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Theorem ttgvalOLD 28899
Description: Obsolete version of ttgval 28898 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 = (-g𝐻)
ttgval.s · = ( ·𝑠𝐻)
ttgval.i 𝐼 = (Itv‘𝐺)
Assertion
Ref Expression
ttgvalOLD (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 ttgvalOLD
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 3499 . . . . 5 (𝐻𝑉𝐻 ∈ V)
4 fveq2 6907 . . . . . . . . . 10 (𝑤 = 𝐻 → (Base‘𝑤) = (Base‘𝐻))
5 ttgval.b . . . . . . . . . 10 𝐵 = (Base‘𝐻)
64, 5eqtr4di 2793 . . . . . . . . 9 (𝑤 = 𝐻 → (Base‘𝑤) = 𝐵)
7 fveq2 6907 . . . . . . . . . . . . . 14 (𝑤 = 𝐻 → (-g𝑤) = (-g𝐻))
8 ttgval.m . . . . . . . . . . . . . 14 = (-g𝐻)
97, 8eqtr4di 2793 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → (-g𝑤) = )
109oveqd 7448 . . . . . . . . . . . 12 (𝑤 = 𝐻 → (𝑧(-g𝑤)𝑥) = (𝑧 𝑥))
11 fveq2 6907 . . . . . . . . . . . . . 14 (𝑤 = 𝐻 → ( ·𝑠𝑤) = ( ·𝑠𝐻))
12 ttgval.s . . . . . . . . . . . . . 14 · = ( ·𝑠𝐻)
1311, 12eqtr4di 2793 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → ( ·𝑠𝑤) = · )
14 eqidd 2736 . . . . . . . . . . . . 13 (𝑤 = 𝐻𝑘 = 𝑘)
159oveqd 7448 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → (𝑦(-g𝑤)𝑥) = (𝑦 𝑥))
1613, 14, 15oveq123d 7452 . . . . . . . . . . . 12 (𝑤 = 𝐻 → (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) = (𝑘 · (𝑦 𝑥)))
1710, 16eqeq12d 2751 . . . . . . . . . . 11 (𝑤 = 𝐻 → ((𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) ↔ (𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
1817rexbidv 3177 . . . . . . . . . 10 (𝑤 = 𝐻 → (∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
196, 18rabeqbidv 3452 . . . . . . . . 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 7508 . . . . . . . 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 3912 . . . . . . 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 7438 . . . . . . . . 9 (𝑤 = 𝐻 → (𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) = (𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩))
236rabeqdv 3449 . . . . . . . . . . 11 (𝑤 = 𝐻 → {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))} = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
246, 6, 23mpoeq123dv 7508 . . . . . . . . . 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 4885 . . . . . . . . 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 7449 . . . . . . . 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 3915 . . . . . . 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 2775 . . . . . 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 28897 . . . . . 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 7464 . . . . . . 7 ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) ∈ V
3130csbex 5317 . . . . . 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 7016 . . . . 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 6921 . . . . . . 7 𝐵 ∈ V
3534, 34mpoex 8103 . . . . . 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 484 . . . . . . 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 7439 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑐 𝑎) = (𝑐 𝑥))
39 oveq2 7439 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝑏 𝑎) = (𝑏 𝑥))
4039oveq2d 7447 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑘 · (𝑏 𝑎)) = (𝑘 · (𝑏 𝑥)))
4138, 40eqeq12d 2751 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑐 𝑎) = (𝑘 · (𝑏 𝑎)) ↔ (𝑐 𝑥) = (𝑘 · (𝑏 𝑥))))
4241rexbidv 3177 . . . . . . . . 9 (𝑎 = 𝑥 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))))
4342rabbidv 3441 . . . . . . . 8 (𝑎 = 𝑥 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))} = {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))})
44 oveq1 7438 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝑏 𝑥) = (𝑦 𝑥))
4544oveq2d 7447 . . . . . . . . . . . 12 (𝑏 = 𝑦 → (𝑘 · (𝑏 𝑥)) = (𝑘 · (𝑦 𝑥)))
4645eqeq2d 2746 . . . . . . . . . . 11 (𝑏 = 𝑦 → ((𝑐 𝑥) = (𝑘 · (𝑏 𝑥)) ↔ (𝑐 𝑥) = (𝑘 · (𝑦 𝑥))))
4746rexbidv 3177 . . . . . . . . . 10 (𝑏 = 𝑦 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))))
4847rabbidv 3441 . . . . . . . . 9 (𝑏 = 𝑦 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))} = {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))})
49 oveq1 7438 . . . . . . . . . . . 12 (𝑐 = 𝑧 → (𝑐 𝑥) = (𝑧 𝑥))
5049eqeq1d 2737 . . . . . . . . . . 11 (𝑐 = 𝑧 → ((𝑐 𝑥) = (𝑘 · (𝑦 𝑥)) ↔ (𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
5150rexbidv 3177 . . . . . . . . . 10 (𝑐 = 𝑧 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
5251cbvrabv 3444 . . . . . . . . 9 {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}
