Theorem fvtransport 32139

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 6653	. 2 ⊢ (⟨𝐴, 𝐵⟩TransportTo⟨𝐶, 𝐷⟩) = (TransportTo‘⟨⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩⟩)
2		opelxpi 5148	. . . . . . 7 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) → ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
3	2	3ad2ant1 1082	. . . . . 6 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
4		opelxpi 5148	. . . . . . 7 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
5	4	3ad2ant2 1083	. . . . . 6 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁)))
6		simp3 1063	. . . . . . 7 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → 𝐶 ≠ 𝐷)
7		op1stg 7180	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
8	7	3ad2ant2 1083	. . . . . . 7 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → (1st ‘⟨𝐶, 𝐷⟩) = 𝐶)
9		op2ndg 7181	. . . . . . . 8 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
10	9	3ad2ant2 1083	. . . . . . 7 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → (2nd ‘⟨𝐶, 𝐷⟩) = 𝐷)
11	6, 8, 10	3netr4d 2871	. . . . . 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 1242	. . . . 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 4408	. . . . . . . . 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 4666	. . . . . . . 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 4408	. . . . . . . . 9 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝐶 ≠ 𝐷) → ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩ = ⟨𝐷, 𝑟⟩)
16	15	breq1d 4663	. . . . . . . 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 747	. . . . . . 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 6613	. . . . . 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 2628	. . . . 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 554	. . . 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 6191	. . . . . . . . 9 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
22	21	sqxpeqd 5141	. . . . . . . 8 ⊢ (𝑛 = 𝑁 → ((𝔼‘𝑛) × (𝔼‘𝑛)) = ((𝔼‘𝑁) × (𝔼‘𝑁)))
23	22	eleq2d 2687	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
24	22	eleq2d 2687	. . . . . . 7 ⊢ (𝑛 = 𝑁 → (⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑁) × (𝔼‘𝑁))))
25	23, 24	3anbi12d 1400	. . . . . 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 6612	. . . . . . 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 2632	. . . . . 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 747	. . . . 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 3309	. . . 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 491	. . 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 4654	. . . . 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 32137	. . . . . 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 2693	. . . . 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 4932	. . . . . 6 ⊢ ⟨𝐴, 𝐵⟩ ∈ V
35		opex 4932	. . . . . 6 ⊢ ⟨𝐶, 𝐷⟩ ∈ V
36		riotaex 6615	. . . . . 6 ⊢ (℩𝑟 ∈ (𝔼‘𝑁)(𝐷 Btwn ⟨𝐶, 𝑟⟩ ∧ ⟨𝐷, 𝑟⟩Cgr⟨𝐴, 𝐵⟩)) ∈ V
37		eleq1 2689	. . . . . . . . . 10 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (𝑝 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐴, 𝐵⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))))
38	37	3anbi1d 1403	. . . . . . . . 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 4657	. . . . . . . . . . . 12 ⊢ (𝑝 = ⟨𝐴, 𝐵⟩ → (⟨(2nd ‘𝑞), 𝑟⟩Cgr𝑝 ↔ ⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))
40	39	anbi2d 740	. . . . . . . . . . 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 6613	. . . . . . . . . 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 2632	. . . . . . . . 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 747	. . . . . . . 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 3052	. . . . . . 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 2689	. . . . . . . . . 10 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (𝑞 ∈ ((𝔼‘𝑛) × (𝔼‘𝑛)) ↔ ⟨𝐶, 𝐷⟩ ∈ ((𝔼‘𝑛) × (𝔼‘𝑛))))
46		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (1st ‘𝑞) = (1st ‘⟨𝐶, 𝐷⟩))
47		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (2nd ‘𝑞) = (2nd ‘⟨𝐶, 𝐷⟩))
48	46, 47	neeq12d 2855	. . . . . . . . . 10 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ((1st ‘𝑞) ≠ (2nd ‘𝑞) ↔ (1st ‘⟨𝐶, 𝐷⟩) ≠ (2nd ‘⟨𝐶, 𝐷⟩)))
49	45, 48	3anbi23d 1402	. . . . . . . . 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 4408	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ⟨(1st ‘𝑞), 𝑟⟩ = ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩)
51	47, 50	breq12d 4666	. . . . . . . . . . . 12 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ((2nd ‘𝑞) Btwn ⟨(1st ‘𝑞), 𝑟⟩ ↔ (2nd ‘⟨𝐶, 𝐷⟩) Btwn ⟨(1st ‘⟨𝐶, 𝐷⟩), 𝑟⟩))
52	47	opeq1d 4408	. . . . . . . . . . . . 13 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → ⟨(2nd ‘𝑞), 𝑟⟩ = ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩)
53	52	breq1d 4663	. . . . . . . . . . . 12 ⊢ (𝑞 = ⟨𝐶, 𝐷⟩ → (⟨(2nd ‘𝑞), 𝑟⟩Cgr⟨𝐴, 𝐵⟩ ↔ ⟨(2nd ‘⟨𝐶, 𝐷⟩), 𝑟⟩Cgr⟨𝐴, 𝐵⟩))
54	51, 53	anbi12d 747	. . . . . . . . . . 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 6613	. . . . . . . . . 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 2632	. . . . . . . . 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 747	. . . . . . . 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 3052	. . . . . . 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 2626	. . . . . . . . 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 740	. . . . . . . 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 3052	. . . . . . 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 6748	. . . . . 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 1424	. . . . 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 286	. . . 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 32138	. . . . 5 ⊢ Fun TransportTo
66		funbrfv 6234	. . . . 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 225	. . 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	syl5eq 2668	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990   ≠ wne 2794  ∃wrex 2913  Vcvv 3200  ⟨cop 4183   class class class wbr 4653   × cxp 5112  Fun wfun 5882  ‘cfv 5888  ℩crio 6610  (class class class)co 6650  {coprab 6651  1^st c1st 7166  2^nd c2nd 7167  ℕcn 11020  𝔼cee 25768   Btwn cbtwn 25769  Cgrccgr 25770  TransportToctransport 32136

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-z 11378  df-uz 11688  df-fz 12327  df-ee 25771  df-transport 32137

This theorem is referenced by:  transportcl  32140  transportprops  32141