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Theorem eenglngeehlnmlem1 48125
Description: Lemma 1 for eenglngeehlnm 48127. (Contributed by AV, 15-Feb-2023.)
Assertion
Ref Expression
eenglngeehlnmlem1 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → ((∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(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)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
Distinct variable group:   𝑖,𝑁,𝑘,𝑙,𝑚,𝑝,𝑡,𝑥,𝑦

Proof of Theorem eenglngeehlnmlem1
StepHypRef Expression
1 oveq2 7432 . . . . . . . 8 (𝑘 = 𝑡 → (1 − 𝑘) = (1 − 𝑡))
21oveq1d 7439 . . . . . . 7 (𝑘 = 𝑡 → ((1 − 𝑘) · (𝑥𝑖)) = ((1 − 𝑡) · (𝑥𝑖)))
3 oveq1 7431 . . . . . . 7 (𝑘 = 𝑡 → (𝑘 · (𝑦𝑖)) = (𝑡 · (𝑦𝑖)))
42, 3oveq12d 7442 . . . . . 6 (𝑘 = 𝑡 → (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))))
54eqeq2d 2737 . . . . 5 (𝑘 = 𝑡 → ((𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
65ralbidv 3168 . . . 4 (𝑘 = 𝑡 → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
76cbvrexvw 3226 . . 3 (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ ∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))))
8 unitssre 13530 . . . 4 (0[,]1) ⊆ ℝ
9 ssrexv 4049 . . . 4 ((0[,]1) ⊆ ℝ → (∃𝑡 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
108, 9mp1i 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)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
117, 10biimtrid 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)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
12 0re 11266 . . . . . . . 8 0 ∈ ℝ
13 1xr 11323 . . . . . . . 8 1 ∈ ℝ*
14 elico2 13442 . . . . . . . 8 ((0 ∈ ℝ ∧ 1 ∈ ℝ*) → (𝑙 ∈ (0[,)1) ↔ (𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1)))
1512, 13, 14mp2an 690 . . . . . . 7 (𝑙 ∈ (0[,)1) ↔ (𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1))
16 simp1 1133 . . . . . . . 8 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → 𝑙 ∈ ℝ)
17 1red 11265 . . . . . . . . 9 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → 1 ∈ ℝ)
1817, 16resubcld 11692 . . . . . . . 8 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → (1 − 𝑙) ∈ ℝ)
19 1cnd 11259 . . . . . . . . 9 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → 1 ∈ ℂ)
2016recnd 11292 . . . . . . . . 9 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → 𝑙 ∈ ℂ)
21 ltne 11361 . . . . . . . . . 10 ((𝑙 ∈ ℝ ∧ 𝑙 < 1) → 1 ≠ 𝑙)
22213adant2 1128 . . . . . . . . 9 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → 1 ≠ 𝑙)
2319, 20, 22subne0d 11630 . . . . . . . 8 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → (1 − 𝑙) ≠ 0)
2416, 18, 23redivcld 12093 . . . . . . 7 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → (𝑙 / (1 − 𝑙)) ∈ ℝ)
2515, 24sylbi 216 . . . . . 6 (𝑙 ∈ (0[,)1) → (𝑙 / (1 − 𝑙)) ∈ ℝ)
2625ad2antlr 725 . . . . 5 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))) → (𝑙 / (1 − 𝑙)) ∈ ℝ)
2726renegcld 11691 . . . 4 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))) → -(𝑙 / (1 − 𝑙)) ∈ ℝ)
28 oveq2 7432 . . . . . . . . 9 (𝑡 = -(𝑙 / (1 − 𝑙)) → (1 − 𝑡) = (1 − -(𝑙 / (1 − 𝑙))))
2928oveq1d 7439 . . . . . . . 8 (𝑡 = -(𝑙 / (1 − 𝑙)) → ((1 − 𝑡) · (𝑥𝑖)) = ((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)))
30 oveq1 7431 . . . . . . . 8 (𝑡 = -(𝑙 / (1 − 𝑙)) → (𝑡 · (𝑦𝑖)) = (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))
3129, 30oveq12d 7442 . . . . . . 7 (𝑡 = -(𝑙 / (1 − 𝑙)) → (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
3231eqeq2d 2737 . . . . . 6 (𝑡 = -(𝑙 / (1 − 𝑙)) → ((𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
3332ralbidv 3168 . . . . 5 (𝑡 = -(𝑙 / (1 − 𝑙)) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
3433adantl 480 . . . 4 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))) ∧ 𝑡 = -(𝑙 / (1 − 𝑙))) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
35 eqcom 2733 . . . . . . . 8 ((𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) = (𝑥𝑖))
