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Theorem brbtwn 28146

Description: The binary relation form of the betweenness predicate. The statement 𝐴 Btwn ⟨𝐵, 𝐶⟩ should be informally read as "𝐴 lies on a line segment between 𝐵 and 𝐶. This exact definition is abstracted away by Tarski's geometry axioms later on. (Contributed by Scott Fenton, 3-Jun-2013.)

**Assertion**
Ref	Expression
brbtwn	⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))

Distinct variable groups: 𝑖,𝑁,𝑡 𝐴,𝑖,𝑡 𝐵,𝑖,𝑡 𝐶,𝑖,𝑡

Proof of Theorem brbtwn Dummy variables 𝑥 𝑦 𝑧 𝑛 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-btwn 28139	. . 3 ⊢ Btwn = ^◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}
2	1	breqi 5153	. 2 ⊢ (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ 𝐴^◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}⟨𝐵, 𝐶⟩)
3		opex 5463	. . . . 5 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
4		brcnvg 5877	. . . . 5 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ ⟨𝐵, 𝐶⟩ ∈ V) → (𝐴^◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}𝐴))
5	3, 4	mpan2 689	. . . 4 ⊢ (𝐴 ∈ (𝔼‘𝑁) → (𝐴^◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}𝐴))
6	5	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (𝐴^◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}⟨𝐵, 𝐶⟩ ↔ ⟨𝐵, 𝐶⟩{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}𝐴))
7		df-br 5148	. . . 4 ⊢ (⟨𝐵, 𝐶⟩{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}𝐴 ↔ ⟨⟨𝐵, 𝐶⟩, 𝐴⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))})
8		eleq1 2821	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
9	8	3anbi1d 1440	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛))))
10		fveq1 6887	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑖) = (𝐵‘𝑖))
11	10	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ((1 − 𝑡) · (𝑦‘𝑖)) = ((1 − 𝑡) · (𝐵‘𝑖)))
12	11	oveq1d 7420	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))))
13	12	eqeq2d 2743	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ (𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖)))))
14	13	rexralbidv 3220	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖)))))
15	9, 14	anbi12d 631	. . . . . . . 8 ⊢ (𝑦 = 𝐵 → (((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖)))) ↔ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))))))
16	15	rexbidv 3178	. . . . . . 7 ⊢ (𝑦 = 𝐵 → (∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖)))) ↔ ∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))))))
17		eleq1 2821	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
18	17	3anbi2d 1441	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛))))
19		fveq1 6887	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐶 → (𝑧‘𝑖) = (𝐶‘𝑖))
20	19	oveq2d 7421	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐶 → (𝑡 · (𝑧‘𝑖)) = (𝑡 · (𝐶‘𝑖)))
21	20	oveq2d 7421	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐶 → (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))
22	21	eqeq2d 2743	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ (𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
23	22	rexralbidv 3220	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
24	18, 23	anbi12d 631	. . . . . . . 8 ⊢ (𝑧 = 𝐶 → (((𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖)))) ↔ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))))
25	24	rexbidv 3178	. . . . . . 7 ⊢ (𝑧 = 𝐶 → (∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖)))) ↔ ∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))))
26		eleq1 2821	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
27	26	3anbi3d 1442	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))))
28		fveq1 6887	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥‘𝑖) = (𝐴‘𝑖))
29	28	eqeq1d 2734	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) ↔ (𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
30	29	rexralbidv 3220	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
31	27, 30	anbi12d 631	. . . . . . . 8 ⊢ (𝑥 = 𝐴 → (((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))) ↔ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))))
32	31	rexbidv 3178	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))) ↔ ∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))))
33	16, 25, 32	eloprabg 7514	. . . . . 6 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → (⟨⟨𝐵, 𝐶⟩, 𝐴⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))} ↔ ∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))))
34		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) → 𝐵 ∈ (𝔼‘𝑛))
35		simp1 1136	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
36		eedimeq 28145	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝑛 = 𝑁)
37	34, 35, 36	syl2anr 597	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))) → 𝑛 = 𝑁)
38		oveq2 7413	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
39	38	raleqdv 3325	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
40	39	rexbidv 3178	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
41	37, 40	syl 17	. . . . . . . . . 10 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))) → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
42	41	biimpd 228	. . . . . . . . 9 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))) → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) → ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
43	42	expimpd 454	. . . . . . . 8 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → (((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))) → ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
44	43	rexlimdvw 3160	. . . . . . 7 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → (∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))) → ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
45		eleenn 28143	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
46	45	3ad2ant1 1133	. . . . . . . 8 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
47		fveq2 6888	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
48	47	eleq2d 2819	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
49	47	eleq2d 2819	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐶 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑁)))
50	47	eleq2d 2819	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
51	48, 49, 50	3anbi123d 1436	. . . . . . . . . . 11 ⊢ (𝑛 = 𝑁 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁))))
52	51, 40	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑛 = 𝑁 → (((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))) ↔ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))))
53	52	rspcev 3612	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))) → ∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
54	53	exp32 421	. . . . . . . 8 ⊢ (𝑁 ∈ ℕ → ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) → ∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))))
55	46, 54	mpcom 38	. . . . . . 7 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) → ∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))))
56	44, 55	impbid 211	. . . . . 6 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → (∃𝑛 ∈ ℕ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))) ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
57	33, 56	bitrd 278	. . . . 5 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → (⟨⟨𝐵, 𝐶⟩, 𝐴⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))} ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
58	57	3comr 1125	. . . 4 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (⟨⟨𝐵, 𝐶⟩, 𝐴⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))} ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
59	7, 58	bitrid 282	. . 3 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (⟨𝐵, 𝐶⟩{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}𝐴 ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
60	6, 59	bitrd 278	. 2 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (𝐴^◡{⟨⟨𝑦, 𝑧⟩, 𝑥⟩ ∣ ∃𝑛 ∈ ℕ ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ∧ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑛)(𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))))}⟨𝐵, 𝐶⟩ ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
61	2, 60	bitrid 282	1 ⊢ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) → (𝐴 Btwn ⟨𝐵, 𝐶⟩ ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3061 ∃wrex 3070 Vcvv 3474 ⟨cop 4633 class class class wbr 5147 ^◡ccnv 5674 ‘cfv 6540 (class class class)co 7405 {coprab 7406 0cc0 11106 1c1 11107 + caddc 11109 · cmul 11111 − cmin 11440 ℕcn 12208 [,]cicc 13323 ...cfz 13480 𝔼cee 28135 Btwn cbtwn 28136

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 2703 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-map 8818 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-nn 12209 df-z 12555 df-uz 12819 df-fz 13481 df-ee 28138 df-btwn 28139

This theorem is referenced by: brbtwn2 28152 axsegcon 28174 ax5seg 28185 axbtwnid 28186 axpasch 28188 axeuclid 28210 axcontlem2 28212 axcontlem4 28214 axcontlem7 28217 axcontlem8 28218 elntg2 28232

W3C validator