eenglngeehlnmlem1 - Mathbox for Alexander van der Vekens

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Theorem eenglngeehlnmlem1 48125

Description: Lemma 1 for eenglngeehlnm 48127. (Contributed by AV, 15-Feb-2023.)

**Assertion**
Ref	Expression
eenglngeehlnmlem1	⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) → ((∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))

Distinct variable group: 𝑖,𝑁,𝑘,𝑙,𝑚,𝑝,𝑡,𝑥,𝑦

**Proof of Theorem eenglngeehlnmlem1**
Step	Hyp	Ref	Expression
1		oveq2 7432	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → (1 − 𝑘) = (1 − 𝑡))
2	1	oveq1d 7439	. . . . . . 7 ⊢ (𝑘 = 𝑡 → ((1 − 𝑘) · (𝑥‘𝑖)) = ((1 − 𝑡) · (𝑥‘𝑖)))
3		oveq1 7431	. . . . . . 7 ⊢ (𝑘 = 𝑡 → (𝑘 · (𝑦‘𝑖)) = (𝑡 · (𝑦‘𝑖)))
4	2, 3	oveq12d 7442	. . . . . 6 ⊢ (𝑘 = 𝑡 → (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))))
5	4	eqeq2d 2737	. . . . 5 ⊢ (𝑘 = 𝑡 → ((𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ↔ (𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
6	5	ralbidv 3168	. . . 4 ⊢ (𝑘 = 𝑡 → (∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
7	6	cbvrexvw 3226	. . 3 ⊢ (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))))
8		unitssre 13530	. . . 4 ⊢ (0[,]1) ⊆ ℝ
9		ssrexv 4049	. . . 4 ⊢ ((0[,]1) ⊆ ℝ → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
10	8, 9	mp1i 13	. . 3 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
11	7, 10	biimtrid 241	. 2 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
12		0re 11266	. . . . . . . 8 ⊢ 0 ∈ ℝ
13		1xr 11323	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
14		elico2 13442	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ^*) → (𝑙 ∈ (0[,)1) ↔ (𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1)))
15	12, 13, 14	mp2an 690	. . . . . . 7 ⊢ (𝑙 ∈ (0[,)1) ↔ (𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1))
16		simp1 1133	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 𝑙 ∈ ℝ)
17		1red 11265	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ∈ ℝ)
18	17, 16	resubcld 11692	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (1 − 𝑙) ∈ ℝ)
19		1cnd 11259	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ∈ ℂ)
20	16	recnd 11292	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 𝑙 ∈ ℂ)
21		ltne 11361	. . . . . . . . . 10 ⊢ ((𝑙 ∈ ℝ ∧ 𝑙 < 1) → 1 ≠ 𝑙)
22	21	3adant2 1128	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ≠ 𝑙)
23	19, 20, 22	subne0d 11630	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (1 − 𝑙) ≠ 0)
24	16, 18, 23	redivcld 12093	. . . . . . 7 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (𝑙 / (1 − 𝑙)) ∈ ℝ)
25	15, 24	sylbi 216	. . . . . 6 ⊢ (𝑙 ∈ (0[,)1) → (𝑙 / (1 − 𝑙)) ∈ ℝ)
26	25	ad2antlr 725	. . . . 5 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))) → (𝑙 / (1 − 𝑙)) ∈ ℝ)
27	26	renegcld 11691	. . . 4 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))) → -(𝑙 / (1 − 𝑙)) ∈ ℝ)
28		oveq2 7432	. . . . . . . . 9 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → (1 − 𝑡) = (1 − -(𝑙 / (1 − 𝑙))))
29	28	oveq1d 7439	. . . . . . . 8 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → ((1 − 𝑡) · (𝑥‘𝑖)) = ((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)))
30		oveq1 7431	. . . . . . . 8 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → (𝑡 · (𝑦‘𝑖)) = (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))
31	29, 30	oveq12d 7442	. . . . . . 7 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))))
32	31	eqeq2d 2737	. . . . . 6 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → ((𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) ↔ (𝑝‘𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
33	32	ralbidv 3168	. . . . 5 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → (∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
34	33	adantl 480	. . . 4 ⊢ ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))) ∧ 𝑡 = -(𝑙 / (1 − 𝑙))) → (∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
35		eqcom 2733	. . . . . . . 8 ⊢ ((𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) = (𝑥‘𝑖))
36		elmapi 8878	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℝ ↑_m (1...𝑁)) → 𝑥:(1...𝑁)⟶ℝ)
37	36	3ad2ant2 1131	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) → 𝑥:(1...𝑁)⟶ℝ)
