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Theorem ttgval 26224
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 3414 . . . . 5 (𝐻𝑉𝐻 ∈ V)
4 fveq2 6446 . . . . . . . . . 10 (𝑤 = 𝐻 → (Base‘𝑤) = (Base‘𝐻))
5 ttgval.b . . . . . . . . . 10 𝐵 = (Base‘𝐻)
64, 5syl6eqr 2832 . . . . . . . . 9 (𝑤 = 𝐻 → (Base‘𝑤) = 𝐵)
7 fveq2 6446 . . . . . . . . . . . . . 14 (𝑤 = 𝐻 → (-g𝑤) = (-g𝐻))
8 ttgval.m . . . . . . . . . . . . . 14 = (-g𝐻)
97, 8syl6eqr 2832 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → (-g𝑤) = )
109oveqd 6939 . . . . . . . . . . . 12 (𝑤 = 𝐻 → (𝑧(-g𝑤)𝑥) = (𝑧 𝑥))
11 fveq2 6446 . . . . . . . . . . . . . 14 (𝑤 = 𝐻 → ( ·𝑠𝑤) = ( ·𝑠𝐻))
12 ttgval.s . . . . . . . . . . . . . 14 · = ( ·𝑠𝐻)
1311, 12syl6eqr 2832 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → ( ·𝑠𝑤) = · )
14 eqidd 2779 . . . . . . . . . . . . 13 (𝑤 = 𝐻𝑘 = 𝑘)
159oveqd 6939 . . . . . . . . . . . . 13 (𝑤 = 𝐻 → (𝑦(-g𝑤)𝑥) = (𝑦 𝑥))
1613, 14, 15oveq123d 6943 . . . . . . . . . . . 12 (𝑤 = 𝐻 → (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) = (𝑘 · (𝑦 𝑥)))
1710, 16eqeq12d 2793 . . . . . . . . . . 11 (𝑤 = 𝐻 → ((𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) ↔ (𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
1817rexbidv 3237 . . . . . . . . . 10 (𝑤 = 𝐻 → (∃𝑘 ∈ (0[,]1)(𝑧(-g𝑤)𝑥) = (𝑘( ·𝑠𝑤)(𝑦(-g𝑤)𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
196, 18rabeqbidv 3392 . . . . . . . . 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 6994 . . . . . . . 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 3758 . . . . . . 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 6929 . . . . . . . . 9 (𝑤 = 𝐻 → (𝑤 sSet ⟨(Itv‘ndx), 𝑖⟩) = (𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩))
23 rabeq 3389 . . . . . . . . . . . 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 6994 . . . . . . . . . 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 4643 . . . . . . . . 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 6940 . . . . . . . 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 4217 . . . . . . 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 2814 . . . . . 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 26223 . . . . . 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 6954 . . . . . . 7 ((𝐻 sSet ⟨(Itv‘ndx), 𝑖⟩) sSet ⟨(LineG‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ (𝑧 ∈ (𝑥𝑖𝑦) ∨ 𝑥 ∈ (𝑧𝑖𝑦) ∨ 𝑦 ∈ (𝑥𝑖𝑧))})⟩) ∈ V
3231csbex 5030 . . . . . 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 6542 . . . . 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))})⟩))
355fvexi 6460 . . . . . . 7 𝐵 ∈ V
3635, 35mpt2ex 7527 . . . . . 6 (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) ∈ V
3736a1i 11 . . . . 5 (𝐻𝑉 → (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}) ∈ V)
38 simpr 479 . . . . . . 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))}))
39 oveq2 6930 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑐 𝑎) = (𝑐 𝑥))
40 oveq2 6930 . . . . . . . . . . . 12 (𝑎 = 𝑥 → (𝑏 𝑎) = (𝑏 𝑥))
4140oveq2d 6938 . . . . . . . . . . 11 (𝑎 = 𝑥 → (𝑘 · (𝑏 𝑎)) = (𝑘 · (𝑏 𝑥)))
4239, 41eqeq12d 2793 . . . . . . . . . 10 (𝑎 = 𝑥 → ((𝑐 𝑎) = (𝑘 · (𝑏 𝑎)) ↔ (𝑐 𝑥) = (𝑘 · (𝑏 𝑥))))
4342rexbidv 3237 . . . . . . . . 9 (𝑎 = 𝑥 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))))
4443rabbidv 3386 . . . . . . . 8 (𝑎 = 𝑥 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))} = {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))})
