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Theorem ttgvalOLD 27526

Description: Obsolete proof of ttgval 27525 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 3459	. . . . 5 ⊢ (𝐻 ∈ 𝑉 → 𝐻 ∈ V)
4		fveq2 6825	. . . . . . . . . 10 ⊢ (𝑤 = 𝐻 → (Base‘𝑤) = (Base‘𝐻))
5		ttgval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐻)
6	4, 5	eqtr4di 2794	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → (Base‘𝑤) = 𝐵)
7		fveq2 6825	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝐻 → (-_g‘𝑤) = (-_g‘𝐻))
8		ttgval.m	. . . . . . . . . . . . . 14 ⊢ − = (-_g‘𝐻)
9	7, 8	eqtr4di 2794	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → (-_g‘𝑤) = − )
10	9	oveqd 7354	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐻 → (𝑧(-_g‘𝑤)𝑥) = (𝑧 − 𝑥))
11		fveq2 6825	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝐻 → ( ·_𝑠 ‘𝑤) = ( ·_𝑠 ‘𝐻))
12		ttgval.s	. . . . . . . . . . . . . 14 ⊢ · = ( ·_𝑠 ‘𝐻)
13	11, 12	eqtr4di 2794	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → ( ·_𝑠 ‘𝑤) = · )
14		eqidd 2737	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → 𝑘 = 𝑘)
15	9	oveqd 7354	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → (𝑦(-_g‘𝑤)𝑥) = (𝑦 − 𝑥))
16	13, 14, 15	oveq123d 7358	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐻 → (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥)) = (𝑘 · (𝑦 − 𝑥)))
17	10, 16	eqeq12d 2752	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐻 → ((𝑧(-_g‘𝑤)𝑥) = (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥)) ↔ (𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
18	17	rexbidv 3171	. . . . . . . . . 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 3420	. . . . . . . . 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 7412	. . . . . . . 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 3847	. . . . . . 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 7344	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → (𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) = (𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩))
23	6	rabeqdv 3418	. . . . . . . . . . 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 7412	. . . . . . . . . 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 4824	. . . . . . . . 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 7355	. . . . . . . 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 3850	. . . . . . 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 2776	. . . . . 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 27524	. . . . . 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 7370	. . . . . . 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 5255	. . . . . 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 6931	. . . . 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 6839	. . . . . . 7 ⊢ 𝐵 ∈ V
35	34, 34	mpoex 7988	. . . . . 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 7345	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑐 − 𝑎) = (𝑐 − 𝑥))
39		oveq2 7345	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (𝑏 − 𝑎) = (𝑏 − 𝑥))
40	39	oveq2d 7353	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑘 · (𝑏 − 𝑎)) = (𝑘 · (𝑏 − 𝑥)))
41	38, 40	eqeq12d 2752	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎)) ↔ (𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥))))
42	41	rexbidv 3171	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥))))
43	42	rabbidv 3411	. . . . . . . 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 7344	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑦 → (𝑏 − 𝑥) = (𝑦 − 𝑥))
45	44	oveq2d 7353	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (𝑘 · (𝑏 − 𝑥)) = (𝑘 · (𝑦 − 𝑥)))
46	45	eqeq2d 2747	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → ((𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥)) ↔ (𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
47	46	rexbidv 3171	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
48	47	rabbidv 3411	. . . . . . . . 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 7344	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑧 → (𝑐 − 𝑥) = (𝑧 − 𝑥))
50	49	eqeq1d 2738	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑧 → ((𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥)) ↔ (𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
51	50	rexbidv 3171	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