38	37	ad2antrr 724	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) → 𝑥:(1...𝑁)⟶ℝ)
39	38	ffvelcdmda 7098	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) ∈ ℝ)
40	39	recnd 11292	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) ∈ ℂ)
41	15, 16	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (0[,)1) → 𝑙 ∈ ℝ)
42	41	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑙 ∈ ℝ)
43	42	recnd 11292	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑙 ∈ ℂ)
44		eldifi 4126	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥}) → 𝑦 ∈ (ℝ ↑_m (1...𝑁)))
45		elmapi 8878	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (ℝ ↑_m (1...𝑁)) → 𝑦:(1...𝑁)⟶ℝ)
46	44, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥}) → 𝑦:(1...𝑁)⟶ℝ)
47	46	3ad2ant3 1132	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) → 𝑦:(1...𝑁)⟶ℝ)
48	47	ad2antrr 724	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) → 𝑦:(1...𝑁)⟶ℝ)
49	48	ffvelcdmda 7098	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦‘𝑖) ∈ ℝ)
50	49	recnd 11292	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦‘𝑖) ∈ ℂ)
51	43, 50	mulcld 11284	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑙 · (𝑦‘𝑖)) ∈ ℂ)
52		1cnd 11259	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → 1 ∈ ℂ)
53	52, 43	subcld 11621	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑙) ∈ ℂ)
54		elmapi 8878	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (ℝ ↑_m (1...𝑁)) → 𝑝:(1...𝑁)⟶ℝ)
55	54	ad2antlr 725	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) → 𝑝:(1...𝑁)⟶ℝ)
56	55	ffvelcdmda 7098	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝‘𝑖) ∈ ℝ)
57	56	recnd 11292	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝‘𝑖) ∈ ℂ)
58	53, 57	mulcld 11284	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − 𝑙) · (𝑝‘𝑖)) ∈ ℂ)
59	40, 51, 58	subadd2d 11640	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) = ((1 − 𝑙) · (𝑝‘𝑖)) ↔ (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) = (𝑥‘𝑖)))
60	35, 59	bitr4id 289	. . . . . . 7 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ ((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) = ((1 − 𝑙) · (𝑝‘𝑖))))
61		eqcom 2733	. . . . . . . . 9 ⊢ (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) = ((1 − 𝑙) · (𝑝‘𝑖)) ↔ ((1 − 𝑙) · (𝑝‘𝑖)) = ((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))))
62	40, 51	subcld 11621	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) ∈ ℂ)
63	15, 22	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (0[,)1) → 1 ≠ 𝑙)
64	63	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → 1 ≠ 𝑙)
65	52, 43, 64	subne0d 11630	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑙) ≠ 0)
66	62, 53, 57, 65	divmuld 12063	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)) = (𝑝‘𝑖) ↔ ((1 − 𝑙) · (𝑝‘𝑖)) = ((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖)))))
67	61, 66	bitr4id 289	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) = ((1 − 𝑙) · (𝑝‘𝑖)) ↔ (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)) = (𝑝‘𝑖)))
68		eqcom 2733	. . . . . . . . . 10 ⊢ ((((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)) = (𝑝‘𝑖) ↔ (𝑝‘𝑖) = (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)))
69	40, 51, 53, 65	divsubdird 12080	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)) = (((𝑥‘𝑖) / (1 − 𝑙)) − ((𝑙 · (𝑦‘𝑖)) / (1 − 𝑙))))
70	40, 53, 65	divrec2d 12045	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥‘𝑖) / (1 − 𝑙)) = ((1 / (1 − 𝑙)) · (𝑥‘𝑖)))
71	43, 50, 53, 65	div23d 12078	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑙 · (𝑦‘𝑖)) / (1 − 𝑙)) = ((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))
72	70, 71	oveq12d 7442	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥‘𝑖) / (1 − 𝑙)) − ((𝑙 · (𝑦‘𝑖)) / (1 − 𝑙))) = (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))))
73	69, 72	eqtrd 2766	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)) = (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))))
74	73	eqeq2d 2737	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝‘𝑖) = (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)) ↔ (𝑝‘𝑖) = (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
