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Theorem eenglngeehlnmlem2 43302
Description: Lemma 2 for eenglngeehlnm 43303. (Contributed by AV, 15-Feb-2023.)
Assertion
Ref Expression
eenglngeehlnmlem2 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) → (∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
Distinct variable group:   𝑖,𝑁,𝑘,𝑙,𝑚,𝑝,𝑡,𝑥,𝑦

Proof of Theorem eenglngeehlnmlem2
StepHypRef Expression
1 0red 10367 . . . 4 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → 0 ∈ ℝ)
2 1red 10364 . . . 4 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → 1 ∈ ℝ)
3 simpr 479 . . . 4 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ)
4 reorelicc 43290 . . . 4 ((0 ∈ ℝ ∧ 1 ∈ ℝ ∧ 𝑡 ∈ ℝ) → (𝑡 < 0 ∨ 𝑡 ∈ (0[,]1) ∨ 1 < 𝑡))
51, 2, 3, 4syl3anc 1494 . . 3 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → (𝑡 < 0 ∨ 𝑡 ∈ (0[,]1) ∨ 1 < 𝑡))
6 0xr 10410 . . . . . . . . . . . 12 0 ∈ ℝ*
76a1i 11 . . . . . . . . . . 11 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → 0 ∈ ℝ*)
8 1xr 10423 . . . . . . . . . . . 12 1 ∈ ℝ*
98a1i 11 . . . . . . . . . . 11 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → 1 ∈ ℝ*)
10 simpl 476 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 𝑡 ∈ ℝ)
1110recnd 10392 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 𝑡 ∈ ℂ)
12 0red 10367 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 0 ∈ ℝ)
13 1red 10364 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 1 ∈ ℝ)
14 simpr 479 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 𝑡 < 0)
15 0lt1 10881 . . . . . . . . . . . . . . . . 17 0 < 1
1615a1i 11 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 0 < 1)
1710, 12, 13, 14, 16lttrd 10524 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 𝑡 < 1)
1810, 17ltned 10499 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 𝑡 ≠ 1)
19 1subrec1sub 43287 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℂ ∧ 𝑡 ≠ 1) → (1 − (1 / (1 − 𝑡))) = (𝑡 / (𝑡 − 1)))
2011, 18, 19syl2anc 579 . . . . . . . . . . . . 13 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − (1 / (1 − 𝑡))) = (𝑡 / (𝑡 − 1)))
2110, 13resubcld 10789 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 − 1) ∈ ℝ)
22 1cnd 10358 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 1 ∈ ℂ)
2311, 22, 18subne0d 10729 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 − 1) ≠ 0)
2410, 21, 23redivcld 11186 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 / (𝑡 − 1)) ∈ ℝ)
2524rexrd 10413 . . . . . . . . . . . . 13 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 / (𝑡 − 1)) ∈ ℝ*)
2620, 25eqeltrd 2906 . . . . . . . . . . . 12 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − (1 / (1 − 𝑡))) ∈ ℝ*)
2726ad4ant23 761 . . . . . . . . . . 11 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (1 − (1 / (1 − 𝑡))) ∈ ℝ*)
2810renegcld 10788 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → -𝑡 ∈ ℝ)
2910, 13sublt0d 10985 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 − 1) < 0 ↔ 𝑡 < 1))
3017, 29mpbird 249 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 − 1) < 0)
3121, 30negelrpd 12155 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → -(𝑡 − 1) ∈ ℝ+)
3210, 12, 14ltled 10511 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 𝑡 ≤ 0)
3310le0neg1d 10930 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 ≤ 0 ↔ 0 ≤ -𝑡))
3432, 33mpbid 224 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 0 ≤ -𝑡)
3528, 31, 34divge0d 12203 . . . . . . . . . . . . 13 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 0 ≤ (-𝑡 / -(𝑡 − 1)))
3621recnd 10392 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 − 1) ∈ ℂ)
3711, 36, 23div2negd 11149 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (-𝑡 / -(𝑡 − 1)) = (𝑡 / (𝑡 − 1)))
3820, 37eqtr4d 2864 . . . . . . . . . . . . 13 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − (1 / (1 − 𝑡))) = (-𝑡 / -(𝑡 − 1)))
3935, 38breqtrrd 4903 . . . . . . . . . . . 12 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 0 ≤ (1 − (1 / (1 − 𝑡))))
4039ad4ant23 761 . . . . . . . . . . 11 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → 0 ≤ (1 − (1 / (1 − 𝑡))))
41 1red 10364 . . . . . . . . . . . 12 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → 1 ∈ ℝ)
4213, 10resubcld 10789 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − 𝑡) ∈ ℝ)
4310, 13posdifd 10946 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 < 1 ↔ 0 < (1 − 𝑡)))
4417, 43mpbid 224 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 0 < (1 − 𝑡))
4542, 44elrpd 12160 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − 𝑡) ∈ ℝ+)
4645ad4ant23 761 . . . . . . . . . . . . 13 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (1 − 𝑡) ∈ ℝ+)
4746rpreccld 12173 . . . . . . . . . . . 12 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (1 / (1 − 𝑡)) ∈ ℝ+)
4841, 47ltsubrpd 12195 . . . . . . . . . . 11 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (1 − (1 / (1 − 𝑡))) < 1)
497, 9, 27, 40, 48elicod 12519 . . . . . . . . . 10 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (1 − (1 / (1 − 𝑡))) ∈ (0[,)1))
50 oveq2 6918 . . . . . . . . . . . . . . 15 (𝑙 = (1 − (1 / (1 − 𝑡))) → (1 − 𝑙) = (1 − (1 − (1 / (1 − 𝑡)))))
