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Theorem ishpg 26539
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 3513 . . . 4 (𝐺 ∈ TarskiG → 𝐺 ∈ V)
3 fveq2 6665 . . . . . . . 8 (𝑔 = 𝐺 → (LineG‘𝑔) = (LineG‘𝐺))
4 ishpg.l . . . . . . . 8 𝐿 = (LineG‘𝐺)
53, 4syl6eqr 2874 . . . . . . 7 (𝑔 = 𝐺 → (LineG‘𝑔) = 𝐿)
65rneqd 5803 . . . . . 6 (𝑔 = 𝐺 → ran (LineG‘𝑔) = ran 𝐿)
7 ishpg.p . . . . . . . 8 𝑃 = (Base‘𝐺)
8 ishpg.i . . . . . . . 8 𝐼 = (Itv‘𝐺)
9 simpl 485 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑝 = 𝑃)
109eqcomd 2827 . . . . . . . . 9 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑃 = 𝑝)
1110difeq1d 4098 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑃𝑑) = (𝑝𝑑))
1211eleq2d 2898 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑎 ∈ (𝑃𝑑) ↔ 𝑎 ∈ (𝑝𝑑)))
1311eleq2d 2898 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑐 ∈ (𝑃𝑑) ↔ 𝑐 ∈ (𝑝𝑑)))
1412, 13anbi12d 632 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ↔ (𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑))))
15 simpr 487 . . . . . . . . . . . . . . 15 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝑖 = 𝐼)
1615eqcomd 2827 . . . . . . . . . . . . . 14 ((𝑝 = 𝑃𝑖 = 𝐼) → 𝐼 = 𝑖)
1716oveqd 7167 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑎𝐼𝑐) = (𝑎𝑖𝑐))
1817eleq2d 2898 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑡 ∈ (𝑎𝐼𝑐) ↔ 𝑡 ∈ (𝑎𝑖𝑐)))
1918rexbidv 3297 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐) ↔ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐)))
2014, 19anbi12d 632 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ↔ ((𝑎 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝑖𝑐))))
2111eleq2d 2898 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑏 ∈ (𝑃𝑑) ↔ 𝑏 ∈ (𝑝𝑑)))
2221, 13anbi12d 632 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ↔ (𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑))))
2316oveqd 7167 . . . . . . . . . . . . 13 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑏𝐼𝑐) = (𝑏𝑖𝑐))
2423eleq2d 2898 . . . . . . . . . . . 12 ((𝑝 = 𝑃𝑖 = 𝐼) → (𝑡 ∈ (𝑏𝐼𝑐) ↔ 𝑡 ∈ (𝑏𝑖𝑐)))
2524rexbidv 3297 . . . . . . . . . . 11 ((𝑝 = 𝑃𝑖 = 𝐼) → (∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐) ↔ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐)))
2622, 25anbi12d 632 . . . . . . . . . 10 ((𝑝 = 𝑃𝑖 = 𝐼) → (((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)) ↔ ((𝑏 ∈ (𝑝𝑑) ∧ 𝑐 ∈ (𝑝𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝑖𝑐))))
2720, 26anbi12d 632 . . . . . . . . 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 3403 . . . . . . . 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 16534 . . . . . . 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 5125 . . . . . 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 5144 . . . . 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 26538 . . . . 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)))}))
334fvexi 6679 . . . . . . 7 𝐿 ∈ V
3433rnex 7611 . . . . . 6 ran 𝐿 ∈ V
3534mptex 6980 . . . . 5 (𝑑 ∈ ran 𝐿 ↦ {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)))}) ∈ V
3631, 32, 35fvmpt 6763 . . . 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)))}))
371, 2, 363syl 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)))}))
38 difeq2 4093 . . . . . . . . . 10 (𝑑 = 𝐷 → (𝑃𝑑) = (𝑃𝐷))
3938eleq2d 2898 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑎 ∈ (𝑃𝑑) ↔ 𝑎 ∈ (𝑃𝐷)))
4038eleq2d 2898 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑐 ∈ (𝑃𝑑) ↔ 𝑐 ∈ (𝑃𝐷)))
4139, 40anbi12d 632 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ↔ (𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
