eenglngeehlnmlem1 - Mathbox for Alexander van der Vekens

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Theorem eenglngeehlnmlem1 44773

Description: Lemma 1 for eenglngeehlnm 44775. (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 7164	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → (1 − 𝑘) = (1 − 𝑡))
2	1	oveq1d 7171	. . . . . . 7 ⊢ (𝑘 = 𝑡 → ((1 − 𝑘) · (𝑥‘𝑖)) = ((1 − 𝑡) · (𝑥‘𝑖)))
3		oveq1 7163	. . . . . . 7 ⊢ (𝑘 = 𝑡 → (𝑘 · (𝑦‘𝑖)) = (𝑡 · (𝑦‘𝑖)))
4	2, 3	oveq12d 7174	. . . . . 6 ⊢ (𝑘 = 𝑡 → (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))))
5	4	eqeq2d 2832	. . . . 5 ⊢ (𝑘 = 𝑡 → ((𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ↔ (𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
6	5	ralbidv 3197	. . . 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 3450	. . 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 12886	. . . 4 ⊢ (0[,]1) ⊆ ℝ
9		ssrexv 4034	. . . 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	syl5bi 244	. 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 10643	. . . . . . . 8 ⊢ 0 ∈ ℝ
13		1xr 10700	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
14		elico2 12801	. . . . . . . 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 1132	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 𝑙 ∈ ℝ)
17		1red 10642	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ∈ ℝ)
18	17, 16	resubcld 11068	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (1 − 𝑙) ∈ ℝ)
19		1cnd 10636	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ∈ ℂ)
20	16	recnd 10669	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 𝑙 ∈ ℂ)
21		ltne 10737	. . . . . . . . . 10 ⊢ ((𝑙 ∈ ℝ ∧ 𝑙 < 1) → 1 ≠ 𝑙)
22	21	3adant2 1127	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ≠ 𝑙)
23	19, 20, 22	subne0d 11006	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (1 − 𝑙) ≠ 0)
24	16, 18, 23	redivcld 11468	. . . . . . 7 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (𝑙 / (1 − 𝑙)) ∈ ℝ)
25	15, 24	sylbi 219	. . . . . 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 11067	. . . 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 7164	. . . . . . . . 9 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → (1 − 𝑡) = (1 − -(𝑙 / (1 − 𝑙))))
29	28	oveq1d 7171	. . . . . . . 8 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → ((1 − 𝑡) · (𝑥‘𝑖)) = ((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)))
30		oveq1 7163	. . . . . . . 8 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → (𝑡 · (𝑦‘𝑖)) = (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))
31	29, 30	oveq12d 7174	. . . . . . 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 2832	. . . . . 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 3197	. . . . 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 484	. . . 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		elmapi 8428	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℝ ↑_m (1...𝑁)) → 𝑥:(1...𝑁)⟶ℝ)
36	35	3ad2ant2 1130	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) → 𝑥:(1...𝑁)⟶ℝ)
37	36	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)
38	37	ffvelrnda 6851	. . . . . . . . . 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)
39	38	recnd 10669	. . . . . . . . 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)
40	15, 16	sylbi 219	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (0[,)1) → 𝑙 ∈ ℝ)
41	40	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)
42	41	recnd 10669	. . . . . . . . . 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)
43		eldifi 4103	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥}) → 𝑦 ∈ (ℝ ↑_m (1...𝑁)))
44		elmapi 8428	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (ℝ ↑_m (1...𝑁)) → 𝑦:(1...𝑁)⟶ℝ)
45	43, 44	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥}) → 𝑦:(1...𝑁)⟶ℝ)
46	45	3ad2ant3 1131	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑_m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑_m (1...𝑁)) ∖ {𝑥})) → 𝑦:(1...𝑁)⟶ℝ)
47	46	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)
48	47	ffvelrnda 6851	. . . . . . . . . . 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)
49	48	recnd 10669	. . . . . . . . . 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)
50	42, 49	mulcld 10661	. . . . . . . . 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)
51		1cnd 10636	. . . . . . . . . . 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)
52	51, 42	subcld 10997	. . . . . . . . . 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)
53		elmapi 8428	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (ℝ ↑_m (1...𝑁)) → 𝑝:(1...𝑁)⟶ℝ)
54	53	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)
55	54	ffvelrnda 6851	. . . . . . . . . . 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)
56	55	recnd 10669	. . . . . . . . . 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)
57	52, 56	mulcld 10661	. . . . . . . . 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)
58	39, 50, 57	subadd2d 11016	. . . . . . . 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)))
59		eqcom 2828	. . . . . . . 8 ⊢ ((𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) = (𝑥‘𝑖))
60	58, 59	syl6rbbr 292	. . . . . . 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	39, 50	subcld 10997	. . . . . . . . . 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)
62	15, 22	sylbi 219	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (0[,)1) → 1 ≠ 𝑙)
