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Theorem metustexhalf 24375
Description: For any element 𝐴 of the filter base generated by the metric 𝐷, the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
Assertion
Ref Expression
metustexhalf (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) → ∃𝑣𝐹 (𝑣𝑣) ⊆ 𝐴)
Distinct variable groups:   𝐷,𝑎   𝑋,𝑎   𝐴,𝑎   𝐹,𝑎,𝑣   𝑣,𝐴   𝑣,𝐷   𝑣,𝐹   𝑣,𝑋

Proof of Theorem metustexhalf
Dummy variables 𝑏 𝑝 𝑞 𝑟 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp-4r 781 . . . 4 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋))
2 simplr 766 . . . . . 6 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝑎 ∈ ℝ+)
32rphalfcld 13024 . . . . 5 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝑎 / 2) ∈ ℝ+)
4 eqidd 2725 . . . . 5 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)(𝑎 / 2))))
5 oveq2 7409 . . . . . . 7 (𝑏 = (𝑎 / 2) → (0[,)𝑏) = (0[,)(𝑎 / 2)))
65imaeq2d 6049 . . . . . 6 (𝑏 = (𝑎 / 2) → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)(𝑎 / 2))))
76rspceeqv 3625 . . . . 5 (((𝑎 / 2) ∈ ℝ+ ∧ (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)(𝑎 / 2)))) → ∃𝑏 ∈ ℝ+ (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)𝑏)))
83, 4, 7syl2anc 583 . . . 4 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ∃𝑏 ∈ ℝ+ (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)𝑏)))
9 metust.1 . . . . . . 7 𝐹 = ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎)))
10 oveq2 7409 . . . . . . . . . 10 (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
1110imaeq2d 6049 . . . . . . . . 9 (𝑎 = 𝑏 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑏)))
1211cbvmptv 5251 . . . . . . . 8 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
1312rneqi 5926 . . . . . . 7 ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
149, 13eqtri 2752 . . . . . 6 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
1514metustel 24369 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → ((𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ↔ ∃𝑏 ∈ ℝ+ (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)𝑏))))
1615biimpar 477 . . . 4 ((𝐷 ∈ (PsMet‘𝑋) ∧ ∃𝑏 ∈ ℝ+ (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)𝑏))) → (𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹)
171, 8, 16syl2anc 583 . . 3 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹)
18 relco 6097 . . . . 5 Rel ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))
1918a1i 11 . . . 4 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → Rel ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))))
20 cossxp 6261 . . . . . . . . . 10 ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2))))
21 cnvimass 6070 . . . . . . . . . . . . . 14 (𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom 𝐷
22 psmetf 24122 . . . . . . . . . . . . . 14 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2321, 22fssdm 6727 . . . . . . . . . . . . 13 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋))
24 dmss 5892 . . . . . . . . . . . . . 14 ((𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → dom (𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom (𝑋 × 𝑋))
25 rnss 5928 . . . . . . . . . . . . . 14 ((𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → ran (𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋))
26 xpss12 5681 . . . . . . . . . . . . . 14 ((dom (𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom (𝑋 × 𝑋) ∧ ran (𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋)) → (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)))
2724, 25, 26syl2anc 583 . . . . . . . . . . . . 13 ((𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)))
2823, 27syl 17 . . . . . . . . . . . 12 (𝐷 ∈ (PsMet‘𝑋) → (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)))
2928adantl 481 . . . . . . . . . . 11 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)))
30 dmxp 5918 . . . . . . . . . . . . 13 (𝑋 ≠ ∅ → dom (𝑋 × 𝑋) = 𝑋)
31 rnxp 6159 . . . . . . . . . . . . 13 (𝑋 ≠ ∅ → ran (𝑋 × 𝑋) = 𝑋)
3230, 31xpeq12d 5697 . . . . . . . . . . . 12 (𝑋 ≠ ∅ → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋))
3332adantr 480 . . . . . . . . . . 11 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋))
3429, 33sseqtrd 4014 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
3520, 34sstrid 3985 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
