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Theorem ttgval 25555
 Description: Define a function to augment a complex 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 3185 . . . . 5 (𝐻𝑉𝐻 ∈ V)
4 fveq2 6103 . . . . . . . . . 10 (𝑤 = 𝐻 → (Base‘𝑤) = (Base‘𝐻))
5 ttgval.b . . . . . . . . . 10 𝐵 = (Base‘𝐻)
64, 5syl6eqr 2662 . . . . . . . . 9 (𝑤 = 𝐻 → (Base‘𝑤) = 𝐵)
7 fveq2 6103 . . . . . . . . . . . . . 14 (𝑤 = 𝐻 → (-g𝑤) = (-g𝐻))
8 ttgval.m . . . . . . . . . . . . . 14 = (-g𝐻)
97, 8syl6eqr 2662 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → (-g𝑤) = )
109oveqd 6566 . . . . . . . . . . . 12 (𝑤 = 𝐻 → (𝑧(-g𝑤)𝑥) = (𝑧 𝑥))
11 fveq2 6103 . . . . . . . . . . . . . 14 (𝑤 = 𝐻 → ( ·𝑠𝑤) = ( ·𝑠𝐻))
12 ttgval.s . . . . . . . . . . . . . 14 · = ( ·𝑠𝐻)
1311, 12syl6eqr 2662 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → ( ·𝑠𝑤) = · )
14 eqidd 2611 . . . . . . . . . . . . 13 (𝑤 = 𝐻𝑘 = 𝑘)
159oveqd 6566 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → (𝑦(-g𝑤)𝑥) = (𝑦 𝑥))
1613, 14, 15oveq123d 6570 . . . . . . . . . . . 12 (𝑤 = 𝐻 → (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) = (𝑘 · (𝑦 𝑥)))
1710, 16eqeq12d 2625 . . . . . . . . . . 11 (𝑤 = 𝐻 → ((𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) ↔ (𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
1817rexbidv 3034 . . . . . . . . . 10 (𝑤 = 𝐻 → (∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
196, 18rabeqbidv 3168 . . . . . . . . 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, 19mpt2eq123dv 6615 . . . . . . . 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 3506 . . . . . . 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 6556 . . . . . . . . 9 (𝑤 = 𝐻 → (𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) = (𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩))
23 rabeq 3166 . . . . . . . . . . . 12 ((Base‘𝑤) = 𝐵 → {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))} = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
246, 23syl 17 . . . . . . . . . . 11 (𝑤 = 𝐻 → {𝑧 ∈ (Base‘𝑤) ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))} = {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})
256, 6, 24mpt2eq123dv 6615 . . . . . . . . . 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))}))
2625opeq2d 4347 . . . . . . . . 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))})⟩)
2722, 26oveq12d 6567 . . . . . . . 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))})⟩))
2827csbeq2dv 3944 . . . . . . 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))})⟩))
2921, 28eqtrd 2644 . . . . . 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))})⟩))
30 df-ttg 25554 . . . . . 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))})⟩))
31 ovex 6577 . . . . . . 7 ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) ∈ V
3231csbex 4721 . . . . . 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
3329, 30, 32fvmpt 6191 . . . . 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))})⟩))
343, 33syl 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))})⟩))
35 fvex 6113 . . . . . . . 8 (Base‘𝐻) ∈ V
365, 35eqeltri 2684 . . . . . . 7 𝐵 ∈ V
3736, 36mpt2ex 7136 . . . . . 6 (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) ∈ V
3837a1i 11 . . . . 5 (𝐻𝑉 → (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) ∈ V)
39 simpr 476 . . . . . . 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))}))
40 oveq2 6557 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑐 𝑎) = (𝑐 𝑥))
41 oveq2 6557 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝑏 𝑎) = (𝑏 𝑥))
4241oveq2d 6565 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑘 · (𝑏 𝑎)) = (𝑘 · (𝑏 𝑥)))
4340, 42eqeq12d 2625 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑐 𝑎) = (𝑘 · (𝑏 𝑎)) ↔ (𝑐 𝑥) = (𝑘 · (𝑏 𝑥))))
4443rexbidv 3034 . . . . . . . . 9 (𝑎 = 𝑥 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))))
4544rabbidv 3164 . . . . . . . 8 (𝑎 = 𝑥 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))} = {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))})
46 oveq1 6556 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝑏 𝑥) = (𝑦 𝑥))
4746oveq2d 6565 . . . . . . . . . . . 12 (𝑏 = 𝑦 → (𝑘 · (𝑏 𝑥)) = (𝑘 · (𝑦 𝑥)))
4847eqeq2d 2620 . . . . . . . . . . 11 (𝑏 = 𝑦 → ((𝑐 𝑥) = (𝑘 · (𝑏 𝑥)) ↔ (𝑐 𝑥) = (𝑘 · (𝑦 𝑥))))
4948rexbidv 3034 . . . . . . . . . 10 (𝑏 = 𝑦 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))))
5049rabbidv 3164 . . . . . . . . 9 (𝑏 = 𝑦 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))} = {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))})
51 oveq1 6556 . . . . . . . . . . . 12 (𝑐 = 𝑧 → (𝑐 𝑥) = (𝑧 𝑥))
5251eqeq1d 2612 . . . . . . . . . . 11 (𝑐 = 𝑧 → ((𝑐 𝑥) = (𝑘 · (𝑦 𝑥)) ↔ (𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
5352rexbidv 3034 . . . . . . . . . 10 (𝑐 = 𝑧 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
5453cbvrabv 3172 . . . . . . . . 9 {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}
5550, 54syl6eq 2660 . . . . . . . 8 (𝑏 = 𝑦 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})
5645, 55cbvmpt2v 6633 . . . . . . 7 (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})
5739, 56syl6eqr 2662 . . . . . 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))}))
58 simpr 476 . . . . . . . . . 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))}))
