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Theorem elntg2 28934

Description: The line definition in the Tarski structure for the Euclidean geometry. In contrast to elntg 28933, the betweenness can be strengthened by excluding 1 resp. 0 from the related intervals (because of 𝑥 ≠ 𝑦). (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)) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))}))

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**
Step	Hyp	Ref	Expression
1		elntg2.1	. . 3 ⊢ 𝑃 = (Base‘(EEG‘𝑁))
2		eqid 2729	. . 3 ⊢ (Itv‘(EEG‘𝑁)) = (Itv‘(EEG‘𝑁))
3	1, 2	elntg 28933	. 2 ⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ (𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝))}))
4		simpl1 1192	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑁 ∈ ℕ)
5		simpl2 1193	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑥 ∈ 𝑃)
6		eldifi 4082	. . . . . . . . 9 ⊢ (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦 ∈ 𝑃)
7	6	3ad2ant3 1135	. . . . . . . 8 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦 ∈ 𝑃)
8	7	adantr 480	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑦 ∈ 𝑃)
9		simpr 484	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ 𝑃)
10	4, 1, 2, 5, 8, 9	ebtwntg 28931	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ 𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦)))
11		eengbas 28930	. . . . . . . . . . . 12 ⊢ (𝑁 ∈ ℕ → (𝔼‘𝑁) = (Base‘(EEG‘𝑁)))
12	1, 11	eqtr4id 2783	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → 𝑃 = (𝔼‘𝑁))
13	12	3ad2ant1 1133	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑃 = (𝔼‘𝑁))
14	13	eleq2d 2814	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ (𝔼‘𝑁)))
15	14	biimpa 476	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑝 ∈ (𝔼‘𝑁))
16	12	eleq2d 2814	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑃 ↔ 𝑥 ∈ (𝔼‘𝑁)))
17	16	biimpa 476	. . . . . . . . . 10 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃) → 𝑥 ∈ (𝔼‘𝑁))
18	17	3adant3 1132	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑥 ∈ (𝔼‘𝑁))
19	18	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑥 ∈ (𝔼‘𝑁))
20	12	eleq2d 2814	. . . . . . . . . . . . 13 ⊢ (𝑁 ∈ ℕ → (𝑦 ∈ 𝑃 ↔ 𝑦 ∈ (𝔼‘𝑁)))
21	20	biimpcd 249	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ 𝑃 → (𝑁 ∈ ℕ → 𝑦 ∈ (𝔼‘𝑁)))
22	21, 6	syl11 33	. . . . . . . . . . 11 ⊢ (𝑁 ∈ ℕ → (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦 ∈ (𝔼‘𝑁)))
23	22	a1d 25	. . . . . . . . . 10 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑃 → (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦 ∈ (𝔼‘𝑁))))
24	23	3imp 1110	. . . . . . . . 9 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦 ∈ (𝔼‘𝑁))
25	24	adantr 480	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑦 ∈ (𝔼‘𝑁))
26		brbtwn 28848	. . . . . . . 8 ⊢ ((𝑝 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖)))))
27	15, 19, 25, 26	syl3anc 1373	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖)))))
28		elntg2.2	. . . . . . . . 9 ⊢ 𝐼 = (1...𝑁)
29	28	raleqi 3287	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))))
30	29	rexbii 3076	. . . . . . 7 ⊢ (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ↔ ∃𝑘 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))))
31	27, 30	bitr4di 289	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑝 Btwn ⟨𝑥, 𝑦⟩ ↔ ∃𝑘 ∈ (0[,]1)∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖)))))
32	10, 31	bitr3d 281	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ↔ ∃𝑘 ∈ (0[,]1)∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖)))))