75	68, 74	bitrid 282	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)) = (𝑝‘𝑖) ↔ (𝑝‘𝑖) = (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
76	43, 53, 65	divcld 12041	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑙 / (1 − 𝑙)) ∈ ℂ)
77	76, 50	mulneg1d 11717	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)) = -((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))
78	77	eqcomd 2732	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → -((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)) = (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))
79	78	oveq2d 7440	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) + -((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))) = (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))))
80	53, 65	reccld 12034	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 / (1 − 𝑙)) ∈ ℂ)
81	80, 40	mulcld 11284	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 / (1 − 𝑙)) · (𝑥‘𝑖)) ∈ ℂ)
82	76, 50	mulcld 11284	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)) ∈ ℂ)
83	81, 82	negsubd 11627	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) + -((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))) = (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))))
84	52, 76	subnegd 11628	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − -(𝑙 / (1 − 𝑙))) = (1 + (𝑙 / (1 − 𝑙))))
85		muldivdir 11958	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℂ ∧ 𝑙 ∈ ℂ ∧ ((1 − 𝑙) ∈ ℂ ∧ (1 − 𝑙) ≠ 0)) → ((((1 − 𝑙) · 1) + 𝑙) / (1 − 𝑙)) = (1 + (𝑙 / (1 − 𝑙))))
86	52, 43, 53, 65, 85	syl112anc 1371	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((1 − 𝑙) · 1) + 𝑙) / (1 − 𝑙)) = (1 + (𝑙 / (1 − 𝑙))))
87	53	mulridd 11281	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − 𝑙) · 1) = (1 − 𝑙))
88	87	oveq1d 7439	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − 𝑙) · 1) + 𝑙) = ((1 − 𝑙) + 𝑙))
89	52, 43	npcand 11625	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − 𝑙) + 𝑙) = 1)
90	88, 89	eqtrd 2766	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − 𝑙) · 1) + 𝑙) = 1)
91	90	oveq1d 7439	. . . . . . . . . . . . . . . 16 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((1 − 𝑙) · 1) + 𝑙) / (1 − 𝑙)) = (1 / (1 − 𝑙)))
92	84, 86, 91	3eqtr2d 2772	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − -(𝑙 / (1 − 𝑙))) = (1 / (1 − 𝑙)))
93	92	eqcomd 2732	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 / (1 − 𝑙)) = (1 − -(𝑙 / (1 − 𝑙))))
94	93	oveq1d 7439	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 / (1 − 𝑙)) · (𝑥‘𝑖)) = ((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)))
95	94	oveq1d 7439	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))))
96	79, 83, 95	3eqtr3d 2774	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))))
97	96	eqeq2d 2737	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝‘𝑖) = (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))) ↔ (𝑝‘𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
98	97	biimpd 228	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝‘𝑖) = (((1 / (1 − 𝑙)) · (𝑥‘𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))) → (𝑝‘𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
99	75, 98	sylbid 239	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)) = (𝑝‘𝑖) → (𝑝‘𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
100	67, 99	sylbid 239	. . . . . . 7 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) = ((1 − 𝑙) · (𝑝‘𝑖)) → (𝑝‘𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
101	60, 100	sylbid 239	. . . . . 6 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) → (𝑝‘𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
102	101	ralimdva 3157	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) → (∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) → ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))))
103	102	imp 405	. . . 4 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))) → ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖))))
104	27, 34, 103	rspcedvd 3610	. . 3 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))))
105	104	rexlimdva2 3147	. 2 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) → (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
