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Theorem eenglngeehlnmlem1 44725
Description: Lemma 1 for eenglngeehlnm 44727. (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 7163 . . . . . . . 8 (𝑘 = 𝑡 → (1 − 𝑘) = (1 − 𝑡))
21oveq1d 7170 . . . . . . 7 (𝑘 = 𝑡 → ((1 − 𝑘) · (𝑥𝑖)) = ((1 − 𝑡) · (𝑥𝑖)))
3 oveq1 7162 . . . . . . 7 (𝑘 = 𝑡 → (𝑘 · (𝑦𝑖)) = (𝑡 · (𝑦𝑖)))
42, 3oveq12d 7173 . . . . . 6 (𝑘 = 𝑡 → (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))))
54eqeq2d 2832 . . . . 5 (𝑘 = 𝑡 → ((𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
65ralbidv 3197 . . . 4 (𝑘 = 𝑡 → (∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
76cbvrexvw 3450 . . 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 12884 . . . 4 (0[,]1) ⊆ ℝ
9 ssrexv 4033 . . . 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, 10syl5bi 244 . 2 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
12 0re 10642 . . . . . . . 8 0 ∈ ℝ
13 1xr 10699 . . . . . . . 8 1 ∈ ℝ*
14 elico2 12799 . . . . . . . 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 1132 . . . . . . . 8 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → 𝑙 ∈ ℝ)
17 1red 10641 . . . . . . . . 9 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → 1 ∈ ℝ)
1817, 16resubcld 11067 . . . . . . . 8 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → (1 − 𝑙) ∈ ℝ)
19 1cnd 10635 . . . . . . . . 9 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → 1 ∈ ℂ)
2016recnd 10668 . . . . . . . . 9 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → 𝑙 ∈ ℂ)
21 ltne 10736 . . . . . . . . . 10 ((𝑙 ∈ ℝ ∧ 𝑙 < 1) → 1 ≠ 𝑙)
22213adant2 1127 . . . . . . . . 9 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → 1 ≠ 𝑙)
2319, 20, 22subne0d 11005 . . . . . . . 8 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → (1 − 𝑙) ≠ 0)
2416, 18, 23redivcld 11467 . . . . . . 7 ((𝑙 ∈ ℝ ∧ 0 ≤ 𝑙𝑙 < 1) → (𝑙 / (1 − 𝑙)) ∈ ℝ)
2515, 24sylbi 219 . . . . . 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 11066 . . . 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 7163 . . . . . . . . 9 (𝑡 = -(𝑙 / (1 − 𝑙)) → (1 − 𝑡) = (1 − -(𝑙 / (1 − 𝑙))))
2928oveq1d 7170 . . . . . . . 8 (𝑡 = -(𝑙 / (1 − 𝑙)) → ((1 − 𝑡) · (𝑥𝑖)) = ((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)))
30 oveq1 7162 . . . . . . . 8 (𝑡 = -(𝑙 / (1 − 𝑙)) → (𝑡 · (𝑦𝑖)) = (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))
3129, 30oveq12d 7173 . . . . . . 7 (𝑡 = -(𝑙 / (1 − 𝑙)) → (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
3231eqeq2d 2832 . . . . . 6 (𝑡 = -(𝑙 / (1 − 𝑙)) → ((𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
3332ralbidv 3197 . . . . 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 484 . . . 4 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(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 elmapi 8427 . . . . . . . . . . . . 13 (𝑥 ∈ (ℝ ↑m (1...𝑁)) → 𝑥:(1...𝑁)⟶ℝ)
36353ad2ant2 1130 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) → 𝑥:(1...𝑁)⟶ℝ)
3736ad2antrr 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)
3837ffvelrnda 6850 . . . . . . . . . 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)
3938recnd 10668 . . . . . . . . 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)
4015, 16sylbi 219 . . . . . . . . . . . 12 (𝑙 ∈ (0[,)1) → 𝑙 ∈ ℝ)
4140ad2antlr 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)
4241recnd 10668 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → 𝑙 ∈ ℂ)
43 eldifi 4102 . . . . . . . . . . . . . . 15 (𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥}) → 𝑦 ∈ (ℝ ↑m (1...𝑁)))
44 elmapi 8427 . . . . . . . . . . . . . . 15 (𝑦 ∈ (ℝ ↑m (1...𝑁)) → 𝑦:(1...𝑁)⟶ℝ)
4543, 44syl 17 . . . . . . . . . . . . . 14 (𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥}) → 𝑦:(1...𝑁)⟶ℝ)
46453ad2ant3 1131 . . . . . . . . . . . . 13 ((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) → 𝑦:(1...𝑁)⟶ℝ)
4746ad2antrr 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)
4847ffvelrnda 6850 . . . . . . . . . . 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)