3635ad3antrrr 727 . . . . . . . 8 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
3736sselda 3974 . . . . . . 7 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
38 opelxp 5702 . . . . . . 7 (⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋) ↔ (𝑝𝑋𝑞𝑋))
3937, 38sylib 217 . . . . . 6 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → (𝑝𝑋𝑞𝑋))
40 simpll 764 . . . . . . 7 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ (𝑝𝑋𝑞𝑋)) → ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))))
41 simprl 768 . . . . . . 7 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ (𝑝𝑋𝑞𝑋)) → 𝑝𝑋)
42 simprr 770 . . . . . . 7 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ (𝑝𝑋𝑞𝑋)) → 𝑞𝑋)
43 simplr 766 . . . . . . 7 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ (𝑝𝑋𝑞𝑋)) → ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))))
44 simplll 772 . . . . . . . . . . . . . . 15 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋))
4544simp1d 1139 . . . . . . . . . . . . . 14 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))))
4645, 1syl 17 . . . . . . . . . . . . 13 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝐷 ∈ (PsMet‘𝑋))
4745, 2syl 17 . . . . . . . . . . . . 13 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ+)
4846, 47jca 511 . . . . . . . . . . . 12 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+))
4944simp2d 1140 . . . . . . . . . . . 12 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝𝑋)
5044simp3d 1141 . . . . . . . . . . . 12 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑞𝑋)
5148, 49, 503jca 1125 . . . . . . . . . . 11 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋))
52 simplr 766 . . . . . . . . . . 11 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟𝑋)
53 simprl 768 . . . . . . . . . . 11 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟)
54 simprr 770 . . . . . . . . . . 11 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)
55 simpll 764 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋))
5655simp1d 1139 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+))
5756simpld 494 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝐷 ∈ (PsMet‘𝑋))
5822ffund 6711 . . . . . . . . . . . . 13 (𝐷 ∈ (PsMet‘𝑋) → Fun 𝐷)
5957, 58syl 17 . . . . . . . . . . . 12 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → Fun 𝐷)
6055simp2d 1140 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝𝑋)
6155simp3d 1141 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑞𝑋)
6260, 61opelxpd 5705 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
6322fdmd 6718 . . . . . . . . . . . . . 14 (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
6457, 63syl 17 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → dom 𝐷 = (𝑋 × 𝑋))
6562, 64eleqtrrd 2828 . . . . . . . . . . . 12 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
66 0xr 11257 . . . . . . . . . . . . . 14 0 ∈ ℝ*
6766a1i 11 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ∈ ℝ*)
6856simprd 495 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ+)
6968rpxrd 13013 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ*)
7057, 22syl 17 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
7170, 62ffvelcdmd 7077 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ ℝ*)
72 psmetge0 24128 . . . . . . . . . . . . . . 15 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝𝑋𝑞𝑋) → 0 ≤ (𝑝𝐷𝑞))
7357, 60, 61, 72syl3anc 1368 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ≤ (𝑝𝐷𝑞))
74 df-ov 7404 . . . . . . . . . . . . . 14 (𝑝𝐷𝑞) = (𝐷‘⟨𝑝, 𝑞⟩)
7573, 74breqtrdi 5179 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ≤ (𝐷‘⟨𝑝, 𝑞⟩))
7674, 71eqeltrid 2829 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) ∈ ℝ*)
77 0red 11213 . . . . . . . . . . . . . . . . . . 19 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ∈ ℝ)
7868rpred 13012 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ)
7978rehalfcld 12455 . . . . . . . . . . . . . . . . . . . 20 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑎 / 2) ∈ ℝ)
8079rexrd 11260 . . . . . . . . . . . . . . . . . . 19 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑎 / 2) ∈ ℝ*)
81 df-ov 7404 . . . . . . . . . . . . . . . . . . . 20 (𝑝𝐷𝑟) = (𝐷‘⟨𝑝, 𝑟⟩)
82 simplr 766 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟𝑋)
8360, 82opelxpd 5705 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑟⟩ ∈ (𝑋 × 𝑋))