36 elmapi 8878 . . . . . . . . . . . . 13 (𝑥 ∈ (ℝ ↑m (1...𝑁)) → 𝑥:(1...𝑁)⟶ℝ)
37363ad2ant2 1131 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) → 𝑥:(1...𝑁)⟶ℝ)
3837ad2antrr 724 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) → 𝑥:(1...𝑁)⟶ℝ)
3938ffvelcdmda 7098 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℝ)
4039recnd 11292 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℂ)
4115, 16sylbi 216 . . . . . . . . . . . 12 (𝑙 ∈ (0[,)1) → 𝑙 ∈ ℝ)
4241ad2antlr 725 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑙 ∈ ℝ)
4342recnd 11292 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑙 ∈ ℂ)
44 eldifi 4126 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥}) → 𝑦 ∈ (ℝ ↑m (1...𝑁)))
45 elmapi 8878 . . . . . . . . . . . . . . 15 (𝑦 ∈ (ℝ ↑m (1...𝑁)) → 𝑦:(1...𝑁)⟶ℝ)
4644, 45syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥}) → 𝑦:(1...𝑁)⟶ℝ)
47463ad2ant3 1132 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) → 𝑦:(1...𝑁)⟶ℝ)
4847ad2antrr 724 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) → 𝑦:(1...𝑁)⟶ℝ)
4948ffvelcdmda 7098 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℝ)
5049recnd 11292 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℂ)
5143, 50mulcld 11284 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑙 · (𝑦𝑖)) ∈ ℂ)
52 1cnd 11259 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → 1 ∈ ℂ)
5352, 43subcld 11621 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑙) ∈ ℂ)
54 elmapi 8878 . . . . . . . . . . . . 13 (𝑝 ∈ (ℝ ↑m (1...𝑁)) → 𝑝:(1...𝑁)⟶ℝ)
5554ad2antlr 725 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) → 𝑝:(1...𝑁)⟶ℝ)
5655ffvelcdmda 7098 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℝ)
5756recnd 11292 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℂ)
5853, 57mulcld 11284 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − 𝑙) · (𝑝𝑖)) ∈ ℂ)
5940, 51, 58subadd2d 11640 . . . . . . . 8 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) = ((1 − 𝑙) · (𝑝𝑖)) ↔ (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) = (𝑥𝑖)))
6035, 59bitr4id 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)) → ((𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ((𝑥𝑖) − (𝑙 · (𝑦𝑖))) = ((1 − 𝑙) · (𝑝𝑖))))
61 eqcom 2733 . . . . . . . . 9 (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) = ((1 − 𝑙) · (𝑝𝑖)) ↔ ((1 − 𝑙) · (𝑝𝑖)) = ((𝑥𝑖) − (𝑙 · (𝑦𝑖))))
6240, 51subcld 11621 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) − (𝑙 · (𝑦𝑖))) ∈ ℂ)
6315, 22sylbi 216 . . . . . . . . . . . 12 (𝑙 ∈ (0[,)1) → 1 ≠ 𝑙)
6463ad2antlr 725 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → 1 ≠ 𝑙)
6552, 43, 64subne0d 11630 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑙) ≠ 0)
6662, 53, 57, 65divmuld 12063 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (𝑝𝑖) ↔ ((1 − 𝑙) · (𝑝𝑖)) = ((𝑥𝑖) − (𝑙 · (𝑦𝑖)))))
6761, 66bitr4id 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)) → (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) = ((1 − 𝑙) · (𝑝𝑖)) ↔ (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (𝑝𝑖)))
68 eqcom 2733 . . . . . . . . . 10 ((((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (𝑝𝑖) ↔ (𝑝𝑖) = (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)))
6940, 51, 53, 65divsubdird 12080 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (((𝑥𝑖) / (1 − 𝑙)) − ((𝑙 · (𝑦𝑖)) / (1 − 𝑙))))
7040, 53, 65divrec2d 12045 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) / (1 − 𝑙)) = ((1 / (1 − 𝑙)) · (𝑥𝑖)))
7143, 50, 53, 65div23d 12078 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑙 · (𝑦𝑖)) / (1 − 𝑙)) = ((𝑙 / (1 − 𝑙)) · (𝑦𝑖)))
7270, 71oveq12d 7442 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) / (1 − 𝑙)) − ((𝑙 · (𝑦𝑖)) / (1 − 𝑙))) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
7369, 72eqtrd 2766 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
