Theorem eenglngeehlnmlem1 45971

Description: Lemma 1 for eenglngeehlnm 45973. (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 7263	. . . . . . . 8 ⊢ (𝑘 = 𝑡 → (1 − 𝑘) = (1 − 𝑡))
2	1	oveq1d 7270	. . . . . . 7 ⊢ (𝑘 = 𝑡 → ((1 − 𝑘) · (𝑥‘𝑖)) = ((1 − 𝑡) · (𝑥‘𝑖)))
3		oveq1 7262	. . . . . . 7 ⊢ (𝑘 = 𝑡 → (𝑘 · (𝑦‘𝑖)) = (𝑡 · (𝑦‘𝑖)))
4	2, 3	oveq12d 7273	. . . . . 6 ⊢ (𝑘 = 𝑡 → (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))))
5	4	eqeq2d 2749	. . . . 5 ⊢ (𝑘 = 𝑡 → ((𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ↔ (𝑝‘𝑖) = (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖)))))
6	5	ralbidv 3120	. . . 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 3373	. . 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 13160	. . . 4 ⊢ (0[,]1) ⊆ ℝ
9		ssrexv 3984	. . . 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 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 10908	. . . . . . . 8 ⊢ 0 ∈ ℝ
13		1xr 10965	. . . . . . . 8 ⊢ 1 ∈ ℝ*
14		elico2 13072	. . . . . . . 8 ⊢ ((0 ∈ ℝ ∧ 1 ∈ ℝ*) → (𝑙 ∈ (0[,)1) ↔ (𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1)))
15	12, 13, 14	mp2an 688	. . . . . . 7 ⊢ (𝑙 ∈ (0[,)1) ↔ (𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1))
16		simp1 1134	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 𝑙 ∈ ℝ)
17		1red 10907	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ∈ ℝ)
18	17, 16	resubcld 11333	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (1 − 𝑙) ∈ ℝ)
19		1cnd 10901	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ∈ ℂ)
20	16	recnd 10934	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 𝑙 ∈ ℂ)
21		ltne 11002	. . . . . . . . . 10 ⊢ ((𝑙 ∈ ℝ ∧ 𝑙 < 1) → 1 ≠ 𝑙)
22	21	3adant2 1129	. . . . . . . . 9 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → 1 ≠ 𝑙)
23	19, 20, 22	subne0d 11271	. . . . . . . 8 ⊢ ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙 ∧ 𝑙 < 1) → (1 − 𝑙) ≠ 0)
24	16, 18, 23	redivcld 11733	. . . . . . 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 723	. . . . 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 11332	. . . 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 7263	. . . . . . . . 9 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → (1 − 𝑡) = (1 − -(𝑙 / (1 − 𝑙))))
29	28	oveq1d 7270	. . . . . . . 8 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → ((1 − 𝑡) · (𝑥‘𝑖)) = ((1 − -(𝑙 / (1 − 𝑙))) · (𝑥‘𝑖)))
30		oveq1 7262	. . . . . . . 8 ⊢ (𝑡 = -(𝑙 / (1 − 𝑙)) → (𝑡 · (𝑦‘𝑖)) = (-(𝑙 / (1 − 𝑙)) · (𝑦‘𝑖)))
31	29, 30	oveq12d 7273	. . . . . . 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 2749	. . . . . 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 3120	. . . . 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 481	. . . 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 2745	. . . . . . . 8 ⊢ ((𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) = (𝑥‘𝑖))
36		elmapi 8595	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (ℝ ↑m (1...𝑁)) → 𝑥:(1...𝑁)⟶ℝ)
37	36	3ad2ant2 1132	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) → 𝑥:(1...𝑁)⟶ℝ)
38	37	ad2antrr 722	. . . . . . . . . . 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 6943	. . . . . . . . . 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 10934	. . . . . . . . 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 723	. . . . . . . . . . 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 10934	. . . . . . . . . 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 4057	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥}) → 𝑦 ∈ (ℝ ↑m (1...𝑁)))
45		elmapi 8595	. . . . . . . . . . . . . . 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 1133	. . . . . . . . . . . . 13 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) → 𝑦:(1...𝑁)⟶ℝ)
48	47	ad2antrr 722	. . . . . . . . . . . 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 6943	. . . . . . . . . . 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 10934	. . . . . . . . . 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 10926	. . . . . . . . 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 10901	. . . . . . . . . . 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 11262	. . . . . . . . . 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 8595	. . . . . . . . . . . . 13 ⊢ (𝑝 ∈ (ℝ ↑m (1...𝑁)) → 𝑝:(1...𝑁)⟶ℝ)
55	54	ad2antlr 723	. . . . . . . . . . . 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 6943	. . . . . . . . . . 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 10934	. . . . . . . . . 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 10926	. . . . . . . . 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 11281	. . . . . . . 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 2745	. . . . . . . . 9 ⊢ (((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))) = ((1 − 𝑙) · (𝑝‘𝑖)) ↔ ((1 − 𝑙) · (𝑝‘𝑖)) = ((𝑥‘𝑖) − (𝑙 · (𝑦‘𝑖))))