5150oveq1d 6925 . . . . . . . . . . . . . 14 (𝑙 = (1 − (1 / (1 − 𝑡))) → ((1 − 𝑙) · (𝑝𝑖)) = ((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)))
52 oveq1 6917 . . . . . . . . . . . . . 14 (𝑙 = (1 − (1 / (1 − 𝑡))) → (𝑙 · (𝑦𝑖)) = ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)))
5351, 52oveq12d 6928 . . . . . . . . . . . . 13 (𝑙 = (1 − (1 / (1 − 𝑡))) → (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) = (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))))
5453eqeq2d 2835 . . . . . . . . . . . 12 (𝑙 = (1 − (1 / (1 − 𝑡))) → ((𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ (𝑥𝑖) = (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)))))
5554ralbidv 3195 . . . . . . . . . . 11 (𝑙 = (1 − (1 / (1 − 𝑡))) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)))))
5655adantl 475 . . . . . . . . . 10 (((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) ∧ 𝑙 = (1 − (1 / (1 − 𝑡)))) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)))))
5722, 11subcld 10720 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − 𝑡) ∈ ℂ)
5810, 17gtned 10498 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 1 ≠ 𝑡)
5922, 11, 58subne0d 10729 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − 𝑡) ≠ 0)
6057, 59reccld 11127 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 / (1 − 𝑡)) ∈ ℂ)
6122, 60nncand 10725 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − (1 − (1 / (1 − 𝑡)))) = (1 / (1 − 𝑡)))
6261, 60eqeltrd 2906 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − (1 − (1 / (1 − 𝑡)))) ∈ ℂ)
6322, 60subcld 10720 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − (1 / (1 − 𝑡))) ∈ ℂ)
6416gt0ne0d 10923 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 1 ≠ 0)
6536, 11subeq0ad 10730 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (((𝑡 − 1) − 𝑡) = 0 ↔ (𝑡 − 1) = 𝑡))
6611, 22, 11sub32d 10752 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 − 1) − 𝑡) = ((𝑡𝑡) − 1))
6766eqeq1d 2827 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (((𝑡 − 1) − 𝑡) = 0 ↔ ((𝑡𝑡) − 1) = 0))
6811subidd 10708 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡𝑡) = 0)
6968oveq1d 6925 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡𝑡) − 1) = (0 − 1))
7069eqeq1d 2827 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (((𝑡𝑡) − 1) = 0 ↔ (0 − 1) = 0))
71 0cnd 10356 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 0 ∈ ℂ)
7271, 22, 71subaddd 10738 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((0 − 1) = 0 ↔ (1 + 0) = 0))
7322addid1d 10562 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 + 0) = 1)
7473eqeq1d 2827 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((1 + 0) = 0 ↔ 1 = 0))
7572, 74bitrd 271 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((0 − 1) = 0 ↔ 1 = 0))
7667, 70, 753bitrd 297 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (((𝑡 − 1) − 𝑡) = 0 ↔ 1 = 0))
7765, 76bitr3d 273 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 − 1) = 𝑡 ↔ 1 = 0))
7877necon3bid 3043 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 − 1) ≠ 𝑡 ↔ 1 ≠ 0))
7964, 78mpbird 249 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 − 1) ≠ 𝑡)
8020eqeq2d 2835 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 = (1 − (1 / (1 − 𝑡))) ↔ 1 = (𝑡 / (𝑡 − 1))))
81 eqcom 2832 . . . . . . . . . . . . . . . . . . . . . . . 24 (1 = (𝑡 / (𝑡 − 1)) ↔ (𝑡 / (𝑡 − 1)) = 1)
8211, 36, 22, 23divmuld 11156 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 / (𝑡 − 1)) = 1 ↔ ((𝑡 − 1) · 1) = 𝑡))
8381, 82syl5bb 275 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 = (𝑡 / (𝑡 − 1)) ↔ ((𝑡 − 1) · 1) = 𝑡))
8436mulid1d 10381 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 − 1) · 1) = (𝑡 − 1))
8584eqeq1d 2827 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (((𝑡 − 1) · 1) = 𝑡 ↔ (𝑡 − 1) = 𝑡))
8680, 83, 853bitrd 297 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 = (1 − (1 / (1 − 𝑡))) ↔ (𝑡 − 1) = 𝑡))
8786necon3bid 3043 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 ≠ (1 − (1 / (1 − 𝑡))) ↔ (𝑡 − 1) ≠ 𝑡))
8879, 87mpbird 249 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 1 ≠ (1 − (1 / (1 − 𝑡))))
8922, 63, 88subne0d 10729 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − (1 − (1 / (1 − 𝑡)))) ≠ 0)
9061oveq1d 6925 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((1 − (1 − (1 / (1 − 𝑡)))) · (1 − 𝑡)) = ((1 / (1 − 𝑡)) · (1 − 𝑡)))
9157, 59recid2d 11130 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((1 / (1 − 𝑡)) · (1 − 𝑡)) = 1)
9290, 91eqtrd 2861 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((1 − (1 − (1 / (1 − 𝑡)))) · (1 − 𝑡)) = 1)
9362, 57, 89, 92mvllmuld 11190 . . . . . . . . . . . . . . . . . 18 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − 𝑡) = (1 / (1 − (1 − (1 / (1 − 𝑡))))))
9493ad4ant23 761 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑡) = (1 / (1 − (1 − (1 / (1 − 𝑡))))))
9594oveq1d 6925 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − 𝑡) · (𝑥𝑖)) = ((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)))
9620oveq1d 6925 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((1 − (1 / (1 − 𝑡))) − 1) = ((𝑡 / (𝑡 − 1)) − 1))
9720, 96oveq12d 6928 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) = ((𝑡 / (𝑡 − 1)) / ((𝑡 / (𝑡 − 1)) − 1)))
