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Theorem brbtwn 28990

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 28982	. . 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 5106	. 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 5421	. . . . 5 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
4		brcnvg 5838	. . . . 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 692	. . . 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 1134	. . 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 5101	. . . 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 2825	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
9	8	3anbi1d 1443	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛))))
10		fveq1 6843	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑖) = (𝐵‘𝑖))
11	10	oveq2d 7386	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ((1 − 𝑡) · (𝑦‘𝑖)) = ((1 − 𝑡) · (𝐵‘𝑖)))
12	11	oveq1d 7385	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))))
13	12	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ (𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖)))))
14	13	rexralbidv 3204	. . . . . . . . 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 633	. . . . . . . 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 3162	. . . . . . 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 2825	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
18	17	3anbi2d 1444	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛))))
19		fveq1 6843	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐶 → (𝑧‘𝑖) = (𝐶‘𝑖))
20	19	oveq2d 7386	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐶 → (𝑡 · (𝑧‘𝑖)) = (𝑡 · (𝐶‘𝑖)))
21	20	oveq2d 7386	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐶 → (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))
22	21	eqeq2d 2748	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ (𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
23	22	rexralbidv 3204	. . . . . . . . 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 633	. . . . . . . 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 3162	. . . . . . 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 2825	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
27	26	3anbi3d 1445	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))))
28		fveq1 6843	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥‘𝑖) = (𝐴‘𝑖))
29	28	eqeq1d 2739	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) ↔ (𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
30	29	rexralbidv 3204	. . . . . . . . 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 633	. . . . . . . 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 3162	. . . . . . 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 7480	. . . . . 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 1137	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) → 𝐵 ∈ (𝔼‘𝑛))
35		simp1 1137	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
36		eedimeq 28989	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝑛 = 𝑁)
37	34, 35, 36	syl2anr 598	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))) → 𝑛 = 𝑁)
38		oveq2 7378	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
39	38	raleqdv 3298	. . . . . . . . . . . 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 3162	. . . . . . . . . . 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 229	. . . . . . . . 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 453	. . . . . . . 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 3144	. . . . . . 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 28987	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
46	45	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
47		fveq2 6844	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
48	47	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
49	47	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐶 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑁)))
50	47	eleq2d 2823	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
51	48, 49, 50	3anbi123d 1439	. . . . . . . . . . 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 633	. . . . . . . . . 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 3578	. . . . . . . . 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 420	. . . . . . . 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 212	. . . . . 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 279	. . . . 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 1126	. . . 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 283	. . 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 279	. 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 283	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 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 ∃wrex 3062 Vcvv 3442 ⟨cop 4588 class class class wbr 5100 ^◡ccnv 5633 ‘cfv 6502 (class class class)co 7370 {coprab 7371 0cc0 11040 1c1 11041 + caddc 11043 · cmul 11045 − cmin 11378 ℕcn 12159 [,]cicc 13278 ...cfz 13437 𝔼cee 28978 Btwn cbtwn 28979

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5245 ax-nul 5255 ax-pow 5314 ax-pr 5381 ax-un 7692 ax-cnex 11096 ax-resscn 11097 ax-1cn 11098 ax-icn 11099 ax-addcl 11100 ax-addrcl 11101 ax-mulcl 11102 ax-mulrcl 11103 ax-mulcom 11104 ax-addass 11105 ax-mulass 11106 ax-distr 11107 ax-i2m1 11108 ax-1ne0 11109 ax-1rid 11110 ax-rnegex 11111 ax-rrecex 11112 ax-cnre 11113 ax-pre-lttri 11114 ax-pre-lttrn 11115 ax-pre-ltadd 11116 ax-pre-mulgt0 11117

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5529 df-eprel 5534 df-po 5542 df-so 5543 df-fr 5587 df-we 5589 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-pred 6269 df-ord 6330 df-on 6331 df-lim 6332 df-suc 6333 df-iota 6458 df-fun 6504 df-fn 6505 df-f 6506 df-f1 6507 df-fo 6508 df-f1o 6509 df-fv 6510 df-riota 7327 df-ov 7373 df-oprab 7374 df-mpo 7375 df-om 7821 df-1st 7945 df-2nd 7946 df-frecs 8235 df-wrecs 8266 df-recs 8315 df-rdg 8353 df-er 8647 df-map 8779 df-en 8898 df-dom 8899 df-sdom 8900 df-pnf 11182 df-mnf 11183 df-xr 11184 df-ltxr 11185 df-le 11186 df-sub 11380 df-neg 11381 df-nn 12160 df-z 12503 df-uz 12766 df-fz 13438 df-ee 28981 df-btwn 28982

This theorem is referenced by: brbtwn2 28996 axsegcon 29018 ax5seg 29029 axbtwnid 29030 axpasch 29032 axeuclid 29054 axcontlem2 29056 axcontlem4 29058 axcontlem7 29061 axcontlem8 29062 elntg2 29076

W3C validator