Theorem fvtransport 34617

Description: Calculate the value of the TransportTo function. This function takes four points, 𝐴 through 𝐷, where 𝐶 and 𝐷 are distinct. It then returns the point that extends 𝐶𝐷 by the length of 𝐴𝐵. (Contributed by Scott Fenton, 18-Oct-2013.) (Revised by Mario Carneiro, 19-Apr-2014.)

Assertion

Ref	Expression
fvtransport	⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (⟨𝐴, 𝐵⟩TransportTo⟨𝐶, 𝐷⟩) = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))

Distinct variable groups:   𝑁,𝑟   𝐴,𝑟   𝐵,𝑟   𝐶,𝑟   𝐷,𝑟

Proof of Theorem fvtransport

Dummy variables 𝑛 𝑝 𝑞 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-ov 7360	. 2 ⊢ (⟨𝐴, 𝐵⟩TransportTo⟨𝐶, 𝐷⟩) = (TransportTo‘⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩)
2		opelxpi 5670	. . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
3	2	3ad2ant1 1133	. . . . . 6 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
4		opelxpi 5670	. . . . . . 7 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
5	4	3ad2ant2 1134	. . . . . 6 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
6		simp3 1138	. . . . . . 7 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → 𝐶 ≠ 𝐷)
7		op1stg 7933	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
8	7	3ad2ant2 1134	. . . . . . 7 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
9		op2ndg 7934	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
10	9	3ad2ant2 1134	. . . . . . 7 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
11	6, 8, 10	3netr4d 3021	. . . . . 6 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩))
12	3, 5, 11	3jca 1128	. . . . 5 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)))
13	8	opeq1d 4836	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ = ⟨𝐶, 𝑟⟩)
14	10, 13	breq12d 5118	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → ((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ↔ 𝐷 Btwn ⟨𝐶, 𝑟⟩))
15	10	opeq1d 4836	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩ = ⟨𝐷, 𝑟⟩)
16	15	breq1d 5115	. . . . . . . 8 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → (⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩ ↔ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩))
17	14, 16	anbi12d 631	. . . . . . 7 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → (((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩) ↔ (𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
18	17	riotabidv 7315	. . . . . 6 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → (℩𝑟 ∈ (𝔼‘𝑁)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
19	18	eqcomd 2742	. . . . 5 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑁)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
20	12, 19	jca 512	. . . 4 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑁)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))))
21		fveq2 6842	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
22	21	sqxpeqd 5665	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑁) × (𝔼‘𝑁)))
23	22	eleq2d 2823	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
24	22	eleq2d 2823	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
25	23, 24	3anbi12d 1437	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ↔ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩))))
26	21	riotaeqdv 7314	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑁)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
27	26	eqeq2d 2747	. . . . . 6 ⊢ (𝑛 = 𝑁 → ((℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) ↔ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑁)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))))
28	25, 27	anbi12d 631	. . . . 5 ⊢ (𝑛 = 𝑁 → (((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))) ↔ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑁)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))))
29	28	rspcev 3581	. . . 4 ⊢ ((𝑁 ∈ ℕ ∧ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑁)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))) → ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))))
30	20, 29	sylan2 593	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))))
31		df-br 5106	. . . . 5 ⊢ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩TransportTo(℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) ↔ ⟨⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩, (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩))⟩ ∈ TransportTo)
32		df-transport 34615	. . . . . 6 ⊢ TransportTo = {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))}
33	32	eleq2i 2829	. . . . 5 ⊢ (⟨⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩, (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩))⟩ ∈ TransportTo ↔ ⟨⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩, (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩))⟩ ∈ {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))})
34		opex 5421	. . . . . 6 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
35		opex 5421	. . . . . 6 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
36		riotaex 7317	. . . . . 6 ⊢ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) ∈ V
37		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))))
38	37	3anbi1d 1440	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ↔ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞))))
39		breq2 5109	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝 ↔ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))
40	39	anbi2d 629	. . . . . . . . . . 11 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝) ↔ ((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
41	40	riotabidv 7315	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
42	41	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)) ↔ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))))