5348, 52eqtrdi 2791 . . . . . . . 8 (𝑏 = 𝑦 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})
5443, 53cbvmpov 7528 . . . . . . 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 2793 . . . . . 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 484 . . . . . . . . . 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, 54eqtrdi 2791 . . . . . . . . 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 4885 . . . . . . . 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 7447 . . . . . . 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 7448 . . . . . . . . . . . 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 2825 . . . . . . . . . . 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 7448 . . . . . . . . . . . 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 2825 . . . . . . . . . . 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 7448 . . . . . . . . . . . 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 2825 . . . . . . . . . . 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 1434 . . . . . . . . . 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 3441 . . . . . . . . 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 7512 . . . . . . . 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 4885 . . . . . . 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 7449 . . . . . 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 591 . . . . 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 3946 . . . 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 2779 . . 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 6911 . . . . . . . . . . . 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 28462 . . . . . . . . . . . . 13 Itv = Slot (Itv‘ndx)
76 1nn0 12540 . . . . . . . . . . . . . . . . 17 1 ∈ ℕ0
77 6nn 12353 . . . . . . . . . . . . . . . . 17 6 ∈ ℕ
7876, 77decnncl 12751 . . . . . . . . . . . . . . . 16 16 ∈ ℕ
7978nnrei 12273 . . . . . . . . . . . . . . 15 16 ∈ ℝ
80 6nn0 12545 . . . . . . . . . . . . . . . 16 6 ∈ ℕ0
81 7nn 12356 . . . . . . . . . . . . . . . 16 7 ∈ ℕ
82 6lt7 12450 . . . . . . . . . . . . . . . 16 6 < 7
8376, 80, 81, 82declt 12759 . . . . . . . . . . . . . . 15 16 < 17
8479, 83ltneii 11372 . . . . . . . . . . . . . 14 16 ≠ 17
85 itvndx 28460 . . . . . . . . . . . . . . 15 (Itv‘ndx) = 16
86 lngndx 28461 . . . . . . . . . . . . . . 15 (LineG‘ndx) = 17
8785, 86neeq12i 3005 . . . . . . . . . . . . . 14 ((Itv‘ndx) ≠ (LineG‘ndx) ↔ 16 ≠ 17)
8884, 87mpbir 231 . . . . . . . . . . . . 13 (Itv‘ndx) ≠ (LineG‘ndx)
8975, 88setsnid 17243 . . . . . . . . . . . 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 2793 . . . . . . . . . . 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 17242 . . . . . . . . . . . 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 691 . . . . . . . . . . 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 2785 . . . . . . . . . 10 (𝐻𝑉𝐼 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}))
9695oveqd 7448 . . . . . . . . 9 (𝐻𝑉 → (𝑥𝐼𝑦) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
9796eleq2d 2825 . . . . . . . 8 (𝐻𝑉 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
9895oveqd 7448 . . . . . . . . 9 (𝐻𝑉 → (𝑧𝐼𝑦) = (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
9998eleq2d 2825 . . . . . . . 8 (𝐻𝑉 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
10095oveqd 7448 . . . . . . . . 9 (𝐻𝑉 → (𝑥𝐼𝑧) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))
101100eleq2d 2825 . . . . . . . 8 (𝐻𝑉 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧)))
10297, 99, 1013orbi123d 1434 . . . . . . 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 3441 . . . . . 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 7512 . . . . 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 4885 . . . 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 7447 . . 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 2778 . 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 511 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 395  w3o 1085   = wceq 1537  wcel 2106  wne 2938  wrex 3068  {crab 3433  Vcvv 3478  csb 3908  cop 4637  cfv 6563  (class class class)co 7431  cmpo 7433  0cc0 11153  1c1 11154  6c6 12323  7c7 12324  cdc 12731  [,]cicc 13387   sSet csts 17197  ndxcnx 17227  Basecbs 17245   ·𝑠 cvsca 17302  -gcsg 18966  Itvcitv 28456  LineGclng 28457  toTGcttg 28896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-dec 12732  df-sets 17198  df-slot 17216  df-ndx 17228  df-itv 28458  df-lng 28459  df-ttg 28897
This theorem is referenced by: (None)
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