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Theorem ax5seglem3 29396

Description: Lemma for ax5seg 29403. Combine congruences for points on a line. (Contributed by Scott Fenton, 11-Jun-2013.)

**Assertion**
Ref	Expression
ax5seglem3	⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))

Distinct variable groups: 𝐴,𝑖,𝑗 𝐵,𝑖,𝑗 𝐶,𝑖,𝑗 𝐷,𝑖,𝑗 𝑖,𝐸,𝑗 𝑖,𝐹,𝑗 𝑖,𝑁,𝑗 𝑆,𝑖,𝑗 𝑇,𝑖,𝑗

**Proof of Theorem ax5seglem3**
Step	Hyp	Ref	Expression
1		1re 11236	. . . . . . . . . 10 ⊢ 1 ∈ ℝ
2		elicc01 13523	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (0[,]1) ↔ (𝑇 ∈ ℝ ∧ 0 ≤ 𝑇 ∧ 𝑇 ≤ 1))
3	2	simp1bi 1163	. . . . . . . . . 10 ⊢ (𝑇 ∈ (0[,]1) → 𝑇 ∈ ℝ)
4		resubcl 11550	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (1 − 𝑇) ∈ ℝ)
5	1, 3, 4	sylancr 599	. . . . . . . . 9 ⊢ (𝑇 ∈ (0[,]1) → (1 − 𝑇) ∈ ℝ)
6	5	ad2antrr 739	. . . . . . . 8 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → (1 − 𝑇) ∈ ℝ)
7	6	3ad2ant2 1152	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (1 − 𝑇) ∈ ℝ)
8		fzfid 14041	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (1...𝑁) ∈ Fin)
9		ax5seglem3a 29395	. . . . . . . . . . . 12 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → (((𝐴‘𝑗) − (𝐶‘𝑗)) ∈ ℝ ∧ ((𝐷‘𝑗) − (𝐹‘𝑗)) ∈ ℝ))
10	9	simpld 500	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → ((𝐴‘𝑗) − (𝐶‘𝑗)) ∈ ℝ)
11	10	resqcld 14193	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → (((𝐴‘𝑗) − (𝐶‘𝑗))↑2) ∈ ℝ)
12	8, 11	fsumrecl 15824	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2) ∈ ℝ)
13	10	sqge0d 14205	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → 0 ≤ (((𝐴‘𝑗) − (𝐶‘𝑗))↑2))
14	8, 11, 13	fsumge0 15886	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → 0 ≤ Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))
15	12, 14	resqrtcld 15509	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) ∈ ℝ)
16	15	3ad2ant1 1151	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) ∈ ℝ)
17	7, 16	remulcld 11267	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((1 − 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) ∈ ℝ)
18		elicc01 13523	. . . . . . . . . . 11 ⊢ (𝑆 ∈ (0[,]1) ↔ (𝑆 ∈ ℝ ∧ 0 ≤ 𝑆 ∧ 𝑆 ≤ 1))
19	18	simp1bi 1163	. . . . . . . . . 10 ⊢ (𝑆 ∈ (0[,]1) → 𝑆 ∈ ℝ)
20		resubcl 11550	. . . . . . . . . 10 ⊢ ((1 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (1 − 𝑆) ∈ ℝ)
21	1, 19, 20	sylancr 599	. . . . . . . . 9 ⊢ (𝑆 ∈ (0[,]1) → (1 − 𝑆) ∈ ℝ)
22	21	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → (1 − 𝑆) ∈ ℝ)
23	22	3ad2ant2 1152	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (1 − 𝑆) ∈ ℝ)
24	9	simprd 501	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → ((𝐷‘𝑗) − (𝐹‘𝑗)) ∈ ℝ)
25	24	resqcld 14193	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → (((𝐷‘𝑗) − (𝐹‘𝑗))↑2) ∈ ℝ)
26	8, 25	fsumrecl 15824	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2) ∈ ℝ)
27	24	sqge0d 14205	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ 𝑗 ∈ (1...𝑁)) → 0 ≤ (((𝐷‘𝑗) − (𝐹‘𝑗))↑2))
28	8, 25, 27	fsumge0 15886	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → 0 ≤ Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))
29	26, 28	resqrtcld 15509	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)) ∈ ℝ)
30	29	3ad2ant1 1151	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)) ∈ ℝ)
31	23, 30	remulcld 11267	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((1 − 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) ∈ ℝ)
32	2	simp3bi 1165	. . . . . . . . . 10 ⊢ (𝑇 ∈ (0[,]1) → 𝑇 ≤ 1)
33		subge0 11755	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ 𝑇 ∈ ℝ) → (0 ≤ (1 − 𝑇) ↔ 𝑇 ≤ 1))
34	1, 3, 33	sylancr 599	. . . . . . . . . 10 ⊢ (𝑇 ∈ (0[,]1) → (0 ≤ (1 − 𝑇) ↔ 𝑇 ≤ 1))
35	32, 34	mpbird 260	. . . . . . . . 9 ⊢ (𝑇 ∈ (0[,]1) → 0 ≤ (1 − 𝑇))
36	35	ad2antrr 739	. . . . . . . 8 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → 0 ≤ (1 − 𝑇))
37	36	3ad2ant2 1152	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 0 ≤ (1 − 𝑇))
38	12, 14	sqrtge0d 15512	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → 0 ≤ (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
39	38	3ad2ant1 1151	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 0 ≤ (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
40	7, 16, 37, 39	mulge0d 11819	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 0 ≤ ((1 − 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))))
41	18	simp3bi 1165	. . . . . . . . . 10 ⊢ (𝑆 ∈ (0[,]1) → 𝑆 ≤ 1)
42		subge0 11755	. . . . . . . . . . 11 ⊢ ((1 ∈ ℝ ∧ 𝑆 ∈ ℝ) → (0 ≤ (1 − 𝑆) ↔ 𝑆 ≤ 1))
43	1, 19, 42	sylancr 599	. . . . . . . . . 10 ⊢ (𝑆 ∈ (0[,]1) → (0 ≤ (1 − 𝑆) ↔ 𝑆 ≤ 1))
44	41, 43	mpbird 260	. . . . . . . . 9 ⊢ (𝑆 ∈ (0[,]1) → 0 ≤ (1 − 𝑆))