7473eqeq2d 2737 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) ↔ (𝑝𝑖) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
7568, 74bitrid 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)) → ((((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (𝑝𝑖) ↔ (𝑝𝑖) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
7643, 53, 65divcld 12041 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑙 / (1 − 𝑙)) ∈ ℂ)
7776, 50mulneg1d 11717 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)) = -((𝑙 / (1 − 𝑙)) · (𝑦𝑖)))
7877eqcomd 2732 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → -((𝑙 / (1 − 𝑙)) · (𝑦𝑖)) = (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))
7978oveq2d 7440 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 − 𝑙)) · (𝑥𝑖)) + -((𝑙 / (1 − 𝑙)) · (𝑦𝑖))) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
8053, 65reccld 12034 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 / (1 − 𝑙)) ∈ ℂ)
8180, 40mulcld 11284 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 / (1 − 𝑙)) · (𝑥𝑖)) ∈ ℂ)
8276, 50mulcld 11284 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑙 / (1 − 𝑙)) · (𝑦𝑖)) ∈ ℂ)
8381, 82negsubd 11627 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 − 𝑙)) · (𝑥𝑖)) + -((𝑙 / (1 − 𝑙)) · (𝑦𝑖))) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
8452, 76subnegd 11628 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − -(𝑙 / (1 − 𝑙))) = (1 + (𝑙 / (1 − 𝑙))))
85 muldivdir 11958 . . . . . . . . . . . . . . . . 17 ((1 ∈ ℂ ∧ 𝑙 ∈ ℂ ∧ ((1 − 𝑙) ∈ ℂ ∧ (1 − 𝑙) ≠ 0)) → ((((1 − 𝑙) · 1) + 𝑙) / (1 − 𝑙)) = (1 + (𝑙 / (1 − 𝑙))))
8652, 43, 53, 65, 85syl112anc 1371 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((1 − 𝑙) · 1) + 𝑙) / (1 − 𝑙)) = (1 + (𝑙 / (1 − 𝑙))))
8753mulridd 11281 . . . . . . . . . . . . . . . . . . 19 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − 𝑙) · 1) = (1 − 𝑙))
8887oveq1d 7439 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − 𝑙) · 1) + 𝑙) = ((1 − 𝑙) + 𝑙))
8952, 43npcand 11625 . . . . . . . . . . . . . . . . . 18 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − 𝑙) + 𝑙) = 1)
9088, 89eqtrd 2766 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − 𝑙) · 1) + 𝑙) = 1)
9190oveq1d 7439 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((1 − 𝑙) · 1) + 𝑙) / (1 − 𝑙)) = (1 / (1 − 𝑙)))
9284, 86, 913eqtr2d 2772 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − -(𝑙 / (1 − 𝑙))) = (1 / (1 − 𝑙)))
9392eqcomd 2732 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 / (1 − 𝑙)) = (1 − -(𝑙 / (1 − 𝑙))))
9493oveq1d 7439 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 / (1 − 𝑙)) · (𝑥𝑖)) = ((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)))
9594oveq1d 7439 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 − 𝑙)) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖))) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
9679, 83, 953eqtr3d 2774 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖))) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
9796eqeq2d 2737 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
9897biimpd 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)) → ((𝑝𝑖) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖))) → (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
9975, 98sylbid 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)) → ((((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (𝑝𝑖) → (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
10067, 99sylbid 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)) → (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) = ((1 − 𝑙) · (𝑝𝑖)) → (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
10160, 100sylbid 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)) → ((𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) → (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
102101ralimdva 3157 . . . . 5 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) → ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
