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Theorem metustexhalf 24570
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 783 . . . 4 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋))
2 simplr 768 . . . . . 6 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → 𝑎 ∈ ℝ+)
32rphalfcld 13090 . . . . 5 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝑎 / 2) ∈ ℝ+)
4 eqidd 2737 . . . . 5 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)(𝑎 / 2))))
5 oveq2 7440 . . . . . . 7 (𝑏 = (𝑎 / 2) → (0[,)𝑏) = (0[,)(𝑎 / 2)))
65imaeq2d 6077 . . . . . 6 (𝑏 = (𝑎 / 2) → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)(𝑎 / 2))))
76rspceeqv 3644 . . . . 5 (((𝑎 / 2) ∈ ℝ+ ∧ (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)(𝑎 / 2)))) → ∃𝑏 ∈ ℝ+ (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)𝑏)))
83, 4, 7syl2anc 584 . . . 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 7440 . . . . . . . . . 10 (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
1110imaeq2d 6077 . . . . . . . . 9 (𝑎 = 𝑏 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑏)))
1211cbvmptv 5254 . . . . . . . 8 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
1312rneqi 5947 . . . . . . 7 ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
149, 13eqtri 2764 . . . . . 6 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
1514metustel 24564 . . . . 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 584 . . 3 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹)
18 relco 6125 . . . . 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 6291 . . . . . . . . . 10 ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2))))
21 cnvimass 6099 . . . . . . . . . . . . . 14 (𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom 𝐷
22 psmetf 24317 . . . . . . . . . . . . . 14 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2321, 22fssdm 6754 . . . . . . . . . . . . 13 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋))
24 dmss 5912 . . . . . . . . . . . . . 14 ((𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → dom (𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom (𝑋 × 𝑋))
25 rnss 5949 . . . . . . . . . . . . . 14 ((𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → ran (𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋))
26 xpss12 5699 . . . . . . . . . . . . . 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 584 . . . . . . . . . . . . 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 5938 . . . . . . . . . . . . 13 (𝑋 ≠ ∅ → dom (𝑋 × 𝑋) = 𝑋)
31 rnxp 6189 . . . . . . . . . . . . 13 (𝑋 ≠ ∅ → ran (𝑋 × 𝑋) = 𝑋)
3230, 31xpeq12d 5715 . . . . . . . . . . . 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 4019 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
3520, 34sstrid 3994 . . . . . . . . 9 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
3635ad3antrrr 730 . . . . . . . 8 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
3736sselda 3982 . . . . . . 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 5720 . . . . . . 7 (⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋) ↔ (𝑝𝑋𝑞𝑋))
3937, 38sylib 218 . . . . . 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 766 . . . . . . 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 770 . . . . . . 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 772 . . . . . . 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 768 . . . . . . 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 774 . . . . . . . . . . . . . . 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 1142 . . . . . . . . . . . . . 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 1143 . . . . . . . . . . . 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 1144 . . . . . . . . . . . 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 1128 . . . . . . . . . . 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 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)) → 𝑟𝑋)
53 simprl 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)) → 𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟)
54 simprr 772 . . . . . . . . . . 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 766 . . . . . . . . . . . . . . 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 1142 . . . . . . . . . . . . . 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 6739 . . . . . . . . . . . . 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 1143 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝𝑋)
6155simp3d 1144 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑞𝑋)
6260, 61opelxpd 5723 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