45	44	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → 0 ≤ (1 − 𝑆))
46	45	3ad2ant2 1152	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 0 ≤ (1 − 𝑆))
47	26, 28	sqrtge0d 15512	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → 0 ≤ (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
48	47	3ad2ant1 1151	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 0 ≤ (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
49	23, 30, 46, 48	mulge0d 11819	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 0 ≤ ((1 − 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))))
50		resqrtth 15346	. . . . . . . . . 10 ⊢ ((Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2) ∈ ℝ ∧ 0 ≤ Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) → ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))
51	12, 14, 50	syl2anc 596	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))
52	51	3ad2ant1 1151	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))
53	52	oveq2d 7433	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑇)↑2) · ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))↑2)) = (((1 − 𝑇)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
54		ax-1cn 11186	. . . . . . . . 9 ⊢ 1 ∈ ℂ
55	3	recnd 11265	. . . . . . . . . . 11 ⊢ (𝑇 ∈ (0[,]1) → 𝑇 ∈ ℂ)
56	55	ad2antrr 739	. . . . . . . . . 10 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → 𝑇 ∈ ℂ)
57	56	3ad2ant2 1152	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝑇 ∈ ℂ)
58		subcl 11484	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → (1 − 𝑇) ∈ ℂ)
59	54, 57, 58	sylancr 599	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (1 − 𝑇) ∈ ℂ)
60	15	recnd 11265	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) ∈ ℂ)
61	60	3ad2ant1 1151	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) ∈ ℂ)
62	59, 61	sqmuld 14226	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))↑2) = (((1 − 𝑇)↑2) · ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))↑2)))
63		resqrtth 15346	. . . . . . . . . . 11 ⊢ ((Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2) ∈ ℝ ∧ 0 ≤ Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)) → ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))
64	26, 28, 63	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))
65	64	3ad2ant1 1151	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))
66	65	oveq2d 7433	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑆)↑2) · ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))↑2)) = (((1 − 𝑆)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
67	19	recnd 11265	. . . . . . . . . . . 12 ⊢ (𝑆 ∈ (0[,]1) → 𝑆 ∈ ℂ)
68	67	ad2antlr 740	. . . . . . . . . . 11 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → 𝑆 ∈ ℂ)
69	68	3ad2ant2 1152	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝑆 ∈ ℂ)
70		subcl 11484	. . . . . . . . . 10 ⊢ ((1 ∈ ℂ ∧ 𝑆 ∈ ℂ) → (1 − 𝑆) ∈ ℂ)
71	54, 69, 70	sylancr 599	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (1 − 𝑆) ∈ ℂ)
72	29	recnd 11265	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)) ∈ ℂ)
73	72	3ad2ant1 1151	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)) ∈ ℂ)
74	71, 73	sqmuld 14226	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))↑2) = (((1 − 𝑆)↑2) · ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))↑2)))
75		simp3r 1221	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)
76		simp122 1325	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝐵 ∈ (𝔼‘𝑁))
77		simp123 1326	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝐶 ∈ (𝔼‘𝑁))
78		simp132 1328	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝐸 ∈ (𝔼‘𝑁))
79		simp133 1329	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝐹 ∈ (𝔼‘𝑁))
80		brcgr 29365	. . . . . . . . . . 11 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩ ↔ Σ𝑗 ∈ (1...𝑁)(((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐸‘𝑗) − (𝐹‘𝑗))↑2)))
81	76, 77, 78, 79, 80	syl22anc 852	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩ ↔ Σ𝑗 ∈ (1...𝑁)(((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐸‘𝑗) − (𝐹‘𝑗))↑2)))
82	75, 81	mpbid 235	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → Σ𝑗 ∈ (1...𝑁)(((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐸‘𝑗) − (𝐹‘𝑗))↑2))
83		simp11 1222	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝑁 ∈ ℕ)
84		simp121 1324	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝐴 ∈ (𝔼‘𝑁))
85		simp2ll 1259	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝑇 ∈ (0[,]1))
86		simp2rl 1261	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))
87		ax5seglem2 29394	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = (((1 − 𝑇)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