8483, 64eleqtrrd 2828 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑟⟩ ∈ dom 𝐷)
85 simprl 768 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟)
86 df-br 5139 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟 ↔ ⟨𝑝, 𝑟⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2))))
8785, 86sylib 217 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑟⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2))))
88 fvimacnv 7044 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝐷 ∧ ⟨𝑝, 𝑟⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑟⟩) ∈ (0[,)(𝑎 / 2)) ↔ ⟨𝑝, 𝑟⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2)))))
8988biimpar 477 . . . . . . . . . . . . . . . . . . . . 21 (((Fun 𝐷 ∧ ⟨𝑝, 𝑟⟩ ∈ dom 𝐷) ∧ ⟨𝑝, 𝑟⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2)))) → (𝐷‘⟨𝑝, 𝑟⟩) ∈ (0[,)(𝑎 / 2)))
9059, 84, 87, 89syl21anc 835 . . . . . . . . . . . . . . . . . . . 20 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷‘⟨𝑝, 𝑟⟩) ∈ (0[,)(𝑎 / 2)))
9181, 90eqeltrid 2829 . . . . . . . . . . . . . . . . . . 19 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2)))
92 elico2 13384 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) → ((𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2))))
9392biimpa 476 . . . . . . . . . . . . . . . . . . . 20 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2)))
9493simp1d 1139 . . . . . . . . . . . . . . . . . . 19 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) ∈ ℝ)
9577, 80, 91, 94syl21anc 835 . . . . . . . . . . . . . . . . . 18 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) ∈ ℝ)
96 df-ov 7404 . . . . . . . . . . . . . . . . . . . 20 (𝑟𝐷𝑞) = (𝐷‘⟨𝑟, 𝑞⟩)
9782, 61opelxpd 5705 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑟, 𝑞⟩ ∈ (𝑋 × 𝑋))
9897, 64eleqtrrd 2828 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑟, 𝑞⟩ ∈ dom 𝐷)
99 simprr 770 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)
100 df-br 5139 . . . . . . . . . . . . . . . . . . . . . 22 (𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞 ↔ ⟨𝑟, 𝑞⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2))))
10199, 100sylib 217 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑟, 𝑞⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2))))
102 fvimacnv 7044 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝐷 ∧ ⟨𝑟, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑟, 𝑞⟩) ∈ (0[,)(𝑎 / 2)) ↔ ⟨𝑟, 𝑞⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2)))))
103102biimpar 477 . . . . . . . . . . . . . . . . . . . . 21 (((Fun 𝐷 ∧ ⟨𝑟, 𝑞⟩ ∈ dom 𝐷) ∧ ⟨𝑟, 𝑞⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2)))) → (𝐷‘⟨𝑟, 𝑞⟩) ∈ (0[,)(𝑎 / 2)))
10459, 98, 101, 103syl21anc 835 . . . . . . . . . . . . . . . . . . . 20 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷‘⟨𝑟, 𝑞⟩) ∈ (0[,)(𝑎 / 2)))
10596, 104eqeltrid 2829 . . . . . . . . . . . . . . . . . . 19 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2)))
106 elico2 13384 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) → ((𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2))))
107106biimpa 476 . . . . . . . . . . . . . . . . . . . 20 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2)))
108107simp1d 1139 . . . . . . . . . . . . . . . . . . 19 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) ∈ ℝ)
10977, 80, 105, 108syl21anc 835 . . . . . . . . . . . . . . . . . 18 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑟𝐷𝑞) ∈ ℝ)
11095, 109rexaddd 13209 . . . . . . . . . . . . . . . . 17 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) = ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)))
11195, 109readdcld 11239 . . . . . . . . . . . . . . . . 17 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)) ∈ ℝ)
112110, 111eqeltrd 2825 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) ∈ ℝ)
113112rexrd 11260 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) ∈ ℝ*)
114 psmettri 24127 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑝𝑋𝑞𝑋𝑟𝑋)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)))
11557, 60, 61, 82, 114syl13anc 1369 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)))
11693simp3d 1141 . . . . . . . . . . . . . . . . . 18 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) < (𝑎 / 2))
11777, 80, 91, 116syl21anc 835 . . . . . . . . . . . . . . . . 17 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) < (𝑎 / 2))
