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Theorem ttgvalOLD 27987

Description: Obsolete proof of ttgval 27986 as of 9-Nov-2024. Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof modification is discouraged.) (New usage is discouraged.)

**Hypotheses**
Ref	Expression
ttgval.n	⊢ 𝐺 = (toTG‘𝐻)
ttgval.b	⊢ 𝐵 = (Base‘𝐻)
ttgval.m	⊢ − = (-_g‘𝐻)
ttgval.s	⊢ · = ( ·_𝑠 ‘𝐻)
ttgval.i	⊢ 𝐼 = (Itv‘𝐺)

**Assertion**
Ref	Expression
ttgvalOLD	⊢ (𝐻 ∈ 𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})))

Distinct variable groups: 𝑥,𝑘,𝑦,𝑧 𝑥,𝐵,𝑦,𝑧 𝑘,𝐻,𝑥,𝑦,𝑧 𝑥,𝑉,𝑦,𝑧 𝑥, − ,𝑦,𝑧 𝑥, · ,𝑦,𝑧 Allowed substitution hints: 𝐵(𝑘) · (𝑘) 𝐺(𝑥,𝑦,𝑧,𝑘) 𝐼(𝑥,𝑦,𝑧,𝑘) − (𝑘) 𝑉(𝑘)

Proof of Theorem ttgvalOLD Dummy variables 𝑎 𝑏 𝑐 𝑖 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ttgval.n	. . . . 5 ⊢ 𝐺 = (toTG‘𝐻)
2	1	a1i 11	. . . 4 ⊢ (𝐻 ∈ 𝑉 → 𝐺 = (toTG‘𝐻))
3		elex 3488	. . . . 5 ⊢ (𝐻 ∈ 𝑉 → 𝐻 ∈ V)
4		fveq2 6875	. . . . . . . . . 10 ⊢ (𝑤 = 𝐻 → (Base‘𝑤) = (Base‘𝐻))
5		ttgval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐻)
6	4, 5	eqtr4di 2789	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → (Base‘𝑤) = 𝐵)
7		fveq2 6875	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝐻 → (-_g‘𝑤) = (-_g‘𝐻))
8		ttgval.m	. . . . . . . . . . . . . 14 ⊢ − = (-_g‘𝐻)
9	7, 8	eqtr4di 2789	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → (-_g‘𝑤) = − )
10	9	oveqd 7407	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐻 → (𝑧(-_g‘𝑤)𝑥) = (𝑧 − 𝑥))
11		fveq2 6875	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝐻 → ( ·_𝑠 ‘𝑤) = ( ·_𝑠 ‘𝐻))
12		ttgval.s	. . . . . . . . . . . . . 14 ⊢ · = ( ·_𝑠 ‘𝐻)
13	11, 12	eqtr4di 2789	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → ( ·_𝑠 ‘𝑤) = · )
14		eqidd 2732	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → 𝑘 = 𝑘)
15	9	oveqd 7407	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → (𝑦(-_g‘𝑤)𝑥) = (𝑦 − 𝑥))
16	13, 14, 15	oveq123d 7411	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐻 → (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥)) = (𝑘 · (𝑦 − 𝑥)))
17	10, 16	eqeq12d 2747	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐻 → ((𝑧(-_g‘𝑤)𝑥) = (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥)) ↔ (𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
18	17	rexbidv 3177	. . . . . . . . . 10 ⊢ (𝑤 = 𝐻 → (∃𝑘 ∈ (0[,]1)(𝑧(-_g‘𝑤)𝑥) = (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
19	6, 18	rabeqbidv 3446	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-_g‘𝑤)𝑥) = (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥))} = {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})
20	6, 6, 19	mpoeq123dv 7465	. . . . . . . 8 ⊢ (𝑤 = 𝐻 → (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-_g‘𝑤)𝑥) = (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}))
21	20	csbeq1d 3890	. . . . . . 7 ⊢ (𝑤 = 𝐻 → ⦋(𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-_g‘𝑤)𝑥) = (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥))}) / 𝑖⦌((𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) = ⦋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) / 𝑖⦌((𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
22		oveq1 7397	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → (𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) = (𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩))
23	6	rabeqdv 3444	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐻 → {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))} = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
24	6, 6, 23	mpoeq123dv 7465	. . . . . . . . . 10 ⊢ (𝑤 = 𝐻 → (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}))
25	24	opeq2d 4870	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩ = ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩)
26	22, 25	oveq12d 7408	. . . . . . . 8 ⊢ (𝑤 = 𝐻 → ((𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) = ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
27	26	csbeq2dv 3893	. . . . . . 7 ⊢ (𝑤 = 𝐻 → ⦋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) / 𝑖⦌((𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) = ⦋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) / 𝑖⦌((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
28	21, 27	eqtrd 2771	. . . . . 6 ⊢ (𝑤 = 𝐻 → ⦋(𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-_g‘𝑤)𝑥) = (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥))}) / 𝑖⦌((𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) = ⦋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) / 𝑖⦌((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
29		df-ttg 27985	. . . . . 6 ⊢ toTG = (𝑤 ∈ V ↦ ⦋(𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-_g‘𝑤)𝑥) = (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥))}) / 𝑖⦌((𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
30		ovex 7423	. . . . . . 7 ⊢ ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) ∈ V