4948recnd 10668 . . . . . . . . . 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)
5042, 49mulcld 10660 . . . . . . . . 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)
51 1cnd 10635 . . . . . . . . . . 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)
5251, 42subcld 10996 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − 𝑙) ∈ ℂ)
53 elmapi 8427 . . . . . . . . . . . . 13 (𝑝 ∈ (ℝ ↑m (1...𝑁)) → 𝑝:(1...𝑁)⟶ℝ)
5453ad2antlr 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)
5554ffvelrnda 6850 . . . . . . . . . . 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)
5655recnd 10668 . . . . . . . . . 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)
5752, 56mulcld 10660 . . . . . . . . 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)
5839, 50, 57subadd2d 11015 . . . . . . . 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)))
59 eqcom 2828 . . . . . . . 8 ((𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) = (𝑥𝑖))
6058, 59syl6rbbr 292 . . . . . . 7 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ((𝑥𝑖) − (𝑙 · (𝑦𝑖))) = ((1 − 𝑙) · (𝑝𝑖))))
6139, 50subcld 10996 . . . . . . . . . 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)
6215, 22sylbi 219 . . . . . . . . . . . 12 (𝑙 ∈ (0[,)1) → 1 ≠ 𝑙)
6362ad2antlr 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)
6451, 42, 63subne0d 11005 . . . . . . . . . 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)
6561, 52, 56, 64divmuld 11437 . . . . . . . . 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)))))
66 eqcom 2828 . . . . . . . . 9 (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) = ((1 − 𝑙) · (𝑝𝑖)) ↔ ((1 − 𝑙) · (𝑝𝑖)) = ((𝑥𝑖) − (𝑙 · (𝑦𝑖))))
6765, 66syl6rbbr 292 . . . . . . . 8 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) = ((1 − 𝑙) · (𝑝𝑖)) ↔ (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (𝑝𝑖)))
68 eqcom 2828 . . . . . . . . . 10 ((((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (𝑝𝑖) ↔ (𝑝𝑖) = (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)))
6939, 50, 52, 64divsubdird 11454 . . . . . . . . . . . 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))))
7039, 52, 64divrec2d 11419 . . . . . . . . . . . . 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)))
7142, 49, 52, 64div23d 11452 . . . . . . . . . . . . 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 7173 . . . . . . . . . . . 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 2856 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
7473eqeq2d 2832 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) ↔ (𝑝𝑖) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
7568, 74syl5bb 285 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (𝑝𝑖) ↔ (𝑝𝑖) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
7642, 52, 64divcld 11415 . . . . . . . . . . . . . . 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, 49mulneg1d 11092 . . . . . . . . . . . . . 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 2827 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → -((𝑙 / (1 − 𝑙)) · (𝑦𝑖)) = (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))
7978oveq2d 7171 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 − 𝑙)) · (𝑥𝑖)) + -((𝑙 / (1 − 𝑙)) · (𝑦𝑖))) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
8052, 64reccld 11408 . . . . . . . . . . . . . 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, 39mulcld 10660 . . . . . . . . . . . . 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, 49mulcld 10660 . . . . . . . . . . . . 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 11002 . . . . . . . . . . . 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))))
8451, 76subnegd 11003 . . . . . . . . . . . . . . . 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 11332 . . . . . . . . . . . . . . . . 17 ((1 ∈ ℂ ∧ 𝑙 ∈ ℂ ∧ ((1 − 𝑙) ∈ ℂ ∧ (1 − 𝑙) ≠ 0)) → ((((1 − 𝑙) · 1) + 𝑙) / (1 − 𝑙)) = (1 + (𝑙 / (1 − 𝑙))))
8651, 42, 52, 64, 85syl112anc 1370 . . . . . . . . . . . . . . . 16 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((1 − 𝑙) · 1) + 𝑙) / (1 − 𝑙)) = (1 + (𝑙 / (1 − 𝑙))))
