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Theorem axsegconlem1 26223

Description: Lemma for axsegcon 26233. 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 26207	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℝ)
2	1	3ad2antl1 1240	. . . . . . . . 9 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℝ)
3		fveere 26207	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝑘 ∈ (1...𝑁)) → (𝐶‘𝑘) ∈ ℝ)
4	3	3ad2antl2 1241	. . . . . . . . . 10 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → (𝐶‘𝑘) ∈ ℝ)
5		fveere 26207	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (𝔼‘𝑁) ∧ 𝑘 ∈ (1...𝑁)) → (𝐷‘𝑘) ∈ ℝ)
6	5	3ad2antl3 1242	. . . . . . . . . 10 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → (𝐷‘𝑘) ∈ ℝ)
7	4, 6	resubcld 10789	. . . . . . . . 9 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐶‘𝑘) − (𝐷‘𝑘)) ∈ ℝ)
8	2, 7	resubcld 10789	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) ∈ ℝ)
9	8	ralrimiva 3175	. . . . . . 7 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ∀𝑘 ∈ (1...𝑁)((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) ∈ ℝ)
10		eleenn 26202	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
11		mptelee 26201	. . . . . . . . 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 17	. . . . . . . 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 1167	. . . . . . 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 249	. . . . . 6 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∈ (𝔼‘𝑁))
15		fveecn 26208	. . . . . . . . 9 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐵‘𝑖) ∈ ℂ)
16	15	3ad2antl1 1240	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (𝐵‘𝑖) ∈ ℂ)
17		fveecn 26208	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐶‘𝑖) ∈ ℂ)
18	17	3ad2antl2 1241	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (𝐶‘𝑖) ∈ ℂ)
19		fveecn 26208	. . . . . . . . 9 ⊢ ((𝐷 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐷‘𝑖) ∈ ℂ)
20	19	3ad2antl3 1242	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (𝐷‘𝑖) ∈ ℂ)
21		1m0e1 11486	. . . . . . . . . . . 12 ⊢ (1 − 0) = 1
22	21	oveq1i 6920	. . . . . . . . . . 11 ⊢ ((1 − 0) · (𝐵‘𝑖)) = (1 · (𝐵‘𝑖))
23		mulid2 10362	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝑖) ∈ ℂ → (1 · (𝐵‘𝑖)) = (𝐵‘𝑖))
24	23	3ad2ant1 1167	. . . . . . . . . . 11 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → (1 · (𝐵‘𝑖)) = (𝐵‘𝑖))
25	22, 24	syl5eq 2873	. . . . . . . . . 10 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((1 − 0) · (𝐵‘𝑖)) = (𝐵‘𝑖))
26		subcl 10607	. . . . . . . . . . . . 13 ⊢ (((𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((𝐶‘𝑖) − (𝐷‘𝑖)) ∈ ℂ)
27		subcl 10607	. . . . . . . . . . . . 13 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ ((𝐶‘𝑖) − (𝐷‘𝑖)) ∈ ℂ) → ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ ℂ)
28	26, 27	sylan2 586	. . . . . . . . . . . 12 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ ((𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ)) → ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ ℂ)
29	28	3impb 1147	. . . . . . . . . . 11 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ ℂ)
30	29	mul02d 10560	. . . . . . . . . 10 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))) = 0)
31	25, 30	oveq12d 6928	. . . . . . . . 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		addid1 10542	. . . . . . . . . 10 ⊢ ((𝐵‘𝑖) ∈ ℂ → ((𝐵‘𝑖) + 0) = (𝐵‘𝑖))
33	32	3ad2ant1 1167	. . . . . . . . 9 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((𝐵‘𝑖) + 0) = (𝐵‘𝑖))
34	31, 33	eqtr2d 2862	. . . . . . . 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 1494	. . . . . . 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 3175	. . . . . 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 10720	. . . . . . . . 9 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝐶‘𝑖) − (𝐷‘𝑖)) ∈ ℂ)
38	16, 37	nncand 10725	. . . . . . . 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 6925	. . . . . . 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 14817	. . . . . 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 12588	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
42		fveq1 6436	. . . . . . . . . . . . . 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 6437	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑖 → (𝐵‘𝑘) = (𝐵‘𝑖))
44		fveq2 6437	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (𝐶‘𝑘) = (𝐶‘𝑖))
45		fveq2 6437	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (𝐷‘𝑘) = (𝐷‘𝑖))
46	44, 45	oveq12d 6928	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑖 → ((𝐶‘𝑘) − (𝐷‘𝑘)) = ((𝐶‘𝑖) − (𝐷‘𝑖)))
47	43, 46	oveq12d 6928	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑖 → ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) = ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))
48		eqid 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))))
49		ovex 6942	. . . . . . . . . . . . . . 15 ⊢ ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ V
50	47, 48, 49	fvmpt 6533	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑁) → ((𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))))‘𝑖) = ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))
51	42, 50	sylan9eq 2881	. . . . . . . . . . . . 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 6926	. . . . . . . . . . . 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 6926	. . . . . . . . . . 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 2835	. . . . . . . . . 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 3194	. . . . . . . . 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 6926	. . . . . . . . . . . 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 6925	. . . . . . . . . . 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 14817	. . . . . . . . . 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 2827	. . . . . . . . 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 624	. . . . . . . 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 6918	. . . . . . . . . . . . 13 ⊢ (𝑡 = 0 → (1 − 𝑡) = (1 − 0))
62	61	oveq1d 6925	. . . . . . . . . . . 12 ⊢ (𝑡 = 0 → ((1 − 𝑡) · (𝐵‘𝑖)) = ((1 − 0) · (𝐵‘𝑖)))
63		oveq1 6917	. . . . . . . . . . . 12 ⊢ (𝑡 = 0 → (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))) = (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))))
64	62, 63	oveq12d 6928	. . . . . . . . . . 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 2835	. . . . . . . . . 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 3195	. . . . . . . . 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 623	. . . . . . . 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 3541	. . . . . . 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 1577	. . . . . 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 870	. . . . 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 1153	. . . 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 705	. . 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 6436	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑖) = (𝐵‘𝑖))
74	73	oveq2d 6926	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → ((1 − 𝑡) · (𝐴‘𝑖)) = ((1 − 𝑡) · (𝐵‘𝑖)))
75	74	oveq1d 6925	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))))
76	75	eqeq2d 2835	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐵‘𝑖) = (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ↔ (𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖)))))
77	76	ralbidv 3195	. . . . 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 623	. . . 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 3267	. . 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	syl5ibr 238	. 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 397	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

