Theorem eenglngeehlnmlem1 45151

Description: Lemma 1 for eenglngeehlnm 45153. (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 7143	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → (1 − 𝑘) = (1 − 𝑡))
2	1	oveq1d 7150	. . . . . . 7 ⊢ (𝑘 = 𝑡 → ((1 − 𝑘) · (𝑥‘𝑖)) = ((1 − 𝑡) · (𝑥‘𝑖)))
3		oveq1 7142	. . . . . . 7 ⊢ (𝑘 = 𝑡 → (𝑘 · (𝑦‘𝑖)) = (𝑡 · (𝑦‘𝑖)))
4	2, 3	oveq12d 7153	. . . . . 6 ⊢ (𝑘 = 𝑡 → (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))))
5	4	eqeq2d 2809	. . . . 5 ⊢ (𝑘 = 𝑡 → ((𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ↔ (𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
6	5	ralbidv 3162	. . . 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 3397	. . 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 12877	. . . 4 ⊢ (0[,]1) ⊆ ℝ
9		ssrexv 3982	. . . 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 245	. 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 10632	. . . . . . . 8 ⊢ 0 ∈ ℝ
13		1xr 10689	. . . . . . . 8 ⊢ 1 ∈ ℝ*
14		elico2 12789	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ*) → (𝑙 ∈ (0[,)1) ↔ (𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1)))
15	12, 13, 14	mp2an 691	. . . . . . 7 ⊢ (𝑙 ∈ (0[,)1) ↔ (𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1))
16		simp1 1133	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 𝑙 ∈ ℝ)
17		1red 10631	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ∈ ℝ)
18	17, 16	resubcld 11057	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (1 − 𝑙) ∈ ℝ)
19		1cnd 10625	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ∈ ℂ)
20	16	recnd 10658	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 𝑙 ∈ ℂ)
21		ltne 10726	. . . . . . . . . 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 10995	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (1 − 𝑙) ≠ 0)
24	16, 18, 23	redivcld 11457	. . . . . . 7 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (𝑙 / (1 − 𝑙)) ∈ ℝ)
25	15, 24	sylbi 220	. . . . . 6 ⊢ (𝑙 ∈ (0[,)1) → (𝑙 / (1 − 𝑙)) ∈ ℝ)
26	25	ad2antlr 726	. . . . 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 11056	. . . 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 7143	. . . . . . . . 9 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → (1 − 𝑡) = (1 − -(𝑙 / (1 − 𝑙))))
29	28	oveq1d 7150	. . . . . . . 8 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → ((1 − 𝑡) · (𝑥‘𝑖)) = ((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)))
30		oveq1 7142	. . . . . . . 8 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → (𝑡 · (𝑦‘𝑖)) = (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))
31	29, 30	oveq12d 7153	. . . . . . 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 2809	. . . . . 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 3162	. . . . 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 485	. . . 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 2805	. . . . . . . 8 ⊢ ((𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) = (𝑥‘𝑖))
36		elmapi 8411	. . . . . . . . . . . . 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 725	. . . . . . . . . . 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	ffvelrnda 6828	. . . . . . . . . 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 10658	. . . . . . . . 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 220	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (0[,)1) → 𝑙 ∈ ℝ)
42	41	ad2antlr 726	. . . . . . . . . . 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 10658	. . . . . . . . . 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 4054	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥}) → 𝑦 ∈ (ℝ ↑m (1...𝑁)))
45		elmapi 8411	. . . . . . . . . . . . . . 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 725	. . . . . . . . . . . 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	ffvelrnda 6828	. . . . . . . . . . 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 10658	. . . . . . . . . 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 10650	. . . . . . . . 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 10625	. . . . . . . . . . 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 10986	. . . . . . . . . 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 8411	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (ℝ ↑m (1...𝑁)) → 𝑝:(1...𝑁)⟶ℝ)
55	54	ad2antlr 726	. . . . . . . . . . . 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	ffvelrnda 6828	. . . . . . . . . . 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 10658	. . . . . . . . . 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 10650	. . . . . . . . 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 11005	. . . . . . . 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 293	. . . . . . 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 2805	. . . . . . . . 9 ⊢ (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) = ((1 − 𝑙) · (𝑝‘𝑖)) ↔ ((1 − 𝑙) · (𝑝‘𝑖)) = ((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))))
62	40, 51	subcld 10986	. . . . . . . . . 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 220	. . . . . . . . . . . 12 ⊢ (𝑙 ∈ (0[,)1) → 1 ≠ 𝑙)
64	63	ad2antlr 726	. . . . . . . . . . 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 10995	. . . . . . . . . 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 11427	. . . . . . . . 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 293	. . . . . . . 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 2805	. . . . . . . . . 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 11444	. . . . . . . . . . . 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 11409	. . . . . . . . . . . . 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 11442	. . . . . . . . . . . . 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 7153	. . . . . . . . . . . 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 2833	. . . . . . . . . . 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 2809	. . . . . . . . . 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 286	. . . . . . . . 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 11405	. . . . . . . . . . . . . . 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 11082	. . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . 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 7151	. . . . . . . . . . . 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 11398	. . . . . . . . . . . . . 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 10650	. . . . . . . . . . . . 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 10650	. . . . . . . . . . . . 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 10992	. . . . . . . . . . . 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 10993	. . . . . . . . . . . . . . . 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 11322	. . . . . . . . . . . . . . . . 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	mulid1d 10647	. . . . . . . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . . . . . . 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 10990	. . . . . . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . 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 2804	. . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . 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 7150	. . . . . . . . . . . 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 2841	. . . . . . . . . . 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 2809	. . . . . . . . . 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 232	. . . . . . . . 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 243	. . . . . . . 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 243	. . . . . . 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 243	. . . . . 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 3144	. . . . 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 410	. . . 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 3574	. . 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 3246	. 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 10677	. . . . . . 7 ⊢ 0 ∈ ℝ*
