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Theorem ishpg 25551
Description: Value of the half-plane relation for a given line 𝐷. (Contributed by Thierry Arnoux, 4-Mar-2020.)
Hypotheses
Ref Expression
ishpg.p 𝑃 = (Base‘𝐺)
ishpg.i 𝐼 = (Itv‘𝐺)
ishpg.l 𝐿 = (LineG‘𝐺)
ishpg.o 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
ishpg.g (𝜑𝐺 ∈ TarskiG)
ishpg.d (𝜑𝐷 ∈ ran 𝐿)
Assertion
Ref Expression
ishpg (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})
Distinct variable groups:   𝐷,𝑎,𝑏,𝑐,𝑡   𝐺,𝑎,𝑏   𝐼,𝑎,𝑏,𝑐,𝑡   𝑃,𝑎,𝑏,𝑐,𝑡
Allowed substitution hints:   𝜑(𝑡,𝑎,𝑏,𝑐)   𝐺(𝑡,𝑐)   𝐿(𝑡,𝑎,𝑏,𝑐)   𝑂(𝑡,𝑎,𝑏,𝑐)

Proof of Theorem ishpg
Dummy variables 𝑑 𝑒 𝑓 𝑔 𝑖 𝑝 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ishpg.g . . . 4 (𝜑𝐺 ∈ TarskiG)
2 elex 3198 . . . 4 (𝐺 ∈ TarskiG → 𝐺 ∈ V)
3 fveq2 6148 . . . . . . . 8 (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4 ishpg.l . . . . . . . 8 𝐿 = (LineG‘𝐺)
53, 4syl6eqr 2673 . . . . . . 7 (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
65rneqd 5313 . . . . . 6 (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7 ishpg.p . . . . . . . 8 𝑃 = (Base‘𝐺)
8 ishpg.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
9 simpl 473 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑝 = 𝑃)
109eqcomd 2627 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑃 = 𝑝)
1110difeq1d 3705 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑃𝑑) = (𝑝𝑑))
1211eleq2d 2684 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑎 ∈ (𝑃𝑑) ↔ 𝑎 ∈ (𝑝𝑑)))
1311eleq2d 2684 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑐 ∈ (𝑃𝑑) ↔ 𝑐 ∈ (𝑝𝑑)))
1412, 13anbi12d 746 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ↔ (𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑))))
15 simpr 477 . . . . . . . . . . . . . . 15 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑖 = 𝐼)
1615eqcomd 2627 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝐼 = 𝑖)
1716oveqd 6621 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑎𝐼𝑐) = (𝑎𝑖𝑐))
1817eleq2d 2684 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑡 ∈ (𝑎𝐼𝑐) ↔ 𝑡 ∈ (𝑎𝑖𝑐)))
1918rexbidv 3045 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐) ↔ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)))
2014, 19anbi12d 746 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ↔ ((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐))))
2111eleq2d 2684 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑏 ∈ (𝑃𝑑) ↔ 𝑏 ∈ (𝑝𝑑)))
2221, 13anbi12d 746 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ↔ (𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑))))
2316oveqd 6621 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑏𝐼𝑐) = (𝑏𝑖𝑐))
2423eleq2d 2684 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑡 ∈ (𝑏𝐼𝑐) ↔ 𝑡 ∈ (𝑏𝑖𝑐)))
2524rexbidv 3045 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐) ↔ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)))
2622, 25anbi12d 746 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)) ↔ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐))))
2720, 26anbi12d 746 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → ((((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐))) ↔ (((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)))))
2810, 27rexeqbidv 3142 . . . . . . . 8 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐))) ↔ ∃𝑐𝑝 (((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)))))
297, 8, 28sbcie2s 15837 . . . . . . 7 (𝑔 = 𝐺 → ([(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]𝑐𝑝 (((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐))) ↔ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))))
3029opabbidv 4678 . . . . . 6 (𝑔 = 𝐺 → {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]𝑐𝑝 (((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))})
316, 30mpteq12dv 4693 . . . . 5 (𝑔 = 𝐺 → (𝑑 ∈ ran (LineG‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]𝑐𝑝 (((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)))}) = (𝑑 ∈ ran 𝐿 ↦ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}))
