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Theorem ishpg 28550
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 3488 . . . 4 (𝐺 ∈ TarskiG → 𝐺 ∈ V)
3 fveq2 6891 . . . . . . . 8 (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4 ishpg.l . . . . . . . 8 𝐿 = (LineG‘𝐺)
53, 4eqtr4di 2785 . . . . . . 7 (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
65rneqd 5934 . . . . . 6 (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7 ishpg.p . . . . . . . 8 𝑃 = (Base‘𝐺)
8 ishpg.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
9 simpl 482 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑝 = 𝑃)
109difeq1d 4117 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑝𝑑) = (𝑃𝑑))
1110eleq2d 2814 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑎 ∈ (𝑝𝑑) ↔ 𝑎 ∈ (𝑃𝑑)))
1210eleq2d 2814 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑐 ∈ (𝑝𝑑) ↔ 𝑐 ∈ (𝑃𝑑)))
1311, 12anbi12d 630 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ↔ (𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑))))
14 simpr 484 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑖 = 𝐼)
1514oveqd 7431 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑎𝑖𝑐) = (𝑎𝐼𝑐))
1615eleq2d 2814 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑡 ∈ (𝑎𝑖𝑐) ↔ 𝑡 ∈ (𝑎𝐼𝑐)))
1716rexbidv 3173 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐) ↔ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)))
1813, 17anbi12d 630 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)) ↔ ((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐))))
1910eleq2d 2814 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑏 ∈ (𝑝𝑑) ↔ 𝑏 ∈ (𝑃𝑑)))
2019, 12anbi12d 630 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ↔ (𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑))))
2114oveqd 7431 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑏𝑖𝑐) = (𝑏𝐼𝑐))
2221eleq2d 2814 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑡 ∈ (𝑏𝑖𝑐) ↔ 𝑡 ∈ (𝑏𝐼𝑐)))
2322rexbidv 3173 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐) ↔ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))
2420, 23anbi12d 630 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)) ↔ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐))))
2518, 24anbi12d 630 . . . . . . . . 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)))))
269, 25rexeqbidv 3338 . . . . . . . 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)))))
277, 8, 26sbcie2s 17121 . . . . . . 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)))))
2827opabbidv 5208 . . . . . 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)))})
296, 28mpteq12dv 5233 . . . . 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)))}))
30 df-hpg 28549 . . . . 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)))}))
314fvexi 6905 . . . . . . 7 𝐿 ∈ V
3231rnex 7912 . . . . . 6 ran 𝐿 ∈ V
3332mptex 7229 . . . . 5 (𝑑 ∈ ran 𝐿 ↦ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}) ∈ V
3429, 30, 33fvmpt 6999 . . . 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)))}))
351, 2, 343syl 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)))}))
36 difeq2 4112 . . . . . . . . . 10 (𝑑 = 𝐷 → (𝑃𝑑) = (𝑃𝐷))
3736eleq2d 2814 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑎 ∈ (𝑃𝑑) ↔ 𝑎 ∈ (𝑃𝐷)))
3836eleq2d 2814 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑐 ∈ (𝑃𝑑) ↔ 𝑐 ∈ (𝑃𝐷)))