8752mulid1d 10657 . . . . . . . . . . . . . . . . . . 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 7170 . . . . . . . . . . . . . . . . . 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))
8951, 42npcand 11000 . . . . . . . . . . . . . . . . . 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 2856 . . . . . . . . . . . . . . . . 17 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 − 𝑙) · 1) + 𝑙) = 1)
9190oveq1d 7170 . . . . . . . . . . . . . . . 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 2862 . . . . . . . . . . . . . . 15 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 − -(𝑙 / (1 − 𝑙))) = (1 / (1 − 𝑙)))
9392eqcomd 2827 . . . . . . . . . . . . . 14 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (1 / (1 − 𝑙)) = (1 − -(𝑙 / (1 − 𝑙))))
9493oveq1d 7170 . . . . . . . . . . . . 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 7170 . . . . . . . . . . . 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 2864 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖))) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
9796eqeq2d 2832 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
9897biimpd 231 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 / (1 − 𝑙)) · (𝑥𝑖)) − ((𝑙 / (1 − 𝑙)) · (𝑦𝑖))) → (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
9975, 98sylbid 242 . . . . . . . 8 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((𝑥𝑖) − (𝑙 · (𝑦𝑖))) / (1 − 𝑙)) = (𝑝𝑖) → (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
10067, 99sylbid 242 . . . . . . 7 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑥𝑖) − (𝑙 · (𝑦𝑖))) = ((1 − 𝑙) · (𝑝𝑖)) → (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
10160, 100sylbid 242 . . . . . 6 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) → (𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
102101ralimdva 3177 . . . . 5 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) → ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖)))))
103102imp 409 . . . 4 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑙 ∈ (0[,)1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))) → ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − -(𝑙 / (1 − 𝑙))) · (𝑥𝑖)) + (-(𝑙 / (1 − 𝑙)) · (𝑦𝑖))))
10427, 34, 103rspcedvd 3625 . . 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 3287 . 2 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
106 0xr 10687 . . . . . . 7 0 ∈ ℝ*
107 1re 10640 . . . . . . 7 1 ∈ ℝ
108 elioc2 12798 . . . . . . 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 1132 . . . . . . 7 ((𝑚 ∈ ℝ ∧ 0 < 𝑚𝑚 ≤ 1) → 𝑚 ∈ ℝ)
111 gt0ne0 11104 . . . . . . . 8 ((𝑚 ∈ ℝ ∧ 0 < 𝑚) → 𝑚 ≠ 0)
1121113adant3 1128 . . . . . . 7 ((𝑚 ∈ ℝ ∧ 0 < 𝑚𝑚 ≤ 1) → 𝑚 ≠ 0)
113110, 112rereccld 11466 . . . . . 6 ((𝑚 ∈ ℝ ∧ 0 < 𝑚𝑚 ≤ 1) → (1 / 𝑚) ∈ ℝ)
114109, 113sylbi 219 . . . . 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 7163 . . . . . . . . 9 (𝑡 = (1 / 𝑚) → (1 − 𝑡) = (1 − (1 / 𝑚)))
117116oveq1d 7170 . . . . . . . 8 (𝑡 = (1 / 𝑚) → ((1 − 𝑡) · (𝑥𝑖)) = ((1 − (1 / 𝑚)) · (𝑥𝑖)))
118 oveq1 7162 . . . . . . . 8 (𝑡 = (1 / 𝑚) → (𝑡 · (𝑦𝑖)) = ((1 / 𝑚) · (𝑦𝑖)))
119117, 118oveq12d 7173 . . . . . . 7 (𝑡 = (1 / 𝑚) → (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖))))
120119eqeq2d 2832 . . . . . 6 (𝑡 = (1 / 𝑚) → ((𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖))) ↔ (𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
121120ralbidv 3197 . . . . 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 484 . . . 4 ((((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ ∀𝑖 ∈ (1...𝑁)(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 2828 . . . . . . . 8 ((𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) = (𝑦𝑖))
12446ad2antrr 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)
125124ffvelrnda 6850 . . . . . . . . . . . . 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 10668 . . . . . . . . . . . 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 10635 . . . . . . . . . . . . . 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 219 . . . . . . . . . . . . . . . 16 (𝑚 ∈ (0(,]1) → 𝑚 ∈ ℝ)