33	4, 1, 2, 9, 8, 5	ebtwntg 28931	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑥 Btwn ⟨𝑝, 𝑦⟩ ↔ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦)))
34		brbtwn 28848	. . . . . . . 8 ⊢ ((𝑥 ∈ (𝔼‘𝑁) ∧ 𝑝 ∈ (𝔼‘𝑁) ∧ 𝑦 ∈ (𝔼‘𝑁)) → (𝑥 Btwn ⟨𝑝, 𝑦⟩ ↔ ∃𝑙 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))))
35	19, 15, 25, 34	syl3anc 1373	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑥 Btwn ⟨𝑝, 𝑦⟩ ↔ ∃𝑙 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))))
36	33, 35	bitr3d 281	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ↔ ∃𝑙 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))))
37		0xr 11162	. . . . . . . . . 10 ⊢ 0 ∈ ℝ^*
38		1xr 11174	. . . . . . . . . 10 ⊢ 1 ∈ ℝ^*
39		0le1 11643	. . . . . . . . . 10 ⊢ 0 ≤ 1
40		snunico 13382	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ((0[,)1) ∪ {1}) = (0[,]1))
41	37, 38, 39, 40	mp3an 1463	. . . . . . . . 9 ⊢ ((0[,)1) ∪ {1}) = (0[,]1)
42	41	eqcomi 2738	. . . . . . . 8 ⊢ (0[,]1) = ((0[,)1) ∪ {1})
43	42	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (0[,]1) = ((0[,)1) ∪ {1}))
44	43	rexeqdv 3290	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (∃𝑙 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ ∃𝑙 ∈ ((0[,)1) ∪ {1})∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))))
45		rexun 4147	. . . . . . 7 ⊢ (∃𝑙 ∈ ((0[,)1) ∪ {1})∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))))
46		eldifsn 4737	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (𝑃 ∖ {𝑥}) ↔ (𝑦 ∈ 𝑃 ∧ 𝑦 ≠ 𝑥))
47		elee 28843	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ (𝔼‘𝑁) ↔ 𝑥:(1...𝑁)⟶ℝ))
48		ffn 6652	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑥:(1...𝑁)⟶ℝ → 𝑥 Fn (1...𝑁))
49	47, 48	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ (𝔼‘𝑁) → 𝑥 Fn (1...𝑁)))
50	16, 49	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑃 → 𝑥 Fn (1...𝑁)))
51	50	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑃 → (𝑁 ∈ ℕ → (𝑥 ∈ 𝑃 → 𝑥 Fn (1...𝑁))))
52	51	3imp 1110	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝑃 ∧ 𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃) → 𝑥 Fn (1...𝑁))
53		elee 28843	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑁 ∈ ℕ → (𝑦 ∈ (𝔼‘𝑁) ↔ 𝑦:(1...𝑁)⟶ℝ))
54		ffn 6652	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝑦:(1...𝑁)⟶ℝ → 𝑦 Fn (1...𝑁))
55	53, 54	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑁 ∈ ℕ → (𝑦 ∈ (𝔼‘𝑁) → 𝑦 Fn (1...𝑁)))
56	20, 55	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑁 ∈ ℕ → (𝑦 ∈ 𝑃 → 𝑦 Fn (1...𝑁)))
57	56	a1i 11	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑥 ∈ 𝑃 → (𝑁 ∈ ℕ → (𝑦 ∈ 𝑃 → 𝑦 Fn (1...𝑁))))
58	57	3imp31 1111	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑦 ∈ 𝑃 ∧ 𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃) → 𝑦 Fn (1...𝑁))
59		eqfnfv 6965	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑥 Fn (1...𝑁) ∧ 𝑦 Fn (1...𝑁)) → (𝑥 = 𝑦 ↔ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))
60	52, 58, 59	syl2anc 584	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑦 ∈ 𝑃 ∧ 𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃) → (𝑥 = 𝑦 ↔ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))
61	60	biimprd 248	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑦 ∈ 𝑃 ∧ 𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖) → 𝑥 = 𝑦))
62		eqcom 2736	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
63	61, 62	imbitrrdi 252	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ 𝑃 ∧ 𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖) → 𝑦 = 𝑥))
