MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elntg2 Structured version   Visualization version   GIF version

Theorem elntg2 26923
Description: The line definition in the Tarski structure for the Euclidean geometry. In contrast to elntg 26922, the betweenness can be strengthened by excluding 1 resp. 0 from the related intervals (because of 𝑥𝑦). (Contributed by AV, 14-Feb-2023.)
Hypotheses
Ref Expression
elntg2.1 𝑃 = (Base‘(EEG‘𝑁))
elntg2.2 𝐼 = (1...𝑁)
Assertion
Ref Expression
elntg2 (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝𝑃 ∣ (∃𝑘 ∈ (0[,]1)∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))}))
Distinct variable groups:   𝑖,𝐼   𝑖,𝑁,𝑘,𝑙,𝑚,𝑝,𝑥,𝑦   𝑃,𝑖,𝑝
Allowed substitution hints:   𝑃(𝑥,𝑦,𝑘,𝑚,𝑙)   𝐼(𝑥,𝑦,𝑘,𝑚,𝑝,𝑙)

Proof of Theorem elntg2
StepHypRef Expression
1 elntg2.1 . . 3 𝑃 = (Base‘(EEG‘𝑁))
2 eqid 2738 . . 3 (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
31, 2elntg 26922 . 2 (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝𝑃 ∣ (𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝))}))
4 simpl1 1192 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑁 ∈ ℕ)
5 simpl2 1193 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑥𝑃)
6 eldifi 4015 . . . . . . . . 9 (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦𝑃)
763ad2ant3 1136 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦𝑃)
87adantr 484 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑦𝑃)
9 simpr 488 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑝𝑃)
104, 1, 2, 5, 8, 9ebtwntg 26920 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ 𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
11 eengbas 26919 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
121, 11eqtr4id 2792 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑃 = (𝔼‘𝑁))
13123ad2ant1 1134 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑃 = (𝔼‘𝑁))
1413eleq2d 2818 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → (𝑝𝑃𝑝 ∈ (𝔼‘𝑁)))
1514biimpa 480 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑝 ∈ (𝔼‘𝑁))
1612eleq2d 2818 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝑥𝑃𝑥 ∈ (𝔼‘𝑁)))
1716biimpa 480 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑥𝑃) → 𝑥 ∈ (𝔼‘𝑁))
18173adant3 1133 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑥 ∈ (𝔼‘𝑁))
1918adantr 484 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑥 ∈ (𝔼‘𝑁))
2012eleq2d 2818 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → (𝑦𝑃𝑦 ∈ (𝔼‘𝑁)))
2120biimpcd 252 . . . . . . . . . . . 12 (𝑦𝑃 → (𝑁 ∈ ℕ → 𝑦 ∈ (𝔼‘𝑁)))
2221, 6syl11 33 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦 ∈ (𝔼‘𝑁)))
2322a1d 25 . . . . . . . . . 10 (𝑁 ∈ ℕ → (𝑥𝑃 → (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦 ∈ (𝔼‘𝑁))))
24233imp 1112 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦 ∈ (𝔼‘𝑁))
2524adantr 484 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑦 ∈ (𝔼‘𝑁))
26 brbtwn 26837 . . . . . . . 8 ((𝑝 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖)))))
2715, 19, 25, 26syl3anc 1372 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖)))))
28 elntg2.2 . . . . . . . . 9 𝐼 = (1...𝑁)
2928raleqi 3313 . . . . . . . 8 (∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))))
3029rexbii 3160 . . . . . . 7 (∃𝑘 ∈ (0[,]1)∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ ∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))))
3127, 30bitr4di 292 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑘 ∈ (0[,]1)∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖)))))
3210, 31bitr3d 284 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ↔ ∃𝑘 ∈ (0[,]1)∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖)))))
334, 1, 2, 9, 8, 5ebtwntg 26920 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑥 Btwn ⟨𝑝, 𝑦⟩ ↔ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦)))
34 brbtwn 26837 . . . . . . . 8 ((𝑥 ∈ (𝔼‘𝑁) ∧ 𝑝 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (𝑥 Btwn ⟨𝑝, 𝑦⟩ ↔ ∃𝑙 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))))
