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Theorem elntg2 27934
Description: The line definition in the Tarski structure for the Euclidean geometry. In contrast to elntg 27933, 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 2736 . . 3 (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
31, 2elntg 27933 . 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 1191 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑁 ∈ ℕ)
5 simpl2 1192 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑥𝑃)
6 eldifi 4086 . . . . . . . . 9 (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦𝑃)
763ad2ant3 1135 . . . . . . . 8 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦𝑃)
87adantr 481 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑦𝑃)
9 simpr 485 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑝𝑃)
104, 1, 2, 5, 8, 9ebtwntg 27931 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ 𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
11 eengbas 27930 . . . . . . . . . . . 12 (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
121, 11eqtr4id 2795 . . . . . . . . . . 11 (𝑁 ∈ ℕ → 𝑃 = (𝔼‘𝑁))
13123ad2ant1 1133 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑃 = (𝔼‘𝑁))
1413eleq2d 2823 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → (𝑝𝑃𝑝 ∈ (𝔼‘𝑁)))
1514biimpa 477 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑝 ∈ (𝔼‘𝑁))
1612eleq2d 2823 . . . . . . . . . . 11 (𝑁 ∈ ℕ → (𝑥𝑃𝑥 ∈ (𝔼‘𝑁)))
1716biimpa 477 . . . . . . . . . 10 ((𝑁 ∈ ℕ ∧ 𝑥𝑃) → 𝑥 ∈ (𝔼‘𝑁))
18173adant3 1132 . . . . . . . . 9 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑥 ∈ (𝔼‘𝑁))
1918adantr 481 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑥 ∈ (𝔼‘𝑁))
2012eleq2d 2823 . . . . . . . . . . . . 13 (𝑁 ∈ ℕ → (𝑦𝑃𝑦 ∈ (𝔼‘𝑁)))
2120biimpcd 248 . . . . . . . . . . . 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 481 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑦 ∈ (𝔼‘𝑁))
26 brbtwn 27848 . . . . . . . 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 1371 . . . . . . 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 3311 . . . . . . . 8 (∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖))))
3029rexbii 3097 . . . . . . 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 288 . . . . . 6 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑘 ∈ (0[,]1)∀𝑖𝐼 (𝑝𝑖) = (((1 − 𝑘) · (𝑥𝑖)) + (𝑘 · (𝑦𝑖)))))
3210, 31bitr3d 280 . . . . 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 27931 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑥 Btwn ⟨𝑝, 𝑦⟩ ↔ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦)))
34 brbtwn 27848 . . . . . . . 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 1371 . . . . . . 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 280 . . . . . 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 11202 . . . . . . . . . 10 0 ∈ ℝ*
38 1xr 11214 . . . . . . . . . 10 1 ∈ ℝ*
39 0le1 11678 . . . . . . . . . 10 0 ≤ 1
40 snunico 13396 . . . . . . . . . 10 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
4137, 38, 39, 40mp3an 1461 . . . . . . . . 9 ((0[,)1) ∪ {1}) = (0[,]1)
4241eqcomi 2745 . . . . . . . 8 (0[,]1) = ((0[,)1) ∪ {1})
4342a1i 11 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (0[,]1) = ((0[,)1) ∪ {1}))
4443rexeqdv 3314 . . . . . 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 4150 . . . . . . 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 4747 . . . . . . . . . . . . . 14 (𝑦 ∈ (𝑃 ∖ {𝑥}) ↔ (𝑦𝑃𝑦𝑥))
47 elee 27843 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → (𝑥 ∈ (𝔼‘𝑁) ↔ 𝑥:(1...𝑁)⟶ℝ))