6322fdmd 6745 . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . 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 11309 . . . . . . . . . . . . . 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 13079 . . . . . . . . . . . . 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 7104 . . . . . . . . . . . . 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 24323 . . . . . . . . . . . . . . 15 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝𝑋𝑞𝑋) → 0 ≤ (𝑝𝐷𝑞))
7357, 60, 61, 72syl3anc 1372 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ≤ (𝑝𝐷𝑞))
74 df-ov 7435 . . . . . . . . . . . . . 14 (𝑝𝐷𝑞) = (𝐷‘⟨𝑝, 𝑞⟩)
7573, 74breqtrdi 5183 . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) ∈ ℝ*)
77 0red 11265 . . . . . . . . . . . . . . . . . . 19 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ∈ ℝ)
7868rpred 13078 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ)
7978rehalfcld 12515 . . . . . . . . . . . . . . . . . . . 20 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑎 / 2) ∈ ℝ)
8079rexrd 11312 . . . . . . . . . . . . . . . . . . 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 7435 . . . . . . . . . . . . . . . . . . . 20 (𝑝𝐷𝑟) = (𝐷‘⟨𝑝, 𝑟⟩)
82 simplr 768 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟𝑋)
8360, 82opelxpd 5723 . . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . . . 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 770 . . . . . . . . . . . . . . . . . . . . . 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 5143 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟 ↔ ⟨𝑝, 𝑟⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2))))
8785, 86sylib 218 . . . . . . . . . . . . . . . . . . . . 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 7072 . . . . . . . . . . . . . . . . . . . . . 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 837 . . . . . . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . . . . . 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 13452 . . . . . . . . . . . . . . . . . . . . 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 1142 . . . . . . . . . . . . . . . . . . 19 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) ∈ ℝ)
9577, 80, 91, 94syl21anc 837 . . . . . . . . . . . . . . . . . 18 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) ∈ ℝ)
96 df-ov 7435 . . . . . . . . . . . . . . . . . . . 20 (𝑟𝐷𝑞) = (𝐷‘⟨𝑟, 𝑞⟩)
9782, 61opelxpd 5723 . . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . . . 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 772 . . . . . . . . . . . . . . . . . . . . . 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 5143 . . . . . . . . . . . . . . . . . . . . . 22 (𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞 ↔ ⟨𝑟, 𝑞⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2))))
10199, 100sylib 218 . . . . . . . . . . . . . . . . . . . . 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 7072 . . . . . . . . . . . . . . . . . . . . . 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 837 . . . . . . . . . . . . . . . . . . . 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 2844 . . . . . . . . . . . . . . . . . . 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 13452 . . . . . . . . . . . . . . . . . . . . 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 1142 . . . . . . . . . . . . . . . . . . 19 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) ∈ ℝ)
10977, 80, 105, 108syl21anc 837 . . . . . . . . . . . . . . . . . 18 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑟𝐷𝑞) ∈ ℝ)
11095, 109rexaddd 13277 . . . . . . . . . . . . . . . . 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 11291 . . . . . . . . . . . . . . . . 17 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)) ∈ ℝ)
112110, 111eqeltrd 2840 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) ∈ ℝ)
113112rexrd 11312 . . . . . . . . . . . . . . 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 24322 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑝𝑋𝑞𝑋𝑟𝑋)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)))
11557, 60, 61, 82, 114syl13anc 1373 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)))
11693simp3d 1144 . . . . . . . . . . . . . . . . . 18 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) < (𝑎 / 2))
11777, 80, 91, 116syl21anc 837 . . . . . . . . . . . . . . . . 17 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) < (𝑎 / 2))
118107simp3d 1144 . . . . . . . . . . . . . . . . . 18 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) < (𝑎 / 2))
11977, 80, 105, 118syl21anc 837 . . . . . . . . . . . . . . . . 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 12516 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)) < 𝑎)
121110, 120eqbrtrd 5164 . . . . . . . . . . . . . . 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 13203 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) < 𝑎)