3519, 15, 25, 34syl3anc 1372 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑥 Btwn ⟨𝑝, 𝑦⟩ ↔ ∃𝑙 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))))
3633, 35bitr3d 284 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ↔ ∃𝑙 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))))
37 0xr 10759 . . . . . . . . . 10 0 ∈ ℝ*
38 1xr 10771 . . . . . . . . . 10 1 ∈ ℝ*
39 0le1 11234 . . . . . . . . . 10 0 ≤ 1
40 snunico 12946 . . . . . . . . . 10 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
4137, 38, 39, 40mp3an 1462 . . . . . . . . 9 ((0[,)1) ∪ {1}) = (0[,]1)
4241eqcomi 2747 . . . . . . . 8 (0[,]1) = ((0[,)1) ∪ {1})
4342a1i 11 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (0[,]1) = ((0[,)1) ∪ {1}))
4443rexeqdv 3316 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∃𝑙 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∃𝑙 ∈ ((0[,)1) ∪ {1})∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))))
45 rexun 4078 . . . . . . 7 (∃𝑙 ∈ ((0[,)1) ∪ {1})∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))))
46 eldifsn 4672 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑃 ∖ {𝑥}) ↔ (𝑦𝑃𝑦𝑥))
47 elee 26832 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → (𝑥 ∈ (𝔼‘𝑁) ↔ 𝑥:(1...𝑁)⟶ℝ))
48 ffn 6498 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥:(1...𝑁)⟶ℝ → 𝑥 Fn (1...𝑁))
4947, 48syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → (𝑥 ∈ (𝔼‘𝑁) → 𝑥 Fn (1...𝑁)))
5016, 49sylbid 243 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ → (𝑥𝑃𝑥 Fn (1...𝑁)))
5150a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝑃 → (𝑁 ∈ ℕ → (𝑥𝑃𝑥 Fn (1...𝑁))))
52513imp 1112 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → 𝑥 Fn (1...𝑁))
53 elee 26832 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → (𝑦 ∈ (𝔼‘𝑁) ↔ 𝑦:(1...𝑁)⟶ℝ))
54 ffn 6498 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦:(1...𝑁)⟶ℝ → 𝑦 Fn (1...𝑁))
5553, 54syl6bi 256 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → (𝑦 ∈ (𝔼‘𝑁) → 𝑦 Fn (1...𝑁)))
5620, 55sylbid 243 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ → (𝑦𝑃𝑦 Fn (1...𝑁)))
5756a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑃 → (𝑁 ∈ ℕ → (𝑦𝑃𝑦 Fn (1...𝑁))))
58573imp31 1113 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → 𝑦 Fn (1...𝑁))
59 eqfnfv 6803 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 Fn (1...𝑁) ∧ 𝑦 Fn (1...𝑁)) → (𝑥 = 𝑦 ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))
6052, 58, 59syl2anc 587 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → (𝑥 = 𝑦 ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))
6160biimprd 251 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖) → 𝑥 = 𝑦))
62 eqcom 2745 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥𝑥 = 𝑦)
6361, 62syl6ibr 255 . . . . . . . . . . . . . . . . . 18 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖) → 𝑦 = 𝑥))
6463necon3ad 2947 . . . . . . . . . . . . . . . . 17 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → (𝑦𝑥 → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))
65643exp 1120 . . . . . . . . . . . . . . . 16 (𝑦𝑃 → (𝑁 ∈ ℕ → (𝑥𝑃 → (𝑦𝑥 → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))))
6665com24 95 . . . . . . . . . . . . . . 15 (𝑦𝑃 → (𝑦𝑥 → (𝑥𝑃 → (𝑁 ∈ ℕ → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))))
6766imp 410 . . . . . . . . . . . . . 14 ((𝑦𝑃𝑦𝑥) → (𝑥𝑃 → (𝑁 ∈ ℕ → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖))))
6846, 67sylbi 220 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑃 ∖ {𝑥}) → (𝑥𝑃 → (𝑁 ∈ ℕ → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖))))
69683imp31 1113 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖))
7069adantr 484 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖))
7112eleq2d 2818 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ → (𝑝𝑃𝑝 ∈ (𝔼‘𝑁)))
72 elee 26832 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ → (𝑝 ∈ (𝔼‘𝑁) ↔ 𝑝:(1...𝑁)⟶ℝ))
7372biimpd 232 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ → (𝑝 ∈ (𝔼‘𝑁) → 𝑝:(1...𝑁)⟶ℝ))
7471, 73sylbid 243 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → (𝑝𝑃𝑝:(1...𝑁)⟶ℝ))
75743ad2ant1 1134 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → (𝑝𝑃𝑝:(1...𝑁)⟶ℝ))