45 oveq1 6929 . . . . . . . . . . . . 13 (𝑏 = 𝑦 → (𝑏 𝑥) = (𝑦 𝑥))
4645oveq2d 6938 . . . . . . . . . . . 12 (𝑏 = 𝑦 → (𝑘 · (𝑏 𝑥)) = (𝑘 · (𝑦 𝑥)))
4746eqeq2d 2788 . . . . . . . . . . 11 (𝑏 = 𝑦 → ((𝑐 𝑥) = (𝑘 · (𝑏 𝑥)) ↔ (𝑐 𝑥) = (𝑘 · (𝑦 𝑥))))
4847rexbidv 3237 . . . . . . . . . 10 (𝑏 = 𝑦 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))))
4948rabbidv 3386 . . . . . . . . 9 (𝑏 = 𝑦 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))} = {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))})
50 oveq1 6929 . . . . . . . . . . . 12 (𝑐 = 𝑧 → (𝑐 𝑥) = (𝑧 𝑥))
5150eqeq1d 2780 . . . . . . . . . . 11 (𝑐 = 𝑧 → ((𝑐 𝑥) = (𝑘 · (𝑦 𝑥)) ↔ (𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
5251rexbidv 3237 . . . . . . . . . 10 (𝑐 = 𝑧 → (∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥)) ↔ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))))
5352cbvrabv 3396 . . . . . . . . 9 {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑦 𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}
5449, 53syl6eq 2830 . . . . . . . 8 (𝑏 = 𝑦 → {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑥) = (𝑘 · (𝑏 𝑥))} = {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})
5544, 54cbvmpt2v 7012 . . . . . . 7 (𝑎𝐵, 𝑏𝐵 ↦ {𝑐𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑐 𝑎) = (𝑘 · (𝑏 𝑎))}) = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})
5638, 55syl6eqr 2832 . . . . . 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))}))
57 simpr 479 . . . . . . . . . 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))}))
5857, 55syl6eq 2830 . . . . . . . . 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))}))
5958opeq2d 4643 . . . . . . . 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))})⟩)
6059oveq2d 6938 . . . . . . 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))})⟩))
6158oveqd 6939 . . . . . . . . . . . 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))
6261eleq2d 2845 . . . . . . . . . . 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)))
6358oveqd 6939 . . . . . . . . . . . 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))
6463eleq2d 2845 . . . . . . . . . . 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)))
6558oveqd 6939 . . . . . . . . . . . 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))
6665eleq2d 2845 . . . . . . . . . . 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)))
6762, 64, 663orbi123d 1508 . . . . . . . . . 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))))
6867rabbidv 3386 . . . . . . . . 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))})
6968mpt2eq3dv 6998 . . . . . . . 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))}))
7069opeq2d 4643 . . . . . . 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))})⟩)
7160, 70oveq12d 6940 . . . . . 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))})⟩))
7256, 71syldan 585 . . . . 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))})⟩))
7337, 72csbied 3778 . . . 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))})⟩))
742, 34, 733eqtrd 2818 . . 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))})⟩))
7574fveq2d 6450 . . . . . . . . . . . 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))})⟩)))
76 itvid 25793 . . . . . . . . . . . . 13 Itv = Slot (Itv‘ndx)
77 1nn0 11660 . . . . . . . . . . . . . . . . 17 1 ∈ ℕ0
78 6nn 11467 . . . . . . . . . . . . . . . . 17 6 ∈ ℕ
