MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ttgval Structured version   Visualization version   GIF version

Theorem ttgval 26667
Description: Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.)
Hypotheses
Ref Expression
ttgval.n 𝐺 = (toTG‘𝐻)
ttgval.b 𝐵 = (Base‘𝐻)
ttgval.m = (-g𝐻)
ttgval.s · = ( ·𝑠𝐻)
ttgval.i 𝐼 = (Itv‘𝐺)
Assertion
Ref Expression
ttgval (𝐻𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃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 ttgval
Dummy variables 𝑎 𝑏 𝑐 𝑖 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ttgval.n . . . . 5 𝐺 = (toTG‘𝐻)
21a1i 11 . . . 4 (𝐻𝑉𝐺 = (toTG‘𝐻))
3 elex 3487 . . . . 5 (𝐻𝑉𝐻 ∈ V)
4 fveq2 6652 . . . . . . . . . 10 (𝑤 = 𝐻 → (Base‘𝑤) = (Base‘𝐻))
5 ttgval.b . . . . . . . . . 10 𝐵 = (Base‘𝐻)
64, 5eqtr4di 2875 . . . . . . . . 9 (𝑤 = 𝐻 → (Base‘𝑤) = 𝐵)
7 fveq2 6652 . . . . . . . . . . . . . 14 (𝑤 = 𝐻 → (-g𝑤) = (-g𝐻))
8 ttgval.m . . . . . . . . . . . . . 14 = (-g𝐻)
97, 8eqtr4di 2875 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → (-g𝑤) = )
109oveqd 7157 . . . . . . . . . . . 12 (𝑤 = 𝐻 → (𝑧(-g𝑤)𝑥) = (𝑧 𝑥))
11 fveq2 6652 . . . . . . . . . . . . . 14 (𝑤 = 𝐻 → ( ·𝑠𝑤) = ( ·𝑠𝐻))
12 ttgval.s . . . . . . . . . . . . . 14 · = ( ·𝑠𝐻)
1311, 12eqtr4di 2875 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → ( ·𝑠𝑤) = · )
14 eqidd 2823 . . . . . . . . . . . . 13 (𝑤 = 𝐻𝑘 = 𝑘)
159oveqd 7157 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → (𝑦(-g𝑤)𝑥) = (𝑦 𝑥))
1613, 14, 15oveq123d 7161 . . . . . . . . . . . 12 (𝑤 = 𝐻 → (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) = (𝑘 · (𝑦 𝑥)))
1710, 16eqeq12d 2838 . . . . . . . . . . 11 (𝑤 = 𝐻 → ((𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) ↔ (𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
1817rexbidv 3283 . . . . . . . . . 10 (𝑤 = 𝐻 → (∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
196, 18rabeqbidv 3461 . . . . . . . . 9 (𝑤 = 𝐻 → {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})
206, 6, 19mpoeq123dv 7213 . . . . . . . 8 (𝑤 = 𝐻 → (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}))
2120csbeq1d 3859 . . . . . . 7 (𝑤 = 𝐻(𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)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))})⟩) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃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 7147 . . . . . . . . 9 (𝑤 = 𝐻 → (𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) = (𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩))
236rabeqdv 3460 . . . . . . . . . . 11 (𝑤 = 𝐻 → {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))} = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
246, 6, 23mpoeq123dv 7213 . . . . . . . . . 10 (𝑤 = 𝐻 → (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}))
2524opeq2d 4785 . . . . . . . . 9 (𝑤 = 𝐻 → ⟨(LineG‘ndx), (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩ = ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩)
2622, 25oveq12d 7158 . . . . . . . 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), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
2726csbeq2dv 3862 . . . . . . 7 (𝑤 = 𝐻(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃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))})⟩) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) / 𝑖((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
2821, 27eqtrd 2857 . . . . . 6 (𝑤 = 𝐻(𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)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))})⟩) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) / 𝑖((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
29 df-ttg 26666 . . . . . 6 toTG = (𝑤 ∈ V ↦ (𝑥 ∈ (Base‘𝑤), 𝑦 ∈ (Base‘𝑤) ↦ {𝑧 ∈ (Base‘𝑤) ∣ ∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)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 7173 . . . . . . 7 ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) ∈ V
3130csbex 5191 . . . . . 6 (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) / 𝑖((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) ∈ V
3228, 29, 31fvmpt 6750 . . . . 5 (𝐻 ∈ V → (toTG‘𝐻) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) / 𝑖((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
333, 32syl 17 . . . 4 (𝐻𝑉 → (toTG‘𝐻) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) / 𝑖((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩))
345fvexi 6666 . . . . . . 7 𝐵 ∈ V
3534, 34mpoex 7764 . . . . . 6 (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) ∈ V
3635a1i 11 . . . . 5 (𝐻𝑉 → (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) ∈ V)
37 simpr 488 . . . . . . 7 ((𝐻𝑉𝑖 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})) → 𝑖 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}))
38 oveq2 7148 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑐 𝑎) = (𝑐 𝑥))
