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Theorem axsegconlem1 29360

Description: Lemma for axsegcon 29370. Handle the degenerate case. (Contributed by Scott Fenton, 7-Jun-2013.)

**Assertion**
Ref	Expression
axsegconlem1	⊢ ((𝐴 = 𝐵 ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)))) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))

Distinct variable groups: 𝑡,𝑁,𝑖,𝑥 𝑡,𝐴,𝑖,𝑥 𝑡,𝐵,𝑖,𝑥 𝑡,𝐶,𝑖,𝑥 𝑡,𝐷,𝑖,𝑥

Proof of Theorem axsegconlem1 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveere 29344	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℝ)
2	1	3ad2antl1 1204	. . . . . . . . 9 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℝ)
3		fveere 29344	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝑘 ∈ (1...𝑁)) → (𝐶‘𝑘) ∈ ℝ)
4	3	3ad2antl2 1205	. . . . . . . . . 10 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → (𝐶‘𝑘) ∈ ℝ)
5		fveere 29344	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (𝔼‘𝑁) ∧ 𝑘 ∈ (1...𝑁)) → (𝐷‘𝑘) ∈ ℝ)
6	5	3ad2antl3 1206	. . . . . . . . . 10 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → (𝐷‘𝑘) ∈ ℝ)
7	4, 6	resubcld 11669	. . . . . . . . 9 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐶‘𝑘) − (𝐷‘𝑘)) ∈ ℝ)
8	2, 7	resubcld 11669	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) ∈ ℝ)
9	8	ralrimiva 3156	. . . . . . 7 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ∀𝑘 ∈ (1...𝑁)((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) ∈ ℝ)
10		eleenn 29339	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
11		mptelee 29337	. . . . . . . . 9 ⊢ (𝑁 ∈ ℕ → ((𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) ∈ ℝ))
12	10, 11	syl 18	. . . . . . . 8 ⊢ (𝐵 ∈ (𝔼‘𝑁) → ((𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) ∈ ℝ))
13	12	3ad2ant1 1151	. . . . . . 7 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ((𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∈ (𝔼‘𝑁) ↔ ∀𝑘 ∈ (1...𝑁)((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) ∈ ℝ))
14	9, 13	mpbird 260	. . . . . 6 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∈ (𝔼‘𝑁))
15		fveecn 29345	. . . . . . . . 9 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐵‘𝑖) ∈ ℂ)
16	15	3ad2antl1 1204	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (𝐵‘𝑖) ∈ ℂ)
17		fveecn 29345	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐶‘𝑖) ∈ ℂ)
18	17	3ad2antl2 1205	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (𝐶‘𝑖) ∈ ℂ)
19		fveecn 29345	. . . . . . . . 9 ⊢ ((𝐷 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐷‘𝑖) ∈ ℂ)
20	19	3ad2antl3 1206	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (𝐷‘𝑖) ∈ ℂ)
21		1m0e1 12387	. . . . . . . . . . . 12 ⊢ (1 − 0) = 1
22	21	oveq1i 7426	. . . . . . . . . . 11 ⊢ ((1 − 0) · (𝐵‘𝑖)) = (1 · (𝐵‘𝑖))
23		mullid 11234	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝑖) ∈ ℂ → (1 · (𝐵‘𝑖)) = (𝐵‘𝑖))
24	23	3ad2ant1 1151	. . . . . . . . . . 11 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → (1 · (𝐵‘𝑖)) = (𝐵‘𝑖))
25	22, 24	eqtrid 2809	. . . . . . . . . 10 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((1 − 0) · (𝐵‘𝑖)) = (𝐵‘𝑖))
26		subcl 11483	. . . . . . . . . . . . 13 ⊢ (((𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((𝐶‘𝑖) − (𝐷‘𝑖)) ∈ ℂ)
27		subcl 11483	. . . . . . . . . . . . 13 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ ((𝐶‘𝑖) − (𝐷‘𝑖)) ∈ ℂ) → ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ ℂ)
28	26, 27	sylan2 605	. . . . . . . . . . . 12 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ ((𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ)) → ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ ℂ)
29	28	3impb 1132	. . . . . . . . . . 11 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ ℂ)
30	29	mul02d 11435	. . . . . . . . . 10 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))) = 0)
31	25, 30	oveq12d 7434	. . . . . . . . 9 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → (((1 − 0) · (𝐵‘𝑖)) + (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))) = ((𝐵‘𝑖) + 0))
32		addrid 11417	. . . . . . . . . 10 ⊢ ((𝐵‘𝑖) ∈ ℂ → ((𝐵‘𝑖) + 0) = (𝐵‘𝑖))
33	32	3ad2ant1 1151	. . . . . . . . 9 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((𝐵‘𝑖) + 0) = (𝐵‘𝑖))