63	62	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)
64	51, 42, 63	subne0d 11006	. . . . . . . . . 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)
65	61, 52, 56, 64	divmuld 11438	. . . . . . . . 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)))))
66		eqcom 2828	. . . . . . . . 9 ⊢ (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) = ((1 − 𝑙) · (𝑝‘𝑖)) ↔ ((1 − 𝑙) · (𝑝‘𝑖)) = ((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))))
67	65, 66	syl6rbbr 292	. . . . . . . 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 2828	. . . . . . . . . 10 ⊢ ((((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)) = (𝑝‘𝑖) ↔ (𝑝‘𝑖) = (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) / (1 − 𝑙)))
69	39, 50, 52, 64	divsubdird 11455	. . . . . . . . . . . 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	39, 52, 64	divrec2d 11420	. . . . . . . . . . . . 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	42, 49, 52, 64	div23d 11453	. . . . . . . . . . . . 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 7174	. . . . . . . . . . . 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 2856	. . . . . . . . . . 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 2832	. . . . . . . . . 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	syl5bb 285	. . . . . . . . 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	42, 52, 64	divcld 11416	. . . . . . . . . . . . . . 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, 49	mulneg1d 11093	. . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . 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 7172	. . . . . . . . . . . 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	52, 64	reccld 11409	. . . . . . . . . . . . . 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, 39	mulcld 10661	. . . . . . . . . . . . 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, 49	mulcld 10661	. . . . . . . . . . . . 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 11003	. . . . . . . . . . . 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	51, 76	subnegd 11004	. . . . . . . . . . . . . . . 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 11333	. . . . . . . . . . . . . . . . 17 ⊢ ((1 ∈ ℂ ∧ 𝑙 ∈ ℂ ∧ ((1 − 𝑙) ∈ ℂ ∧ (1 − 𝑙) ≠ 0)) → ((((1 − 𝑙) · 1) + 𝑙) / (1 − 𝑙)) = (1 + (𝑙 / (1 − 𝑙))))
86	51, 42, 52, 64, 85	syl112anc 1370	. . . . . . . . . . . . . . . 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	52	mulid1d 10658	. . . . . . . . . . . . . . . . . . 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 7171	. . . . . . . . . . . . . . . . . 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	51, 42	npcand 11001	. . . . . . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . . . . . 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 7171	. . . . . . . . . . . . . . . 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 2862	. . . . . . . . . . . . . . 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 2827	. . . . . . . . . . . . . 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 7171	. . . . . . . . . . . . 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 7171	. . . . . . . . . . . 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 2864	. . . . . . . . . . 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 2832	. . . . . . . . . 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 231	. . . . . . . . 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 242	. . . . . . . 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 242	. . . . . . 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 242	. . . . . 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 3177	. . . . 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 409	. . . 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 3626	. . 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 3287	. 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 10688	. . . . . . 7 ⊢ 0 ∈ ℝ^*
107		1re 10641	. . . . . . 7 ⊢ 1 ∈ ℝ
108		elioc2 12800	. . . . . . 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 1132	. . . . . . 7 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → 𝑚 ∈ ℝ)
111		gt0ne0 11105	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚) → 𝑚 ≠ 0)
112	111	3adant3 1128	. . . . . . 7 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → 𝑚 ≠ 0)
113	110, 112	rereccld 11467	. . . . . 6 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → (1 / 𝑚) ∈ ℝ)
114	109, 113	sylbi 219	. . . . 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 7164	. . . . . . . . 9 ⊢ (𝑡 = (1 / 𝑚) → (1 − 𝑡) = (1 − (1 / 𝑚)))
117	116	oveq1d 7171	. . . . . . . 8 ⊢ (𝑡 = (1 / 𝑚) → ((1 − 𝑡) · (𝑥‘𝑖)) = ((1 − (1 / 𝑚)) · (𝑥‘𝑖)))
118		oveq1 7163	. . . . . . . 8 ⊢ (𝑡 = (1 / 𝑚) → (𝑡 · (𝑦‘𝑖)) = ((1 / 𝑚) · (𝑦‘𝑖)))
119	117, 118	oveq12d 7174	. . . . . . 7 ⊢ (𝑡 = (1 / 𝑚) → (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖))))
120	119	eqeq2d 2832	. . . . . 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 3197	. . . . 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 484	. . . 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 2828	. . . . . . . 8 ⊢ ((𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) = (𝑦‘𝑖))
124	46	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	ffvelrnda 6851	. . . . . . . . . . . . 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 10669	. . . . . . . . . . . 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 10636	. . . . . . . . . . . . . 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 219	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (0(,]1) → 𝑚 ∈ ℝ)
