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Theorem elntg2 29000
Description: The line definition in the Tarski structure for the Euclidean geometry. In contrast to elntg 28999, 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 2737 . . 3 (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
31, 2elntg 28999 . 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 4131 . . . . . . . . 9 (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦𝑃)
763ad2ant3 1136 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦𝑃)
87adantr 480 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑦𝑃)
9 simpr 484 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑝𝑃)
104, 1, 2, 5, 8, 9ebtwntg 28997 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ 𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
11 eengbas 28996 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
121, 11eqtr4id 2796 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑃 = (𝔼‘𝑁))
13123ad2ant1 1134 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑃 = (𝔼‘𝑁))
1413eleq2d 2827 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → (𝑝𝑃𝑝 ∈ (𝔼‘𝑁)))
1514biimpa 476 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑝 ∈ (𝔼‘𝑁))
1612eleq2d 2827 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝑥𝑃𝑥 ∈ (𝔼‘𝑁)))
1716biimpa 476 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑥𝑃) → 𝑥 ∈ (𝔼‘𝑁))
18173adant3 1133 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑥 ∈ (𝔼‘𝑁))
1918adantr 480 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑥 ∈ (𝔼‘𝑁))
2012eleq2d 2827 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → (𝑦𝑃𝑦 ∈ (𝔼‘𝑁)))
2120biimpcd 249 . . . . . . . . . . . 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 1111 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦 ∈ (𝔼‘𝑁))
2524adantr 480 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑦 ∈ (𝔼‘𝑁))
26 brbtwn 28914 . . . . . . . 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 1373 . . . . . . 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 3324 . . . . . . . 8 (∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))))
3029rexbii 3094 . . . . . . 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 289 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑘 ∈ (0[,]1)∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖)))))
3210, 31bitr3d 281 . . . . 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 28997 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑥 Btwn ⟨𝑝, 𝑦⟩ ↔ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦)))
34 brbtwn 28914 . . . . . . . 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 1373 . . . . . . 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 281 . . . . . 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 11308 . . . . . . . . . 10 0 ∈ ℝ*
38 1xr 11320 . . . . . . . . . 10 1 ∈ ℝ*
39 0le1 11786 . . . . . . . . . 10 0 ≤ 1
40 snunico 13519 . . . . . . . . . 10 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
4137, 38, 39, 40mp3an 1463 . . . . . . . . 9 ((0[,)1) ∪ {1}) = (0[,]1)
4241eqcomi 2746 . . . . . . . 8 (0[,]1) = ((0[,)1) ∪ {1})
4342a1i 11 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (0[,]1) = ((0[,)1) ∪ {1}))
4443rexeqdv 3327 . . . . . 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 4196 . . . . . . 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 4786 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑃 ∖ {𝑥}) ↔ (𝑦𝑃𝑦𝑥))
47 elee 28909 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → (𝑥 ∈ (𝔼‘𝑁) ↔ 𝑥:(1...𝑁)⟶ℝ))
48 ffn 6736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥:(1...𝑁)⟶ℝ → 𝑥 Fn (1...𝑁))
4947, 48biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → (𝑥 ∈ (𝔼‘𝑁) → 𝑥 Fn (1...𝑁)))
5016, 49sylbid 240 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ → (𝑥𝑃𝑥 Fn (1...𝑁)))
5150a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦𝑃 → (𝑁 ∈ ℕ → (𝑥𝑃𝑥 Fn (1...𝑁))))
52513imp 1111 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → 𝑥 Fn (1...𝑁))
53 elee 28909 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → (𝑦 ∈ (𝔼‘𝑁) ↔ 𝑦:(1...𝑁)⟶ℝ))
54 ffn 6736 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦:(1...𝑁)⟶ℝ → 𝑦 Fn (1...𝑁))
5553, 54biimtrdi 253 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → (𝑦 ∈ (𝔼‘𝑁) → 𝑦 Fn (1...𝑁)))
5620, 55sylbid 240 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑁 ∈ ℕ → (𝑦𝑃𝑦 Fn (1...𝑁)))
5756a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥𝑃 → (𝑁 ∈ ℕ → (𝑦𝑃𝑦 Fn (1...𝑁))))
58573imp31 1112 . . . . . . . . . . . . . . . . . . . . 21 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → 𝑦 Fn (1...𝑁))
59 eqfnfv 7051 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 Fn (1...𝑁) ∧ 𝑦 Fn (1...𝑁)) → (𝑥 = 𝑦 ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))