34	31, 33	eqtr2d 2798	. . . . . . . 8 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → (𝐵‘𝑖) = (((1 − 0) · (𝐵‘𝑖)) + (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))))
35	16, 18, 20, 34	syl3anc 1398	. . . . . . 7 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (𝐵‘𝑖) = (((1 − 0) · (𝐵‘𝑖)) + (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))))
36	35	ralrimiva 3156	. . . . . 6 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 0) · (𝐵‘𝑖)) + (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))))
37	18, 20	subcld 11596	. . . . . . . . 9 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝐶‘𝑖) − (𝐷‘𝑖)) ∈ ℂ)
38	16, 37	nncand 11601	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))) = ((𝐶‘𝑖) − (𝐷‘𝑖)))
39	38	oveq1d 7431	. . . . . . 7 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))↑2) = (((𝐶‘𝑖) − (𝐷‘𝑖))↑2))
40	39	sumeq2dv 15791	. . . . . 6 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))
41		0elunit 13524	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
42		fveq1 6881	. . . . . . . . . . . . . 14 ⊢ (𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) → (𝑥‘𝑖) = ((𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))))‘𝑖))
43		fveq2 6882	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑖 → (𝐵‘𝑘) = (𝐵‘𝑖))
44		fveq2 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (𝐶‘𝑘) = (𝐶‘𝑖))
45		fveq2 6882	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (𝐷‘𝑘) = (𝐷‘𝑖))
46	44, 45	oveq12d 7434	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑖 → ((𝐶‘𝑘) − (𝐷‘𝑘)) = ((𝐶‘𝑖) − (𝐷‘𝑖)))
47	43, 46	oveq12d 7434	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑖 → ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) = ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))
48		eqid 2762	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))))
49		ovex 7449	. . . . . . . . . . . . . . 15 ⊢ ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ V
50	47, 48, 49	fvmpt 6990	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑁) → ((𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))))‘𝑖) = ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))
51	42, 50	sylan9eq 2817	. . . . . . . . . . . . 13 ⊢ ((𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) = ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))
52	51	oveq2d 7432	. . . . . . . . . . . 12 ⊢ ((𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∧ 𝑖 ∈ (1...𝑁)) → (𝑡 · (𝑥‘𝑖)) = (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))))
53	52	oveq2d 7432	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))))
54	53	eqeq2d 2773	. . . . . . . . . 10 ⊢ ((𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ↔ (𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))))))
55	54	ralbidva 3185	. . . . . . . . 9 ⊢ (𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) → (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))))))
56	51	oveq2d 7432	. . . . . . . . . . . 12 ⊢ ((𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∧ 𝑖 ∈ (1...𝑁)) → ((𝐵‘𝑖) − (𝑥‘𝑖)) = ((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))))
57	56	oveq1d 7431	. . . . . . . . . . 11 ⊢ ((𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∧ 𝑖 ∈ (1...𝑁)) → (((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = (((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))↑2))
58	57	sumeq2dv 15791	. . . . . . . . . 10 ⊢ (𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) → Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))↑2))
59	58	eqeq1d 2764	. . . . . . . . 9 ⊢ (𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) → (Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2) ↔ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
60	55, 59	anbi12d 644	. . . . . . . 8 ⊢ (𝑥 = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) → ((∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)) ↔ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))))
61		oveq2 7424	. . . . . . . . . . . . 13 ⊢ (𝑡 = 0 → (1 − 𝑡) = (1 − 0))
62	61	oveq1d 7431	. . . . . . . . . . . 12 ⊢ (𝑡 = 0 → ((1 − 𝑡) · (𝐵‘𝑖)) = ((1 − 0) · (𝐵‘𝑖)))
63		oveq1 7423	. . . . . . . . . . . 12 ⊢ (𝑡 = 0 → (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))) = (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))))
