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Theorem brbtwn 29309

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 29301	. . 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 5117	. 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 5447	. . . . 5 ⊢ ⟨𝐵, 𝐶⟩ ∈ V
4		brcnvg 5867	. . . . 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 704	. . . 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 1151	. . 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 5112	. . . 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 2853	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → (𝑦 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑛)))
9	8	3anbi1d 1468	. . . . . . . . 9 ⊢ (𝑦 = 𝐵 → ((𝑦 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛))))
10		fveq1 6885	. . . . . . . . . . . . 13 ⊢ (𝑦 = 𝐵 → (𝑦‘𝑖) = (𝐵‘𝑖))
11	10	oveq2d 7436	. . . . . . . . . . . 12 ⊢ (𝑦 = 𝐵 → ((1 − 𝑡) · (𝑦‘𝑖)) = ((1 − 𝑡) · (𝐵‘𝑖)))
12	11	oveq1d 7435	. . . . . . . . . . 11 ⊢ (𝑦 = 𝐵 → (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))))
13	12	eqeq2d 2776	. . . . . . . . . 10 ⊢ (𝑦 = 𝐵 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝑦‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ (𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖)))))
14	13	rexralbidv 3233	. . . . . . . . 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 644	. . . . . . . 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 3191	. . . . . . 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 2853	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → (𝑧 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑛)))
18	17	3anbi2d 1469	. . . . . . . . 9 ⊢ (𝑧 = 𝐶 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝑧 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛))))
19		fveq1 6885	. . . . . . . . . . . . 13 ⊢ (𝑧 = 𝐶 → (𝑧‘𝑖) = (𝐶‘𝑖))
20	19	oveq2d 7436	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝐶 → (𝑡 · (𝑧‘𝑖)) = (𝑡 · (𝐶‘𝑖)))
21	20	oveq2d 7436	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐶 → (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))))
22	21	eqeq2d 2776	. . . . . . . . . 10 ⊢ (𝑧 = 𝐶 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑧‘𝑖))) ↔ (𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
23	22	rexralbidv 3233	. . . . . . . . 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 644	. . . . . . . 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 3191	. . . . . . 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 2853	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑛)))
27	26	3anbi3d 1470	. . . . . . . . 9 ⊢ (𝑥 = 𝐴 → ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝑥 ∈ (𝔼‘𝑛)) ↔ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))))
28		fveq1 6885	. . . . . . . . . . 11 ⊢ (𝑥 = 𝐴 → (𝑥‘𝑖) = (𝐴‘𝑖))
29	28	eqeq1d 2767	. . . . . . . . . 10 ⊢ (𝑥 = 𝐴 → ((𝑥‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖))) ↔ (𝐴‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝐶‘𝑖)))))
30	29	rexralbidv 3233	. . . . . . . . 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 644	. . . . . . . 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 3191	. . . . . . 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 7530	. . . . . 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 1154	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛)) → 𝐵 ∈ (𝔼‘𝑛))
35		simp1 1154	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → 𝐵 ∈ (𝔼‘𝑁))
36		eedimeq 29308	. . . . . . . . . . . 12 ⊢ ((𝐵 ∈ (𝔼‘𝑛) ∧ 𝐵 ∈ (𝔼‘𝑁)) → 𝑛 = 𝑁)
37	34, 35, 36	syl2anr 609	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) ∧ (𝐵 ∈ (𝔼‘𝑛) ∧ 𝐶 ∈ (𝔼‘𝑛) ∧ 𝐴 ∈ (𝔼‘𝑛))) → 𝑛 = 𝑁)
38		oveq2 7428	. . . . . . . . . . . . 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 3191	. . . . . . . . . . 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 18	. . . . . . . . . 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 232	. . . . . . . . 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 459	. . . . . . . 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 3173	. . . . . . 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 29306	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
46	45	3ad2ant1 1151	. . . . . . . 8 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐴 ∈ (𝔼‘𝑁)) → 𝑁 ∈ ℕ)
47		fveq2 6886	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑁 → (𝔼‘𝑛) = (𝔼‘𝑁))
48	47	eleq2d 2851	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐵 ∈ (𝔼‘𝑛) ↔ 𝐵 ∈ (𝔼‘𝑁)))
49	47	eleq2d 2851	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐶 ∈ (𝔼‘𝑛) ↔ 𝐶 ∈ (𝔼‘𝑁)))
50	47	eleq2d 2851	. . . . . . . . . . . 12 ⊢ (𝑛 = 𝑁 → (𝐴 ∈ (𝔼‘𝑛) ↔ 𝐴 ∈ (𝔼‘𝑁)))
51	48, 49, 50	3anbi123d 1464	. . . . . . . . . . 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 644	. . . . . . . . . 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 3583	. . . . . . . . 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 426	. . . . . . . 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 39	. . . . . . 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 215	. . . . . 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 282	. . . . 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 1143	. . . 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 286	. . 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 282	. 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 286	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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ∀wral 3081 ∃wrex 3091 Vcvv 3457 ⟨cop 4597 class class class wbr 5111 ^◡ccnv 5662 ‘cfv 6541 (class class class)co 7420 {coprab 7421 0cc0 11120 1c1 11121 + caddc 11123 · cmul 11125 − cmin 11461 ℕcn 12253 [,]cicc 13396 ...cfz 13556 𝔼cee 29297 Btwn cbtwn 29298

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7743 ax-cnex 11176 ax-resscn 11177 ax-1cn 11178 ax-icn 11179 ax-addcl 11180 ax-addrcl 11181 ax-mulcl 11182 ax-mulrcl 11183 ax-mulcom 11184 ax-addass 11185 ax-mulass 11186 ax-distr 11187 ax-i2m1 11188 ax-1ne0 11189 ax-1rid 11190 ax-rnegex 11191 ax-rrecex 11192 ax-cnre 11193 ax-pre-lttri 11194 ax-pre-lttrn 11195 ax-pre-ltadd 11196 ax-pre-mulgt0 11197

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6307 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6497 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7870 df-1st 7993 df-2nd 7994 df-frecs 8285 df-wrecs 8316 df-recs 8365 df-rdg 8404 df-er 8701 df-map 8833 df-en 8951 df-dom 8952 df-sdom 8953 df-pnf 11265 df-mnf 11266 df-xr 11267 df-ltxr 11268 df-le 11269 df-sub 11463 df-neg 11464 df-nn 12254 df-z 12612 df-uz 12884 df-fz 13557 df-ee 29300 df-btwn 29301

This theorem is used by: brbtwn2 29315 axsegcon 29337 ax5seg 29348 axbtwnid 29349 axpasch 29351 axeuclid 29373 axcontlem2 29375 axcontlem4 29377 axcontlem7 29380 axcontlem8 29381 elntg2 29395

W3C validator