98 subdivcomb2 11054 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℂ ∧ 1 ∈ ℂ ∧ ((𝑡 − 1) ∈ ℂ ∧ (𝑡 − 1) ≠ 0)) → ((𝑡 − ((𝑡 − 1) · 1)) / (𝑡 − 1)) = ((𝑡 / (𝑡 − 1)) − 1))
9911, 22, 36, 23, 98syl112anc 1497 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 − ((𝑡 − 1) · 1)) / (𝑡 − 1)) = ((𝑡 / (𝑡 − 1)) − 1))
10084oveq2d 6926 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 − ((𝑡 − 1) · 1)) = (𝑡 − (𝑡 − 1)))
10111, 22nncand 10725 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 − (𝑡 − 1)) = 1)
102100, 101eqtrd 2861 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 − ((𝑡 − 1) · 1)) = 1)
103102oveq1d 6925 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 − ((𝑡 − 1) · 1)) / (𝑡 − 1)) = (1 / (𝑡 − 1)))
10499, 103eqtr3d 2863 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 / (𝑡 − 1)) − 1) = (1 / (𝑡 − 1)))
105104oveq1d 6925 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (((𝑡 / (𝑡 − 1)) − 1) · 𝑡) = ((1 / (𝑡 − 1)) · 𝑡))
10611, 36, 23divrec2d 11138 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 / (𝑡 − 1)) = ((1 / (𝑡 − 1)) · 𝑡))
107105, 106eqtr4d 2864 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (((𝑡 / (𝑡 − 1)) − 1) · 𝑡) = (𝑡 / (𝑡 − 1)))
10820, 63eqeltrrd 2907 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 / (𝑡 − 1)) ∈ ℂ)
109108, 22subcld 10720 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 / (𝑡 − 1)) − 1) ∈ ℂ)
11079necomd 3054 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 𝑡 ≠ (𝑡 − 1))
11111, 36, 23, 110divne1d 11145 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (𝑡 / (𝑡 − 1)) ≠ 1)
112108, 22, 111subne0d 10729 . . . . . . . . . . . . . . . . . . . . 21 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 / (𝑡 − 1)) − 1) ≠ 0)
113108, 109, 11, 112divmuld 11156 . . . . . . . . . . . . . . . . . . . 20 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (((𝑡 / (𝑡 − 1)) / ((𝑡 / (𝑡 − 1)) − 1)) = 𝑡 ↔ (((𝑡 / (𝑡 − 1)) − 1) · 𝑡) = (𝑡 / (𝑡 − 1))))
114107, 113mpbird 249 . . . . . . . . . . . . . . . . . . 19 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((𝑡 / (𝑡 − 1)) / ((𝑡 / (𝑡 − 1)) − 1)) = 𝑡)
11597, 114eqtr2d 2862 . . . . . . . . . . . . . . . . . 18 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → 𝑡 = ((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)))
116115ad4ant23 761 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → 𝑡 = ((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)))
117116oveq1d 6925 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (𝑡 · (𝑦𝑖)) = (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖)))
11895, 117oveq12d 6928 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) = (((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)) + (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖))))
119118eqeq2d 2835 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)) + (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖)))))
120119biimpd 221 . . . . . . . . . . . . 13 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (𝑝𝑖) = (((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)) + (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖)))))
121 elmapi 8149 . . . . . . . . . . . . . . . . . . . . 21 (𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) → 𝑥:(1...𝑁)⟶ℝ)
1221213ad2ant2 1168 . . . . . . . . . . . . . . . . . . . 20 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) → 𝑥:(1...𝑁)⟶ℝ)
123122adantr 474 . . . . . . . . . . . . . . . . . . 19 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) → 𝑥:(1...𝑁)⟶ℝ)
124123ad2antrr 717 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) → 𝑥:(1...𝑁)⟶ℝ)
125124ffvelrnda 6613 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℝ)
126125recnd 10392 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℂ)
127 1cnd 10358 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → 1 ∈ ℂ)
12811ad4ant23 761 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → 𝑡 ∈ ℂ)
129127, 128subcld 10720 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑡) ∈ ℂ)
13058ad4ant23 761 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → 1 ≠ 𝑡)
131127, 128, 130subne0d 10729 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑡) ≠ 0)
132129, 131reccld 11127 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (1 / (1 − 𝑡)) ∈ ℂ)
133127, 132subcld 10720 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (1 − (1 / (1 − 𝑡))) ∈ ℂ)
134 eldifi 3961 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥}) → 𝑦 ∈ (ℝ ↑𝑚 (1...𝑁)))
135 elmapi 8149 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ (ℝ ↑𝑚 (1...𝑁)) → 𝑦:(1...𝑁)⟶ℝ)
136134, 135syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥}) → 𝑦:(1...𝑁)⟶ℝ)
1371363ad2ant3 1169 . . . . . . . . . . . . . . . . . . . . 21 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) → 𝑦:(1...𝑁)⟶ℝ)
138137adantr 474 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) → 𝑦:(1...𝑁)⟶ℝ)
139138ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) → 𝑦:(1...𝑁)⟶ℝ)
140139ffvelrnda 6613 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℝ)
141140recnd 10392 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℂ)
142133, 141mulcld 10384 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) ∈ ℂ)
14362ad4ant23 761 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (1 − (1 − (1 / (1 − 𝑡)))) ∈ ℂ)