62	40, 51	subcld 11262	. . . . . . . . . 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 723	. . . . . . . . . . 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 11271	. . . . . . . . . 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 11703	. . . . . . . . 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 2745	. . . . . . . . . 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 11720	. . . . . . . . . . . 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 11685	. . . . . . . . . . . . 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 11718	. . . . . . . . . . . . 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 7273	. . . . . . . . . . . 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 2778	. . . . . . . . . . 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 2749	. . . . . . . . . 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 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 11681	. . . . . . . . . . . . . . 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 11358	. . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . 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 7271	. . . . . . . . . . . 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 11674	. . . . . . . . . . . . . 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 10926	. . . . . . . . . . . . 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 10926	. . . . . . . . . . . . 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 11268	. . . . . . . . . . . 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 11269	. . . . . . . . . . . . . . . 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 11598	. . . . . . . . . . . . . . . . 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 1372	. . . . . . . . . . . . . . . 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 10923	. . . . . . . . . . . . . . . . . . 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 7270	. . . . . . . . . . . . . . . . . 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 11266	. . . . . . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . . . . . 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 7270	. . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . 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 2744	. . . . . . . . . . . . . 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 7270	. . . . . . . . . . . . 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 7270	. . . . . . . . . . . 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 2786	. . . . . . . . . . 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 2749	. . . . . . . . . 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 3102	. . . . 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 406	. . . 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 3555	. . 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 3215	. 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 10953	. . . . . . 7 ⊢ 0 ∈ ℝ*
107		1re 10906	. . . . . . 7 ⊢ 1 ∈ ℝ
108		elioc2 13071	. . . . . . 7 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1)))
109	106, 107, 108	mp2an 688	. . . . . 6 ⊢ (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1))
110		simp1 1134	. . . . . . 7 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → 𝑚 ∈ ℝ)
111		gt0ne0 11370	. . . . . . . 8 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚) → 𝑚 ≠ 0)
112	111	3adant3 1130	. . . . . . 7 ⊢ ((𝑚 ∈ ℝ ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → 𝑚 ≠ 0)
113	110, 112	rereccld 11732	. . . . . 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 723	. . . 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 7263	. . . . . . . . 9 ⊢ (𝑡 = (1 / 𝑚) → (1 − 𝑡) = (1 − (1 / 𝑚)))
117	116	oveq1d 7270	. . . . . . . 8 ⊢ (𝑡 = (1 / 𝑚) → ((1 − 𝑡) · (𝑥‘𝑖)) = ((1 − (1 / 𝑚)) · (𝑥‘𝑖)))
118		oveq1 7262	. . . . . . . 8 ⊢ (𝑡 = (1 / 𝑚) → (𝑡 · (𝑦‘𝑖)) = ((1 / 𝑚) · (𝑦‘𝑖)))
119	117, 118	oveq12d 7273	. . . . . . 7 ⊢ (𝑡 = (1 / 𝑚) → (((1 − 𝑡) · (𝑥‘𝑖)) + (𝑡 · (𝑦‘𝑖))) = (((1 − (1 / 𝑚)) · (𝑥‘𝑖)) + ((1 / 𝑚) · (𝑦‘𝑖))))
120	119	eqeq2d 2749	. . . . . 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 3120	. . . . 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 481	. . . 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 2745	. . . . . . . 8 ⊢ ((𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) = (𝑦‘𝑖))
124	47	ad2antrr 722	. . . . . . . . . . . . . 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 6943	. . . . . . . . . . . . 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 10934	. . . . . . . . . . . 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 10901	. . . . . . . . . . . . . 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 10934	. . . . . . . . . . . . . . 15 ⊢ (𝑚 ∈ (0(,]1) → 𝑚 ∈ ℂ)
