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Theorem fvtransport 35684
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 7418 . 2 (⟨𝐴, 𝐡⟩TransportTo⟨𝐢, 𝐷⟩) = (TransportToβ€˜βŸ¨βŸ¨π΄, 𝐡⟩, ⟨𝐢, 𝐷⟩⟩)
2 opelxpi 5709 . . . . . . 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 5709 . . . . . . 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 8001 . . . . . . . 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 8002 . . . . . . . 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 3008 . . . . . 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 4875 . . . . . . . . 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 5156 . . . . . . . 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 4875 . . . . . . . . 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 5153 . . . . . . . 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 7373 . . . . . 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 2731 . . . . 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 510 . . . 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 6891 . . . . . . . . 9 (𝑛 = 𝑁 β†’ (π”Όβ€˜π‘›) = (π”Όβ€˜π‘))
2221sqxpeqd 5704 . . . . . . . 8 (𝑛 = 𝑁 β†’ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) = ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘)))
2322eleq2d 2811 . . . . . . 7 (𝑛 = 𝑁 β†’ (⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘›) Γ— (π”Όβ€˜π‘›)) ↔ ⟨𝐴, 𝐡⟩ ∈ ((π”Όβ€˜π‘) Γ— (π”Όβ€˜π‘))))
2422eleq2d 2811 . . . . . . 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 7372 . . . . . . 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 2736 . . . . . 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 3602 . . . 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 591 . . 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 5144 . . . . 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 35682 . . . . . 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 2817 . . . . 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 5460 . . . . . 6 ⟨𝐴, 𝐡⟩ ∈ V
35 opex 5460 . . . . . 6 ⟨𝐢, 𝐷⟩ ∈ V
36 riotaex 7375 . . . . . 6 (β„©π‘Ÿ ∈ (π”Όβ€˜π‘)(𝐷 Btwn ⟨𝐢, π‘ŸβŸ© ∧ ⟨𝐷, π‘ŸβŸ©Cgr⟨𝐴, 𝐡⟩)) ∈ V
37 eleq1 2813 . . . . . . . . . 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 5147 . . . . . . . . . . . 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 7373 . . . . . . . . . 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 2736 . . . . . . . . 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 3169 . . . . . . 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 2813 . . . . . . . . . 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 6891 . . . . . . . . . . 11 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (1st β€˜π‘ž) = (1st β€˜βŸ¨πΆ, 𝐷⟩))
47 fveq2 6891 . . . . . . . . . . 11 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ (2nd β€˜π‘ž) = (2nd β€˜βŸ¨πΆ, 𝐷⟩))
4846, 47neeq12d 2992 . . . . . . . . . 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 4875 . . . . . . . . . . . . 13 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ ⟨(1st β€˜π‘ž), π‘ŸβŸ© = ⟨(1st β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©)
5147, 50breq12d 5156 . . . . . . . . . . . 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 4875 . . . . . . . . . . . . 13 (π‘ž = ⟨𝐢, 𝐷⟩ β†’ ⟨(2nd β€˜π‘ž), π‘ŸβŸ© = ⟨(2nd β€˜βŸ¨πΆ, 𝐷⟩), π‘ŸβŸ©)
5352breq1d 5153 . . . . . . . . . . . 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 7373 . . . . . . . . . 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 2736 . . . . . . . . 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 3169 . . . . . . 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 2729 . . . . . . . . 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 3169 . . . . . . 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 7526 . . . . . 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 296 . . . 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 35683 . . . . 5 Fun TransportTo
66 funbrfv 6942 . . . . 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 2777 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2930  βˆƒwrex 3060  Vcvv 3463  βŸ¨cop 4630   class class class wbr 5143   Γ— cxp 5670  Fun wfun 6536  β€˜cfv 6542  β„©crio 7370  (class class class)co 7415  {coprab 7416  1st c1st 7987  2nd c2nd 7988  β„•cn 12240  π”Όcee 28741   Btwn cbtwn 28742  Cgrccgr 28743  TransportToctransport 35681
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-sep 5294  ax-nul 5301  ax-pow 5359  ax-pr 5423  ax-un 7737  ax-cnex 11192  ax-resscn 11193  ax-1cn 11194  ax-icn 11195  ax-addcl 11196  ax-addrcl 11197  ax-mulcl 11198  ax-mulrcl 11199  ax-mulcom 11200  ax-addass 11201  ax-mulass 11202  ax-distr 11203  ax-i2m1 11204  ax-1ne0 11205  ax-1rid 11206  ax-rnegex 11207  ax-rrecex 11208  ax-cnre 11209  ax-pre-lttri 11210  ax-pre-lttrn 11211  ax-pre-ltadd 11212  ax-pre-mulgt0 11213
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-reu 3365  df-rab 3420  df-v 3465  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-pss 3960  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5227  df-tr 5261  df-id 5570  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  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 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-riota 7371  df-ov 7418  df-oprab 7419  df-mpo 7420  df-om 7868  df-1st 7989  df-2nd 7990  df-frecs 8283  df-wrecs 8314  df-recs 8388  df-rdg 8427  df-er 8721  df-map 8843  df-en 8961  df-dom 8962  df-sdom 8963  df-pnf 11278  df-mnf 11279  df-xr 11280  df-ltxr 11281  df-le 11282  df-sub 11474  df-neg 11475  df-nn 12241  df-z 12587  df-uz 12851  df-fz 13515  df-ee 28744  df-transport 35682
This theorem is referenced by:  transportcl  35685  transportprops  35686
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