88	83, 84, 77, 85, 86, 87	syl122anc 1406	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → Σ𝑗 ∈ (1...𝑁)(((𝐵‘𝑗) − (𝐶‘𝑗))↑2) = (((1 − 𝑇)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
89		simp131 1327	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝐷 ∈ (𝔼‘𝑁))
90		simp2lr 1260	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝑆 ∈ (0[,]1))
91		simp2rr 1262	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))
92		ax5seglem2 29394	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)) ∧ (𝑆 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐸‘𝑗) − (𝐹‘𝑗))↑2) = (((1 − 𝑆)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
93	83, 89, 79, 90, 91, 92	syl122anc 1406	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → Σ𝑗 ∈ (1...𝑁)(((𝐸‘𝑗) − (𝐹‘𝑗))↑2) = (((1 − 𝑆)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
94	82, 88, 93	3eqtr3d 2805	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑇)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) = (((1 − 𝑆)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
95	66, 74, 94	3eqtr4d 2807	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))↑2) = (((1 − 𝑇)↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
96	53, 62, 95	3eqtr4d 2807	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))↑2) = (((1 − 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))↑2))
97	17, 31, 40, 49, 96	sq11d 14326	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((1 − 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) = ((1 − 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))))
98	3	ad2antrr 739	. . . . . . . 8 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → 𝑇 ∈ ℝ)
99	98	3ad2ant2 1152	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝑇 ∈ ℝ)
100	99, 16	remulcld 11267	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (𝑇 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) ∈ ℝ)
101	19	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → 𝑆 ∈ ℝ)
102	101	3ad2ant2 1152	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 𝑆 ∈ ℝ)
103	102, 30	remulcld 11267	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (𝑆 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) ∈ ℝ)
104	2	simp2bi 1164	. . . . . . . . 9 ⊢ (𝑇 ∈ (0[,]1) → 0 ≤ 𝑇)
105	104	ad2antrr 739	. . . . . . . 8 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → 0 ≤ 𝑇)
106	105	3ad2ant2 1152	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 0 ≤ 𝑇)
107	99, 16, 106, 39	mulge0d 11819	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 0 ≤ (𝑇 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))))
108	18	simp2bi 1164	. . . . . . . . 9 ⊢ (𝑆 ∈ (0[,]1) → 0 ≤ 𝑆)
109	108	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → 0 ≤ 𝑆)
110	109	3ad2ant2 1152	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 0 ≤ 𝑆)
111	102, 30, 110, 48	mulge0d 11819	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → 0 ≤ (𝑆 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))))
112	51	oveq2d 7433	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((𝑇↑2) · ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))↑2)) = ((𝑇↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
113	112	3ad2ant1 1151	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((𝑇↑2) · ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))↑2)) = ((𝑇↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
114	57, 61	sqmuld 14226	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((𝑇 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))↑2) = ((𝑇↑2) · ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))↑2)))
115	65	oveq2d 7433	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((𝑆↑2) · ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))↑2)) = ((𝑆↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
116	69, 73	sqmuld 14226	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((𝑆 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))↑2) = ((𝑆↑2) · ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))↑2)))
117		simp3l 1220	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩)
118		brcgr 29365	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁))) → (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ↔ Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐵‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐸‘𝑗))↑2)))
119	84, 76, 89, 78, 118	syl22anc 852	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ↔ Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐵‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐸‘𝑗))↑2)))
120	117, 119	mpbid 235	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐵‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐸‘𝑗))↑2))
