . . . . 7 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ 𝐢 β‰  𝐷)
7 op1stg 7986 . . . . . . . 8 ((𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) β†’ (1st β€˜βŸ¨πΆ, 𝐷⟩) = 𝐢)
873ad2ant2 1131 . . . . . . 7 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ (1st β€˜βŸ¨πΆ, 𝐷⟩) = 𝐢)
9 op2ndg 7987 . . . . . . . 8 ((𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) β†’ (2nd β€˜βŸ¨πΆ, 𝐷⟩) = 𝐷)
1093ad2ant2 1131 . . . . . . 7 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ (2nd β€˜βŸ¨πΆ, 𝐷⟩) = 𝐷)
116, 8, 103netr4d 3012 . . . . . 6 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩))
123, 5, 113jca 1125 . . . . 5 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ (⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)))
138opeq1d 4874 . . . . . . . . 9 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© = ⟨𝐢, π‘ŸβŸ©)
1410, 13breq12d 5154 . . . . . . . 8 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ ((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ↔ 𝐷 Btwn ⟨𝐢, π‘ŸβŸ©))
1510opeq1d 4874 . . . . . . . . 9 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© = ⟨𝐷, π‘ŸβŸ©)
1615breq1d 5151 . . . . . . . 8 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ (⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩ ↔ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))
1714, 16anbi12d 630 . . . . . . 7 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ (((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩) ↔ (𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
1817riotabidv 7363 . . . . . 6 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
1918eqcomd 2732 . . . . 5 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
2012, 19jca 511 . . . 4 (((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷) β†’ ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))))
21 fveq2 6885 . . . . . . . . 9 (𝑛 = 𝑁 β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
2221sqxpeqd 5701 . . . . . . . 8 (𝑛 = 𝑁 β†’ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) = ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)))
2322eleq2d 2813 . . . . . . 7 (𝑛 = 𝑁 β†’ (⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ↔ ⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘))))
2422eleq2d 2813 . . . . . . 7 (𝑛 = 𝑁 β†’ (⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ↔ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘))))
2523, 243anbi12d 1433 . . . . . 6 (𝑛 = 𝑁 β†’ ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ↔ (⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩))))
2621riotaeqdv 7362 . . . . . . 7 (𝑛 = 𝑁 β†’ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
2726eqeq2d 2737 . . . . . 6 (𝑛 = 𝑁 β†’ ((β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) ↔ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))))
2825, 27anbi12d 630 . . . . 5 (𝑛 = 𝑁 β†’ (((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))) ↔ ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))))
2928rspcev 3606 . . . 4 ((𝑁 ∈ β„• ∧ ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))) β†’ βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))))
3020, 29sylan2 592 . . 3 ((𝑁 ∈ β„• ∧ ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷)) β†’ βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))))
31 df-br 5142 . . . . 5 (⟨⟨𝐴, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩TransportTo(β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) ↔ ⟨⟨⟨𝐴, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩, (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))⟩ ∈ TransportTo)
32 df-transport 35535 . . . . . 6 TransportTo = {βŸ¨βŸ¨π‘, π‘žβŸ©, π‘₯⟩ ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr𝑝)))}
3332eleq2i 2819 . . . . 5 (⟨⟨⟨𝐴, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩, (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))⟩ ∈ TransportTo ↔ ⟨⟨⟨𝐴, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩, (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))⟩ ∈ {βŸ¨βŸ¨π‘, π‘žβŸ©, π‘₯⟩ ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr𝑝)))})
34 opex 5457 . . . . . 6 ⟨𝐴, 𝐡⟩ ∈ V
35 opex 5457 . . . . . 6 ⟨𝐢, 𝐷⟩ ∈ V
36 riotaex 7365 . . . . . 6 (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) ∈ V
37 eleq1 2815 . . . . . . . . . 10 (𝑝 = ⟨𝐴, 𝐡⟩ β†’ (𝑝 ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ↔ ⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›))))
38373anbi1d 1436 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐡⟩ β†’ ((𝑝 ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ↔ (⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž))))
39 breq2 5145 . . . . . . . . . . . 12 (𝑝 = ⟨𝐴, 𝐡⟩ β†’ (⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr𝑝 ↔ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))
4039anbi2d 628 . . . . . . . . . . 11 (𝑝 = ⟨𝐴, 𝐡⟩ β†’ (((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr𝑝) ↔ ((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
4140riotabidv 7363 . . . . . . . . . 10 (𝑝 = ⟨𝐴, 𝐡⟩ β†’ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr𝑝)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
4241eqeq2d 2737 . . . . . . . . 9 (𝑝 = ⟨𝐴, 𝐡⟩ β†’ (π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr𝑝)) ↔ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))))
4338, 42anbi12d 630 . . . . . . . 8 (𝑝 = ⟨𝐴, 𝐡⟩ β†’ (((𝑝 ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr𝑝))) ↔ ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))))
4443rexbidv 3172 . . . . . . 7 (𝑝 = ⟨𝐴, 𝐡⟩ β†’ (βˆƒπ‘› ∈ β„• ((𝑝 ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr𝑝))) ↔ βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))))
45 eleq1 2815 . . . . . . . . . 10 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ↔ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›))))
46 fveq2 6885 . . . . . . . . . . 11 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (1st β€˜π‘ž) = (1st β€˜βŸ¨πΆ, 𝐷⟩))