118107simp3d 1141 . . . . . . . . . . . . . . . . . 18 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) < (𝑎 / 2))
11977, 80, 105, 118syl21anc 835 . . . . . . . . . . . . . . . . 17 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑟𝐷𝑞) < (𝑎 / 2))
12095, 109, 78, 117, 119lt2halvesd 12456 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)) < 𝑎)
121110, 120eqbrtrd 5160 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) < 𝑎)
12276, 113, 69, 115, 121xrlelttrd 13135 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) < 𝑎)
12374, 122eqbrtrrid 5174 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷‘⟨𝑝, 𝑞⟩) < 𝑎)
12467, 69, 71, 75, 123elicod 13370 . . . . . . . . . . . 12 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎))
125 fvimacnv 7044 . . . . . . . . . . . . . 14 ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
126125biimpa 476 . . . . . . . . . . . . 13 (((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) ∧ (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎)) → ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
127 df-br 5139 . . . . . . . . . . . . 13 (𝑝(𝐷 “ (0[,)𝑎))𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
128126, 127sylibr 233 . . . . . . . . . . . 12 (((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) ∧ (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎)) → 𝑝(𝐷 “ (0[,)𝑎))𝑞)
12959, 65, 124, 128syl21anc 835 . . . . . . . . . . 11 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝(𝐷 “ (0[,)𝑎))𝑞)
13051, 52, 53, 54, 129syl22anc 836 . . . . . . . . . 10 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝(𝐷 “ (0[,)𝑎))𝑞)
13145simprd 495 . . . . . . . . . . 11 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝐴 = (𝐷 “ (0[,)𝑎)))
132131breqd 5149 . . . . . . . . . 10 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐴𝑞𝑝(𝐷 “ (0[,)𝑎))𝑞))
133130, 132mpbird 257 . . . . . . . . 9 (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝𝐴𝑞)
134 simpr 484 . . . . . . . . . . . . 13 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))))
135 df-br 5139 . . . . . . . . . . . . 13 (𝑝((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))))
136134, 135sylibr 233 . . . . . . . . . . . 12 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → 𝑝((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))𝑞)
137 vex 3470 . . . . . . . . . . . . 13 𝑝 ∈ V
138 vex 3470 . . . . . . . . . . . . 13 𝑞 ∈ V
139137, 138brco 5860 . . . . . . . . . . . 12 (𝑝((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))𝑞 ↔ ∃𝑟(𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞))
140136, 139sylib 217 . . . . . . . . . . 11 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → ∃𝑟(𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞))
14123adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋))
142141, 25syl 17 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋))
14331adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑋 × 𝑋) = 𝑋)
144142, 143sseqtrd 4014 . . . . . . . . . . . . . . . . . . . 20 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋)
145144adantr 480 . . . . . . . . . . . . . . . . . . 19 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟) → ran (𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋)
146 vex 3470 . . . . . . . . . . . . . . . . . . . . 21 𝑟 ∈ V
147137, 146brelrn 5931 . . . . . . . . . . . . . . . . . . . 20 (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟 ∈ ran (𝐷 “ (0[,)(𝑎 / 2))))
148147adantl 481 . . . . . . . . . . . . . . . . . . 19 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟 ∈ ran (𝐷 “ (0[,)(𝑎 / 2))))
149145, 148sseldd 3975 . . . . . . . . . . . . . . . . . 18 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟𝑋)
150149adantrr 714 . . . . . . . . . . . . . . . . 17 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟𝑋)
151150ex 412 . . . . . . . . . . . . . . . 16 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞) → 𝑟𝑋))
152151ancrd 551 . . . . . . . . . . . . . . 15 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞) → (𝑟𝑋 ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞))))
153152eximdv 1912 . . . . . . . . . . . . . 14 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (∃𝑟(𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞) → ∃𝑟(𝑟𝑋 ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞))))