64	63	necon3ad 2938	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ 𝑃 ∧ 𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃) → (𝑦 ≠ 𝑥 → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))
65	64	3exp 1119	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ 𝑃 → (𝑁 ∈ ℕ → (𝑥 ∈ 𝑃 → (𝑦 ≠ 𝑥 → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))))
66	65	com24 95	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ 𝑃 → (𝑦 ≠ 𝑥 → (𝑥 ∈ 𝑃 → (𝑁 ∈ ℕ → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))))
67	66	imp 406	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝑃 ∧ 𝑦 ≠ 𝑥) → (𝑥 ∈ 𝑃 → (𝑁 ∈ ℕ → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖))))
68	46, 67	sylbi 217	. . . . . . . . . . . . 13 ⊢ (𝑦 ∈ (𝑃 ∖ {𝑥}) → (𝑥 ∈ 𝑃 → (𝑁 ∈ ℕ → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖))))
69	68	3imp31 1111	. . . . . . . . . . . 12 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖))
70	69	adantr 480	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖))
71	12	eleq2d 2814	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (𝑝 ∈ 𝑃 ↔ 𝑝 ∈ (𝔼‘𝑁)))
72		elee 28843	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑁 ∈ ℕ → (𝑝 ∈ (𝔼‘𝑁) ↔ 𝑝:(1...𝑁)⟶ℝ))
73	72	biimpd 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (𝑝 ∈ (𝔼‘𝑁) → 𝑝:(1...𝑁)⟶ℝ))
74	71, 73	sylbid 240	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (𝑝 ∈ 𝑃 → 𝑝:(1...𝑁)⟶ℝ))
75	74	3ad2ant1 1133	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → (𝑝 ∈ 𝑃 → 𝑝:(1...𝑁)⟶ℝ))
76	75	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑝:(1...𝑁)⟶ℝ)
77	76	ffvelcdmda 7018	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝‘𝑖) ∈ ℝ)
78	77	recnd 11143	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑝‘𝑖) ∈ ℂ)
79	78	mul02d 11314	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (0 · (𝑝‘𝑖)) = 0)
80	21, 53	mpbidi 241	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑦 ∈ 𝑃 → (𝑁 ∈ ℕ → 𝑦:(1...𝑁)⟶ℝ))
81	80, 6	syl11 33	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦:(1...𝑁)⟶ℝ))
82	81	a1d 25	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑃 → (𝑦 ∈ (𝑃 ∖ {𝑥}) → 𝑦:(1...𝑁)⟶ℝ)))
83	82	3imp 1110	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑦:(1...𝑁)⟶ℝ)
84	83	adantr 480	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑦:(1...𝑁)⟶ℝ)
85	84	ffvelcdmda 7018	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦‘𝑖) ∈ ℝ)
86	85	recnd 11143	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑦‘𝑖) ∈ ℂ)
87	86	mullidd 11133	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (1 · (𝑦‘𝑖)) = (𝑦‘𝑖))
88	79, 87	oveq12d 7367	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((0 · (𝑝‘𝑖)) + (1 · (𝑦‘𝑖))) = (0 + (𝑦‘𝑖)))
89	86	addlidd 11317	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (0 + (𝑦‘𝑖)) = (𝑦‘𝑖))
90	88, 89	eqtrd 2764	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((0 · (𝑝‘𝑖)) + (1 · (𝑦‘𝑖))) = (𝑦‘𝑖))
91	90	eqeq2d 2740	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥‘𝑖) = ((0 · (𝑝‘𝑖)) + (1 · (𝑦‘𝑖))) ↔ (𝑥‘𝑖) = (𝑦‘𝑖)))
92	91	ralbidva 3150	. . . . . . . . . . 11 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = ((0 · (𝑝‘𝑖)) + (1 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖)))
93	70, 92	mtbird 325	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = ((0 · (𝑝‘𝑖)) + (1 · (𝑦‘𝑖))))
94		1re 11115	. . . . . . . . . . 11 ⊢ 1 ∈ ℝ
95		oveq2 7357	. . . . . . . . . . . . . . . . 17 ⊢ (𝑙 = 1 → (1 − 𝑙) = (1 − 1))
96	95	oveq1d 7364	. . . . . . . . . . . . . . . 16 ⊢ (𝑙 = 1 → ((1 − 𝑙) · (𝑝‘𝑖)) = ((1 − 1) · (𝑝‘𝑖)))
97		1m1e0 12200	. . . . . . . . . . . . . . . . 17 ⊢ (1 − 1) = 0
