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Theorem brbtwn 25779

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 25772	. . 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 4659	. 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 4932	. . . . 5 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
4		brcnvg 5303	. . . . 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 707	. . . 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 1082	. . 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 4654	. . . 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 2689	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
9	8	3anbi1d 1403	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛))))
10		fveq1 6190	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑖) = (𝐵‘𝑖))
11	10	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ((1 − 𝑡) · (𝑦‘𝑖)) = ((1 − 𝑡) · (𝐵‘𝑖)))
12	11	oveq1d 6665	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))))
13	12	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ (𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖)))))
14	13	rexralbidv 3058	. . . . . . . . 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 747	. . . . . . . 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 3052	. . . . . . 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 2689	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
18	17	3anbi2d 1404	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛))))
19		fveq1 6190	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐶 → (𝑧‘𝑖) = (𝐶‘𝑖))
20	19	oveq2d 6666	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐶 → (𝑡 · (𝑧‘𝑖)) = (𝑡 · (𝐶‘𝑖)))
21	20	oveq2d 6666	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐶 → (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))
22	21	eqeq2d 2632	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ (𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
23	22	rexralbidv 3058	. . . . . . . . 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 747	. . . . . . . 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 3052	. . . . . . 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 2689	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
27	26	3anbi3d 1405	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))))
28		fveq1 6190	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥‘𝑖) = (𝐴‘𝑖))
29	28	eqeq1d 2624	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) ↔ (𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
30	29	rexralbidv 3058	. . . . . . . . 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 747	. . . . . . . 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 3052	. . . . . . 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 6748	. . . . . 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 1061	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) → 𝐵 ∈ (𝔼‘𝑛))
35		simp1 1061	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
36		eedimeq 25778	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝑛 = 𝑁)
37	34, 35, 36	syl2anr 495	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))) → 𝑛 = 𝑁)
38		oveq2 6658	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
39	38	raleqdv 3144	. . . . . . . . . . . 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 3052	. . . . . . . . . . 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 219	. . . . . . . . 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 629	. . . . . . . 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 3034	. . . . . . 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 25776	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
46	45	3ad2ant1 1082	. . . . . . . 8 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
47		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
48	47	eleq2d 2687	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
49	47	eleq2d 2687	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐶 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑁)))
50	47	eleq2d 2687	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
51	48, 49, 50	3anbi123d 1399	. . . . . . . . . . 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 747	. . . . . . . . . 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 3309	. . . . . . . . 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 631	. . . . . . . 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 202	. . . . . 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 268	. . . . 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 1273	. . . 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	syl5bb 272	. . 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 268	. 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	syl5bb 272	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 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ⟨cop 4183 class class class wbr 4653 ^◡ccnv 5113 ‘cfv 5888 (class class class)co 6650 {coprab 6651 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 ℕcn 11020 [,]cicc 12178 ...cfz 12326 𝔼cee 25768 Btwn cbtwn 25769

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-btwn 25772

This theorem is referenced by: brbtwn2 25785 axsegcon 25807 ax5seg 25818 axbtwnid 25819 axpasch 25821 axeuclid 25843 axcontlem2 25845 axcontlem4 25847 axcontlem7 25850 axcontlem8 25851

W3C validator