129128recnd 10668 . . . . . . . . . . . . . . 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 10996 . . . . . . . . . . . . 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)
13236ad2antrr 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)
133132ffvelrnda 6850 . . . . . . . . . . . . . 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 10668 . . . . . . . . . . . . 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 10660 . . . . . . . . . . . 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 11002 . . . . . . . . . . 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 11092 . . . . . . . . . . . . 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 11012 . . . . . . . . . . . . . 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 7170 . . . . . . . . . . . . 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 2858 . . . . . . . . . . . 12 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → -((1 − 𝑚) · (𝑥𝑖)) = ((𝑚 − 1) · (𝑥𝑖)))
141140oveq2d 7171 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) + -((1 − 𝑚) · (𝑥𝑖))) = ((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))))
142136, 141eqtr3d 2858 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) − ((1 − 𝑚) · (𝑥𝑖))) = ((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))))
143142eqeq1d 2823 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) − ((1 − 𝑚) · (𝑥𝑖))) = (𝑚 · (𝑝𝑖)) ↔ ((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) = (𝑚 · (𝑝𝑖))))
14453ad2antlr 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)
145144ffvelrnda 6850 . . . . . . . . . . . 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 10668 . . . . . . . . . . 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 10660 . . . . . . . . . 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 11014 . . . . . . . . 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 2828 . . . . . . . . . . 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 10996 . . . . . . . . . . . . 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 10660 . . . . . . . . . . . 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 10659 . . . . . . . . . . 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 12779 . . . . . . . . . . . . . 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 616 . . . . . . . . . . . . . . 15 ((𝑚 ∈ ℝ* ∧ 0 < 𝑚) → (0 ∈ ℝ ∧ 0 < 𝑚))
1581573adant3 1128 . . . . . . . . . . . . . 14 ((𝑚 ∈ ℝ* ∧ 0 < 𝑚𝑚 ≤ 1) → (0 ∈ ℝ ∧ 0 < 𝑚))
159 ltne 10736 . . . . . . . . . . . . . 14 ((0 ∈ ℝ ∧ 0 < 𝑚) → 𝑚 ≠ 0)
160158, 159syl 17 . . . . . . . . . . . . 13 ((𝑚 ∈ ℝ* ∧ 0 < 𝑚𝑚 ≤ 1) → 𝑚 ≠ 0)
161155, 160sylbi 219 . . . . . . . . . . . 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 11448 . . . . . . . . . 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 11453 . . . . . . . . . . . 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 11419 . . . . . . . . . . . . 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 11452 . . . . . . . . . . . . . 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 11454 . . . . . . . . . . . . . . 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 7170 . . . . . . . . . . . . . 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 2856 . . . . . . . . . . . . 13 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑚 − 1) · (𝑥𝑖)) / 𝑚) = (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))
170165, 169oveq12d 7173 . . . . . . . . . . . 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 2856 . . . . . . . . . . 11 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) / 𝑚) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖))))
172171eqeq2d 2832 . . . . . . . . . 10 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) / 𝑚) ↔ (𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))))
173150, 163, 1723bitr3d 311 . . . . . . . . 9 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → (((𝑦𝑖) + ((𝑚 − 1) · (𝑥𝑖))) = (𝑚 · (𝑝𝑖)) ↔ (𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))))