103102imp 405 . . . 4 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))) → ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
10427, 34, 103rspcedvd 3610 . . 3 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))))
105104rexlimdva2 3147 . 2 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
106 0xr 11311 . . . . . . 7 0 ∈ ℝ*
107 1re 11264 . . . . . . 7 1 ∈ ℝ
108 elioc2 13441 . . . . . . 7 ((0 ∈ ℝ* ∧ 1 ∈ ℝ) → (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ ∧ 0 < 𝑚𝑚 ≤ 1)))
109106, 107, 108mp2an 690 . . . . . 6 (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ ∧ 0 < 𝑚𝑚 ≤ 1))
110 simp1 1133 . . . . . . 7 ((𝑚 ∈ ℝ ∧ 0 < 𝑚𝑚 ≤ 1) → 𝑚 ∈ ℝ)
111 gt0ne0 11729 . . . . . . . 8 ((𝑚 ∈ ℝ ∧ 0 < 𝑚) → 𝑚 ≠ 0)
1121113adant3 1129 . . . . . . 7 ((𝑚 ∈ ℝ ∧ 0 < 𝑚𝑚 ≤ 1) → 𝑚 ≠ 0)
113110, 112rereccld 12092 . . . . . 6 ((𝑚 ∈ ℝ ∧ 0 < 𝑚𝑚 ≤ 1) → (1 / 𝑚) ∈ ℝ)
114109, 113sylbi 216 . . . . 5 (𝑚 ∈ (0(,]1) → (1 / 𝑚) ∈ ℝ)
115114ad2antlr 725 . . . 4 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))) → (1 / 𝑚) ∈ ℝ)
116 oveq2 7432 . . . . . . . . 9 (𝑡 = (1 / 𝑚) → (1 − 𝑡) = (1 − (1 / 𝑚)))
117116oveq1d 7439 . . . . . . . 8 (𝑡 = (1 / 𝑚) → ((1 − 𝑡) · (𝑥𝑖)) = ((1 − (1 / 𝑚)) · (𝑥𝑖)))
118 oveq1 7431 . . . . . . . 8 (𝑡 = (1 / 𝑚) → (𝑡 · (𝑦𝑖)) = ((1 / 𝑚) · (𝑦𝑖)))
119117, 118oveq12d 7442 . . . . . . 7 (𝑡 = (1 / 𝑚) → (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖))))
120119eqeq2d 2737 . . . . . 6 (𝑡 = (1 / 𝑚) → ((𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
121120ralbidv 3168 . . . . 5 (𝑡 = (1 / 𝑚) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
122121adantl 480 . . . 4 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))) ∧ 𝑡 = (1 / 𝑚)) → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
123 eqcom 2733 . . . . . . . 8 ((𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) = (𝑦𝑖))
12447ad2antrr 724 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) → 𝑦:(1...𝑁)⟶ℝ)
125124ffvelcdmda 7098 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℝ)
126125recnd 11292 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℂ)
127 1cnd 11259 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → 1 ∈ ℂ)
128109, 110sylbi 216 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (0(,]1) → 𝑚 ∈ ℝ)
129128recnd 11292 . . . . . . . . . . . . . . 15 (𝑚 ∈ (0(,]1) → 𝑚 ∈ ℂ)
130129ad2antlr 725 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑚 ∈ ℂ)
131127, 130subcld 11621 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑚) ∈ ℂ)
13237ad2antrr 724 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) → 𝑥:(1...𝑁)⟶ℝ)
133132ffvelcdmda 7098 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℝ)
134133recnd 11292 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℂ)
135131, 134mulcld 11284 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − 𝑚) · (𝑥𝑖)) ∈ ℂ)
136126, 135negsubd 11627 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) + -((1 − 𝑚) · (𝑥𝑖))) = ((𝑦𝑖) − ((1 − 𝑚) · (𝑥𝑖))))
137131, 134mulneg1d 11717 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (-(1 − 𝑚) · (𝑥𝑖)) = -((1 − 𝑚) · (𝑥𝑖)))
138127, 130negsubdi2d 11637 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → -(1 − 𝑚) = (𝑚 − 1))
139138oveq1d 7439 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (-(1 − 𝑚) · (𝑥𝑖)) = ((𝑚 − 1) · (𝑥𝑖)))
140137, 139eqtr3d 2768 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → -((1 − 𝑚) · (𝑥𝑖)) = ((𝑚 − 1) · (𝑥𝑖)))
141140oveq2d 7440 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) + -((1 − 𝑚) · (𝑥𝑖))) = ((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))))
142136, 141eqtr3d 2768 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) − ((1 − 𝑚) · (𝑥𝑖))) = ((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))))
143142eqeq1d 2728 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) − ((1 − 𝑚) · (𝑥𝑖))) = (𝑚 · (𝑝𝑖)) ↔ ((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) = (𝑚 · (𝑝𝑖))))