144 elmapi 8149 . . . . . . . . . . . . . . . . . . . . 21 (𝑝 ∈ (ℝ ↑𝑚 (1...𝑁)) → 𝑝:(1...𝑁)⟶ℝ)
145144adantl 475 . . . . . . . . . . . . . . . . . . . 20 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) → 𝑝:(1...𝑁)⟶ℝ)
146145ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) → 𝑝:(1...𝑁)⟶ℝ)
147146ffvelrnda 6613 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℝ)
148147recnd 10392 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℂ)
149143, 148mulcld 10384 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) ∈ ℂ)
150126, 142, 149subadd2d 10739 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) = ((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) ↔ (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) = (𝑥𝑖)))
151 eqcom 2832 . . . . . . . . . . . . . . 15 ((𝑥𝑖) = (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) ↔ (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) = (𝑥𝑖))
152150, 151syl6rbbr 282 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) = (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) ↔ ((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) = ((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖))))
153 eqcom 2832 . . . . . . . . . . . . . . 15 (((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) = ((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) ↔ ((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) = ((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))))
154126, 142subcld 10720 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) ∈ ℂ)
15589ad4ant23 761 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (1 − (1 − (1 / (1 − 𝑡)))) ≠ 0)
156154, 143, 148, 155divmuld 11156 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) / (1 − (1 − (1 / (1 − 𝑡))))) = (𝑝𝑖) ↔ ((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) = ((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)))))
157 eqcom 2832 . . . . . . . . . . . . . . . . 17 ((((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) / (1 − (1 − (1 / (1 − 𝑡))))) = (𝑝𝑖) ↔ (𝑝𝑖) = (((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) / (1 − (1 − (1 / (1 − 𝑡))))))
158126, 142, 143, 155divsubdird 11173 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) / (1 − (1 − (1 / (1 − 𝑡))))) = (((𝑥𝑖) / (1 − (1 − (1 / (1 − 𝑡))))) − (((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / (1 − (1 − (1 / (1 − 𝑡)))))))
159126, 143, 155divcld 11134 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) / (1 − (1 − (1 / (1 − 𝑡))))) ∈ ℂ)
160142, 143, 155divcld 11134 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / (1 − (1 − (1 / (1 − 𝑡))))) ∈ ℂ)
161159, 160negsubd 10726 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) / (1 − (1 − (1 / (1 − 𝑡))))) + -(((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / (1 − (1 − (1 / (1 − 𝑡)))))) = (((𝑥𝑖) / (1 − (1 − (1 / (1 − 𝑡))))) − (((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / (1 − (1 − (1 / (1 − 𝑡)))))))
162126, 143, 155divrec2d 11138 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) / (1 − (1 − (1 / (1 − 𝑡))))) = ((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)))
163142, 143, 155divneg2d 11148 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → -(((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / (1 − (1 − (1 / (1 − 𝑡))))) = (((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / -(1 − (1 − (1 / (1 − 𝑡))))))
164127, 133negsubdi2d 10736 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → -(1 − (1 − (1 / (1 − 𝑡)))) = ((1 − (1 / (1 − 𝑡))) − 1))
165164oveq2d 6926 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / -(1 − (1 − (1 / (1 − 𝑡))))) = (((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / ((1 − (1 / (1 − 𝑡))) − 1)))
166133, 127subcld 10720 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − (1 / (1 − 𝑡))) − 1) ∈ ℂ)
16788necomd 3054 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → (1 − (1 / (1 − 𝑡))) ≠ 1)
16863, 22, 167subne0d 10729 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑡 ∈ ℝ ∧ 𝑡 < 0) → ((1 − (1 / (1 − 𝑡))) − 1) ≠ 0)
169168ad4ant23 761 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − (1 / (1 − 𝑡))) − 1) ≠ 0)
170133, 141, 166, 169div23d 11171 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / ((1 − (1 / (1 − 𝑡))) − 1)) = (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖)))
171163, 165, 1703eqtrd 2865 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → -(((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / (1 − (1 − (1 / (1 − 𝑡))))) = (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖)))
172162, 171oveq12d 6928 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) / (1 − (1 − (1 / (1 − 𝑡))))) + -(((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)) / (1 − (1 − (1 / (1 − 𝑡)))))) = (((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)) + (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖))))
173158, 161, 1723eqtr2d 2867 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) / (1 − (1 − (1 / (1 − 𝑡))))) = (((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)) + (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖))))