48 ffn 6668 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑥:(1...𝑁)⟶ℝ → 𝑥 Fn (1...𝑁))
4947, 48syl6bi 252 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → (𝑥 ∈ (𝔼‘𝑁) → 𝑥 Fn (1...𝑁)))
5016, 49sylbid 239 . . . . . . . . . . . . . . . . . . . . . . 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 27843 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑁 ∈ ℕ → (𝑦 ∈ (𝔼‘𝑁) ↔ 𝑦:(1...𝑁)⟶ℝ))
54 ffn 6668 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦:(1...𝑁)⟶ℝ → 𝑦 Fn (1...𝑁))
5553, 54syl6bi 252 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑁 ∈ ℕ → (𝑦 ∈ (𝔼‘𝑁) → 𝑦 Fn (1...𝑁)))
5620, 55sylbid 239 . . . . . . . . . . . . . . . . . . . . . . 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 6982 . . . . . . . . . . . . . . . . . . . . 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 247 . . . . . . . . . . . . . . . . . . 19 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖) → 𝑥 = 𝑦))
62 eqcom 2743 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑥𝑥 = 𝑦)
6361, 62syl6ibr 251 . . . . . . . . . . . . . . . . . 18 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖) → 𝑦 = 𝑥))
6463necon3ad 2956 . . . . . . . . . . . . . . . . 17 ((𝑦𝑃𝑁 ∈ ℕ ∧ 𝑥𝑃) → (𝑦𝑥 → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))
65643exp 1119 . . . . . . . . . . . . . . . 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 407 . . . . . . . . . . . . . 14 ((𝑦𝑃𝑦𝑥) → (𝑥𝑃 → (𝑁 ∈ ℕ → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖))))
6846, 67sylbi 216 . . . . . . . . . . . . 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 481 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖))
7112eleq2d 2823 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ → (𝑝𝑃𝑝 ∈ (𝔼‘𝑁)))
72 elee 27843 . . . . . . . . . . . . . . . . . . . . . 22 (𝑁 ∈ ℕ → (𝑝 ∈ (𝔼‘𝑁) ↔ 𝑝:(1...𝑁)⟶ℝ))
7372biimpd 228 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ → (𝑝 ∈ (𝔼‘𝑁) → 𝑝:(1...𝑁)⟶ℝ))
7471, 73sylbid 239 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → (𝑝𝑃𝑝:(1...𝑁)⟶ℝ))
75743ad2ant1 1133 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → (𝑝𝑃𝑝:(1...𝑁)⟶ℝ))
7675imp 407 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑝:(1...𝑁)⟶ℝ)
7776ffvelcdmda 7035 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℝ)
7877recnd 11183 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝𝑖) ∈ ℂ)
7978mul02d 11353 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (0 · (𝑝𝑖)) = 0)
8021, 53mpbidi 240 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . 18 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑦:(1...𝑁)⟶ℝ)
8584ffvelcdmda 7035 . . . . . . . . . . . . . . . . 17 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℝ)
8685recnd 11183 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦𝑖) ∈ ℂ)
8786mulid2d 11173 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (1 · (𝑦𝑖)) = (𝑦𝑖))
8879, 87oveq12d 7375 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) = (0 + (𝑦𝑖)))
8986addid2d 11356 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (0 + (𝑦𝑖)) = (𝑦𝑖))
9088, 89eqtrd 2776 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) = (𝑦𝑖))
9190eqeq2d 2747 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) ↔ (𝑥𝑖) = (𝑦𝑖)))
9291ralbidva 3172 . . . . . . . . . . 11 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖)))
9370, 92mtbird 324 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))))
94 1re 11155 . . . . . . . . . . 11 1 ∈ ℝ
95 oveq2 7365 . . . . . . . . . . . . . . . . 17 (𝑙 = 1 → (1 − 𝑙) = (1 − 1))
9695oveq1d 7372 . . . . . . . . . . . . . . . 16 (𝑙 = 1 → ((1 − 𝑙) · (𝑝𝑖)) = ((1 − 1) · (𝑝𝑖)))
97 1m1e0 12225 . . . . . . . . . . . . . . . . 17 (1 − 1) = 0