32 df-hpg 25550 . . . . 5 hpG = (𝑔 ∈ V ↦ (𝑑 ∈ ran (LineG‘𝑔) ↦ {⟨𝑎, 𝑏⟩ ∣ [(Base‘𝑔) / 𝑝][(Itv‘𝑔) / 𝑖]𝑐𝑝 (((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ∧ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)))}))
33 fvex 6158 . . . . . . . 8 (LineG‘𝐺) ∈ V
344, 33eqeltri 2694 . . . . . . 7 𝐿 ∈ V
3534rnex 7047 . . . . . 6 ran 𝐿 ∈ V
3635mptex 6440 . . . . 5 (𝑑 ∈ ran 𝐿 ↦ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}) ∈ V
3731, 32, 36fvmpt 6239 . . . 4 (𝐺 ∈ V → (hpG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}))
381, 2, 373syl 18 . . 3 (𝜑 → (hpG‘𝐺) = (𝑑 ∈ ran 𝐿 ↦ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}))
39 difeq2 3700 . . . . . . . . . 10 (𝑑 = 𝐷 → (𝑃𝑑) = (𝑃𝐷))
4039eleq2d 2684 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑎 ∈ (𝑃𝑑) ↔ 𝑎 ∈ (𝑃𝐷)))
4139eleq2d 2684 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑐 ∈ (𝑃𝑑) ↔ 𝑐 ∈ (𝑃𝐷)))
4240, 41anbi12d 746 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ↔ (𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
43 id 22 . . . . . . . . 9 (𝑑 = 𝐷𝑑 = 𝐷)
4443rexeqdv 3134 . . . . . . . 8 (𝑑 = 𝐷 → (∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
4542, 44anbi12d 746 . . . . . . 7 (𝑑 = 𝐷 → (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐))))
4639eleq2d 2684 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑏 ∈ (𝑃𝑑) ↔ 𝑏 ∈ (𝑃𝐷)))
4746, 41anbi12d 746 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ↔ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
4843rexeqdv 3134 . . . . . . . 8 (𝑑 = 𝐷 → (∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
4947, 48anbi12d 746 . . . . . . 7 (𝑑 = 𝐷 → (((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)) ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
5045, 49anbi12d 746 . . . . . 6 (𝑑 = 𝐷 → ((((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐))) ↔ (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))))
5150rexbidv 3045 . . . . 5 (𝑑 = 𝐷 → (∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐))) ↔ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))))
5251opabbidv 4678 . . . 4 (𝑑 = 𝐷 → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))})
5352adantl 482 . . 3 ((𝜑𝑑 = 𝐷) → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))})
54 ishpg.d . . 3 (𝜑𝐷 ∈ ran 𝐿)
55 df-xp 5080 . . . . . 6 (𝑃 × 𝑃) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑃𝑏𝑃)}
56 fvex 6158 . . . . . . . 8 (Base‘𝐺) ∈ V
577, 56eqeltri 2694 . . . . . . 7 𝑃 ∈ V
5857, 57xpex 6915 . . . . . 6 (𝑃 × 𝑃) ∈ V
5955, 58eqeltrri 2695 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑃𝑏𝑃)} ∈ V
60 eldifi 3710 . . . . . . . . . . . 12 (𝑎 ∈ (𝑃𝐷) → 𝑎𝑃)
61 eldifi 3710 . . . . . . . . . . . 12 (𝑏 ∈ (𝑃𝐷) → 𝑏𝑃)
6260, 61anim12i 589 . . . . . . . . . . 11 ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) → (𝑎𝑃𝑏𝑃))
6362adantrr 752 . . . . . . . . . 10 ((𝑎 ∈ (𝑃𝐷) ∧ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))) → (𝑎𝑃𝑏𝑃))
6463adantlr 750 . . . . . . . . 9 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))) → (𝑎𝑃𝑏𝑃))
6564adantlr 750 . . . . . . . 8 ((((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))) → (𝑎𝑃𝑏𝑃))
6665adantrr 752 . . . . . . 7 ((((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))) → (𝑎𝑃𝑏𝑃))
6766rexlimivw 3022 . . . . . 6 (∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))) → (𝑎𝑃𝑏𝑃))
6867ssopab2i 4963 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ⊆ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑃𝑏𝑃)}
6959, 68ssexi 4763 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ∈ V
7069a1i 11 . . 3 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ∈ V)
7138, 53, 54, 70fvmptd 6245 . 2 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))})
72 vex 3189 . . . . . . 7 𝑎 ∈ V
73 vex 3189 . . . . . . 7 𝑐 ∈ V
74 eleq1 2686 . . . . . . . . 9 (𝑒 = 𝑎 → (𝑒 ∈ (𝑃𝐷) ↔ 𝑎 ∈ (𝑃𝐷)))
7574anbi1d 740 . . . . . . . 8 (𝑒 = 𝑎 → ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷))))
76 oveq1 6611 . . . . . . . . . 10 (𝑒 = 𝑎 → (𝑒𝐼𝑓) = (𝑎𝐼𝑓))
7776eleq2d 2684 . . . . . . . . 9 (𝑒 = 𝑎 → (𝑡 ∈ (𝑒𝐼𝑓) ↔ 𝑡 ∈ (𝑎𝐼𝑓)))
7877rexbidv 3045 . . . . . . . 8 (𝑒 = 𝑎 → (∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓)))