42 rexeq 3407 . . . . . . . 8 (𝑑 = 𝐷 → (∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
4341, 42anbi12d 632 . . . . . . 7 (𝑑 = 𝐷 → (((𝑎 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑎𝐼𝑐)) ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐))))
4438eleq2d 2898 . . . . . . . . 9 (𝑑 = 𝐷 → (𝑏 ∈ (𝑃𝑑) ↔ 𝑏 ∈ (𝑃𝐷)))
4544, 40anbi12d 632 . . . . . . . 8 (𝑑 = 𝐷 → ((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ↔ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
46 rexeq 3407 . . . . . . . 8 (𝑑 = 𝐷 → (∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
4745, 46anbi12d 632 . . . . . . 7 (𝑑 = 𝐷 → (((𝑏 ∈ (𝑃𝑑) ∧ 𝑐 ∈ (𝑃𝑑)) ∧ ∃𝑡𝑑 𝑡 ∈ (𝑏𝐼𝑐)) ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
4843, 47anbi12d 632 . . . . . 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)))))
4948rexbidv 3297 . . . . 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)))))
5049opabbidv 5125 . . . 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)))})
5150adantl 484 . . 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)))})
52 ishpg.d . . 3 (𝜑𝐷 ∈ ran 𝐿)
53 df-xp 5556 . . . . . 6 (𝑃 × 𝑃) = {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑃𝑏𝑃)}
547fvexi 6679 . . . . . . 7 𝑃 ∈ V
5554, 54xpex 7470 . . . . . 6 (𝑃 × 𝑃) ∈ V
5653, 55eqeltrri 2910 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑃𝑏𝑃)} ∈ V
57 eldifi 4103 . . . . . . . . . 10 (𝑎 ∈ (𝑃𝐷) → 𝑎𝑃)
58 eldifi 4103 . . . . . . . . . 10 (𝑏 ∈ (𝑃𝐷) → 𝑏𝑃)
5957, 58anim12i 614 . . . . . . . . 9 ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) → (𝑎𝑃𝑏𝑃))
6059ad2ant2r 745 . . . . . . . 8 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))) → (𝑎𝑃𝑏𝑃))
6160ad2ant2r 745 . . . . . . 7 ((((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))) → (𝑎𝑃𝑏𝑃))
6261rexlimivw 3282 . . . . . 6 (∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))) → (𝑎𝑃𝑏𝑃))
6362ssopab2i 5430 . . . . 5 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ⊆ {⟨𝑎, 𝑏⟩ ∣ (𝑎𝑃𝑏𝑃)}
6456, 63ssexi 5219 . . . 4 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ∈ V
6564a1i 11 . . 3 (𝜑 → {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))} ∈ V)
6637, 51, 52, 65fvmptd 6770 . 2 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))})
67 vex 3498 . . . . . 6 𝑎 ∈ V
68 vex 3498 . . . . . 6 𝑐 ∈ V
69 eleq1w 2895 . . . . . . . 8 (𝑒 = 𝑎 → (𝑒 ∈ (𝑃𝐷) ↔ 𝑎 ∈ (𝑃𝐷)))
7069anbi1d 631 . . . . . . 7 (𝑒 = 𝑎 → ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷))))
71 oveq1 7157 . . . . . . . . 9 (𝑒 = 𝑎 → (𝑒𝐼𝑓) = (𝑎𝐼𝑓))
7271eleq2d 2898 . . . . . . . 8 (𝑒 = 𝑎 → (𝑡 ∈ (𝑒𝐼𝑓) ↔ 𝑡 ∈ (𝑎𝐼𝑓)))
7372rexbidv 3297 . . . . . . 7 (𝑒 = 𝑎 → (∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓)))
7470, 73anbi12d 632 . . . . . 6 (𝑒 = 𝑎 → (((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓)) ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓))))
75 eleq1w 2895 . . . . . . . 8 (𝑓 = 𝑐 → (𝑓 ∈ (𝑃𝐷) ↔ 𝑐 ∈ (𝑃𝐷)))
7675anbi2d 630 . . . . . . 7 (𝑓 = 𝑐 → ((𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
77 oveq2 7158 . . . . . . . . 9 (𝑓 = 𝑐 → (𝑎𝐼𝑓) = (𝑎𝐼𝑐))
7877eleq2d 2898 . . . . . . . 8 (𝑓 = 𝑐 → (𝑡 ∈ (𝑎𝐼𝑓) ↔ 𝑡 ∈ (𝑎𝐼𝑐)))
7978rexbidv 3297 . . . . . . 7 (𝑓 = 𝑐 → (∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
8076, 79anbi12d 632 . . . . . 6 (𝑓 = 𝑐 → (((𝑎 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑓)) ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐))))
81 ishpg.o . . . . . . 7 𝑂 = {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))}
82 simpl 485 . . . . . . . . . . 11 ((𝑎 = 𝑒𝑏 = 𝑓) → 𝑎 = 𝑒)
8382eleq1d 2897 . . . . . . . . . 10 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑎 ∈ (𝑃𝐷) ↔ 𝑒 ∈ (𝑃𝐷)))
84 simpr 487 . . . . . . . . . . 11 ((𝑎 = 𝑒𝑏 = 𝑓) → 𝑏 = 𝑓)