39 oveq2 7148 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝑏 𝑎) = (𝑏 𝑥))
4039oveq2d 7156 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑘 · (𝑏 𝑎)) = (𝑘 · (𝑏 𝑥)))
4138, 40eqeq12d 2838 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑐 𝑎) = (𝑘 · (𝑏 𝑎)) ↔ (𝑐 𝑥) = (𝑘 · (𝑏 𝑥))))
4241rexbidv 3283 . . . . . . . . 9 (𝑎 = 𝑥 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))))
4342rabbidv 3455 . . . . . . . 8 (𝑎 = 𝑥 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))} = {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))})
44 oveq1 7147 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝑏 𝑥) = (𝑦 𝑥))
4544oveq2d 7156 . . . . . . . . . . . 12 (𝑏 = 𝑦 → (𝑘 · (𝑏 𝑥)) = (𝑘 · (𝑦 𝑥)))
4645eqeq2d 2833 . . . . . . . . . . 11 (𝑏 = 𝑦 → ((𝑐 𝑥) = (𝑘 · (𝑏 𝑥)) ↔ (𝑐 𝑥) = (𝑘 · (𝑦 𝑥))))
4746rexbidv 3283 . . . . . . . . . 10 (𝑏 = 𝑦 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))))
4847rabbidv 3455 . . . . . . . . 9 (𝑏 = 𝑦 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))} = {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))})
49 oveq1 7147 . . . . . . . . . . . 12 (𝑐 = 𝑧 → (𝑐 𝑥) = (𝑧 𝑥))
5049eqeq1d 2824 . . . . . . . . . . 11 (𝑐 = 𝑧 → ((𝑐 𝑥) = (𝑘 · (𝑦 𝑥)) ↔ (𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
5150rexbidv 3283 . . . . . . . . . 10 (𝑐 = 𝑧 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
5251cbvrabv 3467 . . . . . . . . 9 {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}
5348, 52syl6eq 2873 . . . . . . . 8 (𝑏 = 𝑦 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})
5443, 53cbvmpov 7233 . . . . . . 7 (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})
5537, 54eqtr4di 2875 . . . . . 6 ((𝐻𝑉𝑖 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})) → 𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))}))
56 simpr 488 . . . . . . . . . 10 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → 𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))}))
5756, 54syl6eq 2873 . . . . . . . . 9 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → 𝑖 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}))
5857opeq2d 4785 . . . . . . . 8 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → ⟨(Itv‘ndx), 𝑖⟩ = ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩)
5958oveq2d 7156 . . . . . . 7 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → (𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) = (𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩))
6057oveqd 7157 . . . . . . . . . . . 12 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → (𝑥𝑖𝑦) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
6160eleq2d 2899 . . . . . . . . . . 11 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → (𝑧 ∈ (𝑥𝑖𝑦) ↔ 𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
6257oveqd 7157 . . . . . . . . . . . 12 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → (𝑧𝑖𝑦) = (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
6362eleq2d 2899 . . . . . . . . . . 11 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → (𝑥 ∈ (𝑧𝑖𝑦) ↔ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
6457oveqd 7157 . . . . . . . . . . . 12 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → (𝑥𝑖𝑧) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))
6564eleq2d 2899 . . . . . . . . . . 11 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → (𝑦 ∈ (𝑥𝑖𝑧) ↔ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧)))
6661, 63, 653orbi123d 1432 . . . . . . . . . 10 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → ((𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧)) ↔ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))))
6766rabbidv 3455 . . . . . . . . 9 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))} = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})
6867mpoeq3dv 7217 . . . . . . . 8 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))}))
6968opeq2d 4785 . . . . . . 7 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩ = ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})⟩)
7059, 69oveq12d 7158 . . . . . 6 ((𝐻𝑉𝑖 = (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))})) → ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})⟩))
7155, 70syldan 594 . . . . 5 ((𝐻𝑉𝑖 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})) → ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})⟩))
7236, 71csbied 3891 . . . 4 (𝐻𝑉(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) / 𝑖((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})⟩))
732, 33, 723eqtrd 2861 . . 3 (𝐻𝑉𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})⟩))
7473fveq2d 6656 . . . . . . . . . . . 12 (𝐻𝑉 → (Itv‘𝐺) = (Itv‘((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})⟩)))
75 itvid 26234 . . . . . . . . . . . . 13 Itv = Slot (Itv‘ndx)