31	30	csbex 5301	. . . . . 6 ⊢ ⦋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) / 𝑖⦌((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) ∈ V
32	28, 29, 31	fvmpt 6981	. . . . 5 ⊢ (𝐻 ∈ V → (toTG‘𝐻) = ⦋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) / 𝑖⦌((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
33	3, 32	syl 17	. . . 4 ⊢ (𝐻 ∈ 𝑉 → (toTG‘𝐻) = ⦋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) / 𝑖⦌((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
34	5	fvexi 6889	. . . . . . 7 ⊢ 𝐵 ∈ V
35	34, 34	mpoex 8045	. . . . . 6 ⊢ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) ∈ V
36	35	a1i 11	. . . . 5 ⊢ (𝐻 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) ∈ V)
37		simpr 485	. . . . . . 7 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})) → 𝑖 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}))
38		oveq2 7398	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑐 − 𝑎) = (𝑐 − 𝑥))
39		oveq2 7398	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (𝑏 − 𝑎) = (𝑏 − 𝑥))
40	39	oveq2d 7406	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑘 · (𝑏 − 𝑎)) = (𝑘 · (𝑏 − 𝑥)))
41	38, 40	eqeq12d 2747	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎)) ↔ (𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥))))
42	41	rexbidv 3177	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥))))
43	42	rabbidv 3437	. . . . . . . 8 ⊢ (𝑎 = 𝑥 → {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))} = {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥))})
44		oveq1 7397	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑦 → (𝑏 − 𝑥) = (𝑦 − 𝑥))
45	44	oveq2d 7406	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (𝑘 · (𝑏 − 𝑥)) = (𝑘 · (𝑦 − 𝑥)))
46	45	eqeq2d 2742	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → ((𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥)) ↔ (𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
47	46	rexbidv 3177	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
48	47	rabbidv 3437	. . . . . . . . 9 ⊢ (𝑏 = 𝑦 → {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥))} = {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})
49		oveq1 7397	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑧 → (𝑐 − 𝑥) = (𝑧 − 𝑥))
50	49	eqeq1d 2733	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑧 → ((𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥)) ↔ (𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
51	50	rexbidv 3177	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
52	51	cbvrabv 3439	. . . . . . . . 9 ⊢ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥))} = {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}
53	48, 52	eqtrdi 2787	. . . . . . . 8 ⊢ (𝑏 = 𝑦 → {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥))} = {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})
54	43, 53	cbvmpov 7485	. . . . . . 7 ⊢ (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})
55	37, 54	eqtr4di 2789	. . . . . 6 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})) → 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))}))
56		simpr 485	. . . . . . . . . 10 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))}))
57	56, 54	eqtrdi 2787	. . . . . . . . 9 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → 𝑖 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}))
58	57	opeq2d 4870	. . . . . . . 8 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → ⟨(Itv‘ndx), 𝑖⟩ = ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩)
59	58	oveq2d 7406	. . . . . . 7 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → (𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) = (𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩))
60	57	oveqd 7407	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → (𝑥𝑖𝑦) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦))
61	60	eleq2d 2818	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → (𝑧 ∈ (𝑥𝑖𝑦) ↔ 𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦)))
62	57	oveqd 7407	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → (𝑧𝑖𝑦) = (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦))
63	62	eleq2d 2818	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → (𝑥 ∈ (𝑧𝑖𝑦) ↔ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦)))
64	57	oveqd 7407	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → (𝑥𝑖𝑧) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))
65	64	eleq2d 2818	. . . . . . . . . . 11 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → (𝑦 ∈ (𝑥𝑖𝑧) ↔ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧)))
66	61, 63, 65	3orbi123d 1435	. . . . . . . . . 10 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → ((𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ↔ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))))
