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Theorem axsegconlem1 28866

Description: Lemma for axsegcon 28876. 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 28850	. . . . . . . . . 10 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℝ)
2	1	3ad2antl1 1186	. . . . . . . . 9 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → (𝐵‘𝑘) ∈ ℝ)
3		fveere 28850	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝑘 ∈ (1...𝑁)) → (𝐶‘𝑘) ∈ ℝ)
4	3	3ad2antl2 1187	. . . . . . . . . 10 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → (𝐶‘𝑘) ∈ ℝ)
5		fveere 28850	. . . . . . . . . . 11 ⊢ ((𝐷 ∈ (𝔼‘𝑁) ∧ 𝑘 ∈ (1...𝑁)) → (𝐷‘𝑘) ∈ ℝ)
6	5	3ad2antl3 1188	. . . . . . . . . 10 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → (𝐷‘𝑘) ∈ ℝ)
7	4, 6	resubcld 11548	. . . . . . . . 9 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐶‘𝑘) − (𝐷‘𝑘)) ∈ ℝ)
8	2, 7	resubcld 11548	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑘 ∈ (1...𝑁)) → ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) ∈ ℝ)
9	8	ralrimiva 3121	. . . . . . 7 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → ∀𝑘 ∈ (1...𝑁)((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) ∈ ℝ)
10		eleenn 28845	. . . . . . . . 9 ⊢ (𝐵 ∈ (𝔼‘𝑁) → 𝑁 ∈ ℕ)
11		mptelee 28844	. . . . . . . . 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 1133	. . . . . . 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 257	. . . . . 6 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) → (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) ∈ (𝔼‘𝑁))
15		fveecn 28851	. . . . . . . . 9 ⊢ ((𝐵 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐵‘𝑖) ∈ ℂ)
16	15	3ad2antl1 1186	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (𝐵‘𝑖) ∈ ℂ)
17		fveecn 28851	. . . . . . . . 9 ⊢ ((𝐶 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐶‘𝑖) ∈ ℂ)
18	17	3ad2antl2 1187	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (𝐶‘𝑖) ∈ ℂ)
19		fveecn 28851	. . . . . . . . 9 ⊢ ((𝐷 ∈ (𝔼‘𝑁) ∧ 𝑖 ∈ (1...𝑁)) → (𝐷‘𝑖) ∈ ℂ)
20	19	3ad2antl3 1188	. . . . . . . 8 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → (𝐷‘𝑖) ∈ ℂ)
21		1m0e1 12244	. . . . . . . . . . . 12 ⊢ (1 − 0) = 1
22	21	oveq1i 7359	. . . . . . . . . . 11 ⊢ ((1 − 0) · (𝐵‘𝑖)) = (1 · (𝐵‘𝑖))
23		mullid 11114	. . . . . . . . . . . 12 ⊢ ((𝐵‘𝑖) ∈ ℂ → (1 · (𝐵‘𝑖)) = (𝐵‘𝑖))
24	23	3ad2ant1 1133	. . . . . . . . . . 11 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → (1 · (𝐵‘𝑖)) = (𝐵‘𝑖))
25	22, 24	eqtrid 2776	. . . . . . . . . 10 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((1 − 0) · (𝐵‘𝑖)) = (𝐵‘𝑖))
26		subcl 11362	. . . . . . . . . . . . 13 ⊢ (((𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((𝐶‘𝑖) − (𝐷‘𝑖)) ∈ ℂ)
27		subcl 11362	. . . . . . . . . . . . 13 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ ((𝐶‘𝑖) − (𝐷‘𝑖)) ∈ ℂ) → ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ ℂ)
28	26, 27	sylan2 593	. . . . . . . . . . . 12 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ ((𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ)) → ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ ℂ)
29	28	3impb 1114	. . . . . . . . . . 11 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ ℂ)
30	29	mul02d 11314	. . . . . . . . . 10 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))) = 0)
31	25, 30	oveq12d 7367	. . . . . . . . 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 11296	. . . . . . . . . 10 ⊢ ((𝐵‘𝑖) ∈ ℂ → ((𝐵‘𝑖) + 0) = (𝐵‘𝑖))
33	32	3ad2ant1 1133	. . . . . . . . 9 ⊢ (((𝐵‘𝑖) ∈ ℂ ∧ (𝐶‘𝑖) ∈ ℂ ∧ (𝐷‘𝑖) ∈ ℂ) → ((𝐵‘𝑖) + 0) = (𝐵‘𝑖))
34	31, 33	eqtr2d 2765	. . . . . . . 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 1373	. . . . . . 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 3121	. . . . . 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 11475	. . . . . . . . 9 ⊢ (((𝐵 ∈ (𝔼‘𝑁) ∧ 𝐶 ∈ (𝔼‘𝑁) ∧ 𝐷 ∈ (𝔼‘𝑁)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝐶‘𝑖) − (𝐷‘𝑖)) ∈ ℂ)
38	16, 37	nncand 11480	. . . . . . . 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 7364	. . . . . . 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 15609	. . . . . 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 13372	. . . . . . 7 ⊢ 0 ∈ (0[,]1)
