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Theorem ttgval 28883
Description: Define a function to augment a subcomplex Hilbert space with betweenness and a line definition. (Contributed by Thierry Arnoux, 25-Mar-2019.) (Proof shortened by AV, 9-Nov-2024.)
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 3501 . . . . 5 (𝐻𝑉𝐻 ∈ V)
4 fveq2 6906 . . . . . . . . 9 (𝑤 = 𝐻 → (Base‘𝑤) = (Base‘𝐻))
5 ttgval.b . . . . . . . . 9 𝐵 = (Base‘𝐻)
64, 5eqtr4di 2795 . . . . . . . 8 (𝑤 = 𝐻 → (Base‘𝑤) = 𝐵)
7 fveq2 6906 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → (-g𝑤) = (-g𝐻))
8 ttgval.m . . . . . . . . . . . . 13 = (-g𝐻)
97, 8eqtr4di 2795 . . . . . . . . . . . 12 (𝑤 = 𝐻 → (-g𝑤) = )
109oveqd 7448 . . . . . . . . . . 11 (𝑤 = 𝐻 → (𝑧(-g𝑤)𝑥) = (𝑧 𝑥))
11 fveq2 6906 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → ( ·𝑠𝑤) = ( ·𝑠𝐻))
12 ttgval.s . . . . . . . . . . . . 13 · = ( ·𝑠𝐻)
1311, 12eqtr4di 2795 . . . . . . . . . . . 12 (𝑤 = 𝐻 → ( ·𝑠𝑤) = · )
14 eqidd 2738 . . . . . . . . . . . 12 (𝑤 = 𝐻𝑘 = 𝑘)
159oveqd 7448 . . . . . . . . . . . 12 (𝑤 = 𝐻 → (𝑦(-g𝑤)𝑥) = (𝑦 𝑥))
1613, 14, 15oveq123d 7452 . . . . . . . . . . 11 (𝑤 = 𝐻 → (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) = (𝑘 · (𝑦 𝑥)))
1710, 16eqeq12d 2753 . . . . . . . . . 10 (𝑤 = 𝐻 → ((𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) ↔ (𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
1817rexbidv 3179 . . . . . . . . 9 (𝑤 = 𝐻 → (∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
196, 18rabeqbidv 3455 . . . . . . . 8 (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 . . . . . . 7 (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))}))
21 oveq1 7438 . . . . . . . 8 (𝑤 = 𝐻 → (𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) = (𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩))
226rabeqdv 3452 . . . . . . . . . 10 (𝑤 = 𝐻 → {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))} = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
236, 6, 22mpoeq123dv 7508 . . . . . . . . 9 (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))}))
2423opeq2d 4880 . . . . . . . 8 (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))})⟩)
2521, 24oveq12d 7449 . . . . . . 7 (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))})⟩))
2620, 25csbeq12dv 3908 . . . . . 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))})⟩))
27 df-ttg 28882 . . . . . 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))})⟩))
28 ovex 7464 . . . . . . 7 ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) ∈ V
2928csbex 5311 . . . . . 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
3026, 27, 29fvmpt 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))})⟩))
313, 30syl 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))})⟩))
325fvexi 6920 . . . . . . 7 𝐵 ∈ V
3332, 32mpoex 8104 . . . . . 6 (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) ∈ V
3433a1i 11 . . . . 5 (𝐻𝑉 → (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) ∈ V)
35 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))}))
36 oveq2 7439 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑐 𝑎) = (𝑐 𝑥))
37 oveq2 7439 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝑏 𝑎) = (𝑏 𝑥))
3837oveq2d 7447 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑘 · (𝑏 𝑎)) = (𝑘 · (𝑏 𝑥)))
3936, 38eqeq12d 2753 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑐 𝑎) = (𝑘 · (𝑏 𝑎)) ↔ (𝑐 𝑥) = (𝑘 · (𝑏 𝑥))))
4039rexbidv 3179 . . . . . . . . 9 (𝑎 = 𝑥 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))))
4140rabbidv 3444 . . . . . . . 8 (𝑎 = 𝑥 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))} = {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))})
42 oveq1 7438 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝑏 𝑥) = (𝑦 𝑥))
4342oveq2d 7447 . . . . . . . . . . . 12 (𝑏 = 𝑦 → (𝑘 · (𝑏 𝑥)) = (𝑘 · (𝑦 𝑥)))
4443eqeq2d 2748 . . . . . . . . . . 11 (𝑏 = 𝑦 → ((𝑐 𝑥) = (𝑘 · (𝑏 𝑥)) ↔ (𝑐 𝑥) = (𝑘 · (𝑦 𝑥))))
4544rexbidv 3179 . . . . . . . . . 10 (𝑏 = 𝑦 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))))
4645rabbidv 3444 . . . . . . . . 9 (𝑏 = 𝑦 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))} = {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))})
47 oveq1 7438 . . . . . . . . . . . 12 (𝑐 = 𝑧 → (𝑐 𝑥) = (𝑧 𝑥))
4847eqeq1d 2739 . . . . . . . . . . 11 (𝑐 = 𝑧 → ((𝑐 𝑥) = (𝑘 · (𝑦 𝑥)) ↔ (𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
4948rexbidv 3179 . . . . . . . . . 10 (𝑐 = 𝑧 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
5049cbvrabv 3447 . . . . . . . . 9 {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}