130	129	ad2antlr 723	. . . . . . . . . . . . . 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 11262	. . . . . . . . . . . . 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 722	. . . . . . . . . . . . . . 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 6943	. . . . . . . . . . . . . 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 10934	. . . . . . . . . . . . 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 10926	. . . . . . . . . . . 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 11268	. . . . . . . . . . 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 11358	. . . . . . . . . . . . 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 11278	. . . . . . . . . . . . . 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 7270	. . . . . . . . . . . . 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 2780	. . . . . . . . . . . 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 7271	. . . . . . . . . . 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 2780	. . . . . . . . . 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 2740	. . . . . . . . 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 723	. . . . . . . . . . . . 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 6943	. . . . . . . . . . . 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 10934	. . . . . . . . . . 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 10926	. . . . . . . . . 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 11280	. . . . . . . . 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 2745	. . . . . . . . . . 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 11262	. . . . . . . . . . . . 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 10926	. . . . . . . . . . . 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 10925	. . . . . . . . . . 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 13050	. . . . . . . . . . . . . 14 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ* ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1)))
155	106, 13, 154	mp2an 688	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ* ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1))
156	12	a1i 11	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 ∈ ℝ* → 0 ∈ ℝ)
157	156	anim1i 614	. . . . . . . . . . . . . . 15 ⊢ ((𝑚 ∈ ℝ* ∧ 0 < 𝑚) → (0 ∈ ℝ ∧ 0 < 𝑚))
158	157	3adant3 1130	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ ℝ* ∧ 0 < 𝑚 ∧ 𝑚 ≤ 1) → (0 ∈ ℝ ∧ 0 < 𝑚))
159		ltne 11002	. . . . . . . . . . . . . 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 723	. . . . . . . . . . 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 11714	. . . . . . . . . 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 11719	. . . . . . . . . . . 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 11685	. . . . . . . . . . . . 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 11718	. . . . . . . . . . . . . 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 11720	. . . . . . . . . . . . . . 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 7270	. . . . . . . . . . . . . 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 2778	. . . . . . . . . . . . 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 7273	. . . . . . . . . . . 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 2778	. . . . . . . . . . 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 2749	. . . . . . . . . 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	syl5bb 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 11674	. . . . . . . . . . 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 10926	. . . . . . . . . 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 11262	. . . . . . . . . . 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 10926	. . . . . . . . . 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 11679	. . . . . . . . . . . . 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 7270	. . . . . . . . . . . 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 7270	. . . . . . . . . . 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 7271	. . . . . . . . . 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 11119	. . . . . . . . 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 2749	. . . . . . . 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 3102	. . . . 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 406	. . . 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 3555	. . 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 3215	. 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)))))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∨ w3o 1084   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108   ≠ wne 2942  ∀wral 3063  ∃wrex 3064   ∖ cdif 3880   ⊆ wss 3883  {csn 4558   class class class wbr 5070  ⟶wf 6414  ‘cfv 6418  (class class class)co 7255   ↑_m cmap 8573  ℂcc 10800  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   · cmul 10807  ℝ^*cxr 10939   < clt 10940   ≤ cle 10941   − cmin 11135  -cneg 11136   / cdiv 11562  ℕcn 11903  (,]cioc 13009  [,)cico 13010  [,]cicc 13011  ...cfz 13168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-po 5494  df-so 5495  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-1st 7804  df-2nd 7805  df-er 8456  df-map 8575  df-en 8692  df-dom 8693  df-sdom 8694  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-ioc 13013  df-ico 13014  df-icc 13015

This theorem is referenced by:  eenglngeehlnm  45973