43	38, 42	anbi12d 631	. . . . . . . 8 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))))
44	43	rexbidv 3175	. . . . . . 7 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝))) ↔ ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))))
45		eleq1 2825	. . . . . . . . . 10 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))))
46		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (1st ‘𝑞) = (1st ‘⟨𝐶, 𝐷⟩))
47		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑞) = (2nd ‘⟨𝐶, 𝐷⟩))
48	46, 47	neeq12d 3005	. . . . . . . . . 10 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ((1st ‘𝑞) ≠ (2nd ‘𝑞) ↔ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)))
49	45, 48	3anbi23d 1439	. . . . . . . . 9 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ↔ (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩))))
50	46	opeq1d 4836	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ⟨(1st ‘𝑞), 𝑟⟩ = ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩)
51	47, 50	breq12d 5118	. . . . . . . . . . . 12 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ↔ (2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩))
52	47	opeq1d 4836	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ⟨(2nd ‘𝑞), 𝑟⟩ = ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩)
53	52	breq1d 5115	. . . . . . . . . . . 12 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩ ↔ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))
54	51, 53	anbi12d 631	. . . . . . . . . . 11 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩) ↔ ((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
55	54	riotabidv 7315	. . . . . . . . . 10 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
56	55	eqeq2d 2747	. . . . . . . . 9 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) ↔ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))))
57	49, 56	anbi12d 631	. . . . . . . 8 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))) ↔ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))))
58	57	rexbidv 3175	. . . . . . 7 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))) ↔ ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))))
59		eqeq1 2740	. . . . . . . . 9 ⊢ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) → (𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) ↔ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))))
60	59	anbi2d 629	. . . . . . . 8 ⊢ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) → (((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))) ↔ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))))
61	60	rexbidv 3175	. . . . . . 7 ⊢ (𝑥 = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) → (∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))) ↔ ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))))
62	44, 58, 61	eloprabg 7466	. . . . . 6 ⊢ ((⟨𝐴, 𝐵⟩ ∈ V ∧ ⟨𝐶, 𝐷⟩ ∈ V ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) ∈ V) → (⟨⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩, (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩))⟩ ∈ {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))} ↔ ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))))
63	34, 35, 36, 62	mp3an 1461	. . . . 5 ⊢ (⟨⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩, (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩))⟩ ∈ {⟨⟨𝑝, 𝑞⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ 𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘𝑞) ≠ (2nd ‘𝑞)) ∧ 𝑥 = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ∧ ⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝)))} ↔ ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))))
64	31, 33, 63	3bitri 296	. . . 4 ⊢ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩TransportTo(℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) ↔ ∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))))
65		funtransport 34616	. . . . 5 ⊢ Fun TransportTo
66		funbrfv 6893	. . . . 5 ⊢ (Fun TransportTo → (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩TransportTo(℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) → (TransportTo‘⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩) = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩))))
67	65, 66	ax-mp 5	. . . 4 ⊢ (⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩TransportTo(℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) → (TransportTo‘⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩) = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
68	64, 67	sylbir 234	. . 3 ⊢ (∃𝑛 ∈ ℕ ((⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ∧ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)) ∧ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) = (℩𝑟 ∈ (𝔼‘𝑛)((2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩ ∧ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))) → (TransportTo‘⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩) = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
69	30, 68	syl 17	. 2 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (TransportTo‘⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩) = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))
70	1, 69	eqtrid 2788	1 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷)) → (⟨𝐴, 𝐵⟩TransportTo⟨𝐶, 𝐷⟩) = (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ≠ wne 2943  ∃wrex 3073  Vcvv 3445  ⟨cop 4592   class class class wbr 5105   × cxp 5631  Fun wfun 6490  ‘cfv 6496  ℩crio 7312  (class class class)co 7357  {coprab 7358  1^st c1st 7919  2^nd c2nd 7920  ℕcn 12153  𝔼cee 27837   Btwn cbtwn 27838  Cgrccgr 27839  TransportToctransport 34614

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-z 12500  df-uz 12764  df-fz 13425  df-ee 27840  df-transport 34615

This theorem is referenced by:  transportcl  34618  transportprops  34619