121		ax5seglem1 29393	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝑇 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐵‘𝑗))↑2) = ((𝑇↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
122	83, 84, 77, 85, 86, 121	syl122anc 1406	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐵‘𝑗))↑2) = ((𝑇↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
123		ax5seglem1 29393	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁)) ∧ (𝑆 ∈ (0[,]1) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐸‘𝑗))↑2) = ((𝑆↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
124	83, 89, 79, 90, 91, 123	syl122anc 1406	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐸‘𝑗))↑2) = ((𝑆↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
125	120, 122, 124	3eqtr3d 2805	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((𝑇↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) = ((𝑆↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
126	115, 116, 125	3eqtr4d 2807	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((𝑆 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))↑2) = ((𝑇↑2) · Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
127	113, 114, 126	3eqtr4d 2807	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((𝑇 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))↑2) = ((𝑆 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))↑2))
128	100, 103, 107, 111, 127	sq11d 14326	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (𝑇 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) = (𝑆 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))))
129	97, 128	oveq12d 7435	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) + (𝑇 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))) = (((1 − 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) + (𝑆 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))))
130	59, 57, 61	adddird 11262	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑇) + 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) = (((1 − 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) + (𝑇 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))))
131	71, 69, 73	adddird 11262	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑆) + 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) = (((1 − 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) + (𝑆 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))))
132	129, 130, 131	3eqtr4d 2807	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑇) + 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) = (((1 − 𝑆) + 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))))
133		npcan 11494	. . . . . . . . 9 ⊢ ((1 ∈ ℂ ∧ 𝑇 ∈ ℂ) → ((1 − 𝑇) + 𝑇) = 1)
134	54, 55, 133	sylancr 599	. . . . . . . 8 ⊢ (𝑇 ∈ (0[,]1) → ((1 − 𝑇) + 𝑇) = 1)
135	134	adantr 486	. . . . . . 7 ⊢ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) → ((1 − 𝑇) + 𝑇) = 1)
136	135	oveq1d 7432	. . . . . 6 ⊢ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) → (((1 − 𝑇) + 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) = (1 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))))
137	136	adantr 486	. . . . 5 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → (((1 − 𝑇) + 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) = (1 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))))
138	137	3ad2ant2 1152	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑇) + 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) = (1 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))))
139	60	mullidd 11255	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (1 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) = (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
140	139	3ad2ant1 1151	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (1 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) = (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
141	138, 140	eqtrd 2797	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑇) + 𝑇) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2))) = (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)))
142		npcan 11494	. . . . . . . 8 ⊢ ((1 ∈ ℂ ∧ 𝑆 ∈ ℂ) → ((1 − 𝑆) + 𝑆) = 1)
143	54, 67, 142	sylancr 599	. . . . . . 7 ⊢ (𝑆 ∈ (0[,]1) → ((1 − 𝑆) + 𝑆) = 1)
144	143	oveq1d 7432	. . . . . 6 ⊢ (𝑆 ∈ (0[,]1) → (((1 − 𝑆) + 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) = (1 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))))
145	144	ad2antlr 740	. . . . 5 ⊢ (((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) → (((1 − 𝑆) + 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) = (1 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))))