14454ad2antlr 725 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) → 𝑝:(1...𝑁)⟶ℝ)
145144ffvelcdmda 7098 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℝ)
146145recnd 11292 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℂ)
147130, 146mulcld 11284 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑚 · (𝑝𝑖)) ∈ ℂ)
148126, 135, 147subaddd 11639 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) − ((1 − 𝑚) · (𝑥𝑖))) = (𝑚 · (𝑝𝑖)) ↔ (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) = (𝑦𝑖)))
149 eqcom 2733 . . . . . . . . . . 11 ((((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) / 𝑚) = (𝑝𝑖) ↔ (𝑝𝑖) = (((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) / 𝑚))
150149a1i 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)) → ((((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) / 𝑚) = (𝑝𝑖) ↔ (𝑝𝑖) = (((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) / 𝑚)))
151130, 127subcld 11621 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑚 − 1) ∈ ℂ)
152151, 134mulcld 11284 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑚 − 1) · (𝑥𝑖)) ∈ ℂ)
153126, 152addcld 11283 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) ∈ ℂ)
154 elioc1 13420 . . . . . . . . . . . . . 14 ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ* ∧ 0 < 𝑚𝑚 ≤ 1)))
155106, 13, 154mp2an 690 . . . . . . . . . . . . 13 (𝑚 ∈ (0(,]1) ↔ (𝑚 ∈ ℝ* ∧ 0 < 𝑚𝑚 ≤ 1))
15612a1i 11 . . . . . . . . . . . . . . . 16 (𝑚 ∈ ℝ* → 0 ∈ ℝ)
157156anim1i 613 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℝ* ∧ 0 < 𝑚) → (0 ∈ ℝ ∧ 0 < 𝑚))
1581573adant3 1129 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℝ* ∧ 0 < 𝑚𝑚 ≤ 1) → (0 ∈ ℝ ∧ 0 < 𝑚))
159 ltne 11361 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ 0 < 𝑚) → 𝑚 ≠ 0)
160158, 159syl 17 . . . . . . . . . . . . 13 ((𝑚 ∈ ℝ* ∧ 0 < 𝑚𝑚 ≤ 1) → 𝑚 ≠ 0)
161155, 160sylbi 216 . . . . . . . . . . . 12 (𝑚 ∈ (0(,]1) → 𝑚 ≠ 0)
162161ad2antlr 725 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑚 ≠ 0)
163153, 146, 130, 162divmul2d 12074 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) / 𝑚) = (𝑝𝑖) ↔ ((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) = (𝑚 · (𝑝𝑖))))
164126, 152, 130, 162divdird 12079 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) / 𝑚) = (((𝑦𝑖) / 𝑚) + (((𝑚 − 1) · (𝑥𝑖)) / 𝑚)))
165126, 130, 162divrec2d 12045 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) / 𝑚) = ((1 / 𝑚) · (𝑦𝑖)))
166151, 134, 130, 162div23d 12078 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑚 − 1) · (𝑥𝑖)) / 𝑚) = (((𝑚 − 1) / 𝑚) · (𝑥𝑖)))
167130, 127, 130, 162divsubdird 12080 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑚 − 1) / 𝑚) = ((𝑚 / 𝑚) − (1 / 𝑚)))
168167oveq1d 7439 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑚 − 1) / 𝑚) · (𝑥𝑖)) = (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))
169166, 168eqtrd 2766 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑚 − 1) · (𝑥𝑖)) / 𝑚) = (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))
170165, 169oveq12d 7442 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) / 𝑚) + (((𝑚 − 1) · (𝑥𝑖)) / 𝑚)) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖))))
171164, 170eqtrd 2766 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) / 𝑚) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖))))
172171eqeq2d 2737 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) / 𝑚) ↔ (𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))))
173150, 163, 1723bitr3d 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)) → (((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) = (𝑚 · (𝑝𝑖)) ↔ (𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))))
174143, 148, 1733bitr3d 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) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) = (𝑦𝑖) ↔ (𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))))