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 ∧ w3a 1111 = wceq 1656 ∈ wcel 2164 ∀wral 3117 ∃wrex 3118 ↦ cmpt 4954 ‘cfv 6127 (class class class)co 6910 ℂcc 10257 ℝcr 10258 0cc0 10259 1c1 10260 + caddc 10262 · cmul 10264 − cmin 10592 ℕcn 11357 2c2 11413 [,]cicc 12473 ...cfz 12626 ↑cexp 13161 Σcsu 14800 𝔼cee 26194

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 ax-un 7214 ax-cnex 10315 ax-resscn 10316 ax-1cn 10317 ax-icn 10318 ax-addcl 10319 ax-addrcl 10320 ax-mulcl 10321 ax-mulrcl 10322 ax-mulcom 10323 ax-addass 10324 ax-mulass 10325 ax-distr 10326 ax-i2m1 10327 ax-1ne0 10328 ax-1rid 10329 ax-rnegex 10330 ax-rrecex 10331 ax-cnre 10332 ax-pre-lttri 10333 ax-pre-lttrn 10334 ax-pre-ltadd 10335 ax-pre-mulgt0 10336

This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-fal 1670 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-iun 4744 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-res 5358 df-ima 5359 df-pred 5924 df-ord 5970 df-on 5971 df-lim 5972 df-suc 5973 df-iota 6090 df-fun 6129 df-fn 6130 df-f 6131 df-f1 6132 df-fo 6133 df-f1o 6134 df-fv 6135 df-riota 6871 df-ov 6913 df-oprab 6914 df-mpt2 6915 df-om 7332 df-1st 7433 df-2nd 7434 df-wrecs 7677 df-recs 7739 df-rdg 7777 df-er 8014 df-map 8129 df-en 8229 df-dom 8230 df-sdom 8231 df-pnf 10400 df-mnf 10401 df-xr 10402 df-ltxr 10403 df-le 10404 df-sub 10594 df-neg 10595 df-nn 11358 df-n0 11626 df-z 11712 df-uz 11976 df-icc 12477 df-fz 12627 df-seq 13103 df-sum 14801 df-ee 26197

This theorem is referenced by: axsegcon 26233

W3C validator