106		0xr 11311	. . . . . . 7 ⊢ 0 ∈ ℝ^*
107		1re 11264	. . . . . . 7 ⊢ 1 ∈ ℝ
108		elioc2 13441	. . . . . . 7 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ) → (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1)))
109	106, 107, 108	mp2an 690	. . . . . 6 ⊢ (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1))
110		simp1 1133	. . . . . . 7 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → 𝑚 ∈ ℝ)
111		gt0ne0 11729	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚) → 𝑚 ≠ 0)
112	111	3adant3 1129	. . . . . . 7 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → 𝑚 ≠ 0)
113	110, 112	rereccld 12092	. . . . . 6 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → (1 / 𝑚) ∈ ℝ)
114	109, 113	sylbi 216	. . . . 5 ⊢ (𝑚 ∈ (0(,]1) → (1 / 𝑚) ∈ ℝ)
115	114	ad2antlr 725	. . . 4 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))) → (1 / 𝑚) ∈ ℝ)
116		oveq2 7432	. . . . . . . . 9 ⊢ (𝑡 = (1 / 𝑚) → (1 − 𝑡) = (1 − (1 / 𝑚)))
117	116	oveq1d 7439	. . . . . . . 8 ⊢ (𝑡 = (1 / 𝑚) → ((1 − 𝑡) · (𝑥‘𝑖)) = ((1 − (1 / 𝑚)) · (𝑥‘𝑖)))
118		oveq1 7431	. . . . . . . 8 ⊢ (𝑡 = (1 / 𝑚) → (𝑡 · (𝑦‘𝑖)) = ((1 / 𝑚) · (𝑦‘𝑖)))
119	117, 118	oveq12d 7442	. . . . . . 7 ⊢ (𝑡 = (1 / 𝑚) → (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖))))
120	119	eqeq2d 2737	. . . . . 6 ⊢ (𝑡 = (1 / 𝑚) → ((𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) ↔ (𝑝‘𝑖) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖)))))
121	120	ralbidv 3168	. . . . 5 ⊢ (𝑡 = (1 / 𝑚) → (∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖)))))
122	121	adantl 480	. . . 4 ⊢ ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))) ∧ 𝑡 = (1 / 𝑚)) → (∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖)))))
123		eqcom 2733	. . . . . . . 8 ⊢ ((𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) = (𝑦‘𝑖))
124	47	ad2antrr 724	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) → 𝑦:(1...𝑁)⟶ℝ)
125	124	ffvelcdmda 7098	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦‘𝑖) ∈ ℝ)
126	125	recnd 11292	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦‘𝑖) ∈ ℂ)
127		1cnd 11259	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → 1 ∈ ℂ)
128	109, 110	sylbi 216	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (0(,]1) → 𝑚 ∈ ℝ)
129	128	recnd 11292	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (0(,]1) → 𝑚 ∈ ℂ)
130	129	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑚 ∈ ℂ)
131	127, 130	subcld 11621	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑚) ∈ ℂ)
132	37	ad2antrr 724	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) → 𝑥:(1...𝑁)⟶ℝ)
133	132	ffvelcdmda 7098	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) ∈ ℝ)
134	133	recnd 11292	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) ∈ ℂ)
135	131, 134	mulcld 11284	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − 𝑚) · (𝑥‘𝑖)) ∈ ℂ)
136	126, 135	negsubd 11627	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦‘𝑖) + -((1 − 𝑚) · (𝑥‘𝑖))) = ((𝑦‘𝑖) − ((1 − 𝑚) · (𝑥‘𝑖))))
137	131, 134	mulneg1d 11717	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (-(1 − 𝑚) · (𝑥‘𝑖)) = -((1 − 𝑚) · (𝑥‘𝑖)))
138	127, 130	negsubdi2d 11637	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → -(1 − 𝑚) = (𝑚 − 1))
139	138	oveq1d 7439	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (-(1 − 𝑚) · (𝑥‘𝑖)) = ((𝑚 − 1) · (𝑥‘𝑖)))
140	137, 139	eqtr3d 2768	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → -((1 − 𝑚) · (𝑥‘𝑖)) = ((𝑚 − 1) · (𝑥‘𝑖)))
141	140	oveq2d 7440	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦‘𝑖) + -((1 − 𝑚) · (𝑥‘𝑖))) = ((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))))
142	136, 141	eqtr3d 2768	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦‘𝑖) − ((1 − 𝑚) · (𝑥‘𝑖))) = ((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))))
143	142	eqeq1d 2728	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦‘𝑖) − ((1 − 𝑚) · (𝑥‘𝑖))) = (𝑚 · (𝑝‘𝑖)) ↔ ((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) = (𝑚 · (𝑝‘𝑖))))
144	54	ad2antlr 725	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) → 𝑝:(1...𝑁)⟶ℝ)
145	144	ffvelcdmda 7098	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝‘𝑖) ∈ ℝ)
146	145	recnd 11292	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝‘𝑖) ∈ ℂ)