6052, 58, 59syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → (𝑥 = 𝑦 ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))
6160biimprd 248 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖) → 𝑥 = 𝑦))
62 eqcom 2744 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥𝑥 = 𝑦)
6361, 62imbitrrdi 252 . . . . . . . . . . . . . . . . . 18 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖) → 𝑦 = 𝑥))
6463necon3ad 2953 . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . 14 ((𝑦𝑃𝑦𝑥) → (𝑥𝑃 → (𝑁 ∈ ℕ → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖))))
6846, 67sylbi 217 . . . . . . . . . . . . 13 (𝑦 ∈ (𝑃 ∖ {𝑥}) → (𝑥𝑃 → (𝑁 ∈ ℕ → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖))))
69683imp31 1112 . . . . . . . . . . . 12 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖))
7069adantr 480 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖))
7112eleq2d 2827 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ → (𝑝𝑃𝑝 ∈ (𝔼‘𝑁)))
72 elee 28909 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ → (𝑝 ∈ (𝔼‘𝑁) ↔ 𝑝:(1...𝑁)⟶ℝ))
7372biimpd 229 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ → (𝑝 ∈ (𝔼‘𝑁) → 𝑝:(1...𝑁)⟶ℝ))
7471, 73sylbid 240 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → (𝑝𝑃𝑝:(1...𝑁)⟶ℝ))
75743ad2ant1 1134 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → (𝑝𝑃𝑝:(1...𝑁)⟶ℝ))
7675imp 406 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑝:(1...𝑁)⟶ℝ)
7776ffvelcdmda 7104 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℝ)
7877recnd 11289 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℂ)
7978mul02d 11459 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (0 · (𝑝𝑖)) = 0)
8021, 53mpbidi 241 . . . . . . . . . . . . . . . . . . . . . 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 1111 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦:(1...𝑁)⟶ℝ)
8483adantr 480 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑦:(1...𝑁)⟶ℝ)
8584ffvelcdmda 7104 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℝ)
8685recnd 11289 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℂ)
8786mullidd 11279 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (1 · (𝑦𝑖)) = (𝑦𝑖))
8879, 87oveq12d 7449 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) = (0 + (𝑦𝑖)))
8986addlidd 11462 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (0 + (𝑦𝑖)) = (𝑦𝑖))
9088, 89eqtrd 2777 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) = (𝑦𝑖))
9190eqeq2d 2748 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) ↔ (𝑥𝑖) = (𝑦𝑖)))
9291ralbidva 3176 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))
9370, 92mtbird 325 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))))
94 1re 11261 . . . . . . . . . . 11 1 ∈ ℝ
95 oveq2 7439 . . . . . . . . . . . . . . . . 17 (𝑙 = 1 → (1 − 𝑙) = (1 − 1))
9695oveq1d 7446 . . . . . . . . . . . . . . . 16 (𝑙 = 1 → ((1 − 𝑙) · (𝑝𝑖)) = ((1 − 1) · (𝑝𝑖)))
97 1m1e0 12338 . . . . . . . . . . . . . . . . 17 (1 − 1) = 0
9897oveq1i 7441 . . . . . . . . . . . . . . . 16 ((1 − 1) · (𝑝𝑖)) = (0 · (𝑝𝑖))
9996, 98eqtrdi 2793 . . . . . . . . . . . . . . 15 (𝑙 = 1 → ((1 − 𝑙) · (𝑝𝑖)) = (0 · (𝑝𝑖)))
100 oveq1 7438 . . . . . . . . . . . . . . 15 (𝑙 = 1 → (𝑙 · (𝑦𝑖)) = (1 · (𝑦𝑖)))
10199, 100oveq12d 7449 . . . . . . . . . . . . . 14 (𝑙 = 1 → (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))))
102101eqeq2d 2748 . . . . . . . . . . . . 13 (𝑙 = 1 → ((𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ (𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖)))))
103102ralbidv 3178 . . . . . . . . . . . 12 (𝑙 = 1 → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖)))))
104103rexsng 4676 . . . . . . . . . . 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 329 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))
10728raleqi 3324 . . . . . . . . . . 11 (∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))
108107rexbii 3094 . . . . . . . . . 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 937 . . . . . . . . . 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, 109bitrid 283 . . . . . . . . 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 871 . . . . . . . 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 288 . . . . . . 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, 113bitrid 283 . . . . . 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 305 . . . . 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 28997 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑦 Btwn ⟨𝑥, 𝑝⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝)))
117 brbtwn 28914 . . . . . . . 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 1373 . . . . . . 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 281 . . . . . 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 13520 . . . . . . . . . 10 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