146	145	3ad2ant2 1152	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑆) + 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) = (1 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))))
147	72	mullidd 11255	. . . . 5 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → (1 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) = (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
148	147	3ad2ant1 1151	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (1 · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) = (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
149	146, 148	eqtrd 2797	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (((1 − 𝑆) + 𝑆) · (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) = (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
150	132, 141, 149	3eqtr3d 2805	. 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → (√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) = (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
151		sqrt11 15353	. . . 4 ⊢ (((Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2) ∈ ℝ ∧ 0 ≤ Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) ∧ (Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2) ∈ ℝ ∧ 0 ≤ Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))) → ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) = (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)) ↔ Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
152	12, 14, 26, 28, 151	syl22anc 852	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) → ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) = (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)) ↔ Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
153	152	3ad2ant1 1151	. 2 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → ((√‘Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2)) = (√‘Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)) ↔ Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2)))
154	150, 153	mpbid 235	1 ⊢ (((𝑁 ∈ ℕ ∧ (𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁)) ∧ (𝐷 ∈ (𝔼‘𝑁) ∧ 𝐸 ∈ (𝔼‘𝑁) ∧ 𝐹 ∈ (𝔼‘𝑁))) ∧ ((𝑇 ∈ (0[,]1) ∧ 𝑆 ∈ (0[,]1)) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑇) · (𝐴‘𝑖)) + (𝑇 · (𝐶‘𝑖))) ∧ ∀𝑖 ∈ (1...𝑁)(𝐸‘𝑖) = (((1 − 𝑆) · (𝐷‘𝑖)) + (𝑆 · (𝐹‘𝑖))))) ∧ (⟨𝐴, 𝐵⟩Cgr⟨𝐷, 𝐸⟩ ∧ ⟨𝐵, 𝐶⟩Cgr⟨𝐸, 𝐹⟩)) → Σ𝑗 ∈ (1...𝑁)(((𝐴‘𝑗) − (𝐶‘𝑗))↑2) = Σ𝑗 ∈ (1...𝑁)(((𝐷‘𝑗) − (𝐹‘𝑗))↑2))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3078 ⟨cop 4593 class class class wbr 5107 ‘cfv 6537 (class class class)co 7417 ℂcc 11126 ℝcr 11127 0cc0 11128 1c1 11129 + caddc 11131 · cmul 11133 ≤ cle 11272 − cmin 11469 ℕcn 12261 2c2 12323 [,]cicc 13405 ...cfz 13565 ↑cexp 14129 √csqrt 15324 Σcsu 15777 𝔼cee 29352 Cgrccgr 29354

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 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2215 ax-ext 2734 ax-rep 5236 ax-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7740 ax-inf2 9624 ax-cnex 11184 ax-resscn 11185 ax-1cn 11186 ax-icn 11187 ax-addcl 11188 ax-addrcl 11189 ax-mulcl 11190 ax-mulrcl 11191 ax-mulcom 11192 ax-addass 11193 ax-mulass 11194 ax-distr 11195 ax-i2m1 11196 ax-1ne0 11197 ax-1rid 11198 ax-rnegex 11199 ax-rrecex 11200 ax-cnre 11201 ax-pre-lttri 11202 ax-pre-lttrn 11203 ax-pre-ltadd 11204 ax-pre-mulgt0 11205 ax-pre-sup 11206

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 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rmo 3367 df-reu 3368 df-rab 3415 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4283 df-if 4486 df-pw 4562 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4871 df-int 4911 df-iun 4956 df-br 5108 df-opab 5172 df-mpt 5191 df-tr 5217 df-id 5554 df-eprel 5559 df-po 5567 df-so 5568 df-fr 5612 df-se 5613 df-we 5614 df-xp 5665 df-rel 5666 df-cnv 5667 df-co 5668 df-dm 5669 df-rn 5670 df-res 5671 df-ima 5672 df-pred 6303 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6493 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-isom 6546 df-riota 7374 df-ov 7420 df-oprab 7421 df-mpo 7422 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-1o 8459 df-er 8700 df-map 8832 df-en 8957 df-dom 8958 df-sdom 8959 df-fin 8960 df-sup 9416 df-oi 9486 df-card 9948 df-pnf 11273 df-mnf 11274 df-xr 11275 df-ltxr 11276 df-le 11277 df-sub 11471 df-neg 11472 df-div 11900 df-nn 12262 df-2 12331 df-3 12332 df-n0 12533 df-z 12620 df-uz 12892 df-rp 13047 df-ico 13408 df-icc 13409 df-fz 13566 df-fzo 13714 df-seq 14070 df-exp 14130 df-hash 14399 df-cj 15190 df-re 15191 df-im 15192 df-sqrt 15326 df-abs 15327 df-clim 15579 df-sum 15778 df-ee 29355 df-cgr 29357

This theorem is used by: ax5seglem6 29399 ax5seg 29403

W3C validator