147	130, 146	mulcld 11284	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑚 · (𝑝‘𝑖)) ∈ ℂ)
148	126, 135, 147	subaddd 11639	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦‘𝑖) − ((1 − 𝑚) · (𝑥‘𝑖))) = (𝑚 · (𝑝‘𝑖)) ↔ (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) = (𝑦‘𝑖)))
149		eqcom 2733	. . . . . . . . . . 11 ⊢ ((((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) / 𝑚) = (𝑝‘𝑖) ↔ (𝑝‘𝑖) = (((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) / 𝑚))
150	149	a1i 11	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) / 𝑚) = (𝑝‘𝑖) ↔ (𝑝‘𝑖) = (((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) / 𝑚)))
151	130, 127	subcld 11621	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑚 − 1) ∈ ℂ)
152	151, 134	mulcld 11284	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑚 − 1) · (𝑥‘𝑖)) ∈ ℂ)
153	126, 152	addcld 11283	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) ∈ ℂ)
154		elioc1 13420	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^) → (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ^ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1)))
155	106, 13, 154	mp2an 690	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ^* ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1))
156	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℝ^* → 0 ∈ ℝ)
157	156	anim1i 613	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℝ^* ∧ 0 < 𝑚) → (0 ∈ ℝ ∧ 0 < 𝑚))
158	157	3adant3 1129	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℝ^* ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → (0 ∈ ℝ ∧ 0 < 𝑚))
159		ltne 11361	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ ∧ 0 < 𝑚) → 𝑚 ≠ 0)
160	158, 159	syl 17	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ ℝ^* ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → 𝑚 ≠ 0)
161	155, 160	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0(,]1) → 𝑚 ≠ 0)
162	161	ad2antlr 725	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑚 ≠ 0)
163	153, 146, 130, 162	divmul2d 12074	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) / 𝑚) = (𝑝‘𝑖) ↔ ((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) = (𝑚 · (𝑝‘𝑖))))
164	126, 152, 130, 162	divdird 12079	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) / 𝑚) = (((𝑦‘𝑖) / 𝑚) + (((𝑚 − 1) · (𝑥‘𝑖)) / 𝑚)))
165	126, 130, 162	divrec2d 12045	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦‘𝑖) / 𝑚) = ((1 / 𝑚) · (𝑦‘𝑖)))
166	151, 134, 130, 162	div23d 12078	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑚 − 1) · (𝑥‘𝑖)) / 𝑚) = (((𝑚 − 1) / 𝑚) · (𝑥‘𝑖)))
167	130, 127, 130, 162	divsubdird 12080	. . . . . . . . . . . . . . 15 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑚 − 1) / 𝑚) = ((𝑚 / 𝑚) − (1 / 𝑚)))
168	167	oveq1d 7439	. . . . . . . . . . . . . 14 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑚 − 1) / 𝑚) · (𝑥‘𝑖)) = (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖)))
169	166, 168	eqtrd 2766	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑚 − 1) · (𝑥‘𝑖)) / 𝑚) = (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖)))
170	165, 169	oveq12d 7442	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦‘𝑖) / 𝑚) + (((𝑚 − 1) · (𝑥‘𝑖)) / 𝑚)) = (((1 / 𝑚) · (𝑦‘𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖))))
171	164, 170	eqtrd 2766	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) / 𝑚) = (((1 / 𝑚) · (𝑦‘𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖))))
172	171	eqeq2d 2737	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝‘𝑖) = (((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) / 𝑚) ↔ (𝑝‘𝑖) = (((1 / 𝑚) · (𝑦‘𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖)))))
173	150, 163, 172	3bitr3d 308	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦‘𝑖) + ((𝑚 − 1) · (𝑥‘𝑖))) = (𝑚 · (𝑝‘𝑖)) ↔ (𝑝‘𝑖) = (((1 / 𝑚) · (𝑦‘𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖)))))
174	143, 148, 173	3bitr3d 308	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) = (𝑦‘𝑖) ↔ (𝑝‘𝑖) = (((1 / 𝑚) · (𝑦‘𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖)))))
175	123, 174	bitrid 282	. . . . . . 7 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ (𝑝‘𝑖) = (((1 / 𝑚) · (𝑦‘𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖)))))