52	51	cbvrabv 3413	. . . . . . . . 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 2792	. . . . . . . 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 7432	. . . . . . 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 2794	. . . . . 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 2792	. . . . . . . . 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 4824	. . . . . . . 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 7353	. . . . . . 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 7354	. . . . . . . . . . . 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 2822	. . . . . . . . . . 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 7354	. . . . . . . . . . . 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 2822	. . . . . . . . . . 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 7354	. . . . . . . . . . . 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 2822	. . . . . . . . . . 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 1434	. . . . . . . . . 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 3411	. . . . . . . . 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 7416	. . . . . . . 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 4824	. . . . . . 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 7355	. . . . . 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 3881	. . . 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 2780	. . 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 6829	. . . . . . . . . . . 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 27089	. . . . . . . . . . . . 13 ⊢ Itv = Slot (Itv‘ndx)
76		1nn0 12350	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ₀
77		6nn 12163	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℕ
78	76, 77	decnncl 12558	. . . . . . . . . . . . . . . 16 ⊢ ;16 ∈ ℕ
79	78	nnrei 12083	. . . . . . . . . . . . . . 15 ⊢ ;16 ∈ ℝ
80		6nn0 12355	. . . . . . . . . . . . . . . 16 ⊢ 6 ∈ ℕ₀
81		7nn 12166	. . . . . . . . . . . . . . . 16 ⊢ 7 ∈ ℕ
82		6lt7 12260	. . . . . . . . . . . . . . . 16 ⊢ 6 < 7
83	76, 80, 81, 82	declt 12566	. . . . . . . . . . . . . . 15 ⊢ ;16 < ;17
84	79, 83	ltneii 11189	. . . . . . . . . . . . . 14 ⊢ ;16 ≠ ;17
85		itvndx 27087	. . . . . . . . . . . . . . 15 ⊢ (Itv‘ndx) = ;16
86		lngndx 27088	. . . . . . . . . . . . . . 15 ⊢ (LineG‘ndx) = ;17
87	85, 86	neeq12i 3007	. . . . . . . . . . . . . 14 ⊢ ((Itv‘ndx) ≠ (LineG‘ndx) ↔ ;16 ≠ ;17)
88	84, 87	mpbir 230	. . . . . . . . . . . . 13 ⊢ (Itv‘ndx) ≠ (LineG‘ndx)
89	75, 88	setsnid 17007	. . . . . . . . . . . 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 2794	. . . . . . . . . . 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 17006	. . . . . . . . . . . 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 688	. . . . . . . . . . 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 2786	. . . . . . . . . 10 ⊢ (𝐻 ∈ 𝑉 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}))
96	95	oveqd 7354	. . . . . . . . 9 ⊢ (𝐻 ∈ 𝑉 → (𝑥𝐼𝑦) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦))
97	96	eleq2d 2822	. . . . . . . 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 7354	. . . . . . . . 9 ⊢ (𝐻 ∈ 𝑉 → (𝑧𝐼𝑦) = (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦))
99	98	eleq2d 2822	. . . . . . . 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 7354	. . . . . . . . 9 ⊢ (𝐻 ∈ 𝑉 → (𝑥𝐼𝑧) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))
101	100	eleq2d 2822	. . . . . . . 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 1434	. . . . . . 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 3411	. . . . . 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 7416	. . . . 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 4824	. . . 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 7353	. . 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 2779	. 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 1085 = wceq 1540 ∈ wcel 2105 ≠ wne 2940 ∃wrex 3070 {crab 3403 Vcvv 3441 ⦋csb 3843 ⟨cop 4579 ‘cfv 6479 (class class class)co 7337 ∈ cmpo 7339 0cc0 10972 1c1 10973 6c6 12133 7c7 12134 cdc 12538 [,]cicc 13183 sSet csts 16961 ndxcnx 16991 Basecbs 17009 ·_𝑠 cvsca 17063 -_gcsg 18675 Itvcitv 27083 LineGclng 27084 toTGcttg 27523

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-op 4580 df-uni 4853 df-iun 4943 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-om 7781 df-1st 7899 df-2nd 7900 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-er 8569 df-en 8805 df-dom 8806 df-sdom 8807 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-7 12142 df-8 12143 df-9 12144 df-n0 12335 df-dec 12539 df-sets 16962 df-slot 16980 df-ndx 16992 df-itv 27085 df-lng 27086 df-ttg 27524

This theorem is referenced by: (None)

W3C validator