98	97	oveq1i 7359	. . . . . . . . . . . . . . . 16 ⊢ ((1 − 1) · (𝑝‘𝑖)) = (0 · (𝑝‘𝑖))
99	96, 98	eqtrdi 2780	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 1 → ((1 − 𝑙) · (𝑝‘𝑖)) = (0 · (𝑝‘𝑖)))
100		oveq1 7356	. . . . . . . . . . . . . . 15 ⊢ (𝑙 = 1 → (𝑙 · (𝑦‘𝑖)) = (1 · (𝑦‘𝑖)))
101	99, 100	oveq12d 7367	. . . . . . . . . . . . . 14 ⊢ (𝑙 = 1 → (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) = ((0 · (𝑝‘𝑖)) + (1 · (𝑦‘𝑖))))
102	101	eqeq2d 2740	. . . . . . . . . . . . 13 ⊢ (𝑙 = 1 → ((𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ (𝑥‘𝑖) = ((0 · (𝑝‘𝑖)) + (1 · (𝑦‘𝑖)))))
103	102	ralbidv 3152	. . . . . . . . . . . 12 ⊢ (𝑙 = 1 → (∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = ((0 · (𝑝‘𝑖)) + (1 · (𝑦‘𝑖)))))
104	103	rexsng 4628	. . . . . . . . . . 11 ⊢ (1 ∈ ℝ → (∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = ((0 · (𝑝‘𝑖)) + (1 · (𝑦‘𝑖)))))
105	94, 104	ax-mp 5	. . . . . . . . . 10 ⊢ (∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = ((0 · (𝑝‘𝑖)) + (1 · (𝑦‘𝑖))))
106	93, 105	sylnibr 329	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → ¬ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))))
107	28	raleqi 3287	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))))
108	107	rexbii 3076	. . . . . . . . . 10 ⊢ (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))))
109		biorf 936	. . . . . . . . . 10 ⊢ (¬ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) → (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ (∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))))))
110	108, 109	bitrid 283	. . . . . . . . 9 ⊢ (¬ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) → (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ (∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))))))
111	106, 110	syl 17	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ (∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))))))
112		orcom 870	. . . . . . . 8 ⊢ ((∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))) ↔ (∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))))
113	111, 112	bitr2di 288	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → ((∃𝑙 ∈ (0[,)1)∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ {1}∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))) ↔ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))))
114	45, 113	bitrid 283	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (∃𝑙 ∈ ((0[,)1) ∪ {1})∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ↔ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))))
115	36, 44, 114	3bitrd 305	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ↔ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖)))))
116	4, 1, 2, 5, 9, 8	ebtwntg 28931	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑦 Btwn ⟨𝑥, 𝑝⟩ ↔ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝)))
117		brbtwn 28848	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝔼‘𝑁) ∧ 𝑥 ∈ (𝔼‘𝑁) ∧ 𝑝 ∈ (𝔼‘𝑁)) → (𝑦 Btwn ⟨𝑥, 𝑝⟩ ↔ ∃𝑚 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))))
118	25, 19, 15, 117	syl3anc 1373	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑦 Btwn ⟨𝑥, 𝑝⟩ ↔ ∃𝑚 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))))
119	116, 118	bitr3d 281	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝) ↔ ∃𝑚 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))))
120		snunioc 13383	. . . . . . . . . 10 ⊢ ((0 ∈ ℝ^* ∧ 1 ∈ ℝ^* ∧ 0 ≤ 1) → ({0} ∪ (0(,]1)) = (0[,]1))
121	37, 38, 39, 120	mp3an 1463	. . . . . . . . 9 ⊢ ({0} ∪ (0(,]1)) = (0[,]1)
122	121	eqcomi 2738	. . . . . . . 8 ⊢ (0[,]1) = ({0} ∪ (0(,]1))
