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Theorem brbtwn 27170

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 27163	. . 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 5076	. 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 5373	. . . . 5 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
4		brcnvg 5777	. . . . 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 687	. . . 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 1131	. . 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 5071	. . . 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 2826	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
9	8	3anbi1d 1438	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛))))
10		fveq1 6755	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑖) = (𝐵‘𝑖))
11	10	oveq2d 7271	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ((1 − 𝑡) · (𝑦‘𝑖)) = ((1 − 𝑡) · (𝐵‘𝑖)))
12	11	oveq1d 7270	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))))
13	12	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ (𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖)))))
14	13	rexralbidv 3229	. . . . . . . . 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 630	. . . . . . . 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 3225	. . . . . . 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 2826	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
18	17	3anbi2d 1439	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛))))
19		fveq1 6755	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐶 → (𝑧‘𝑖) = (𝐶‘𝑖))
20	19	oveq2d 7271	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐶 → (𝑡 · (𝑧‘𝑖)) = (𝑡 · (𝐶‘𝑖)))
21	20	oveq2d 7271	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐶 → (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))
22	21	eqeq2d 2749	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ (𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
23	22	rexralbidv 3229	. . . . . . . . 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 630	. . . . . . . 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 3225	. . . . . . 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 2826	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
27	26	3anbi3d 1440	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))))
28		fveq1 6755	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥‘𝑖) = (𝐴‘𝑖))
29	28	eqeq1d 2740	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) ↔ (𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
30	29	rexralbidv 3229	. . . . . . . . 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 630	. . . . . . . 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 3225	. . . . . . 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 7362	. . . . . 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 1134	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) → 𝐵 ∈ (𝔼‘𝑛))
35		simp1 1134	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
36		eedimeq 27169	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝑛 = 𝑁)
37	34, 35, 36	syl2anr 596	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))) → 𝑛 = 𝑁)
38		oveq2 7263	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (1...𝑛) = (1...𝑁))
39	38	raleqdv 3339	. . . . . . . . . . . 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 3225	. . . . . . . . . . 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 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 3218	. . . . . . 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 27167	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
46	45	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
47		fveq2 6756	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
48	47	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
49	47	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐶 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑁)))
50	47	eleq2d 2824	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
51	48, 49, 50	3anbi123d 1434	. . . . . . . . . . 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 630	. . . . . . . . . 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 3552	. . . . . . . . 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 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 1123	. . . 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 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	syl5bb 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 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ∀wral 3063 ∃wrex 3064 Vcvv 3422 ⟨cop 4564 class class class wbr 5070 ^◡ccnv 5579 ‘cfv 6418 (class class class)co 7255 {coprab 7256 0cc0 10802 1c1 10803 + caddc 10805 · cmul 10807 − cmin 11135 ℕcn 11903 [,]cicc 13011 ...cfz 13168 𝔼cee 27159 Btwn cbtwn 27160

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-1st 7804 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-map 8575 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-nn 11904 df-z 12250 df-uz 12512 df-fz 13169 df-ee 27162 df-btwn 27163

This theorem is referenced by: brbtwn2 27176 axsegcon 27198 ax5seg 27209 axbtwnid 27210 axpasch 27212 axeuclid 27234 axcontlem2 27236 axcontlem4 27238 axcontlem7 27241 axcontlem8 27242 elntg2 27256

W3C validator