7675imp 410 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑝:(1...𝑁)⟶ℝ)
7776ffvelrnda 6855 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℝ)
7877recnd 10740 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℂ)
7978mul02d 10909 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (0 · (𝑝𝑖)) = 0)
8021, 53mpbidi 244 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝑃 → (𝑁 ∈ ℕ → 𝑦:(1...𝑁)⟶ℝ))
8180, 6syl11 33 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ → (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦:(1...𝑁)⟶ℝ))
8281a1d 25 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → (𝑥𝑃 → (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦:(1...𝑁)⟶ℝ)))
83823imp 1112 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦:(1...𝑁)⟶ℝ)
8483adantr 484 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑦:(1...𝑁)⟶ℝ)
8584ffvelrnda 6855 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℝ)
8685recnd 10740 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℂ)
8786mulid2d 10730 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (1 · (𝑦𝑖)) = (𝑦𝑖))
8879, 87oveq12d 7182 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) = (0 + (𝑦𝑖)))
8986addid2d 10912 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (0 + (𝑦𝑖)) = (𝑦𝑖))
9088, 89eqtrd 2773 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) = (𝑦𝑖))
9190eqeq2d 2749 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) ↔ (𝑥𝑖) = (𝑦𝑖)))
9291ralbidva 3108 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))
9370, 92mtbird 328 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))))
94 1re 10712 . . . . . . . . . . 11 1 ∈ ℝ
95 oveq2 7172 . . . . . . . . . . . . . . . . 17 (𝑙 = 1 → (1 − 𝑙) = (1 − 1))
9695oveq1d 7179 . . . . . . . . . . . . . . . 16 (𝑙 = 1 → ((1 − 𝑙) · (𝑝𝑖)) = ((1 − 1) · (𝑝𝑖)))
97 1m1e0 11781 . . . . . . . . . . . . . . . . 17 (1 − 1) = 0
9897oveq1i 7174 . . . . . . . . . . . . . . . 16 ((1 − 1) · (𝑝𝑖)) = (0 · (𝑝𝑖))
9996, 98eqtrdi 2789 . . . . . . . . . . . . . . 15 (𝑙 = 1 → ((1 − 𝑙) · (𝑝𝑖)) = (0 · (𝑝𝑖)))
100 oveq1 7171 . . . . . . . . . . . . . . 15 (𝑙 = 1 → (𝑙 · (𝑦𝑖)) = (1 · (𝑦𝑖)))
10199, 100oveq12d 7182 . . . . . . . . . . . . . 14 (𝑙 = 1 → (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))))
102101eqeq2d 2749 . . . . . . . . . . . . 13 (𝑙 = 1 → ((𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ (𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖)))))
103102ralbidv 3109 . . . . . . . . . . . 12 (𝑙 = 1 → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖)))))
104103rexsng 4562 . . . . . . . . . . 11 (1 ∈ ℝ → (∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖)))))
10594, 104ax-mp 5 . . . . . . . . . 10 (∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))))
10693, 105sylnibr 332 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))
10728raleqi 3313 . . . . . . . . . . 11 (∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))
108107rexbii 3160 . . . . . . . . . 10 (∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))
109 biorf 936 . . . . . . . . . 10 (¬ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) → (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ (∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))))
110108, 109syl5bb 286 . . . . . . . . 9 (¬ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) → (∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ (∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))))
111106, 110syl 17 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ (∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))))
112 orcom 869 . . . . . . . 8 ((∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))) ↔ (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))))
113111, 112bitr2di 291 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ((∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))) ↔ ∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))))
11445, 113syl5bb 286 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∃𝑙 ∈ ((0[,)1) ∪ {1})∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))))