7975, 78anbi12d 746 . . . . . . 7 (𝑒 = 𝑎 → (((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓)) ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓))))
80 eleq1 2686 . . . . . . . . 9 (𝑓 = 𝑐 → (𝑓 ∈ (𝑃𝐷) ↔ 𝑐 ∈ (𝑃𝐷)))
8180anbi2d 739 . . . . . . . 8 (𝑓 = 𝑐 → ((𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
82 oveq2 6612 . . . . . . . . . 10 (𝑓 = 𝑐 → (𝑎𝐼𝑓) = (𝑎𝐼𝑐))
8382eleq2d 2684 . . . . . . . . 9 (𝑓 = 𝑐 → (𝑡 ∈ (𝑎𝐼𝑓) ↔ 𝑡 ∈ (𝑎𝐼𝑐)))
8483rexbidv 3045 . . . . . . . 8 (𝑓 = 𝑐 → (∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
8581, 84anbi12d 746 . . . . . . 7 (𝑓 = 𝑐 → (((𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓)) ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐))))
86 ishpg.o . . . . . . . 8 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
87 simpl 473 . . . . . . . . . . . 12 ((𝑎 = 𝑒𝑏 = 𝑓) → 𝑎 = 𝑒)
8887eleq1d 2683 . . . . . . . . . . 11 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑎 ∈ (𝑃𝐷) ↔ 𝑒 ∈ (𝑃𝐷)))
89 simpr 477 . . . . . . . . . . . 12 ((𝑎 = 𝑒𝑏 = 𝑓) → 𝑏 = 𝑓)
9089eleq1d 2683 . . . . . . . . . . 11 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑏 ∈ (𝑃𝐷) ↔ 𝑓 ∈ (𝑃𝐷)))
9188, 90anbi12d 746 . . . . . . . . . 10 ((𝑎 = 𝑒𝑏 = 𝑓) → ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ↔ (𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷))))
92 oveq12 6613 . . . . . . . . . . . 12 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑎𝐼𝑏) = (𝑒𝐼𝑓))
9392eleq2d 2684 . . . . . . . . . . 11 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑒𝐼𝑓)))
9493rexbidv 3045 . . . . . . . . . 10 ((𝑎 = 𝑒𝑏 = 𝑓) → (∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓)))
9591, 94anbi12d 746 . . . . . . . . 9 ((𝑎 = 𝑒𝑏 = 𝑓) → (((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓))))
9695cbvopabv 4684 . . . . . . . 8 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓))}
9786, 96eqtri 2643 . . . . . . 7 𝑂 = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓))}
9872, 73, 79, 85, 97brab 4958 . . . . . 6 (𝑎𝑂𝑐 ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
99 vex 3189 . . . . . . 7 𝑏 ∈ V
100 eleq1 2686 . . . . . . . . 9 (𝑒 = 𝑏 → (𝑒 ∈ (𝑃𝐷) ↔ 𝑏 ∈ (𝑃𝐷)))
101100anbi1d 740 . . . . . . . 8 (𝑒 = 𝑏 → ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷))))
102 oveq1 6611 . . . . . . . . . 10 (𝑒 = 𝑏 → (𝑒𝐼𝑓) = (𝑏𝐼𝑓))
103102eleq2d 2684 . . . . . . . . 9 (𝑒 = 𝑏 → (𝑡 ∈ (𝑒𝐼𝑓) ↔ 𝑡 ∈ (𝑏𝐼𝑓)))
104103rexbidv 3045 . . . . . . . 8 (𝑒 = 𝑏 → (∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓)))
105101, 104anbi12d 746 . . . . . . 7 (𝑒 = 𝑏 → (((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓)) ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓))))
10680anbi2d 739 . . . . . . . 8 (𝑓 = 𝑐 → ((𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
107 oveq2 6612 . . . . . . . . . 10 (𝑓 = 𝑐 → (𝑏𝐼𝑓) = (𝑏𝐼𝑐))
108107eleq2d 2684 . . . . . . . . 9 (𝑓 = 𝑐 → (𝑡 ∈ (𝑏𝐼𝑓) ↔ 𝑡 ∈ (𝑏𝐼𝑐)))
109108rexbidv 3045 . . . . . . . 8 (𝑓 = 𝑐 → (∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
110106, 109anbi12d 746 . . . . . . 7 (𝑓 = 𝑐 → (((𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓)) ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
11199, 73, 105, 110, 97brab 4958 . . . . . 6 (𝑏𝑂𝑐 ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
11298, 111anbi12i 732 . . . . 5 ((𝑎𝑂𝑐𝑏𝑂𝑐) ↔ (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
113112rexbii 3034 . . . 4 (∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐) ↔ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
114113opabbii 4679 . . 3 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))}
115114a1i 11 . 2 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))})
11671, 115eqtr4d 2658 1 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wcel 1987  wrex 2908  Vcvv 3186  [wsbc 3417  cdif 3552   class class class wbr 4613  {copab 4672  cmpt 4673   × cxp 5072  ran crn 5075  cfv 5847  (class class class)co 6604  Basecbs 15781  TarskiGcstrkg 25229  Itvcitv 25235  LineGclng 25236  hpGchpg 25549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-hpg 25550
This theorem is referenced by:  hpgbr  25552
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