7977, 78decnncl 11866 . . . . . . . . . . . . . . . 16 16 ∈ ℕ
8079nnrei 11384 . . . . . . . . . . . . . . 15 16 ∈ ℝ
81 6nn0 11665 . . . . . . . . . . . . . . . 16 6 ∈ ℕ0
82 7nn 11471 . . . . . . . . . . . . . . . 16 7 ∈ ℕ
83 6lt7 11568 . . . . . . . . . . . . . . . 16 6 < 7
8477, 81, 82, 83declt 11874 . . . . . . . . . . . . . . 15 16 < 17
8580, 84ltneii 10489 . . . . . . . . . . . . . 14 16 ≠ 17
86 itvndx 25791 . . . . . . . . . . . . . . 15 (Itv‘ndx) = 16
87 lngndx 25792 . . . . . . . . . . . . . . 15 (LineG‘ndx) = 17
8886, 87neeq12i 3035 . . . . . . . . . . . . . 14 ((Itv‘ndx) ≠ (LineG‘ndx) ↔ 16 ≠ 17)
8985, 88mpbir 223 . . . . . . . . . . . . 13 (Itv‘ndx) ≠ (LineG‘ndx)
9076, 89setsnid 16311 . . . . . . . . . . . 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))})⟩))
9175, 90syl6eqr 2832 . . . . . . . . . . 11 (𝐻𝑉 → (Itv‘𝐺) = (Itv‘(𝐻 sSet ⟨(Itv‘ndx), (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})⟩)))
92 ttgval.i . . . . . . . . . . . 12 𝐼 = (Itv‘𝐺)
9392a1i 11 . . . . . . . . . . 11 (𝐻𝑉𝐼 = (Itv‘𝐺))
9476setsid 16310 . . . . . . . . . . . 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))})⟩)))
9536, 94mpan2 681 . . . . . . . . . . 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))})⟩)))
9691, 93, 953eqtr4d 2824 . . . . . . . . . 10 (𝐻𝑉𝐼 = (𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))}))
9796oveqd 6939 . . . . . . . . 9 (𝐻𝑉 → (𝑥𝐼𝑦) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
9897eleq2d 2845 . . . . . . . 8 (𝐻𝑉 → (𝑧 ∈ (𝑥𝐼𝑦) ↔ 𝑧 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
9996oveqd 6939 . . . . . . . . 9 (𝐻𝑉 → (𝑧𝐼𝑦) = (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦))
10099eleq2d 2845 . . . . . . . 8 (𝐻𝑉 → (𝑥 ∈ (𝑧𝐼𝑦) ↔ 𝑥 ∈ (𝑧(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑦)))
10196oveqd 6939 . . . . . . . . 9 (𝐻𝑉 → (𝑥𝐼𝑧) = (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧))
102101eleq2d 2845 . . . . . . . 8 (𝐻𝑉 → (𝑦 ∈ (𝑥𝐼𝑧) ↔ 𝑦 ∈ (𝑥(𝑥𝐵, 𝑦𝐵 ↦ {𝑧𝐵 ∣ ∃𝑘 ∈ (0[,]1)(𝑧 𝑥) = (𝑘 · (𝑦 𝑥))})𝑧)))
10398, 100, 1023orbi123d 1508 . . . . . . 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))))
104103rabbidv 3386 . . . . . 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))})
105104mpt2eq3dv 6998 . . . . 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))}))
106105opeq2d 4643 . . . 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))})⟩)
107106oveq2d 6938 . . 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))})⟩))
10874, 107eqtr4d 2817 . 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))})⟩))
109108, 96jca 507 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 386  w3o 1070   = wceq 1601  wcel 2107  wne 2969  wrex 3091  {crab 3094  Vcvv 3398  csb 3751  cop 4404  cfv 6135  (class class class)co 6922  cmpt2 6924  0cc0 10272  1c1 10273  6c6 11434  7c7 11435  cdc 11845  [,]cicc 12490  ndxcnx 16252   sSet csts 16253  Basecbs 16255   ·𝑠 cvsca 16342  -gcsg 17811  Itvcitv 25787  LineGclng 25788  toTGcttg 26222
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-fal 1615  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-nn 11375  df-2 11438  df-3 11439  df-4 11440  df-5 11441  df-6 11442  df-7 11443  df-8 11444  df-9 11445  df-n0 11643  df-dec 11846  df-ndx 16258  df-slot 16259  df-sets 16262  df-itv 25789  df-lng 25790  df-ttg 26223
This theorem is referenced by:  ttglem  26225  ttgitvval  26231
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