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Theorem ttgvalOLD 27227

Description: Obsolete proof of ttgval 27226 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 3449	. . . . 5 ⊢ (𝐻 ∈ 𝑉 → 𝐻 ∈ V)
4		fveq2 6769	. . . . . . . . . 10 ⊢ (𝑤 = 𝐻 → (Base‘𝑤) = (Base‘𝐻))
5		ttgval.b	. . . . . . . . . 10 ⊢ 𝐵 = (Base‘𝐻)
6	4, 5	eqtr4di 2798	. . . . . . . . 9 ⊢ (𝑤 = 𝐻 → (Base‘𝑤) = 𝐵)
7		fveq2 6769	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝐻 → (-_g‘𝑤) = (-_g‘𝐻))
8		ttgval.m	. . . . . . . . . . . . . 14 ⊢ − = (-_g‘𝐻)
9	7, 8	eqtr4di 2798	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → (-_g‘𝑤) = − )
10	9	oveqd 7286	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐻 → (𝑧(-_g‘𝑤)𝑥) = (𝑧 − 𝑥))
11		fveq2 6769	. . . . . . . . . . . . . 14 ⊢ (𝑤 = 𝐻 → ( ·_𝑠 ‘𝑤) = ( ·_𝑠 ‘𝐻))
12		ttgval.s	. . . . . . . . . . . . . 14 ⊢ · = ( ·_𝑠 ‘𝐻)
13	11, 12	eqtr4di 2798	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → ( ·_𝑠 ‘𝑤) = · )
14		eqidd 2741	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → 𝑘 = 𝑘)
15	9	oveqd 7286	. . . . . . . . . . . . 13 ⊢ (𝑤 = 𝐻 → (𝑦(-_g‘𝑤)𝑥) = (𝑦 − 𝑥))
16	13, 14, 15	oveq123d 7290	. . . . . . . . . . . 12 ⊢ (𝑤 = 𝐻 → (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥)) = (𝑘 · (𝑦 − 𝑥)))
17	10, 16	eqeq12d 2756	. . . . . . . . . . 11 ⊢ (𝑤 = 𝐻 → ((𝑧(-_g‘𝑤)𝑥) = (𝑘( ·_𝑠 ‘𝑤)(𝑦(-_g‘𝑤)𝑥)) ↔ (𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
18	17	rexbidv 3228	. . . . . . . . . 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 3419	. . . . . . . . 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 7342	. . . . . . . 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 3841	. . . . . . 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 7276	. . . . . . . . 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 7342	. . . . . . . . . 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 4817	. . . . . . . . 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 7287	. . . . . . . 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 3844	. . . . . . 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 2780	. . . . . 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 27225	. . . . . 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 7302	. . . . . . 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 5239	. . . . . 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 6870	. . . . 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 6783	. . . . . . 7 ⊢ 𝐵 ∈ V
35	34, 34	mpoex 7907	. . . . . 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 7277	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑐 − 𝑎) = (𝑐 − 𝑥))
39		oveq2 7277	. . . . . . . . . . . 12 ⊢ (𝑎 = 𝑥 → (𝑏 − 𝑎) = (𝑏 − 𝑥))
40	39	oveq2d 7285	. . . . . . . . . . 11 ⊢ (𝑎 = 𝑥 → (𝑘 · (𝑏 − 𝑎)) = (𝑘 · (𝑏 − 𝑥)))
41	38, 40	eqeq12d 2756	. . . . . . . . . 10 ⊢ (𝑎 = 𝑥 → ((𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎)) ↔ (𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥))))
42	41	rexbidv 3228	. . . . . . . . 9 ⊢ (𝑎 = 𝑥 → (∃𝑘 ∈ (0[,]1)(𝑐 − 𝑎) = (𝑘 · (𝑏 − 𝑎)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥))))
43	42	rabbidv 3413	. . . . . . . 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 7276	. . . . . . . . . . . . 13 ⊢ (𝑏 = 𝑦 → (𝑏 − 𝑥) = (𝑦 − 𝑥))
45	44	oveq2d 7285	. . . . . . . . . . . 12 ⊢ (𝑏 = 𝑦 → (𝑘 · (𝑏 − 𝑥)) = (𝑘 · (𝑦 − 𝑥)))
46	45	eqeq2d 2751	. . . . . . . . . . 11 ⊢ (𝑏 = 𝑦 → ((𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥)) ↔ (𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
47	46	rexbidv 3228	. . . . . . . . . 10 ⊢ (𝑏 = 𝑦 → (∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑏 − 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
48	47	rabbidv 3413	. . . . . . . . 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 7276	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑧 → (𝑐 − 𝑥) = (𝑧 − 𝑥))
50	49	eqeq1d 2742	. . . . . . . . . . 11 ⊢ (𝑐 = 𝑧 → ((𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥)) ↔ (𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
51	50	rexbidv 3228	. . . . . . . . . 10 ⊢ (𝑐 = 𝑧 → (∃𝑘 ∈ (0[,]1)(𝑐 − 𝑥) = (𝑘 · (𝑦 − 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))))
52	51	cbvrabv 3425	. . . . . . . . 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 2796	. . . . . . . 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 7362	. . . . . . 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 2798	. . . . . 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 2796	. . . . . . . . 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 4817	. . . . . . . 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 7285	. . . . . . 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 7286	. . . . . . . . . . . 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 2826	. . . . . . . . . . 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 7286	. . . . . . . . . . . 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 2826	. . . . . . . . . . 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 7286	. . . . . . . . . . . 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 2826	. . . . . . . . . . 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 3413	. . . . . . . . 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 7346	. . . . . . . 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 4817	. . . . . . 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 7287	. . . . . 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 3875	. . . 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 2784	. . 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 6773	. . . . . . . . . . . 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 26790	. . . . . . . . . . . . 13 ⊢ Itv = Slot (Itv‘ndx)
76		1nn0 12241	. . . . . . . . . . . . . . . . 17 ⊢ 1 ∈ ℕ₀
77		6nn 12054	. . . . . . . . . . . . . . . . 17 ⊢ 6 ∈ ℕ
78	76, 77	decnncl 12448	. . . . . . . . . . . . . . . 16 ⊢ ;16 ∈ ℕ
79	78	nnrei 11974	. . . . . . . . . . . . . . 15 ⊢ ;16 ∈ ℝ
80		6nn0 12246	. . . . . . . . . . . . . . . 16 ⊢ 6 ∈ ℕ₀
81		7nn 12057	. . . . . . . . . . . . . . . 16 ⊢ 7 ∈ ℕ
82		6lt7 12151	. . . . . . . . . . . . . . . 16 ⊢ 6 < 7
83	76, 80, 81, 82	declt 12456	. . . . . . . . . . . . . . 15 ⊢ ;16 < ;17
84	79, 83	ltneii 11080	. . . . . . . . . . . . . 14 ⊢ ;16 ≠ ;17
85		itvndx 26788	. . . . . . . . . . . . . . 15 ⊢ (Itv‘ndx) = ;16
86		lngndx 26789	. . . . . . . . . . . . . . 15 ⊢ (LineG‘ndx) = ;17
87	85, 86	neeq12i 3012	. . . . . . . . . . . . . 14 ⊢ ((Itv‘ndx) ≠ (LineG‘ndx) ↔ ;16 ≠ ;17)
88	84, 87	mpbir 230	. . . . . . . . . . . . 13 ⊢ (Itv‘ndx) ≠ (LineG‘ndx)
89	75, 88	setsnid 16900	. . . . . . . . . . . 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 2798	. . . . . . . . . . 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 16899	. . . . . . . . . . . 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 2790	. . . . . . . . . 10 ⊢ (𝐻 ∈ 𝑉 → 𝐼 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))}))
96	95	oveqd 7286	. . . . . . . . 9 ⊢ (𝐻 ∈ 𝑉 → (𝑥𝐼𝑦) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦))
97	96	eleq2d 2826	. . . . . . . 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 7286	. . . . . . . . 9 ⊢ (𝐻 ∈ 𝑉 → (𝑧𝐼𝑦) = (𝑧(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑦))
99	98	eleq2d 2826	. . . . . . . 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 7286	. . . . . . . . 9 ⊢ (𝐻 ∈ 𝑉 → (𝑥𝐼𝑧) = (𝑥(𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ {𝑧 ∈ 𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 − 𝑥) = (𝑘 · (𝑦 − 𝑥))})𝑧))
101	100	eleq2d 2826	. . . . . . . 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 3413	. . . . . 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 7346	. . . . 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 4817	. . . 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 7285	. . 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 2783	. 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 1542 ∈ wcel 2110 ≠ wne 2945 ∃wrex 3067 {crab 3070 Vcvv 3431 ⦋csb 3837 ⟨cop 4573 ‘cfv 6431 (class class class)co 7269 ∈ cmpo 7271 0cc0 10864 1c1 10865 6c6 12024 7c7 12025 cdc 12428 [,]cicc 13073 sSet csts 16854 ndxcnx 16884 Basecbs 16902 ·_𝑠 cvsca 16956 -_gcsg 18569 Itvcitv 26784 LineGclng 26785 toTGcttg 27224