42		fveq1 6821	. . . . . . . . . . . . . 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 6822	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑖 → (𝐵‘𝑘) = (𝐵‘𝑖))
44		fveq2 6822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (𝐶‘𝑘) = (𝐶‘𝑖))
45		fveq2 6822	. . . . . . . . . . . . . . . . 17 ⊢ (𝑘 = 𝑖 → (𝐷‘𝑘) = (𝐷‘𝑖))
46	44, 45	oveq12d 7367	. . . . . . . . . . . . . . . 16 ⊢ (𝑘 = 𝑖 → ((𝐶‘𝑘) − (𝐷‘𝑘)) = ((𝐶‘𝑖) − (𝐷‘𝑖)))
47	43, 46	oveq12d 7367	. . . . . . . . . . . . . . 15 ⊢ (𝑘 = 𝑖 → ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))) = ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))
48		eqid 2729	. . . . . . . . . . . . . . 15 ⊢ (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘)))) = (𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))))
49		ovex 7382	. . . . . . . . . . . . . . 15 ⊢ ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))) ∈ V
50	47, 48, 49	fvmpt 6930	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ (1...𝑁) → ((𝑘 ∈ (1...𝑁) ↦ ((𝐵‘𝑘) − ((𝐶‘𝑘) − (𝐷‘𝑘))))‘𝑖) = ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖))))
51	42, 50	sylan9eq 2784	. . . . . . . . . . . . 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 7365	. . . . . . . . . . . 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 7365	. . . . . . . . . . 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 2740	. . . . . . . . . 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 3150	. . . . . . . . 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 7365	. . . . . . . . . . . 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 7364	. . . . . . . . . . 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 15609	. . . . . . . . . 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 2731	. . . . . . . . 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 632	. . . . . . . 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 7357	. . . . . . . . . . . . 13 ⊢ (𝑡 = 0 → (1 − 𝑡) = (1 − 0))
62	61	oveq1d 7364	. . . . . . . . . . . 12 ⊢ (𝑡 = 0 → ((1 − 𝑡) · (𝐵‘𝑖)) = ((1 − 0) · (𝐵‘𝑖)))
63		oveq1 7356	. . . . . . . . . . . 12 ⊢ (𝑡 = 0 → (𝑡 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))) = (0 · ((𝐵‘𝑖) − ((𝐶‘𝑖) − (𝐷‘𝑖)))))
64	62, 63	oveq12d 7367	. . . . . . . . . . 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 2740	. . . . . . . . . 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 3152	. . . . . . . . 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 631	. . . . . . . 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 3590	. . . . . . 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 1451	. . . . . 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 836	. . . . 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 1120	. . . 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 714	. . 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 6821	. . . . . . . . 9 ⊢ (𝐴 = 𝐵 → (𝐴‘𝑖) = (𝐵‘𝑖))
74	73	oveq2d 7365	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → ((1 − 𝑡) · (𝐴‘𝑖)) = ((1 − 𝑡) · (𝐵‘𝑖)))
75	74	oveq1d 7364	. . . . . . 7 ⊢ (𝐴 = 𝐵 → (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖))))
76	75	eqeq2d 2740	. . . . . 6 ⊢ (𝐴 = 𝐵 → ((𝐵‘𝑖) = (((1 − 𝑡) · (𝐴‘𝑖)) + (𝑡 · (𝑥‘𝑖))) ↔ (𝐵‘𝑖) = (((1 − 𝑡) · (𝐵‘𝑖)) + (𝑡 · (𝑥‘𝑖)))))
77	76	ralbidv 3152	. . . . 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 631	. . . 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 3194	. . 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 246	. 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 406	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ∀wral 3044 ∃wrex 3053 ↦ cmpt 5173 ‘cfv 6482 (class class class)co 7349 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 − cmin 11347 ℕcn 12128 2c2 12183 [,]cicc 13251 ...cfz 13410 ↑cexp 13968 Σcsu 15593 𝔼cee 28837

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-er 8625 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-n0 12385 df-z 12472 df-uz 12736 df-icc 13255 df-fz 13411 df-seq 13909 df-sum 15594 df-ee 28840

This theorem is referenced by: axsegcon 28876

W3C validator