12374, 122eqbrtrrid 5178 . . . . . . . . . . . . 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 13438 . . . . . . . . . . . 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 7072 . . . . . . . . . . . . . 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 5143 . . . . . . . . . . . . 13 (𝑝(𝐷 “ (0[,)𝑎))𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
128126, 127sylibr 234 . . . . . . . . . . . 12 (((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) ∧ (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎)) → 𝑝(𝐷 “ (0[,)𝑎))𝑞)
12959, 65, 124, 128syl21anc 837 . . . . . . . . . . 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 838 . . . . . . . . . 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 5153 . . . . . . . . . 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 5143 . . . . . . . . . . . . 13 (𝑝((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2))))𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))))
136134, 135sylibr 234 . . . . . . . . . . . 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 3483 . . . . . . . . . . . . 13 𝑝 ∈ V
138 vex 3483 . . . . . . . . . . . . 13 𝑞 ∈ V
139137, 138brco 5880 . . . . . . . . . . . 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 218 . . . . . . . . . . 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 4019 . . . . . . . . . . . . . . . . . . . 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 3483 . . . . . . . . . . . . . . . . . . . . 21 𝑟 ∈ V
147137, 146brelrn 5952 . . . . . . . . . . . . . . . . . . . 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 3983 . . . . . . . . . . . . . . . . . 18 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟𝑋)
150149adantrr 717 . . . . . . . . . . . . . . . . 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 1916 . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . 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 1133 . . . . . . . . . . . 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 3070 . . . . . . . . . 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 234 . . . . . . . . 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 3161 . . . . . . . 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 5143 . . . . . . . 8 (𝑝𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴)
162160, 161sylib 218 . . . . . . 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 1374 . . . . . 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 687 . . . . 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 5797 . . 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 5874 . . . . 5 (𝑣 = (𝐷 “ (0[,)(𝑎 / 2))) → (𝑣𝑣) = ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))))
169168sseq1d 4014 . . . 4 (𝑣 = (𝐷 “ (0[,)(𝑎 / 2))) → ((𝑣𝑣) ⊆ 𝐴 ↔ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴))
170169rspcev 3621 . . 3 (((𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ∧ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴) → ∃𝑣𝐹 (𝑣𝑣) ⊆ 𝐴)
17117, 166, 170syl2anc 584 . 2 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → ∃𝑣𝐹 (𝑣𝑣) ⊆ 𝐴)
1729metustel 24564 . . . 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 3161 1 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) → ∃𝑣𝐹 (𝑣𝑣) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  wne 2939  wrex 3069  wss 3950  c0 4332  cop 4631   class class class wbr 5142  cmpt 5224   × cxp 5682  ccnv 5683  dom cdm 5684  ran crn 5685  cima 5687  ccom 5688  Rel wrel 5689  Fun wfun 6554  wf 6556  cfv 6560  (class class class)co 7432  cr 11155  0cc0 11156   + caddc 11159  *cxr 11295   < clt 11296  cle 11297   / cdiv 11921  2c2 12322  +crp 13035   +𝑒 cxad 13153  [,)cico 13390  PsMetcpsmet 21349
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756  ax-cnex 11212  ax-resscn 11213  ax-1cn 11214  ax-icn 11215  ax-addcl 11216  ax-addrcl 11217  ax-mulcl 11218  ax-mulrcl 11219  ax-mulcom 11220  ax-addass 11221  ax-mulass 11222  ax-distr 11223  ax-i2m1 11224  ax-1ne0 11225  ax-1rid 11226  ax-rnegex 11227  ax-rrecex 11228  ax-cnre 11229  ax-pre-lttri 11230  ax-pre-lttrn 11231  ax-pre-ltadd 11232  ax-pre-mulgt0 11233
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-riota 7389  df-ov 7435  df-oprab 7436  df-mpo 7437  df-om 7889  df-1st 8015  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-recs 8412  df-rdg 8451  df-er 8746  df-map 8869  df-en 8987  df-dom 8988  df-sdom 8989  df-pnf 11298  df-mnf 11299  df-xr 11300  df-ltxr 11301  df-le 11302  df-sub 11495  df-neg 11496  df-div 11922  df-nn 12268  df-2 12330  df-rp 13036  df-xneg 13155  df-xadd 13156  df-xmul 13157  df-ico 13394  df-psmet 21357
This theorem is referenced by:  metust  24572
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