8584eleq1d 2897 . . . . . . . . . 10 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑏 ∈ (𝑃𝐷) ↔ 𝑓 ∈ (𝑃𝐷)))
8683, 85anbi12d 632 . . . . . . . . 9 ((𝑎 = 𝑒𝑏 = 𝑓) → ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ↔ (𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷))))
87 oveq12 7159 . . . . . . . . . . 11 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑎𝐼𝑏) = (𝑒𝐼𝑓))
8887eleq2d 2898 . . . . . . . . . 10 ((𝑎 = 𝑒𝑏 = 𝑓) → (𝑡 ∈ (𝑎𝐼𝑏) ↔ 𝑡 ∈ (𝑒𝐼𝑓)))
8988rexbidv 3297 . . . . . . . . 9 ((𝑎 = 𝑒𝑏 = 𝑓) → (∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓)))
9086, 89anbi12d 632 . . . . . . . 8 ((𝑎 = 𝑒𝑏 = 𝑓) → (((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏)) ↔ ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓))))
9190cbvopabv 5131 . . . . . . 7 {⟨𝑎, 𝑏⟩ ∣ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑏 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑏))} = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓))}
9281, 91eqtri 2844 . . . . . 6 𝑂 = {⟨𝑒, 𝑓⟩ ∣ ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓))}
9367, 68, 74, 80, 92brab 5423 . . . . 5 (𝑎𝑂𝑐 ↔ ((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)))
94 vex 3498 . . . . . 6 𝑏 ∈ V
95 eleq1w 2895 . . . . . . . 8 (𝑒 = 𝑏 → (𝑒 ∈ (𝑃𝐷) ↔ 𝑏 ∈ (𝑃𝐷)))
9695anbi1d 631 . . . . . . 7 (𝑒 = 𝑏 → ((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷))))
97 oveq1 7157 . . . . . . . . 9 (𝑒 = 𝑏 → (𝑒𝐼𝑓) = (𝑏𝐼𝑓))
9897eleq2d 2898 . . . . . . . 8 (𝑒 = 𝑏 → (𝑡 ∈ (𝑒𝐼𝑓) ↔ 𝑡 ∈ (𝑏𝐼𝑓)))
9998rexbidv 3297 . . . . . . 7 (𝑒 = 𝑏 → (∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓)))
10096, 99anbi12d 632 . . . . . 6 (𝑒 = 𝑏 → (((𝑒 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑒𝐼𝑓)) ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓))))
10175anbi2d 630 . . . . . . 7 (𝑓 = 𝑐 → ((𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ↔ (𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷))))
102 oveq2 7158 . . . . . . . . 9 (𝑓 = 𝑐 → (𝑏𝐼𝑓) = (𝑏𝐼𝑐))
103102eleq2d 2898 . . . . . . . 8 (𝑓 = 𝑐 → (𝑡 ∈ (𝑏𝐼𝑓) ↔ 𝑡 ∈ (𝑏𝐼𝑐)))
104103rexbidv 3297 . . . . . . 7 (𝑓 = 𝑐 → (∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓) ↔ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
105101, 104anbi12d 632 . . . . . 6 (𝑓 = 𝑐 → (((𝑏 ∈ (𝑃𝐷) ∧ 𝑓 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑓)) ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
10694, 68, 100, 105, 92brab 5423 . . . . 5 (𝑏𝑂𝑐 ↔ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))
10793, 106anbi12i 628 . . . 4 ((𝑎𝑂𝑐𝑏𝑂𝑐) ↔ (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
108107rexbii 3247 . . 3 (∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐) ↔ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐))))
109108opabbii 5126 . 2 {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)} = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (((𝑎 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑎𝐼𝑐)) ∧ ((𝑏 ∈ (𝑃𝐷) ∧ 𝑐 ∈ (𝑃𝐷)) ∧ ∃𝑡𝐷 𝑡 ∈ (𝑏𝐼𝑐)))}
11066, 109syl6eqr 2874 1 (𝜑 → ((hpG‘𝐺)‘𝐷) = {⟨𝑎, 𝑏⟩ ∣ ∃𝑐𝑃 (𝑎𝑂𝑐𝑏𝑂𝑐)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398   = wceq 1533  wcel 2110  wrex 3139  Vcvv 3495  [wsbc 3772  cdif 3933   class class class wbr 5059  {copab 5121  cmpt 5139   × cxp 5548  ran crn 5551  cfv 6350  (class class class)co 7150  Basecbs 16477  TarskiGcstrkg 26210  Itvcitv 26216  LineGclng 26217  hpGchpg 26537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pow 5259  ax-pr 5322  ax-un 7455
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-hpg 26538
This theorem is referenced by:  hpgbr  26540
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