67	66	rabbidv 3437	. . . . . . . . 9 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))} = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})
68	67	mpoeq3dv 7469	. . . . . . . 8 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))}))
69	68	opeq2d 4870	. . . . . . 7 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩ = ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})⟩)
70	59, 69	oveq12d 7408	. . . . . 6 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑎 ∈ 𝐵, 𝑏 ∈ 𝐵 ↦ {𝑐 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎))})) → ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})⟩))
71	55, 70	syldan 591	. . . . 5 ⊢ ((𝐻 ∈ 𝑉 ∧ 𝑖 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})) → ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})⟩))
72	36, 71	csbied 3924	. . . 4 ⊢ (𝐻 ∈ 𝑉 → ⦋(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) / 𝑖⦌((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})⟩))
73	2, 33, 72	3eqtrd 2775	. . 3 ⊢ (𝐻 ∈ 𝑉 → 𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})⟩))
74	73	fveq2d 6879	. . . . . . . . . . . 12 ⊢ (𝐻 ∈ 𝑉 → (Itv‘𝐺) = (Itv‘((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})⟩)))
75		itvid 27550	. . . . . . . . . . . . 13 ⊢ Itv = Slot (Itv‘ndx)
76		1nn0 12467	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ₀
77		6nn 12280	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℕ
78	76, 77	decnncl 12676	. . . . . . . . . . . . . . . 16 ⊢ ;16 ∈ ℕ
79	78	nnrei 12200	. . . . . . . . . . . . . . 15 ⊢ ;16 ∈ ℝ
80		6nn0 12472	. . . . . . . . . . . . . . . 16 ⊢ 6 ∈ ℕ₀
81		7nn 12283	. . . . . . . . . . . . . . . 16 ⊢ 7 ∈ ℕ
82		6lt7 12377	. . . . . . . . . . . . . . . 16 ⊢ 6 < 7
83	76, 80, 81, 82	declt 12684	. . . . . . . . . . . . . . 15 ⊢ ;16 < ;17
84	79, 83	ltneii 11306	. . . . . . . . . . . . . 14 ⊢ ;16 ≠ ;17
85		itvndx 27548	. . . . . . . . . . . . . . 15 ⊢ (Itv‘ndx) = ;16
86		lngndx 27549	. . . . . . . . . . . . . . 15 ⊢ (LineG‘ndx) = ;17
87	85, 86	neeq12i 3006	. . . . . . . . . . . . . 14 ⊢ ((Itv‘ndx) ≠ (LineG‘ndx) ↔ ;16 ≠ ;17)
88	84, 87	mpbir 230	. . . . . . . . . . . . 13 ⊢ (Itv‘ndx) ≠ (LineG‘ndx)
89	75, 88	setsnid 17121	. . . . . . . . . . . 12 ⊢ (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩)) = (Itv‘((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})⟩))
90	74, 89	eqtr4di 2789	. . . . . . . . . . 11 ⊢ (𝐻 ∈ 𝑉 → (Itv‘𝐺) = (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩)))
91		ttgval.i	. . . . . . . . . . . 12 ⊢ 𝐼 = (Itv‘𝐺)
92	91	a1i 11	. . . . . . . . . . 11 ⊢ (𝐻 ∈ 𝑉 → 𝐼 = (Itv‘𝐺))
93	75	setsid 17120	. . . . . . . . . . . 12 ⊢ ((𝐻 ∈ 𝑉 ∧ (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) ∈ V) → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) = (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩)))
94	35, 93	mpan2 689	. . . . . . . . . . 11 ⊢ (𝐻 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}) = (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩)))
95	90, 92, 94	3eqtr4d 2781	. . . . . . . . . 10 ⊢ (𝐻 ∈ 𝑉 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}))
96	95	oveqd 7407	. . . . . . . . 9 ⊢ (𝐻 ∈ 𝑉 → (𝑥𝐼𝑦) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦))
97	96	eleq2d 2818	. . . . . . . 8 ⊢ (𝐻 ∈ 𝑉 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦)))
98	95	oveqd 7407	. . . . . . . . 9 ⊢ (𝐻 ∈ 𝑉 → (𝑧𝐼𝑦) = (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦))
99	98	eleq2d 2818	. . . . . . . 8 ⊢ (𝐻 ∈ 𝑉 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦)))
100	95	oveqd 7407	. . . . . . . . 9 ⊢ (𝐻 ∈ 𝑉 → (𝑥𝐼𝑧) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))
101	100	eleq2d 2818	. . . . . . . 8 ⊢ (𝐻 ∈ 𝑉 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧)))
102	97, 99, 101	3orbi123d 1435	. . . . . . 7 ⊢ (𝐻 ∈ 𝑉 → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))))
103	102	rabbidv 3437	. . . . . 6 ⊢ (𝐻 ∈ 𝑉 → {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} = {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})
104	103	mpoeq3dv 7469	. . . . 5 ⊢ (𝐻 ∈ 𝑉 → (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))}))
105	104	opeq2d 4870	. . . 4 ⊢ (𝐻 ∈ 𝑉 → ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩ = ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})⟩)
106	105	oveq2d 7406	. . 3 ⊢ (𝐻 ∈ 𝑉 → ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))})⟩))
107	73, 106	eqtr4d 2774	. 2 ⊢ (𝐻 ∈ 𝑉 → 𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩))
108	107, 95	jca 512	1 ⊢ (𝐻 ∈ 𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∨ w3o 1086 = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∃wrex 3069 {crab 3429 Vcvv 3470 ⦋csb 3886 ⟨cop 4625 ‘cfv 6529 (class class class)co 7390 ∈ cmpo 7392 0cc0 11089 1c1 11090 6c6 12250 7c7 12251 cdc 12656 [,]cicc 13306 sSet csts 17075 ndxcnx 17105 Basecbs 17123 ·_𝑠 cvsca 17180 -_gcsg 18793 Itvcitv 27544 LineGclng 27545 toTGcttg 27984