5958, 56syl6eq 2660 . . . . . . . . 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))}))
6059opeq2d 4347 . . . . . . . 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))})⟩)
6160oveq2d 6565 . . . . . . 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))})⟩))
6259oveqd 6566 . . . . . . . . . . . 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))
6362eleq2d 2673 . . . . . . . . . . 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)))
6459oveqd 6566 . . . . . . . . . . . 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))
6564eleq2d 2673 . . . . . . . . . . 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)))
6659oveqd 6566 . . . . . . . . . . . 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))
6766eleq2d 2673 . . . . . . . . . . 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)))
6863, 65, 673orbi123d 1390 . . . . . . . . . 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))))
6968rabbidv 3164 . . . . . . . . 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))})
7069mpt2eq3dv 6619 . . . . . . . 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))}))
7170opeq2d 4347 . . . . . . 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))})⟩)
7261, 71oveq12d 6567 . . . . . 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))})⟩))
7357, 72syldan 486 . . . . 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))})⟩))
7438, 73csbied 3526 . . . 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))})⟩))
752, 34, 743eqtrd 2648 . . 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))})⟩))
7675fveq2d 6107 . . . . . . . . . . . 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))})⟩)))
77 itvid 25141 . . . . . . . . . . . . 13 Itv = Slot (Itv‘ndx)
78 1nn0 11185 . . . . . . . . . . . . . . . . 17 1 ∈ ℕ0
79 6nn 11066 . . . . . . . . . . . . . . . . 17 6 ∈ ℕ
8078, 79decnncl 11394 . . . . . . . . . . . . . . . 16 16 ∈ ℕ
8180nnrei 10906 . . . . . . . . . . . . . . 15 16 ∈ ℝ
82 6nn0 11190 . . . . . . . . . . . . . . . 16 6 ∈ ℕ0
83 7nn 11067 . . . . . . . . . . . . . . . 16 7 ∈ ℕ
84 6lt7 11086 . . . . . . . . . . . . . . . 16 6 < 7
8578, 82, 83, 84declt 11406 . . . . . . . . . . . . . . 15 16 < 17
8681, 85ltneii 10029 . . . . . . . . . . . . . 14 16 ≠ 17
87 itvndx 25139 . . . . . . . . . . . . . . 15 (Itv‘ndx) = 16
88 lngndx 25140 . . . . . . . . . . . . . . 15 (LineG‘ndx) = 17
8987, 88neeq12i 2848 . . . . . . . . . . . . . 14 ((Itv‘ndx) ≠ (LineG‘ndx) ↔ 16 ≠ 17)
9086, 89mpbir 220 . . . . . . . . . . . . 13 (Itv‘ndx) ≠ (LineG‘ndx)
9177, 90setsnid 15743 . . . . . . . . . . . 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))})⟩))
9276, 91syl6eqr 2662 . . . . . . . . . . 11 (𝐻𝑉 → (Itv‘𝐺) = (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩)))
93 ttgval.i . . . . . . . . . . . 12 𝐼 = (Itv‘𝐺)
9493a1i 11 . . . . . . . . . . 11 (𝐻𝑉𝐼 = (Itv‘𝐺))
9577setsid 15742 . . . . . . . . . . . 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))})⟩)))
9637, 95mpan2 703 . . . . . . . . . . 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))})⟩)))
9792, 94, 963eqtr4d 2654 . . . . . . . . . 10 (𝐻𝑉𝐼 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}))
9897oveqd 6566 . . . . . . . . 9 (𝐻𝑉 → (𝑥𝐼𝑦) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
9998eleq2d 2673 . . . . . . . 8 (𝐻𝑉 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
10097oveqd 6566 . . . . . . . . 9 (𝐻𝑉 → (𝑧𝐼𝑦) = (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
101100eleq2d 2673 . . . . . . . 8 (𝐻𝑉 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
10297oveqd 6566 . . . . . . . . 9 (𝐻𝑉 → (𝑥𝐼𝑧) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))
103102eleq2d 2673 . . . . . . . 8 (𝐻𝑉 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧)))
10499, 101, 1033orbi123d 1390 . . . . . . 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))))
105104rabbidv 3164 . . . . . 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))})
106105mpt2eq3dv 6619 . . . . 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))}))
107106opeq2d 4347 . . . 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))})⟩)
108107oveq2d 6565 . . 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))})⟩))
10975, 108eqtr4d 2647 . 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))})⟩))
110109, 97jca 553 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 383   ∨ w3o 1030   = wceq 1475   ∈ wcel 1977   ≠ wne 2780  ∃wrex 2897  {crab 2900  Vcvv 3173  ⦋csb 3499  ⟨cop 4131  ‘cfv 5804  (class class class)co 6549   ↦ cmpt2 6551  0cc0 9815  1c1 9816  6c6 10951  7c7 10952  ;cdc 11369  [,]cicc 12049  ndxcnx 15692   sSet csts 15693  Basecbs 15695   ·𝑠 cvsca 15772  -gcsg 17247  Itvcitv 25135  LineGclng 25136  toTGcttg 25553 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891  ax-pre-mulgt0 9892 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-om 6958  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-nn 10898  df-2 10956  df-3 10957  df-4 10958  df-5 10959  df-6 10960  df-7 10961  df-8 10962  df-9 10963  df-n0 11170  df-dec 11370  df-ndx 15698  df-slot 15699  df-sets 15701  df-itv 25137  df-lng 25138  df-ttg 25554 This theorem is referenced by:  ttglem  25556  ttgitvval  25562
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