3937, 38anbi12d 630 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ↔ (𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
40 rexeq 3316 . . . . . . . 8 (𝑑 = 𝐷 → (∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
4139, 40anbi12d 630 . . . . . . 7 (𝑑 = 𝐷 → (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐))))
4236eleq2d 2814 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑏 ∈ (𝑃𝑑) ↔ 𝑏 ∈ (𝑃𝐷)))
4342, 38anbi12d 630 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ↔ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
44 rexeq 3316 . . . . . . . 8 (𝑑 = 𝐷 → (∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
4543, 44anbi12d 630 . . . . . . 7 (𝑑 = 𝐷 → (((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)) ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
4641, 45anbi12d 630 . . . . . 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)))))
4746rexbidv 3173 . . . . 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)))))
4847opabbidv 5208 . . . 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)))})
4948adantl 481 . . 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)))})
50 ishpg.d . . 3 (𝜑𝐷 ∈ ran 𝐿)
51 df-xp 5678 . . . . . 6 (𝑃 × 𝑃) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑃𝑏𝑃)}
527fvexi 6905 . . . . . . 7 𝑃 ∈ V
5352, 52xpex 7749 . . . . . 6 (𝑃 × 𝑃) ∈ V
5451, 53eqeltrri 2825 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑃𝑏𝑃)} ∈ V
55 eldifi 4122 . . . . . . . . . 10 (𝑎 ∈ (𝑃𝐷) → 𝑎𝑃)
56 eldifi 4122 . . . . . . . . . 10 (𝑏 ∈ (𝑃𝐷) → 𝑏𝑃)
5755, 56anim12i 612 . . . . . . . . 9 ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) → (𝑎𝑃𝑏𝑃))
5857ad2ant2r 746 . . . . . . . 8 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))) → (𝑎𝑃𝑏𝑃))
5958ad2ant2r 746 . . . . . . 7 ((((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))) → (𝑎𝑃𝑏𝑃))
6059rexlimivw 3146 . . . . . 6 (∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))) → (𝑎𝑃𝑏𝑃))
6160ssopab2i 5546 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ⊆ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑃𝑏𝑃)}
6254, 61ssexi 5316 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ∈ V
6362a1i 11 . . 3 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ∈ V)
6435, 49, 50, 63fvmptd 7006 . 2 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))})
65 vex 3473 . . . . . 6 𝑎 ∈ V
66 vex 3473 . . . . . 6 𝑐 ∈ V
67 eleq1w 2811 . . . . . . . 8 (𝑒 = 𝑎 → (𝑒 ∈ (𝑃𝐷) ↔ 𝑎 ∈ (𝑃𝐷)))
6867anbi1d 629 . . . . . . 7 (𝑒 = 𝑎 → ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷))))
69 oveq1 7421 . . . . . . . . 9 (𝑒 = 𝑎 → (𝑒𝐼𝑓) = (𝑎𝐼𝑓))
7069eleq2d 2814 . . . . . . . 8 (𝑒 = 𝑎 → (𝑡 ∈ (𝑒𝐼𝑓) ↔ 𝑡 ∈ (𝑎𝐼𝑓)))
7170rexbidv 3173 . . . . . . 7 (𝑒 = 𝑎 → (∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓)))
7268, 71anbi12d 630 . . . . . 6 (𝑒 = 𝑎 → (((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓)) ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓))))
73 eleq1w 2811 . . . . . . . 8 (𝑓 = 𝑐 → (𝑓 ∈ (𝑃𝐷) ↔ 𝑐 ∈ (𝑃𝐷)))
7473anbi2d 628 . . . . . . 7 (𝑓 = 𝑐 → ((𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
75 oveq2 7422 . . . . . . . . 9 (𝑓 = 𝑐 → (𝑎𝐼𝑓) = (𝑎𝐼𝑐))
7675eleq2d 2814 . . . . . . . 8 (𝑓 = 𝑐 → (𝑡 ∈ (𝑎𝐼𝑓) ↔ 𝑡 ∈ (𝑎𝐼𝑐)))
7776rexbidv 3173 . . . . . . 7 (𝑓 = 𝑐 → (∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
7874, 77anbi12d 630 . . . . . 6 (𝑓 = 𝑐 → (((𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓)) ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐))))