64	62, 63	oveq12d 7434	. . . . . . . . . . 11 ⊢ (𝑡 = 0 → (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))) = (((1 − 0) · (𝐵‘𝑖)) + (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))))
65	64	eqeq2d 2773	. . . . . . . . . 10 ⊢ (𝑡 = 0 → ((𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))) ↔ (𝐵‘𝑖) = (((1 − 0) · (𝐵‘𝑖)) + (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))))))
66	65	ralbidv 3187	. . . . . . . . 9 ⊢ (𝑡 = 0 → (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))) ↔ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 0) · (𝐵‘𝑖)) + (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))))))
67	66	anbi1d 643	. . . . . . . 8 ⊢ (𝑡 = 0 → ((∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)) ↔ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 0) · (𝐵‘𝑖)) + (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))))
68	60, 67	rspc2ev 3592	. . . . . . 7 ⊢ (((𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∈ (𝔼‘𝑁) ∧ 0 ∈ (0[,]1) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 0) · (𝐵‘𝑖)) + (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
69	41, 68	mp3an2 1478	. . . . . 6 ⊢ (((𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∈ (𝔼‘𝑁) ∧ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 0) · (𝐵‘𝑖)) + (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
70	14, 36, 40, 69	syl12anc 850	. . . . 5 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
71	70	3expb 1138	. . . 4 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
72	71	adantll 727	. . 3 ⊢ (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)))
73		fveq1 6881	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑖) = (𝐵‘𝑖))
74	73	oveq2d 7432	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → ((1 − 𝑡) · (𝐴‘𝑖)) = ((1 − 𝑡) · (𝐵‘𝑖)))
75	74	oveq1d 7431	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))))
76	75	eqeq2d 2773	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐵‘𝑖) = (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ↔ (𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖)))))
77	76	ralbidv 3187	. . . . 5 ⊢ (𝐴 = 𝐵 → (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖)))))
78	77	anbi1d 643	. . . 4 ⊢ (𝐴 = 𝐵 → ((∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)) ↔ (∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))))
79	78	2rexbidv 3229	. . 3 ⊢ (𝐴 = 𝐵 → (∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2)) ↔ ∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))))
80	72, 79	imbitrrid 249	. 2 ⊢ (𝐴 = 𝐵 → (((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁))) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑2))))
81	80	imp 412	1 ⊢ ((𝐴 = 𝐵 ∧ ((𝐴 ∈ (𝔼‘𝑁) ∧ 𝐵 ∈ (𝔼‘𝑁)) ∧ (𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)))) → ∃𝑥 ∈ (𝔼‘𝑁)∃𝑡 ∈ (0[,]1)(∀𝑖 ∈ (1...𝑁)(𝐵‘𝑖) = (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ∧ Σ𝑖 ∈ (1...𝑁)(((𝐵‘𝑖) − (𝑥‘𝑖))↑2) = Σ𝑖 ∈ (1...𝑁)(((𝐶‘𝑖) − (𝐷‘𝑖))↑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 ∃wrex 3088 ↦ cmpt 5190 ‘cfv 6537 (class class class)co 7416 ℂcc 11125 ℝcr 11126 0cc0 11127 1c1 11128 + caddc 11130 · cmul 11132 − cmin 11468 ℕcn 12260 2c2 12322 [,]cicc 13403 ...cfz 13563 ↑cexp 14127 Σcsu 15775 𝔼cee 29330

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-sep 5255 ax-nul 5267 ax-pow 5334 ax-pr 5402 ax-un 7739 ax-cnex 11183 ax-resscn 11184 ax-1cn 11185 ax-icn 11186 ax-addcl 11187 ax-addrcl 11188 ax-mulcl 11189 ax-mulrcl 11190 ax-mulcom 11191 ax-addass 11192 ax-mulass 11193 ax-distr 11194 ax-i2m1 11195 ax-1ne0 11196 ax-1rid 11197 ax-rnegex 11198 ax-rrecex 11199 ax-cnre 11200 ax-pre-lttri 11201 ax-pre-lttrn 11202 ax-pre-ltadd 11203 ax-pre-mulgt0 11204

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-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-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-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-riota 7373 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7866 df-1st 7989 df-2nd 7990 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8699 df-map 8831 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11470 df-neg 11471 df-nn 12261 df-n0 12532 df-z 12619 df-uz 12891 df-icc 13407 df-fz 13564 df-seq 14068 df-sum 15776 df-ee 29333

This theorem is used by: axsegcon 29370

W3C validator