174143, 148, 1733bitr3d 311 . . . . . . . 8 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) = (𝑦𝑖) ↔ (𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))))
175123, 174syl5bb 285 . . . . . . 7 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖)))))
176130, 162reccld 11408 . . . . . . . . . . 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 10660 . . . . . . . . . 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 10996 . . . . . . . . . . 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 10660 . . . . . . . . . 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 11413 . . . . . . . . . . . . 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 7170 . . . . . . . . . . . 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 7170 . . . . . . . . . . 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 7171 . . . . . . . . . 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 10853 . . . . . . . . 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 2832 . . . . . . . 8 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖))) ↔ (𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
186185biimpd 231 . . . . . . 7 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑝𝑖) = (((1 / 𝑚) · (𝑦𝑖)) + (((𝑚 / 𝑚) − (1 / 𝑚)) · (𝑥𝑖))) → (𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
187175, 186sylbid 242 . . . . . 6 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) → (𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
188187ralimdva 3177 . . . . 5 ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) → (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) → ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖)))))
189188imp 409 . . . 4 (((((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) ∧ 𝑚 ∈ (0(,]1)) ∧ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))) → ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − (1 / 𝑚)) · (𝑥𝑖)) + ((1 / 𝑚) · (𝑦𝑖))))
190115, 122, 189rspcedvd 3625 . . 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 3287 . 2 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → (∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) → ∃𝑡 ∈ ℝ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑡) · (𝑥𝑖)) + (𝑡 · (𝑦𝑖)))))
19211, 105, 1913jaod 1424 1 (((𝑁 ∈ ℕ ∧ 𝑥 ∈ (ℝ ↑m (1...𝑁)) ∧ 𝑦 ∈ ((ℝ ↑m (1...𝑁)) ∖ {𝑥})) ∧ 𝑝 ∈ (ℝ ↑m (1...𝑁))) → ((∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(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 208  wa 398  w3o 1082  w3a 1083   = wceq 1533  wcel 2110  wne 3016  wral 3138  wrex 3139  cdif 3932  wss 3935  {csn 4566   class class class wbr 5065  wf 6350  cfv 6354  (class class class)co 7155  m cmap 8405  cc 10534  cr 10535  0cc0 10536  1c1 10537   + caddc 10539   · cmul 10541  *cxr 10673   < clt 10674  cle 10675  cmin 10869  -cneg 10870   / cdiv 11296  cn 11637  (,]cioc 12738  [,)cico 12739  [,]cicc 12740  ...cfz 12891
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pow 5265  ax-pr 5329  ax-un 7460  ax-cnex 10592  ax-resscn 10593  ax-1cn 10594  ax-icn 10595  ax-addcl 10596  ax-addrcl 10597  ax-mulcl 10598  ax-mulrcl 10599  ax-mulcom 10600  ax-addass 10601  ax-mulass 10602  ax-distr 10603  ax-i2m1 10604  ax-1ne0 10605  ax-1rid 10606  ax-rnegex 10607  ax-rrecex 10608  ax-cnre 10609  ax-pre-lttri 10610  ax-pre-lttrn 10611  ax-pre-ltadd 10612  ax-pre-mulgt0 10613
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-csb 3883  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-pw 4540  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-iun 4920  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-po 5473  df-so 5474  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-1st 7688  df-2nd 7689  df-er 8288  df-map 8407  df-en 8509  df-dom 8510  df-sdom 8511  df-pnf 10676  df-mnf 10677  df-xr 10678  df-ltxr 10679  df-le 10680  df-sub 10871  df-neg 10872  df-div 11297  df-ioc 12742  df-ico 12743  df-icc 12744
This theorem is referenced by:  eenglngeehlnm  44727
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