175123, 174bitrid 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)) → ((𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))))
176130, 162reccld 12034 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 / 𝑚) ∈ ℂ)
177176, 126mulcld 11284 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 / 𝑚) · (𝑦𝑖)) ∈ ℂ)
178127, 176subcld 11621 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − (1 / 𝑚)) ∈ ℂ)
179178, 134mulcld 11284 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((1 − (1 / 𝑚)) · (𝑥𝑖)) ∈ ℂ)
180130, 162dividd 12039 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (𝑚 / 𝑚) = 1)
181180oveq1d 7439 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑚 / 𝑚) − (1 / 𝑚)) = (1 − (1 / 𝑚)))
182181oveq1d 7439 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)) = ((1 − (1 / 𝑚)) · (𝑥𝑖)))
183182oveq2d 7440 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖))) = (((1 / 𝑚) · (𝑦𝑖)) + ((1 − (1 / 𝑚)) · (𝑥𝑖))))
184177, 179, 183comraddd 11478 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖))) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖))))
185184eqeq2d 2737 . . . . . . . 8 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖))) ↔ (𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
186185biimpd 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)) → ((𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖))) → (𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
187175, 186sylbid 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)) → ((𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) → (𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
188187ralimdva 3157 . . . . 5 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) → (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) → ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
189188imp 405 . . . 4 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))) → ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖))))
190115, 122, 189rspcedvd 3610 . . 3 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))))
191190rexlimdva2 3147 . 2 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → (∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
19211, 105, 1913jaod 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)(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)))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  w3o 1083  w3a 1084   = wceq 1534  wcel 2099  wne 2930  wral 3051  wrex 3060  cdif 3944  wss 3947  {csn 4633   class class class wbr 5153  wf 6550  cfv 6554  (class class class)co 7424  m cmap 8855  cc 11156  cr 11157  0cc0 11158  1c1 11159   + caddc 11161   · cmul 11163  *cxr 11297   < clt 11298  cle 11299  cmin 11494  -cneg 11495   / cdiv 11921  cn 12264  (,]cioc 13379  [,)cico 13380  [,]cicc 13381  ...cfz 13538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215  ax-1cn 11216  ax-icn 11217  ax-addcl 11218  ax-addrcl 11219  ax-mulcl 11220  ax-mulrcl 11221  ax-mulcom 11222  ax-addass 11223  ax-mulass 11224  ax-distr 11225  ax-i2m1 11226  ax-1ne0 11227  ax-1rid 11228  ax-rnegex 11229  ax-rrecex 11230  ax-cnre 11231  ax-pre-lttri 11232  ax-pre-lttrn 11233  ax-pre-ltadd 11234  ax-pre-mulgt0 11235
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-nel 3037  df-ral 3052  df-rex 3061  df-rmo 3364  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-po 5594  df-so 5595  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-riota 7380  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-er 8734  df-map 8857  df-en 8975  df-dom 8976  df-sdom 8977  df-pnf 11300  df-mnf 11301  df-xr 11302  df-ltxr 11303  df-le 11304  df-sub 11496  df-neg 11497  df-div 11922  df-ioc 13383  df-ico 13384  df-icc 13385
This theorem is referenced by:  eenglngeehlnm  48127
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