174173eqeq2d 2835 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) / (1 − (1 − (1 / (1 − 𝑡))))) ↔ (𝑝𝑖) = (((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)) + (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖)))))
175157, 174syl5bb 275 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) / (1 − (1 − (1 / (1 − 𝑡))))) = (𝑝𝑖) ↔ (𝑝𝑖) = (((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)) + (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖)))))
176156, 175bitr3d 273 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) = ((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)) + (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖)))))
177153, 176syl5bb 275 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) − ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) = ((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) ↔ (𝑝𝑖) = (((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)) + (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖)))))
178152, 177bitrd 271 . . . . . . . . . . . . 13 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) = (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 / (1 − (1 − (1 / (1 − 𝑡))))) · (𝑥𝑖)) + (((1 − (1 / (1 − 𝑡))) / ((1 − (1 / (1 − 𝑡))) − 1)) · (𝑦𝑖)))))
179120, 178sylibrd 251 . . . . . . . . . . . 12 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (𝑥𝑖) = (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)))))
180179ralimdva 3171 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖)))))
181180imp 397 . . . . . . . . . 10 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − (1 − (1 / (1 − 𝑡)))) · (𝑝𝑖)) + ((1 − (1 / (1 − 𝑡))) · (𝑦𝑖))))
18249, 56, 181rspcedvd 3533 . . . . . . . . 9 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))
1831823mix2d 1440 . . . . . . . 8 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 𝑡 < 0) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))
184183exp31 412 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → (𝑡 < 0 → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))))
185184com23 86 . . . . . 6 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (𝑡 < 0 → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))))
186185imp 397 . . . . 5 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (𝑡 < 0 → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
187 simpr 479 . . . . . . . 8 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) ∧ 𝑡 ∈ (0[,]1)) → 𝑡 ∈ (0[,]1))
188 oveq2 6918 . . . . . . . . . . . . 13 (𝑘 = 𝑡 → (1 − 𝑘) = (1 − 𝑡))
189188oveq1d 6925 . . . . . . . . . . . 12 (𝑘 = 𝑡 → ((1 − 𝑘) · (𝑥𝑖)) = ((1 − 𝑡) · (𝑥𝑖)))
190 oveq1 6917 . . . . . . . . . . . 12 (𝑘 = 𝑡 → (𝑘 · (𝑦𝑖)) = (𝑡 · (𝑦𝑖)))
191189, 190oveq12d 6928 . . . . . . . . . . 11 (𝑘 = 𝑡 → (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))))
192191eqeq2d 2835 . . . . . . . . . 10 (𝑘 = 𝑡 → ((𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
193192ralbidv 3195 . . . . . . . . 9 (𝑘 = 𝑡 → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
194193adantl 475 . . . . . . . 8 (((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) ∧ 𝑡 ∈ (0[,]1)) ∧ 𝑘 = 𝑡) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
195 simplr 785 . . . . . . . 8 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) ∧ 𝑡 ∈ (0[,]1)) → ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))))
196187, 194, 195rspcedvd 3533 . . . . . . 7 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) ∧ 𝑡 ∈ (0[,]1)) → ∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))))
1971963mix1d 1439 . . . . . 6 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) ∧ 𝑡 ∈ (0[,]1)) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))
198197ex 403 . . . . 5 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (𝑡 ∈ (0[,]1) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
199 1red 10364 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → 1 ∈ ℝ)
200 simpl 476 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → 𝑡 ∈ ℝ)
201 0red 10367 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → 0 ∈ ℝ)
20215a1i 11 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → 0 < 1)
203 simpr 479 . . . . . . . . . . . . . . . 16 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → 1 < 𝑡)
204201, 199, 200, 202, 203lttrd 10524 . . . . . . . . . . . . . . 15 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → 0 < 𝑡)
205204gt0ne0d 10923 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → 𝑡 ≠ 0)
206199, 200, 205redivcld 11186 . . . . . . . . . . . . 13 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → (1 / 𝑡) ∈ ℝ)
207200, 204recgt0d 11295 . . . . . . . . . . . . 13 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → 0 < (1 / 𝑡))
208 recgt1i 11257 . . . . . . . . . . . . . 14 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → (0 < (1 / 𝑡) ∧ (1 / 𝑡) < 1))
209206adantr 474 . . . . . . . . . . . . . . 15 (((𝑡 ∈ ℝ ∧ 1 < 𝑡) ∧ (0 < (1 / 𝑡) ∧ (1 / 𝑡) < 1)) → (1 / 𝑡) ∈ ℝ)
210 1red 10364 . . . . . . . . . . . . . . 15 (((𝑡 ∈ ℝ ∧ 1 < 𝑡) ∧ (0 < (1 / 𝑡) ∧ (1 / 𝑡) < 1)) → 1 ∈ ℝ)
211 simprr 789 . . . . . . . . . . . . . . 15 (((𝑡 ∈ ℝ ∧ 1 < 𝑡) ∧ (0 < (1 / 𝑡) ∧ (1 / 𝑡) < 1)) → (1 / 𝑡) < 1)
212209, 210, 211ltled 10511 . . . . . . . . . . . . . 14 (((𝑡 ∈ ℝ ∧ 1 < 𝑡) ∧ (0 < (1 / 𝑡) ∧ (1 / 𝑡) < 1)) → (1 / 𝑡) ≤ 1)