12137, 38, 39, 120mp3an 1463 . . . . . . . . 9 ({0} ∪ (0(,]1)) = (0[,]1)
122121eqcomi 2746 . . . . . . . 8 (0[,]1) = ({0} ∪ (0(,]1))
123122a1i 11 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (0[,]1) = ({0} ∪ (0(,]1)))
124123rexeqdv 3327 . . . . . 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 4196 . . . . . . 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 2744 . . . . . . . . . . . 12 ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑦𝑖) = (𝑥𝑖))
127126ralbii 3093 . . . . . . . . . . 11 (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (𝑥𝑖))
12870, 127sylnib 328 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (𝑥𝑖))
12916biimpd 229 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ → (𝑥𝑃𝑥 ∈ (𝔼‘𝑁)))
130129, 47sylibd 239 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → (𝑥𝑃𝑥:(1...𝑁)⟶ℝ))
131130imp 406 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ ∧ 𝑥𝑃) → 𝑥:(1...𝑁)⟶ℝ)
1321313adant3 1133 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑥:(1...𝑁)⟶ℝ)
133132adantr 480 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑥:(1...𝑁)⟶ℝ)
134133ffvelcdmda 7104 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℝ)
135134recnd 11289 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℂ)
136135mullidd 11279 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (1 · (𝑥𝑖)) = (𝑥𝑖))
137136, 79oveq12d 7449 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) = ((𝑥𝑖) + 0))
138135addridd 11461 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) + 0) = (𝑥𝑖))
139137, 138eqtrd 2777 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) = (𝑥𝑖))
140139eqeq2d 2748 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) ↔ (𝑦𝑖) = (𝑥𝑖)))
141140ralbidva 3176 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (𝑥𝑖)))
142128, 141mtbird 325 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))))
143 0re 11263 . . . . . . . . . 10 0 ∈ ℝ
144 oveq2 7439 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (1 − 𝑚) = (1 − 0))
145144oveq1d 7446 . . . . . . . . . . . . . . 15 (𝑚 = 0 → ((1 − 𝑚) · (𝑥𝑖)) = ((1 − 0) · (𝑥𝑖)))
146 1m0e1 12387 . . . . . . . . . . . . . . . 16 (1 − 0) = 1
147146oveq1i 7441 . . . . . . . . . . . . . . 15 ((1 − 0) · (𝑥𝑖)) = (1 · (𝑥𝑖))
148145, 147eqtrdi 2793 . . . . . . . . . . . . . 14 (𝑚 = 0 → ((1 − 𝑚) · (𝑥𝑖)) = (1 · (𝑥𝑖)))
149 oveq1 7438 . . . . . . . . . . . . . 14 (𝑚 = 0 → (𝑚 · (𝑝𝑖)) = (0 · (𝑝𝑖)))
150148, 149oveq12d 7449 . . . . . . . . . . . . 13 (𝑚 = 0 → (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))))
151150eqeq2d 2748 . . . . . . . . . . . 12 (𝑚 = 0 → ((𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖)))))
152151ralbidv 3178 . . . . . . . . . . 11 (𝑚 = 0 → (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖)))))
153152rexsng 4676 . . . . . . . . . 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 329 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))
15628raleqi 3324 . . . . . . . . . 10 (∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))
157156rexbii 3094 . . . . . . . . 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 937 . . . . . . . . 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, 158bitrid 283 . . . . . . . 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 290 . . . . . 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 305 . . . . 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 1437 . . . 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 3443 . . 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 7510 . 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 2777 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 206  wa 395  wo 848  w3o 1086  w3a 1087   = wceq 1540  wcel 2108  wne 2940  wral 3061  wrex 3070  {crab 3436  cdif 3948  cun 3949  {csn 4626  cop 4632   class class class wbr 5143   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  cmpo 7433  cr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160  *cxr 11294  cle 11296  cmin 11492  cn 12266  (,]cioc 13388  [,)cico 13389  [,]cicc 13390  ...cfz 13547  Basecbs 17247  Itvcitv 28441  LineGclng 28442  𝔼cee 28903   Btwn cbtwn 28904  EEGceeng 28992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-er 8745  df-map 8868  df-en 8986  df-dom 8987  df-sdom 8988  df-fin 8989  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-z 12614  df-dec 12734  df-uz 12879  df-ioc 13392  df-ico 13393  df-icc 13394  df-fz 13548  df-seq 14043  df-sum 15723  df-struct 17184  df-slot 17219  df-ndx 17231  df-base 17248  df-ds 17319  df-itv 28443  df-lng 28444  df-ee 28906  df-btwn 28907  df-eeng 28993
This theorem is referenced by:  eenglngeehlnm  48660
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