47 fveq2 6885 . . . . . . . . . . 11 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (2nd β€˜π‘ž) = (2nd β€˜βŸ¨πΆ, 𝐷⟩))
4846, 47neeq12d 2996 . . . . . . . . . 10 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ ((1st β€˜π‘ž) β‰  (2nd β€˜π‘ž) ↔ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)))
4945, 483anbi23d 1435 . . . . . . . . 9 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ↔ (⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩))))
5046opeq1d 4874 . . . . . . . . . . . . 13 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ ⟨(1st β€˜π‘ž), π‘ŸβŸ© = ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©)
5147, 50breq12d 5154 . . . . . . . . . . . 12 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ ((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ↔ (2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©))
5247opeq1d 4874 . . . . . . . . . . . . 13 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ ⟨(2nd β€˜π‘ž), π‘ŸβŸ© = ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©)
5352breq1d 5151 . . . . . . . . . . . 12 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩ ↔ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))
5451, 53anbi12d 630 . . . . . . . . . . 11 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩) ↔ ((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
5554riotabidv 7363 . . . . . . . . . 10 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
5655eqeq2d 2737 . . . . . . . . 9 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) ↔ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))))
5749, 56anbi12d 630 . . . . . . . 8 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))) ↔ ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))))
5857rexbidv 3172 . . . . . . 7 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))) ↔ βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))))
59 eqeq1 2730 . . . . . . . . 9 (π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) β†’ (π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) ↔ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))))
6059anbi2d 628 . . . . . . . 8 (π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) β†’ (((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))) ↔ ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))))
6160rexbidv 3172 . . . . . . 7 (π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) β†’ (βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))) ↔ βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))))
6244, 58, 61eloprabg 7514 . . . . . 6 ((⟨𝐴, 𝐡⟩ ∈ V ∧ ⟨𝐢, 𝐷⟩ ∈ V ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) ∈ V) β†’ (⟨⟨⟨𝐴, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩, (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))⟩ ∈ {βŸ¨βŸ¨π‘, π‘žβŸ©, π‘₯⟩ ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr𝑝)))} ↔ βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))))
6334, 35, 36, 62mp3an 1457 . . . . 5 (⟨⟨⟨𝐴, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩, (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))⟩ ∈ {βŸ¨βŸ¨π‘, π‘žβŸ©, π‘₯⟩ ∣ βˆƒπ‘› ∈ β„• ((𝑝 ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ π‘ž ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜π‘ž) β‰  (2nd β€˜π‘ž)) ∧ π‘₯ = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜π‘ž) Btwn ⟨(1st β€˜π‘ž), π‘ŸβŸ© ∧ ⟨(2nd β€˜π‘ž), π‘ŸβŸ©Cgr𝑝)))} ↔ βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))))
6431, 33, 633bitri 297 . . . 4 (⟨⟨𝐴, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩TransportTo(β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) ↔ βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))))
65 funtransport 35536 . . . . 5 Fun TransportTo
66 funbrfv 6936 . . . . 5 (Fun TransportTo β†’ (⟨⟨𝐴, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩TransportTo(β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) β†’ (TransportToβ€˜βŸ¨βŸ¨π΄, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))))
6765, 66ax-mp 5 . . . 4 (⟨⟨𝐴, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩TransportTo(β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) β†’ (TransportToβ€˜βŸ¨βŸ¨π΄, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
6864, 67sylbir 234 . . 3 (βˆƒπ‘› ∈ β„• ((⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ ⟨𝐢, 𝐷⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ∧ (1st β€˜βŸ¨πΆ, 𝐷⟩) β‰  (2nd β€˜βŸ¨πΆ, 𝐷⟩)) ∧ (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘›)((2nd β€˜βŸ¨πΆ, 𝐷⟩) Btwn ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ© ∧ ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩))) β†’ (TransportToβ€˜βŸ¨βŸ¨π΄, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
6930, 68syl 17 . 2 ((𝑁 ∈ β„• ∧ ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷)) β†’ (TransportToβ€˜βŸ¨βŸ¨π΄, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
701, 69eqtrid 2778 1 ((𝑁 ∈ β„• ∧ ((𝐴 ∈ (π”Όβ€˜π‘) ∧ 𝐡 ∈ (π”Όβ€˜π‘)) ∧ (𝐢 ∈ (π”Όβ€˜π‘) ∧ 𝐷 ∈ (π”Όβ€˜π‘)) ∧ 𝐢 β‰  𝐷)) β†’ (⟨𝐴, 𝐡⟩TransportTo⟨𝐢, 𝐷⟩) = (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2934  βˆƒwrex 3064  Vcvv 3468  βŸ¨cop 4629   class class class wbr 5141   Γ— cxp 5667  Fun wfun 6531  β€˜cfv 6537  β„©crio 7360  (class class class)co 7405  {coprab 7406  1st c1st 7972  2nd c2nd 7973  β„•cn 12216  π”Όcee 28654   Btwn cbtwn 28655  Cgrccgr 28656  TransportToctransport 35534
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7722  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-nel 3041  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-pred 6294  df-ord 6361  df-on 6362  df-lim 6363  df-suc 6364  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-om 7853  df-1st 7974  df-2nd 7975  df-frecs 8267  df-wrecs 8298  df-recs 8372  df-rdg 8411  df-er 8705  df-map 8824  df-en 8942  df-dom 8943  df-sdom 8944  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-z 12563  df-uz 12827  df-fz 13491  df-ee 28657  df-transport 35535
This theorem is referenced by:  transportcl  35538  transportprops  35539
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