154153ad3antrrr 727 . . . . . . . . . . . . 13 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (∃𝑟(𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞) → ∃𝑟(𝑟𝑋 ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞))))
1551543ad2ant1 1130 . . . . . . . . . . . 12 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) → (∃𝑟(𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞) → ∃𝑟(𝑟𝑋 ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞))))
156155adantr 480 . . . . . . . . . . 11 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → (∃𝑟(𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞) → ∃𝑟(𝑟𝑋 ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞))))
157140, 156mpd 15 . . . . . . . . . 10 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → ∃𝑟(𝑟𝑋 ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)))
158 df-rex 3063 . . . . . . . . . 10 (∃𝑟𝑋 (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞) ↔ ∃𝑟(𝑟𝑋 ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)))
159157, 158sylibr 233 . . . . . . . . 9 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → ∃𝑟𝑋 (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞))
160133, 159r19.29a 3154 . . . . . . . 8 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → 𝑝𝐴𝑞)
161 df-br 5139 . . . . . . . 8 (𝑝𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴)
162160, 161sylib 217 . . . . . . 7 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ 𝑝𝑋𝑞𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
16340, 41, 42, 43, 162syl31anc 1370 . . . . . 6 (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) ∧ (𝑝𝑋𝑞𝑋)) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
16439, 163mpdan 684 . . . . 5 ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
165164ex 412 . . . 4 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴))
16619, 165relssdv 5778 . . 3 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴)
167 id 22 . . . . . 6 (𝑣 = (𝐷 “ (0[,)(𝑎 / 2))) → 𝑣 = (𝐷 “ (0[,)(𝑎 / 2))))
168167, 167coeq12d 5854 . . . . 5 (𝑣 = (𝐷 “ (0[,)(𝑎 / 2))) → (𝑣𝑣) = ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))))
169168sseq1d 4005 . . . 4 (𝑣 = (𝐷 “ (0[,)(𝑎 / 2))) → ((𝑣𝑣) ⊆ 𝐴 ↔ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴))
170169rspcev 3604 . . 3 (((𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ∧ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴) → ∃𝑣𝐹 (𝑣𝑣) ⊆ 𝐴)
17117, 166, 170syl2anc 583 . 2 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ∃𝑣𝐹 (𝑣𝑣) ⊆ 𝐴)
1729metustel 24369 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
173172adantl 481 . . 3 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
174173biimpa 476 . 2 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
175171, 174r19.29a 3154 1 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) → ∃𝑣𝐹 (𝑣𝑣) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wne 2932  wrex 3062  wss 3940  c0 4314  cop 4626   class class class wbr 5138  cmpt 5221   × cxp 5664  ccnv 5665  dom cdm 5666  ran crn 5667  cima 5669  ccom 5670  Rel wrel 5671  Fun wfun 6527  wf 6529  cfv 6533  (class class class)co 7401  cr 11104  0cc0 11105   + caddc 11108  *cxr 11243   < clt 11244  cle 11245   / cdiv 11867  2c2 12263  +crp 12970   +𝑒 cxad 13086  [,)cico 13322  PsMetcpsmet 21207
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718  ax-cnex 11161  ax-resscn 11162  ax-1cn 11163  ax-icn 11164  ax-addcl 11165  ax-addrcl 11166  ax-mulcl 11167  ax-mulrcl 11168  ax-mulcom 11169  ax-addass 11170  ax-mulass 11171  ax-distr 11172  ax-i2m1 11173  ax-1ne0 11174  ax-1rid 11175  ax-rnegex 11176  ax-rrecex 11177  ax-cnre 11178  ax-pre-lttri 11179  ax-pre-lttrn 11180  ax-pre-ltadd 11181  ax-pre-mulgt0 11182
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-nel 3039  df-ral 3054  df-rex 3063  df-rmo 3368  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-iun 4989  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-po 5578  df-so 5579  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-riota 7357  df-ov 7404  df-oprab 7405  df-mpo 7406  df-1st 7968  df-2nd 7969  df-er 8698  df-map 8817  df-en 8935  df-dom 8936  df-sdom 8937  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11442  df-neg 11443  df-div 11868  df-2 12271  df-rp 12971  df-xneg 13088  df-xadd 13089  df-xmul 13090  df-ico 13326  df-psmet 21215
This theorem is referenced by:  metust  24377
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