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 2702 ax-rep 5275 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7705 ax-cnex 11145 ax-resscn 11146 ax-1cn 11147 ax-icn 11148 ax-addcl 11149 ax-addrcl 11150 ax-mulcl 11151 ax-mulrcl 11152 ax-mulcom 11153 ax-addass 11154 ax-mulass 11155 ax-distr 11156 ax-i2m1 11157 ax-1ne0 11158 ax-1rid 11159 ax-rnegex 11160 ax-rrecex 11161 ax-cnre 11162 ax-pre-lttri 11163 ax-pre-lttrn 11164 ax-pre-ltadd 11165 ax-pre-mulgt0 11166

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 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3376 df-rab 3430 df-v 3472 df-sbc 3771 df-csb 3887 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-pss 3960 df-nul 4316 df-if 4520 df-pw 4595 df-sn 4620 df-pr 4622 df-op 4626 df-uni 4899 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6286 df-ord 6353 df-on 6354 df-lim 6355 df-suc 6356 df-iota 6481 df-fun 6531 df-fn 6532 df-f 6533 df-f1 6534 df-fo 6535 df-f1o 6536 df-fv 6537 df-riota 7346 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7836 df-1st 7954 df-2nd 7955 df-frecs 8245 df-wrecs 8276 df-recs 8350 df-rdg 8389 df-er 8683 df-en 8920 df-dom 8921 df-sdom 8922 df-pnf 11229 df-mnf 11230 df-xr 11231 df-ltxr 11232 df-le 11233 df-sub 11425 df-neg 11426 df-nn 12192 df-2 12254 df-3 12255 df-4 12256 df-5 12257 df-6 12258 df-7 12259 df-8 12260 df-9 12261 df-n0 12452 df-dec 12657 df-sets 17076 df-slot 17094 df-ndx 17106 df-itv 27546 df-lng 27547 df-ttg 27985

This theorem is referenced by: (None)

W3C validator