9897oveq1i 7367 . . . . . . . . . . . . . . . 16 ((1 − 1) · (𝑝𝑖)) = (0 · (𝑝𝑖))
9996, 98eqtrdi 2792 . . . . . . . . . . . . . . 15 (𝑙 = 1 → ((1 − 𝑙) · (𝑝𝑖)) = (0 · (𝑝𝑖)))
100 oveq1 7364 . . . . . . . . . . . . . . 15 (𝑙 = 1 → (𝑙 · (𝑦𝑖)) = (1 · (𝑦𝑖)))
10199, 100oveq12d 7375 . . . . . . . . . . . . . 14 (𝑙 = 1 → (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖))))
102101eqeq2d 2747 . . . . . . . . . . . . 13 (𝑙 = 1 → ((𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ (𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖)))))
103102ralbidv 3174 . . . . . . . . . . . 12 (𝑙 = 1 → (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = ((0 · (𝑝𝑖)) + (1 · (𝑦𝑖)))))
104103rexsng 4635 . . . . . . . . . . 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 328 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))
10728raleqi 3311 . . . . . . . . . . 11 (∀𝑖𝐼 (𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (((1 − 𝑙) · (𝑝𝑖)) + (𝑙 · (𝑦𝑖))))
108107rexbii 3097 . . . . . . . . . 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 935 . . . . . . . . . 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 282 . . . . . . . . 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 868 . . . . . . . 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 287 . . . . . . 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 282 . . . . . 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 304 . . . . 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 27931 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (𝑦 Btwn ⟨𝑥, 𝑝⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝)))
117 brbtwn 27848 . . . . . . . 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 1371 . . . . . . 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 280 . . . . . 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 13397 . . . . . . . . . 10 ((0 ∈ ℝ* ∧ 1 ∈ ℝ* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
12137, 38, 39, 120mp3an 1461 . . . . . . . . 9 ({0} ∪ (0(,]1)) = (0[,]1)
122121eqcomi 2745 . . . . . . . 8 (0[,]1) = ({0} ∪ (0(,]1))
123122a1i 11 . . . . . . 7 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (0[,]1) = ({0} ∪ (0(,]1)))
124123rexeqdv 3314 . . . . . 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 4150 . . . . . . 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 2743 . . . . . . . . . . . 12 ((𝑥𝑖) = (𝑦𝑖) ↔ (𝑦𝑖) = (𝑥𝑖))
127126ralbii 3096 . . . . . . . . . . 11 (∀𝑖 ∈ (1...𝑁)(𝑥𝑖) = (𝑦𝑖) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (𝑥𝑖))
12870, 127sylnib 327 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (𝑥𝑖))
12916biimpd 228 . . . . . . . . . . . . . . . . . . . . 21 (𝑁 ∈ ℕ → (𝑥𝑃𝑥 ∈ (𝔼‘𝑁)))
130129, 47sylibd 238 . . . . . . . . . . . . . . . . . . . 20 (𝑁 ∈ ℕ → (𝑥𝑃𝑥:(1...𝑁)⟶ℝ))
131130imp 407 . . . . . . . . . . . . . . . . . . 19 ((𝑁 ∈ ℕ ∧ 𝑥𝑃) → 𝑥:(1...𝑁)⟶ℝ)
1321313adant3 1132 . . . . . . . . . . . . . . . . . 18 ((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑥:(1...𝑁)⟶ℝ)
133132adantr 481 . . . . . . . . . . . . . . . . 17 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → 𝑥:(1...𝑁)⟶ℝ)
134133ffvelcdmda 7035 . . . . . . . . . . . . . . . 16 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℝ)
135134recnd 11183 . . . . . . . . . . . . . . 15 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥𝑖) ∈ ℂ)
136135mulid2d 11173 . . . . . . . . . . . . . 14 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (1 · (𝑥𝑖)) = (𝑥𝑖))
137136, 79oveq12d 7375 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) = ((𝑥𝑖) + 0))