123	122	a1i 11	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (0[,]1) = ({0} ∪ (0(,]1)))
124	123	rexeqdv 3290	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (∃𝑚 ∈ (0[,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ ∃𝑚 ∈ ({0} ∪ (0(,]1))∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))))
125		rexun 4147	. . . . . . 7 ⊢ (∃𝑚 ∈ ({0} ∪ (0(,]1))∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))))
126		eqcom 2736	. . . . . . . . . . . 12 ⊢ ((𝑥‘𝑖) = (𝑦‘𝑖) ↔ (𝑦‘𝑖) = (𝑥‘𝑖))
127	126	ralbii 3075	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ (1...𝑁)(𝑥‘𝑖) = (𝑦‘𝑖) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (𝑥‘𝑖))
128	70, 127	sylnib 328	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (𝑥‘𝑖))
129	16	biimpd 229	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑃 → 𝑥 ∈ (𝔼‘𝑁)))
130	129, 47	sylibd 239	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑃 → 𝑥:(1...𝑁)⟶ℝ))
131	130	imp 406	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃) → 𝑥:(1...𝑁)⟶ℝ)
132	131	3adant3 1132	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → 𝑥:(1...𝑁)⟶ℝ)
133	132	adantr 480	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → 𝑥:(1...𝑁)⟶ℝ)
134	133	ffvelcdmda 7018	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) ∈ ℝ)
135	134	recnd 11143	. . . . . . . . . . . . . . 15 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (𝑥‘𝑖) ∈ ℂ)
136	135	mullidd 11133	. . . . . . . . . . . . . 14 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → (1 · (𝑥‘𝑖)) = (𝑥‘𝑖))
137	136, 79	oveq12d 7367	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((1 · (𝑥‘𝑖)) + (0 · (𝑝‘𝑖))) = ((𝑥‘𝑖) + 0))
138	135	addridd 11316	. . . . . . . . . . . . 13 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑥‘𝑖) + 0) = (𝑥‘𝑖))
139	137, 138	eqtrd 2764	. . . . . . . . . . . 12 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((1 · (𝑥‘𝑖)) + (0 · (𝑝‘𝑖))) = (𝑥‘𝑖))
140	139	eqeq2d 2740	. . . . . . . . . . 11 ⊢ ((((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) ∧ 𝑖 ∈ (1...𝑁)) → ((𝑦‘𝑖) = ((1 · (𝑥‘𝑖)) + (0 · (𝑝‘𝑖))) ↔ (𝑦‘𝑖) = (𝑥‘𝑖)))
141	140	ralbidva 3150	. . . . . . . . . 10 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = ((1 · (𝑥‘𝑖)) + (0 · (𝑝‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (𝑥‘𝑖)))
142	128, 141	mtbird 325	. . . . . . . . 9 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → ¬ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = ((1 · (𝑥‘𝑖)) + (0 · (𝑝‘𝑖))))
143		0re 11117	. . . . . . . . . 10 ⊢ 0 ∈ ℝ
144		oveq2 7357	. . . . . . . . . . . . . . . 16 ⊢ (𝑚 = 0 → (1 − 𝑚) = (1 − 0))
145	144	oveq1d 7364	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 0 → ((1 − 𝑚) · (𝑥‘𝑖)) = ((1 − 0) · (𝑥‘𝑖)))
146		1m0e1 12244	. . . . . . . . . . . . . . . 16 ⊢ (1 − 0) = 1
147	146	oveq1i 7359	. . . . . . . . . . . . . . 15 ⊢ ((1 − 0) · (𝑥‘𝑖)) = (1 · (𝑥‘𝑖))
148	145, 147	eqtrdi 2780	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → ((1 − 𝑚) · (𝑥‘𝑖)) = (1 · (𝑥‘𝑖)))
149		oveq1 7356	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 0 → (𝑚 · (𝑝‘𝑖)) = (0 · (𝑝‘𝑖)))
150	148, 149	oveq12d 7367	. . . . . . . . . . . . 13 ⊢ (𝑚 = 0 → (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) = ((1 · (𝑥‘𝑖)) + (0 · (𝑝‘𝑖))))
151	150	eqeq2d 2740	. . . . . . . . . . . 12 ⊢ (𝑚 = 0 → ((𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ (𝑦‘𝑖) = ((1 · (𝑥‘𝑖)) + (0 · (𝑝‘𝑖)))))
152	151	ralbidv 3152	. . . . . . . . . . 11 ⊢ (𝑚 = 0 → (∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = ((1 · (𝑥‘𝑖)) + (0 · (𝑝‘𝑖)))))