5146, 50eqtrdi 2793 . . . . . . . 8 (𝑏 = 𝑦 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})
5241, 51cbvmpov 7528 . . . . . . 7 (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})
5335, 52eqtr4di 2795 . . . . . 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))}))
54 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))}))
5554, 52eqtrdi 2793 . . . . . . . . 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))}))
5655opeq2d 4880 . . . . . . . 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))})⟩)
5756oveq2d 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))})⟩))
5855oveqd 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))
5958eleq2d 2827 . . . . . . . . . . 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)))
6055oveqd 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))
6160eleq2d 2827 . . . . . . . . . . 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)))
6255oveqd 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))
6362eleq2d 2827 . . . . . . . . . . 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)))
6459, 61, 633orbi123d 1437 . . . . . . . . . 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))))
6564rabbidv 3444 . . . . . . . . 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))})
6665mpoeq3dv 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))}))
6766opeq2d 4880 . . . . . . 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))})⟩)
6857, 67oveq12d 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))})⟩))
6953, 68syldan 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))})⟩))
7034, 69csbied 3935 . . . 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))})⟩))
712, 31, 703eqtrd 2781 . . 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))})⟩))
7271fveq2d 6910 . . . . . . . . . . . 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))})⟩)))
73 itvid 28447 . . . . . . . . . . . . 13 Itv = Slot (Itv‘ndx)
74 lngndxnitvndx 28451 . . . . . . . . . . . . . 14 (LineG‘ndx) ≠ (Itv‘ndx)
7574necomi 2995 . . . . . . . . . . . . 13 (Itv‘ndx) ≠ (LineG‘ndx)
7673, 75setsnid 17245 . . . . . . . . . . . 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))})⟩))
7772, 76eqtr4di 2795 . . . . . . . . . . 11 (𝐻𝑉 → (Itv‘𝐺) = (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩)))
78 ttgval.i . . . . . . . . . . . 12 𝐼 = (Itv‘𝐺)
7978a1i 11 . . . . . . . . . . 11 (𝐻𝑉𝐼 = (Itv‘𝐺))
8073setsid 17244 . . . . . . . . . . . 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))})⟩)))
8133, 80mpan2 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))})⟩)))
8277, 79, 813eqtr4d 2787 . . . . . . . . . 10 (𝐻𝑉𝐼 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}))
8382oveqd 7448 . . . . . . . . 9 (𝐻𝑉 → (𝑥𝐼𝑦) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
8483eleq2d 2827 . . . . . . . 8 (𝐻𝑉 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
8582oveqd 7448 . . . . . . . . 9 (𝐻𝑉 → (𝑧𝐼𝑦) = (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
8685eleq2d 2827 . . . . . . . 8 (𝐻𝑉 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
8782oveqd 7448 . . . . . . . . 9 (𝐻𝑉 → (𝑥𝐼𝑧) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))
8887eleq2d 2827 . . . . . . . 8 (𝐻𝑉 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧)))
8984, 86, 883orbi123d 1437 . . . . . . 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))))
9089rabbidv 3444 . . . . . 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))})
9190mpoeq3dv 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))}))
9291opeq2d 4880 . . . 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))})⟩)
9392oveq2d 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))})⟩))
9471, 93eqtr4d 2780 . 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))})⟩))
9594, 82jca 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 1086   = wceq 1540  wcel 2108  wrex 3070  {crab 3436  Vcvv 3480  csb 3899  cop 4632  cfv 6561  (class class class)co 7431  cmpo 7433  0cc0 11155  1c1 11156  [,]cicc 13390   sSet csts 17200  ndxcnx 17230  Basecbs 17247   ·𝑠 cvsca 17301  -gcsg 18953  Itvcitv 28441  LineGclng 28442  toTGcttg 28881
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-nn 12267  df-2 12329  df-3 12330  df-4 12331  df-5 12332  df-6 12333  df-7 12334  df-8 12335  df-9 12336  df-n0 12527  df-dec 12734  df-sets 17201  df-slot 17219  df-ndx 17231  df-itv 28443  df-lng 28444  df-ttg 28882
This theorem is referenced by:  ttglem  28885  ttglemOLD  28886  ttgitvval  28896
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