79 ishpg.o . . . . . . 7 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
80 simpl 482 . . . . . . . . . . 11 ((𝑎 = 𝑒𝑏 = 𝑓) → 𝑎 = 𝑒)
8180eleq1d 2813 . . . . . . . . . 10 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑎 ∈ (𝑃𝐷) ↔ 𝑒 ∈ (𝑃𝐷)))
82 simpr 484 . . . . . . . . . . 11 ((𝑎 = 𝑒𝑏 = 𝑓) → 𝑏 = 𝑓)
8382eleq1d 2813 . . . . . . . . . 10 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑏 ∈ (𝑃𝐷) ↔ 𝑓 ∈ (𝑃𝐷)))
8481, 83anbi12d 630 . . . . . . . . 9 ((𝑎 = 𝑒𝑏 = 𝑓) → ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ↔ (𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷))))
85 oveq12 7423 . . . . . . . . . . 11 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑎𝐼𝑏) = (𝑒𝐼𝑓))
8685eleq2d 2814 . . . . . . . . . 10 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑒𝐼𝑓)))
8786rexbidv 3173 . . . . . . . . 9 ((𝑎 = 𝑒𝑏 = 𝑓) → (∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓)))
8884, 87anbi12d 630 . . . . . . . 8 ((𝑎 = 𝑒𝑏 = 𝑓) → (((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓))))
8988cbvopabv 5215 . . . . . . 7 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓))}
9079, 89eqtri 2755 . . . . . 6 𝑂 = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓))}
9165, 66, 72, 78, 90brab 5539 . . . . 5 (𝑎𝑂𝑐 ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
92 vex 3473 . . . . . 6 𝑏 ∈ V
93 eleq1w 2811 . . . . . . . 8 (𝑒 = 𝑏 → (𝑒 ∈ (𝑃𝐷) ↔ 𝑏 ∈ (𝑃𝐷)))
9493anbi1d 629 . . . . . . 7 (𝑒 = 𝑏 → ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷))))
95 oveq1 7421 . . . . . . . . 9 (𝑒 = 𝑏 → (𝑒𝐼𝑓) = (𝑏𝐼𝑓))
9695eleq2d 2814 . . . . . . . 8 (𝑒 = 𝑏 → (𝑡 ∈ (𝑒𝐼𝑓) ↔ 𝑡 ∈ (𝑏𝐼𝑓)))
9796rexbidv 3173 . . . . . . 7 (𝑒 = 𝑏 → (∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓)))
9894, 97anbi12d 630 . . . . . 6 (𝑒 = 𝑏 → (((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓)) ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓))))
9973anbi2d 628 . . . . . . 7 (𝑓 = 𝑐 → ((𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
100 oveq2 7422 . . . . . . . . 9 (𝑓 = 𝑐 → (𝑏𝐼𝑓) = (𝑏𝐼𝑐))
101100eleq2d 2814 . . . . . . . 8 (𝑓 = 𝑐 → (𝑡 ∈ (𝑏𝐼𝑓) ↔ 𝑡 ∈ (𝑏𝐼𝑐)))
102101rexbidv 3173 . . . . . . 7 (𝑓 = 𝑐 → (∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
10399, 102anbi12d 630 . . . . . 6 (𝑓 = 𝑐 → (((𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓)) ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
10492, 66, 98, 103, 90brab 5539 . . . . 5 (𝑏𝑂𝑐 ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
10591, 104anbi12i 626 . . . 4 ((𝑎𝑂𝑐𝑏𝑂𝑐) ↔ (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
106105rexbii 3089 . . 3 (∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐) ↔ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
107106opabbii 5209 . 2 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))}
10864, 107eqtr4di 2785 1 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1534  wcel 2099  wrex 3065  Vcvv 3469  [wsbc 3774  cdif 3941   class class class wbr 5142  {copab 5204  cmpt 5225   × cxp 5670  ran crn 5673  cfv 6542  (class class class)co 7414  Basecbs 17171  TarskiGcstrkg 28218  Itvcitv 28224  LineGclng 28225  hpGchpg 28548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-reu 3372  df-rab 3428  df-v 3471  df-sbc 3775  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-hpg 28549
This theorem is referenced by:  hpgbr  28551
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