11536, 44, 1143bitrd 308 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ↔ ∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖)))))
1164, 1, 2, 5, 9, 8ebtwntg 26920 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑦 Btwn ⟨𝑥, 𝑝⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝)))
117 brbtwn 26837 . . . . . . . 8 ((𝑦 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑝 ∈ (𝔼‘𝑁)) → (𝑦 Btwn ⟨𝑥, 𝑝⟩ ↔ ∃𝑚 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))
11825, 19, 15, 117syl3anc 1372 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑦 Btwn ⟨𝑥, 𝑝⟩ ↔ ∃𝑚 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))
119116, 118bitr3d 284 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝) ↔ ∃𝑚 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))
120 snunioc 12947 . . . . . . . . . 10 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
12137, 38, 39, 120mp3an 1462 . . . . . . . . 9 ({0} ∪ (0(,]1)) = (0[,]1)
122121eqcomi 2747 . . . . . . . 8 (0[,]1) = ({0} ∪ (0(,]1))
123122a1i 11 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (0[,]1) = ({0} ∪ (0(,]1)))
124123rexeqdv 3316 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∃𝑚 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∃𝑚 ∈ ({0} ∪ (0(,]1))∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))
125 rexun 4078 . . . . . . 7 (∃𝑚 ∈ ({0} ∪ (0(,]1))∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))
126 eqcom 2745 . . . . . . . . . . . 12 ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑦𝑖) = (𝑥𝑖))
127126ralbii 3080 . . . . . . . . . . 11 (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (𝑥𝑖))
12870, 127sylnib 331 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (𝑥𝑖))
12916biimpd 232 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ → (𝑥𝑃𝑥 ∈ (𝔼‘𝑁)))
130129, 47sylibd 242 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → (𝑥𝑃𝑥:(1...𝑁)⟶ℝ))
131130imp 410 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ ∧ 𝑥𝑃) → 𝑥:(1...𝑁)⟶ℝ)
1321313adant3 1133 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑥:(1...𝑁)⟶ℝ)
133132adantr 484 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑥:(1...𝑁)⟶ℝ)
134133ffvelrnda 6855 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℝ)
135134recnd 10740 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℂ)
136135mulid2d 10730 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (1 · (𝑥𝑖)) = (𝑥𝑖))
137136, 79oveq12d 7182 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) = ((𝑥𝑖) + 0))
138135addid1d 10911 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) + 0) = (𝑥𝑖))
139137, 138eqtrd 2773 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) = (𝑥𝑖))
140139eqeq2d 2749 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) ↔ (𝑦𝑖) = (𝑥𝑖)))
141140ralbidva 3108 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (𝑥𝑖)))
142128, 141mtbird 328 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))))
143 0re 10714 . . . . . . . . . 10 0 ∈ ℝ
144 oveq2 7172 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (1 − 𝑚) = (1 − 0))
145144oveq1d 7179 . . . . . . . . . . . . . . 15 (𝑚 = 0 → ((1 − 𝑚) · (𝑥𝑖)) = ((1 − 0) · (𝑥𝑖)))
146 1m0e1 11830 . . . . . . . . . . . . . . . 16 (1 − 0) = 1
147146oveq1i 7174 . . . . . . . . . . . . . . 15 ((1 − 0) · (𝑥𝑖)) = (1 · (𝑥𝑖))
148145, 147eqtrdi 2789 . . . . . . . . . . . . . 14 (𝑚 = 0 → ((1 − 𝑚) · (𝑥𝑖)) = (1 · (𝑥𝑖)))
149 oveq1 7171 . . . . . . . . . . . . . 14 (𝑚 = 0 → (𝑚 · (𝑝𝑖)) = (0 · (𝑝𝑖)))
150148, 149oveq12d 7182 . . . . . . . . . . . . 13 (𝑚 = 0 → (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))))
151150eqeq2d 2749 . . . . . . . . . . . 12 (𝑚 = 0 → ((𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖)))))
152151ralbidv 3109 . . . . . . . . . . 11 (𝑚 = 0 → (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖)))))
153152rexsng 4562 . . . . . . . . . 10 (0 ∈ ℝ → (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖)))))
154143, 153ax-mp 5 . . . . . . . . 9 (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))))
155142, 154sylnibr 332 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))