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7580 ax-cnex 10920 ax-resscn 10921 ax-1cn 10922 ax-icn 10923 ax-addcl 10924 ax-addrcl 10925 ax-mulcl 10926 ax-mulrcl 10927 ax-mulcom 10928 ax-addass 10929 ax-mulass 10930 ax-distr 10931 ax-i2m1 10932 ax-1ne0 10933 ax-1rid 10934 ax-rnegex 10935 ax-rrecex 10936 ax-cnre 10937 ax-pre-lttri 10938 ax-pre-lttrn 10939 ax-pre-ltadd 10940 ax-pre-mulgt0 10941

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-nel 3052 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-tr 5197 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6200 df-ord 6267 df-on 6268 df-lim 6269 df-suc 6270 df-iota 6389 df-fun 6433 df-fn 6434 df-f 6435 df-f1 6436 df-fo 6437 df-f1o 6438 df-fv 6439 df-riota 7226 df-ov 7272 df-oprab 7273 df-mpo 7274 df-om 7702 df-1st 7818 df-2nd 7819 df-frecs 8082 df-wrecs 8113 df-recs 8187 df-rdg 8226 df-er 8473 df-en 8709 df-dom 8710 df-sdom 8711 df-pnf 11004 df-mnf 11005 df-xr 11006 df-ltxr 11007 df-le 11008 df-sub 11199 df-neg 11200 df-nn 11966 df-2 12028 df-3 12029 df-4 12030 df-5 12031 df-6 12032 df-7 12033 df-8 12034 df-9 12035 df-n0 12226 df-dec 12429 df-sets 16855 df-slot 16873 df-ndx 16885 df-itv 26786 df-lng 26787 df-ttg 27225

This theorem is referenced by: (None)

W3C validator