138135addid1d 11355 . . . . . . . . . . . . 13 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥𝑖) + 0) = (𝑥𝑖))
139137, 138eqtrd 2776 . . . . . . . . . . . 12 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) = (𝑥𝑖))
140139eqeq2d 2747 . . . . . . . . . . 11 ((((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) ↔ (𝑦𝑖) = (𝑥𝑖)))
141140ralbidva 3172 . . . . . . . . . 10 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (𝑥𝑖)))
142128, 141mtbird 324 . . . . . . . . 9 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))))
143 0re 11157 . . . . . . . . . 10 0 ∈ ℝ
144 oveq2 7365 . . . . . . . . . . . . . . . 16 (𝑚 = 0 → (1 − 𝑚) = (1 − 0))
145144oveq1d 7372 . . . . . . . . . . . . . . 15 (𝑚 = 0 → ((1 − 𝑚) · (𝑥𝑖)) = ((1 − 0) · (𝑥𝑖)))
146 1m0e1 12274 . . . . . . . . . . . . . . . 16 (1 − 0) = 1
147146oveq1i 7367 . . . . . . . . . . . . . . 15 ((1 − 0) · (𝑥𝑖)) = (1 · (𝑥𝑖))
148145, 147eqtrdi 2792 . . . . . . . . . . . . . 14 (𝑚 = 0 → ((1 − 𝑚) · (𝑥𝑖)) = (1 · (𝑥𝑖)))
149 oveq1 7364 . . . . . . . . . . . . . 14 (𝑚 = 0 → (𝑚 · (𝑝𝑖)) = (0 · (𝑝𝑖)))
150148, 149oveq12d 7375 . . . . . . . . . . . . 13 (𝑚 = 0 → (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖))))
151150eqeq2d 2747 . . . . . . . . . . . 12 (𝑚 = 0 → ((𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ (𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖)))))
152151ralbidv 3174 . . . . . . . . . . 11 (𝑚 = 0 → (∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = ((1 · (𝑥𝑖)) + (0 · (𝑝𝑖)))))
153152rexsng 4635 . . . . . . . . . 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 328 . . . . . . . 8 (((𝑁 ∈ ℕ ∧ 𝑥𝑃𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝𝑃) → ¬ ∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))
15628raleqi 3311 . . . . . . . . . 10 (∀𝑖𝐼 (𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦𝑖) = (((1 − 𝑚) · (𝑥𝑖)) + (𝑚 · (𝑝𝑖))))
157156rexbii 3097 . . . . . . . . 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 935 . . . . . . . . 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 282 . . . . . . . 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 289 . . . . . 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 304 . . . . 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 1435 . . . 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 3414 . . 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 7434 . 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 2776 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 205  wa 396  wo 845  w3o 1086  w3a 1087   = wceq 1541  wcel 2106  wne 2943  wral 3064  wrex 3073  {crab 3407  cdif 3907  cun 3908  {csn 4586  cop 4592   class class class wbr 5105   Fn wfn 6491  wf 6492  cfv 6496  (class class class)co 7357  cmpo 7359  cr 11050  0cc0 11051  1c1 11052   + caddc 11054   · cmul 11056  *cxr 11188  cle 11190  cmin 11385  cn 12153  (,]cioc 13265  [,)cico 13266  [,]cicc 13267  ...cfz 13424  Basecbs 17083  Itvcitv 27375  LineGclng 27376  𝔼cee 27837   Btwn cbtwn 27838  EEGceeng 27926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-er 8648  df-map 8767  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-nn 12154  df-2 12216  df-3 12217  df-4 12218  df-5 12219  df-6 12220  df-7 12221  df-8 12222  df-9 12223  df-n0 12414  df-z 12500  df-dec 12619  df-uz 12764  df-ioc 13269  df-ico 13270  df-icc 13271  df-fz 13425  df-seq 13907  df-sum 15571  df-struct 17019  df-slot 17054  df-ndx 17066  df-base 17084  df-ds 17155  df-itv 27377  df-lng 27378  df-ee 27840  df-btwn 27841  df-eeng 27927
This theorem is referenced by:  eenglngeehlnm  46815
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