213208, 212mpdan 678 . . . . . . . . . . . . 13 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → (1 / 𝑡) ≤ 1)
214206, 207, 2133jca 1162 . . . . . . . . . . . 12 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → ((1 / 𝑡) ∈ ℝ ∧ 0 < (1 / 𝑡) ∧ (1 / 𝑡) ≤ 1))
215 1re 10363 . . . . . . . . . . . . . 14 1 ∈ ℝ
2166, 215pm3.2i 464 . . . . . . . . . . . . 13 (0 ∈ ℝ* ∧ 1 ∈ ℝ)
217 elioc2 12531 . . . . . . . . . . . . 13 ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → ((1 / 𝑡) ∈ (0(,]1) ↔ ((1 / 𝑡) ∈ ℝ ∧ 0 < (1 / 𝑡) ∧ (1 / 𝑡) ≤ 1)))
218216, 217mp1i 13 . . . . . . . . . . . 12 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → ((1 / 𝑡) ∈ (0(,]1) ↔ ((1 / 𝑡) ∈ ℝ ∧ 0 < (1 / 𝑡) ∧ (1 / 𝑡) ≤ 1)))
219214, 218mpbird 249 . . . . . . . . . . 11 ((𝑡 ∈ ℝ ∧ 1 < 𝑡) → (1 / 𝑡) ∈ (0(,]1))
220219ad4ant23 761 . . . . . . . . . 10 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (1 / 𝑡) ∈ (0(,]1))
221 oveq2 6918 . . . . . . . . . . . . . . 15 (𝑚 = (1 / 𝑡) → (1 − 𝑚) = (1 − (1 / 𝑡)))
222221oveq1d 6925 . . . . . . . . . . . . . 14 (𝑚 = (1 / 𝑡) → ((1 − 𝑚) · (𝑥𝑖)) = ((1 − (1 / 𝑡)) · (𝑥𝑖)))
223 oveq1 6917 . . . . . . . . . . . . . 14 (𝑚 = (1 / 𝑡) → (𝑚 · (𝑝𝑖)) = ((1 / 𝑡) · (𝑝𝑖)))
224222, 223oveq12d 6928 . . . . . . . . . . . . 13 (𝑚 = (1 / 𝑡) → (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) = (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖))))
225224eqeq2d 2835 . . . . . . . . . . . 12 (𝑚 = (1 / 𝑡) → ((𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (𝑦𝑖) = (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖)))))
226225ralbidv 3195 . . . . . . . . . . 11 (𝑚 = (1 / 𝑡) → (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖)))))
227226adantl 475 . . . . . . . . . 10 (((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) ∧ 𝑚 = (1 / 𝑡)) → (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖)))))
228 simplr 785 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) → 𝑡 ∈ ℝ)
229228recnd 10392 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) → 𝑡 ∈ ℂ)
230229adantr 474 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → 𝑡 ∈ ℂ)
231204ex 403 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 ∈ ℝ → (1 < 𝑡 → 0 < 𝑡))
232 gt0ne0 10824 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑡 ∈ ℝ ∧ 0 < 𝑡) → 𝑡 ≠ 0)
233232ex 403 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑡 ∈ ℝ → (0 < 𝑡𝑡 ≠ 0))
234231, 233syld 47 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑡 ∈ ℝ → (1 < 𝑡𝑡 ≠ 0))
235234adantl 475 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → (1 < 𝑡𝑡 ≠ 0))
236235imp 397 . . . . . . . . . . . . . . . . . . . . 21 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) → 𝑡 ≠ 0)
237236adantr 474 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → 𝑡 ≠ 0)
238230, 237reccld 11127 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑡) ∈ ℂ)
239 1cnd 10358 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → 1 ∈ ℂ)
240230, 237recne0d 11128 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑡) ≠ 0)
241238, 239, 238, 240divsubdird 11173 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / 𝑡) − 1) / (1 / 𝑡)) = (((1 / 𝑡) / (1 / 𝑡)) − (1 / (1 / 𝑡))))
242238, 240dividd 11132 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((1 / 𝑡) / (1 / 𝑡)) = 1)
243230, 237recrecd 11131 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (1 / (1 / 𝑡)) = 𝑡)
244242, 243oveq12d 6928 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / 𝑡) / (1 / 𝑡)) − (1 / (1 / 𝑡))) = (1 − 𝑡))
245241, 244eqtr2d 2862 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑡) = (((1 / 𝑡) − 1) / (1 / 𝑡)))
246245oveq1d 6925 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − 𝑡) · (𝑥𝑖)) = ((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)))
247229, 236recrecd 11131 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) → (1 / (1 / 𝑡)) = 𝑡)
248247eqcomd 2831 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) → 𝑡 = (1 / (1 / 𝑡)))
249248adantr 474 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → 𝑡 = (1 / (1 / 𝑡)))
250249oveq1d 6925 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (𝑡 · (𝑦𝑖)) = ((1 / (1 / 𝑡)) · (𝑦𝑖)))
251246, 250oveq12d 6928 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) = (((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)) + ((1 / (1 / 𝑡)) · (𝑦𝑖))))
252251eqeq2d 2835 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) ↔ (𝑝𝑖) = (((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)) + ((1 / (1 / 𝑡)) · (𝑦𝑖)))))
253252biimpd 221 . . . . . . . . . . . . 13 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (𝑝𝑖) = (((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)) + ((1 / (1 / 𝑡)) · (𝑦𝑖)))))
254138ad2antrr 717 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) → 𝑦:(1...𝑁)⟶ℝ)
255254ffvelrnda 6613 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℝ)
256255recnd 10392 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℂ)
257239, 238subcld 10720 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (1 − (1 / 𝑡)) ∈ ℂ)
258123ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) → 𝑥:(1...𝑁)⟶ℝ)
259258ffvelrnda 6613 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℝ)
260259recnd 10392 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℂ)