76 1nn0 11901 . . . . . . . . . . . . . . . . 17 1 ∈ ℕ0
77 6nn 11714 . . . . . . . . . . . . . . . . 17 6 ∈ ℕ
7876, 77decnncl 12106 . . . . . . . . . . . . . . . 16 16 ∈ ℕ
7978nnrei 11634 . . . . . . . . . . . . . . 15 16 ∈ ℝ
80 6nn0 11906 . . . . . . . . . . . . . . . 16 6 ∈ ℕ0
81 7nn 11717 . . . . . . . . . . . . . . . 16 7 ∈ ℕ
82 6lt7 11811 . . . . . . . . . . . . . . . 16 6 < 7
8376, 80, 81, 82declt 12114 . . . . . . . . . . . . . . 15 16 < 17
8479, 83ltneii 10742 . . . . . . . . . . . . . 14 16 ≠ 17
85 itvndx 26232 . . . . . . . . . . . . . . 15 (Itv‘ndx) = 16
86 lngndx 26233 . . . . . . . . . . . . . . 15 (LineG‘ndx) = 17
8785, 86neeq12i 3077 . . . . . . . . . . . . . 14 ((Itv‘ndx) ≠ (LineG‘ndx) ↔ 16 ≠ 17)
8884, 87mpbir 234 . . . . . . . . . . . . 13 (Itv‘ndx) ≠ (LineG‘ndx)
8975, 88setsnid 16530 . . . . . . . . . . . 12 (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩)) = (Itv‘((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})⟩))
9074, 89eqtr4di 2875 . . . . . . . . . . 11 (𝐻𝑉 → (Itv‘𝐺) = (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩)))
91 ttgval.i . . . . . . . . . . . 12 𝐼 = (Itv‘𝐺)
9291a1i 11 . . . . . . . . . . 11 (𝐻𝑉𝐼 = (Itv‘𝐺))
9375setsid 16529 . . . . . . . . . . . 12 ((𝐻𝑉 ∧ (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) ∈ V) → (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) = (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩)))
9435, 93mpan2 690 . . . . . . . . . . 11 (𝐻𝑉 → (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) = (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩)))
9590, 92, 943eqtr4d 2867 . . . . . . . . . 10 (𝐻𝑉𝐼 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}))
9695oveqd 7157 . . . . . . . . 9 (𝐻𝑉 → (𝑥𝐼𝑦) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
9796eleq2d 2899 . . . . . . . 8 (𝐻𝑉 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
9895oveqd 7157 . . . . . . . . 9 (𝐻𝑉 → (𝑧𝐼𝑦) = (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
9998eleq2d 2899 . . . . . . . 8 (𝐻𝑉 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
10095oveqd 7157 . . . . . . . . 9 (𝐻𝑉 → (𝑥𝐼𝑧) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))
101100eleq2d 2899 . . . . . . . 8 (𝐻𝑉 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧)))
10297, 99, 1013orbi123d 1432 . . . . . . 7 (𝐻𝑉 → ((𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧)) ↔ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))))
103102rabbidv 3455 . . . . . 6 (𝐻𝑉 → {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))} = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})
104103mpoeq3dv 7217 . . . . 5 (𝐻𝑉 → (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))}))
105104opeq2d 4785 . . . 4 (𝐻𝑉 → ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩ = ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})⟩)
106105oveq2d 7156 . . 3 (𝐻𝑉 → ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦) ∨ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))})⟩))
10773, 106eqtr4d 2860 . 2 (𝐻𝑉𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩))
108107, 95jca 515 1 (𝐻𝑉 → (𝐺 = ((𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝐼𝑦) ∨ 𝑥 ∈ (𝑧𝐼𝑦) ∨ 𝑦 ∈ (𝑥𝐼𝑧))})⟩) ∧ 𝐼 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  w3o 1083   = wceq 1538  wcel 2114  wne 3011  wrex 3131  {crab 3134  Vcvv 3469  csb 3855  cop 4545  cfv 6334  (class class class)co 7140  cmpo 7142  0cc0 10526  1c1 10527  6c6 11684  7c7 11685  cdc 12086  [,]cicc 12729  ndxcnx 16471   sSet csts 16472  Basecbs 16474   ·𝑠 cvsca 16560  -gcsg 18096  Itvcitv 26228  LineGclng 26229  toTGcttg 26665
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 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5243  ax-pr 5307  ax-un 7446  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-nel 3116  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-pss 3927  df-nul 4266  df-if 4440  df-pw 4513  df-sn 4540  df-pr 4542  df-tp 4544  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5437  df-eprel 5442  df-po 5451  df-so 5452  df-fr 5491  df-we 5493  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-pred 6126  df-ord 6172  df-on 6173  df-lim 6174  df-suc 6175  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-riota 7098  df-ov 7143  df-oprab 7144  df-mpo 7145  df-om 7566  df-1st 7675  df-2nd 7676  df-wrecs 7934  df-recs 7995  df-rdg 8033  df-er 8276  df-en 8497  df-dom 8498  df-sdom 8499  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-nn 11626  df-2 11688  df-3 11689  df-4 11690  df-5 11691  df-6 11692  df-7 11693  df-8 11694  df-9 11695  df-n0 11886  df-dec 12087  df-ndx 16477  df-slot 16478  df-sets 16481  df-itv 26230  df-lng 26231  df-ttg 26666
This theorem is referenced by:  ttglem  26668  ttgitvval  26674
  Copyright terms: Public domain W3C validator