107		1re 10630	. . . . . . 7 ⊢ 1 ∈ ℝ
108		elioc2 12788	. . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1)))
109	106, 107, 108	mp2an 691	. . . . . 6 ⊢ (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1))
110		simp1 1133	. . . . . . 7 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → 𝑚 ∈ ℝ)
111		gt0ne0 11094	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚) → 𝑚 ≠ 0)
112	111	3adant3 1129	. . . . . . 7 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → 𝑚 ≠ 0)
113	110, 112	rereccld 11456	. . . . . 6 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → (1 / 𝑚) ∈ ℝ)
114	109, 113	sylbi 220	. . . . 5 ⊢ (𝑚 ∈ (0(,]1) → (1 / 𝑚) ∈ ℝ)
115	114	ad2antlr 726	. . . 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 7143	. . . . . . . . 9 ⊢ (𝑡 = (1 / 𝑚) → (1 − 𝑡) = (1 − (1 / 𝑚)))
117	116	oveq1d 7150	. . . . . . . 8 ⊢ (𝑡 = (1 / 𝑚) → ((1 − 𝑡) · (𝑥‘𝑖)) = ((1 − (1 / 𝑚)) · (𝑥‘𝑖)))
118		oveq1 7142	. . . . . . . 8 ⊢ (𝑡 = (1 / 𝑚) → (𝑡 · (𝑦‘𝑖)) = ((1 / 𝑚) · (𝑦‘𝑖)))
119	117, 118	oveq12d 7153	. . . . . . 7 ⊢ (𝑡 = (1 / 𝑚) → (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖))))
120	119	eqeq2d 2809	. . . . . 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 3162	. . . . 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 485	. . . 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 2805	. . . . . . . 8 ⊢ ((𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) = (𝑦‘𝑖))
124	47	ad2antrr 725	. . . . . . . . . . . . . 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 6828	. . . . . . . . . . . . 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 10658	. . . . . . . . . . . 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 10625	. . . . . . . . . . . . . 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 220	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ (0(,]1) → 𝑚 ∈ ℝ)
129	128	recnd 10658	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (0(,]1) → 𝑚 ∈ ℂ)
130	129	ad2antlr 726	. . . . . . . . . . . . . 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 10986	. . . . . . . . . . . . 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 725	. . . . . . . . . . . . . . 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 6828	. . . . . . . . . . . . . 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 10658	. . . . . . . . . . . . 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 10650	. . . . . . . . . . . 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 10992	. . . . . . . . . . 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 11082	. . . . . . . . . . . . 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 11002	. . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . 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 2835	. . . . . . . . . . . 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 7151	. . . . . . . . . . 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 2835	. . . . . . . . . 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 2800	. . . . . . . . 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 726	. . . . . . . . . . . . 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 6828	. . . . . . . . . . . 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 10658	. . . . . . . . . . 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 10650	. . . . . . . . . 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 11004	. . . . . . . . 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 2805	. . . . . . . . . . 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 10986	. . . . . . . . . . . . 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 10650	. . . . . . . . . . . 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 10649	. . . . . . . . . . 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 12768	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ* ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1)))
155	106, 13, 154	mp2an 691	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ* ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1))
156	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℝ* → 0 ∈ ℝ)
157	156	anim1i 617	. . . . . . . . . . . . . . 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 10726	. . . . . . . . . . . . . 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 220	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (0(,]1) → 𝑚 ≠ 0)
162	161	ad2antlr 726	. . . . . . . . . . 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 11438	. . . . . . . . . 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 11443	. . . . . . . . . . . 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 11409	. . . . . . . . . . . . 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 11442	. . . . . . . . . . . . . 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 11444	. . . . . . . . . . . . . . 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 7150	. . . . . . . . . . . . . 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 2833	. . . . . . . . . . . . 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 7153	. . . . . . . . . . . 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 2833	. . . . . . . . . . 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 2809	. . . . . . . . . 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 312	. . . . . . . . 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 312	. . . . . . . 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 286	. . . . . . 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 11398	. . . . . . . . . . 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 10650	. . . . . . . . . 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 10986	. . . . . . . . . . 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 10650	. . . . . . . . . 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 11403	. . . . . . . . . . . . 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 7150	. . . . . . . . . . . 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 7150	. . . . . . . . . . 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 7151	. . . . . . . . . 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 10843	. . . . . . . . 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 2809	. . . . . . . 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 232	. . . . . . 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 243	. . . . . 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 3144	. . . . 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 410	. . . 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 3574	. . 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 3246	. 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 1425	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)))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ w3o 1083   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ⊆ wss 3881  {csn 4525   class class class wbr 5030  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ↑_m cmap 8389  ℂcc 10524  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531  ℝ^*cxr 10663   < clt 10664   ≤ cle 10665   − cmin 10859  -cneg 10860   / cdiv 11286  ℕcn 11625  (,]cioc 12727  [,)cico 12728  [,]cicc 12729  ...cfz 12885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-1st 7671  df-2nd 7672  df-er 8272  df-map 8391  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-ioc 12731  df-ico 12732  df-icc 12733

This theorem is referenced by:  eenglngeehlnm  45153