261257, 260mulcld 10384 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − (1 / 𝑡)) · (𝑥𝑖)) ∈ ℂ)
262145ad2antrr 717 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) → 𝑝:(1...𝑁)⟶ℝ)
263262ffvelrnda 6613 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℝ)
264263recnd 10392 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℂ)
265238, 264mulcld 10384 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((1 / 𝑡) · (𝑝𝑖)) ∈ ℂ)
266256, 261, 265subaddd 10738 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) = ((1 / 𝑡) · (𝑝𝑖)) ↔ (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖))) = (𝑦𝑖)))
267 eqcom 2832 . . . . . . . . . . . . . . 15 ((𝑦𝑖) = (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖))) ↔ (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖))) = (𝑦𝑖))
268266, 267syl6rbbr 282 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) = (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖))) ↔ ((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) = ((1 / 𝑡) · (𝑝𝑖))))
269256, 261subcld 10720 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) ∈ ℂ)
270269, 238, 264, 240divmuld 11156 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) / (1 / 𝑡)) = (𝑝𝑖) ↔ ((1 / 𝑡) · (𝑝𝑖)) = ((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖)))))
271 eqcom 2832 . . . . . . . . . . . . . . 15 (((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) = ((1 / 𝑡) · (𝑝𝑖)) ↔ ((1 / 𝑡) · (𝑝𝑖)) = ((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))))
272270, 271syl6rbbr 282 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) = ((1 / 𝑡) · (𝑝𝑖)) ↔ (((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) / (1 / 𝑡)) = (𝑝𝑖)))
273 eqcom 2832 . . . . . . . . . . . . . . 15 ((((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) / (1 / 𝑡)) = (𝑝𝑖) ↔ (𝑝𝑖) = (((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) / (1 / 𝑡)))
274256, 261, 238, 240divsubdird 11173 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) / (1 / 𝑡)) = (((𝑦𝑖) / (1 / 𝑡)) − (((1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡))))
275256, 238, 240divcld 11134 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) / (1 / 𝑡)) ∈ ℂ)
276261, 238, 240divcld 11134 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡)) ∈ ℂ)
277275, 276negsubd 10726 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) / (1 / 𝑡)) + -(((1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡))) = (((𝑦𝑖) / (1 / 𝑡)) − (((1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡))))
278261, 238, 240divnegd 11147 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → -(((1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡)) = (-((1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡)))
279257, 260mulneg1d 10814 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (-(1 − (1 / 𝑡)) · (𝑥𝑖)) = -((1 − (1 / 𝑡)) · (𝑥𝑖)))
280279eqcomd 2831 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → -((1 − (1 / 𝑡)) · (𝑥𝑖)) = (-(1 − (1 / 𝑡)) · (𝑥𝑖)))
281280oveq1d 6925 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (-((1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡)) = ((-(1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡)))
282239, 238negsubdi2d 10736 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → -(1 − (1 / 𝑡)) = ((1 / 𝑡) − 1))
283282oveq1d 6925 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (-(1 − (1 / 𝑡)) · (𝑥𝑖)) = (((1 / 𝑡) − 1) · (𝑥𝑖)))
284283oveq1d 6925 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((-(1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡)) = ((((1 / 𝑡) − 1) · (𝑥𝑖)) / (1 / 𝑡)))
285238, 239subcld 10720 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((1 / 𝑡) − 1) ∈ ℂ)
286285, 260, 238, 240div23d 11171 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((((1 / 𝑡) − 1) · (𝑥𝑖)) / (1 / 𝑡)) = ((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)))
287284, 286eqtrd 2861 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((-(1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡)) = ((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)))
288278, 281, 2873eqtrd 2865 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → -(((1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡)) = ((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)))
289288oveq2d 6926 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) / (1 / 𝑡)) + -(((1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡))) = (((𝑦𝑖) / (1 / 𝑡)) + ((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖))))
290256, 238, 240divrec2d 11138 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) / (1 / 𝑡)) = ((1 / (1 / 𝑡)) · (𝑦𝑖)))
291290oveq1d 6925 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) / (1 / 𝑡)) + ((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖))) = (((1 / (1 / 𝑡)) · (𝑦𝑖)) + ((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖))))
292238, 240reccld 11127 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (1 / (1 / 𝑡)) ∈ ℂ)
293292, 256mulcld 10384 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((1 / (1 / 𝑡)) · (𝑦𝑖)) ∈ ℂ)
294285, 238, 240divcld 11134 . . . . . . . . . . . . . . . . . . . 20 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / 𝑡) − 1) / (1 / 𝑡)) ∈ ℂ)