15628raleqi 3313 . . . . . . . . . 10 (∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))
157156rexbii 3160 . . . . . . . . 9 (∃𝑚 ∈ (0(,]1)∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))
158 biorf 936 . . . . . . . . 9 (¬ ∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) → (∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
159157, 158syl5bb 286 . . . . . . . 8 (¬ ∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) → (∃𝑚 ∈ (0(,]1)∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
160155, 159syl 17 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∃𝑚 ∈ (0(,]1)∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
161125, 160bitr4id 293 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∃𝑚 ∈ ({0} ∪ (0(,]1))∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∃𝑚 ∈ (0(,]1)∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))
162119, 124, 1613bitrd 308 . . . . 5 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝) ↔ ∃𝑚 ∈ (0(,]1)∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖)))))
16332, 115, 1623orbi123d 1436 . . . 4 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ((𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝)) ↔ (∃𝑘 ∈ (0[,]1)∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))))
164163rabbidva 3378 . . 3 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → {𝑝𝑃 ∣ (𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝))} = {𝑝𝑃 ∣ (∃𝑘 ∈ (0[,]1)∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))})
165164mpoeq3dva 7239 . 2 (𝑁 ∈ ℕ → (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝𝑃 ∣ (𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝))}) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝𝑃 ∣ (∃𝑘 ∈ (0[,]1)∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))}))
1663, 165eqtrd 2773 1 (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝𝑃 ∣ (∃𝑘 ∈ (0[,]1)∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))}))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 846  w3o 1087  w3a 1088   = wceq 1542  wcel 2113  wne 2934  wral 3053  wrex 3054  {crab 3057  cdif 3838  cun 3839  {csn 4513  cop 4519   class class class wbr 5027   Fn wfn 6328  wf 6329  cfv 6333  (class class class)co 7164  cmpo 7166  cr 10607  0cc0 10608  1c1 10609   + caddc 10611   · cmul 10613  *cxr 10745  cle 10747  cmin 10941  cn 11709  (,]cioc 12815  [,)cico 12816  [,]cicc 12817  ...cfz 12974  Basecbs 16579  Itvcitv 26374  LineGclng 26375  𝔼cee 26826   Btwn cbtwn 26827  EEGceeng 26915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473  ax-cnex 10664  ax-resscn 10665  ax-1cn 10666  ax-icn 10667  ax-addcl 10668  ax-addrcl 10669  ax-mulcl 10670  ax-mulrcl 10671  ax-mulcom 10672  ax-addass 10673  ax-mulass 10674  ax-distr 10675  ax-i2m1 10676  ax-1ne0 10677  ax-1rid 10678  ax-rnegex 10679  ax-rrecex 10680  ax-cnre 10681  ax-pre-lttri 10682  ax-pre-lttrn 10683  ax-pre-ltadd 10684  ax-pre-mulgt0 10685
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-pss 3860  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-tp 4518  df-op 4520  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-tr 5134  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6123  df-ord 6169  df-on 6170  df-lim 6171  df-suc 6172  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7121  df-ov 7167  df-oprab 7168  df-mpo 7169  df-om 7594  df-1st 7707  df-2nd 7708  df-wrecs 7969  df-recs 8030  df-rdg 8068  df-1o 8124  df-er 8313  df-map 8432  df-en 8549  df-dom 8550  df-sdom 8551  df-fin 8552  df-pnf 10748  df-mnf 10749  df-xr 10750  df-ltxr 10751  df-le 10752  df-sub 10943  df-neg 10944  df-nn 11710  df-2 11772  df-3 11773  df-4 11774  df-5 11775  df-6 11776  df-7 11777  df-8 11778  df-9 11779  df-n0 11970  df-z 12056  df-dec 12173  df-uz 12318  df-ioc 12819  df-ico 12820  df-icc 12821  df-fz 12975  df-seq 13454  df-sum 15129  df-struct 16581  df-ndx 16582  df-slot 16583  df-base 16585  df-ds 16683  df-itv 26376  df-lng 26377  df-ee 26829  df-btwn 26830  df-eeng 26916
This theorem is referenced by:  eenglngeehlnm  45603
  Copyright terms: Public domain W3C validator