176	130, 162	reccld 12034	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑚) ∈ ℂ)
177	176, 126	mulcld 11284	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 / 𝑚) · (𝑦‘𝑖)) ∈ ℂ)
178	127, 176	subcld 11621	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − (1 / 𝑚)) ∈ ℂ)
179	178, 134	mulcld 11284	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − (1 / 𝑚)) · (𝑥‘𝑖)) ∈ ℂ)
180	130, 162	dividd 12039	. . . . . . . . . . . . 13 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑚 / 𝑚) = 1)
181	180	oveq1d 7439	. . . . . . . . . . . 12 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑚 / 𝑚) − (1 / 𝑚)) = (1 − (1 / 𝑚)))
182	181	oveq1d 7439	. . . . . . . . . . 11 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖)) = ((1 − (1 / 𝑚)) · (𝑥‘𝑖)))
183	182	oveq2d 7440	. . . . . . . . . 10 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / 𝑚) · (𝑦‘𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖))) = (((1 / 𝑚) · (𝑦‘𝑖)) + ((1 − (1 / 𝑚)) · (𝑥‘𝑖))))
184	177, 179, 183	comraddd 11478	. . . . . . . . 9 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / 𝑚) · (𝑦‘𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖))) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖))))
185	184	eqeq2d 2737	. . . . . . . 8 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝‘𝑖) = (((1 / 𝑚) · (𝑦‘𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖))) ↔ (𝑝‘𝑖) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖)))))
186	185	biimpd 228	. . . . . . 7 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝‘𝑖) = (((1 / 𝑚) · (𝑦‘𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥‘𝑖))) → (𝑝‘𝑖) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖)))))
187	175, 186	sylbid 239	. . . . . 6 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) → (𝑝‘𝑖) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖)))))
188	187	ralimdva 3157	. . . . 5 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) → (∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) → ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖)))))
189	188	imp 405	. . . 4 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))) → ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖))))
190	115, 122, 189	rspcedvd 3610	. . 3 ⊢ (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))))
191	190	rexlimdva2 3147	. 2 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) → (∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
192	11, 105, 191	3jaod 1426	1 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑_m (1...𝑁))) → ((∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 394 ∨ w3o 1083 ∧ w3a 1084 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∀wral 3051 ∃wrex 3060 ∖ cdif 3944 ⊆ wss 3947 {csn 4633 class class class wbr 5153 ⟶wf 6550 ‘cfv 6554 (class class class)co 7424 ↑_m cmap 8855 ℂcc 11156 ℝcr 11157 0cc0 11158 1c1 11159 + caddc 11161 · cmul 11163 ℝ^*cxr 11297 < clt 11298 ≤ cle 11299 − cmin 11494 -cneg 11495 / cdiv 11921 ℕcn 12264 (,]cioc 13379 [,)cico 13380 [,]cicc 13381 ...cfz 13538

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5304 ax-nul 5311 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11214 ax-resscn 11215 ax-1cn 11216 ax-icn 11217 ax-addcl 11218 ax-addrcl 11219 ax-mulcl 11220 ax-mulrcl 11221 ax-mulcom 11222 ax-addass 11223 ax-mulass 11224 ax-distr 11225 ax-i2m1 11226 ax-1ne0 11227 ax-1rid 11228 ax-rnegex 11229 ax-rrecex 11230 ax-cnre 11231 ax-pre-lttri 11232 ax-pre-lttrn 11233 ax-pre-ltadd 11234 ax-pre-mulgt0 11235

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3364 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-iun 5003 df-br 5154 df-opab 5216 df-mpt 5237 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6506 df-fun 6556 df-fn 6557 df-f 6558 df-f1 6559 df-fo 6560 df-f1o 6561 df-fv 6562 df-riota 7380 df-ov 7427 df-oprab 7428 df-mpo 7429 df-1st 8003 df-2nd 8004 df-er 8734 df-map 8857 df-en 8975 df-dom 8976 df-sdom 8977 df-pnf 11300 df-mnf 11301 df-xr 11302 df-ltxr 11303 df-le 11304 df-sub 11496 df-neg 11497 df-div 11922 df-ioc 13383 df-ico 13384 df-icc 13385

This theorem is referenced by: eenglngeehlnm 48127

W3C validator