295294, 260mulcld 10384 . . . . . . . . . . . . . . . . . . 19 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)) ∈ ℂ)
296293, 295addcomd 10564 . . . . . . . . . . . . . . . . . 18 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 / 𝑡)) · (𝑦𝑖)) + ((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖))) = (((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)) + ((1 / (1 / 𝑡)) · (𝑦𝑖))))
297289, 291, 2963eqtrd 2865 . . . . . . . . . . . . . . . . 17 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) / (1 / 𝑡)) + -(((1 − (1 / 𝑡)) · (𝑥𝑖)) / (1 / 𝑡))) = (((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)) + ((1 / (1 / 𝑡)) · (𝑦𝑖))))
298274, 277, 2973eqtr2d 2867 . . . . . . . . . . . . . . . 16 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) / (1 / 𝑡)) = (((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)) + ((1 / (1 / 𝑡)) · (𝑦𝑖))))
299298eqeq2d 2835 . . . . . . . . . . . . . . 15 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) / (1 / 𝑡)) ↔ (𝑝𝑖) = (((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)) + ((1 / (1 / 𝑡)) · (𝑦𝑖)))))
300273, 299syl5bb 275 . . . . . . . . . . . . . 14 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑦𝑖) − ((1 − (1 / 𝑡)) · (𝑥𝑖))) / (1 / 𝑡)) = (𝑝𝑖) ↔ (𝑝𝑖) = (((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)) + ((1 / (1 / 𝑡)) · (𝑦𝑖)))))
301268, 272, 3003bitrd 297 . . . . . . . . . . . . 13 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) = (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖))) ↔ (𝑝𝑖) = (((((1 / 𝑡) − 1) / (1 / 𝑡)) · (𝑥𝑖)) + ((1 / (1 / 𝑡)) · (𝑦𝑖)))))
302253, 301sylibrd 251 . . . . . . . . . . . 12 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (𝑦𝑖) = (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖)))))
303302ralimdva 3171 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖)))))
304303imp 397 . . . . . . . . . 10 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − (1 / 𝑡)) · (𝑥𝑖)) + ((1 / 𝑡) · (𝑝𝑖))))
305220, 227, 304rspcedvd 3533 . . . . . . . . 9 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))
3063053mix3d 1441 . . . . . . . 8 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ 1 < 𝑡) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))
307306exp31 412 . . . . . . 7 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → (1 < 𝑡 → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))))
308307com23 86 . . . . . 6 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (1 < 𝑡 → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))))
309308imp 397 . . . . 5 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → (1 < 𝑡 → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
310186, 198, 3093jaod 1557 . . . 4 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) ∧ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))) → ((𝑡 < 0 ∨ 𝑡 ∈ (0[,]1) ∨ 1 < 𝑡) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
311310ex 403 . . 3 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → ((𝑡 < 0 ∨ 𝑡 ∈ (0[,]1) ∨ 1 < 𝑡) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))))
3125, 311mpid 44 . 2 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) ∧ 𝑡 ∈ ℝ) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
313312rexlimdva 3240 1 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑𝑚 (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑𝑚 (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑𝑚 (1...𝑁))) → (∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3o 1110  w3a 1111   = wceq 1656  wcel 2164  wne 2999  wral 3117  wrex 3118  cdif 3795  {csn 4399   class class class wbr 4875  wf 6123  cfv 6127  (class class class)co 6910  𝑚 cmap 8127  cc 10257  cr 10258  0cc0 10259  1c1 10260   + caddc 10262   · cmul 10264  *cxr 10397   < clt 10398  cle 10399  cmin 10592  -cneg 10593   / cdiv 11016  cn 11357  +crp 12119  (,]cioc 12471  [,)cico 12472  [,]cicc 12473  ...cfz 12626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pow 5067  ax-pr 5129  ax-un 7214  ax-cnex 10315  ax-resscn 10316  ax-1cn 10317  ax-icn 10318  ax-addcl 10319  ax-addrcl 10320  ax-mulcl 10321  ax-mulrcl 10322  ax-mulcom 10323  ax-addass 10324  ax-mulass 10325  ax-distr 10326  ax-i2m1 10327  ax-1ne0 10328  ax-1rid 10329  ax-rnegex 10330  ax-rrecex 10331  ax-cnre 10332  ax-pre-lttri 10333  ax-pre-lttrn 10334  ax-pre-ltadd 10335  ax-pre-mulgt0 10336
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-nel 3103  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-csb 3758  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-pw 4382  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-iun 4744  df-br 4876  df-opab 4938  df-mpt 4955  df-id 5252  df-po 5265  df-so 5266  df-xp 5352  df-rel 5353  df-cnv 5354  df-co 5355  df-dm 5356  df-rn 5357  df-res 5358  df-ima 5359  df-iota 6090  df-fun 6129  df-fn 6130  df-f 6131  df-f1 6132  df-fo 6133  df-f1o 6134  df-fv 6135  df-riota 6871  df-ov 6913  df-oprab 6914  df-mpt2 6915  df-1st 7433  df-2nd 7434  df-er 8014  df-map 8129  df-en 8229  df-dom 8230  df-sdom 8231  df-pnf 10400  df-mnf 10401  df-xr 10402  df-ltxr 10403  df-le 10404  df-sub 10594  df-neg 10595  df-div 11017  df-rp 12120  df-ioc 12475  df-ico 12476  df-icc 12477
This theorem is referenced by:  eenglngeehlnm  43303
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