129	128	recnd 10669	. . . . . . . . . . . . . . 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 10997	. . . . . . . . . . . . 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	36	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	ffvelrnda 6851	. . . . . . . . . . . . . 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 10669	. . . . . . . . . . . . 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 10661	. . . . . . . . . . . 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 11003	. . . . . . . . . . 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 11093	. . . . . . . . . . . . 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 11013	. . . . . . . . . . . . . 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 7171	. . . . . . . . . . . . 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 2858	. . . . . . . . . . . 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 7172	. . . . . . . . . . 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 2858	. . . . . . . . . 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 2823	. . . . . . . . 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	53	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	ffvelrnda 6851	. . . . . . . . . . . 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 10669	. . . . . . . . . . 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 10661	. . . . . . . . . 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 11015	. . . . . . . . 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 2828	. . . . . . . . . . 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 10997	. . . . . . . . . . . . 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 10661	. . . . . . . . . . . 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 10660	. . . . . . . . . . 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 12781	. . . . . . . . . . . . . 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 616	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℝ^* ∧ 0 < 𝑚) → (0 ∈ ℝ ∧ 0 < 𝑚))
158	157	3adant3 1128	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℝ^* ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → (0 ∈ ℝ ∧ 0 < 𝑚))
159		ltne 10737	. . . . . . . . . . . . . 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 219	. . . . . . . . . . . 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 11449	. . . . . . . . . 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 11454	. . . . . . . . . . . 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 11420	. . . . . . . . . . . . 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 11453	. . . . . . . . . . . . . 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 11455	. . . . . . . . . . . . . . 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 7171	. . . . . . . . . . . . . 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 2856	. . . . . . . . . . . . 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 7174	. . . . . . . . . . . 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 2856	. . . . . . . . . . 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 2832	. . . . . . . . . 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 311	. . . . . . . . 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 311	. . . . . . . 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	syl5bb 285	. . . . . . 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 11409	. . . . . . . . . . 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 10661	. . . . . . . . . 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 10997	. . . . . . . . . . 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 10661	. . . . . . . . . 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 11414	. . . . . . . . . . . . 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 7171	. . . . . . . . . . . 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 7171	. . . . . . . . . . 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 7172	. . . . . . . . . 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 10854	. . . . . . . . 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 2832	. . . . . . . 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 231	. . . . . . 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 242	. . . . . 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 3177	. . . . 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 409	. . . 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 3626	. . 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 3287	. 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 1424	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 208 ∧ wa 398 ∨ w3o 1082 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∀wral 3138 ∃wrex 3139 ∖ cdif 3933 ⊆ wss 3936 {csn 4567 class class class wbr 5066 ⟶wf 6351 ‘cfv 6355 (class class class)co 7156 ↑_m cmap 8406 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ℝ^*cxr 10674 < clt 10675 ≤ cle 10676 − cmin 10870 -cneg 10871 / cdiv 11297 ℕcn 11638 (,]cioc 12740 [,)cico 12741 [,]cicc 12742 ...cfz 12893

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-ioc 12744 df-ico 12745 df-icc 12746

This theorem is referenced by: eenglngeehlnm 44775

W3C validator