153	152	rexsng 4628	. . . . . . . . . 10 ⊢ (0 ∈ ℝ → (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = ((1 · (𝑥‘𝑖)) + (0 · (𝑝‘𝑖)))))
154	143, 153	ax-mp 5	. . . . . . . . 9 ⊢ (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = ((1 · (𝑥‘𝑖)) + (0 · (𝑝‘𝑖))))
155	142, 154	sylnibr 329	. . . . . . . 8 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → ¬ ∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))
156	28	raleqi 3287	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ ∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))
157	156	rexbii 3076	. . . . . . . . 9 ⊢ (∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))
158		biorf 936	. . . . . . . . 9 ⊢ (¬ ∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) → (∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))))
159	157, 158	bitrid 283	. . . . . . . 8 ⊢ (¬ ∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) → (∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))))
160	155, 159	syl 17	. . . . . . 7 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ (∃𝑚 ∈ {0}∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))))
161	125, 160	bitr4id 290	. . . . . 6 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (∃𝑚 ∈ ({0} ∪ (0(,]1))∀𝑖 ∈ (1...𝑁)(𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))) ↔ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))))
162	119, 124, 161	3bitrd 305	. . . . 5 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → (𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝) ↔ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖)))))
163	32, 115, 162	3orbi123d 1437	. . . 4 ⊢ (((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) ∧ 𝑝 ∈ 𝑃) → ((𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝)) ↔ (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))))
164	163	rabbidva 3401	. . 3 ⊢ ((𝑁 ∈ ℕ ∧ 𝑥 ∈ 𝑃 ∧ 𝑦 ∈ (𝑃 ∖ {𝑥})) → {𝑝 ∈ 𝑃 ∣ (𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝))} = {𝑝 ∈ 𝑃 ∣ (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))})
165	164	mpoeq3dva 7426	. 2 ⊢ (𝑁 ∈ ℕ → (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ (𝑝 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑥 ∈ (𝑝(Itv‘(EEG‘𝑁))𝑦) ∨ 𝑦 ∈ (𝑥(Itv‘(EEG‘𝑁))𝑝))}) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))}))
166	3, 165	eqtrd 2764	1 ⊢ (𝑁 ∈ ℕ → (LineG‘(EEG‘𝑁)) = (𝑥 ∈ 𝑃, 𝑦 ∈ (𝑃 ∖ {𝑥}) ↦ {𝑝 ∈ 𝑃 ∣ (∃𝑘 ∈ (0[,]1)∀𝑖 ∈ 𝐼 (𝑝‘𝑖) = (((1 − 𝑘) · (𝑥‘𝑖)) + (𝑘 · (𝑦‘𝑖))) ∨ ∃𝑙 ∈ (0[,)1)∀𝑖 ∈ 𝐼 (𝑥‘𝑖) = (((1 − 𝑙) · (𝑝‘𝑖)) + (𝑙 · (𝑦‘𝑖))) ∨ ∃𝑚 ∈ (0(,]1)∀𝑖 ∈ 𝐼 (𝑦‘𝑖) = (((1 − 𝑚) · (𝑥‘𝑖)) + (𝑚 · (𝑝‘𝑖))))}))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3394 ∖ cdif 3900 ∪ cun 3901 {csn 4577 ⟨cop 4583 class class class wbr 5092 Fn wfn 6477 ⟶wf 6478 ‘cfv 6482 (class class class)co 7349 ∈ cmpo 7351 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 ℝ^*cxr 11148 ≤ cle 11150 − cmin 11347 ℕcn 12128 (,]cioc 13249 [,)cico 13250 [,]cicc 13251 ...cfz 13410 Basecbs 17120 Itvcitv 28382 LineGclng 28383 𝔼cee 28837 Btwn cbtwn 28838 EEGceeng 28926

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 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-er 8625 df-map 8755 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-nn 12129 df-2 12191 df-3 12192 df-4 12193 df-5 12194 df-6 12195 df-7 12196 df-8 12197 df-9 12198 df-n0 12385 df-z 12472 df-dec 12592 df-uz 12736 df-ioc 13253 df-ico 13254 df-icc 13255 df-fz 13411 df-seq 13909 df-sum 15594 df-struct 17058 df-slot 17093 df-ndx 17105 df-base 17121 df-ds 17183 df-itv 28384 df-lng 28385 df-ee 28840 df-btwn 28841 df-eeng 28927

This theorem is referenced by: eenglngeehlnm 48744

W3C validator