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Theorem metustexhalf 23712
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 12784 . . . . 5 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝑎 / 2) ∈ ℝ+)
4 eqidd 2739 . . . . 5 (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (𝐷 “ (0[,)𝑎))) → (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)(𝑎 / 2))))
5 oveq2 7283 . . . . . . 7 (𝑏 = (𝑎 / 2) → (0[,)𝑏) = (0[,)(𝑎 / 2)))
65imaeq2d 5969 . . . . . 6 (𝑏 = (𝑎 / 2) → (𝐷 “ (0[,)𝑏)) = (𝐷 “ (0[,)(𝑎 / 2))))
76rspceeqv 3575 . . . . 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 7283 . . . . . . . . . 10 (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
1110imaeq2d 5969 . . . . . . . . 9 (𝑎 = 𝑏 → (𝐷 “ (0[,)𝑎)) = (𝐷 “ (0[,)𝑏)))
1211cbvmptv 5187 . . . . . . . 8 (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
1312rneqi 5846 . . . . . . 7 ran (𝑎 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
149, 13eqtri 2766 . . . . . 6 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (𝐷 “ (0[,)𝑏)))
1514metustel 23706 . . . . 5 (𝐷 ∈ (PsMet‘𝑋) → ((𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ↔ ∃𝑏 ∈ ℝ+ (𝐷 “ (0[,)(𝑎 / 2))) = (𝐷 “ (0[,)𝑏))))
1615biimpar 478 . . . 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 6148 . . . . 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 6175 . . . . . . . . . 10 ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2))))
21 cnvimass 5989 . . . . . . . . . . . . . 14 (𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom 𝐷
22 psmetf 23459 . . . . . . . . . . . . . 14 (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
2321, 22fssdm 6620 . . . . . . . . . . . . 13 (𝐷 ∈ (PsMet‘𝑋) → (𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋))
24 dmss 5811 . . . . . . . . . . . . . 14 ((𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → dom (𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom (𝑋 × 𝑋))
25 rnss 5848 . . . . . . . . . . . . . 14 ((𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → ran (𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋))
26 xpss12 5604 . . . . . . . . . . . . . 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 482 . . . . . . . . . . 11 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)))
30 dmxp 5838 . . . . . . . . . . . . 13 (𝑋 ≠ ∅ → dom (𝑋 × 𝑋) = 𝑋)
31 rnxp 6073 . . . . . . . . . . . . 13 (𝑋 ≠ ∅ → ran (𝑋 × 𝑋) = 𝑋)
3230, 31xpeq12d 5620 . . . . . . . . . . . 12 (𝑋 ≠ ∅ → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋))
3332adantr 481 . . . . . . . . . . 11 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋))
3429, 33sseqtrd 3961 . . . . . . . . . 10 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝐷 “ (0[,)(𝑎 / 2))) × ran (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
3520, 34sstrid 3932 . . . . . . . . 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 3921 . . . . . . 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 5625 . . . . . . 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 1141 . . . . . . . . . . . . . 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 512 . . . . . . . . . . . 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 1142 . . . . . . . . . . . 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 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)) → 𝑞𝑋)
5148, 49, 503jca 1127 . . . . . . . . . . 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 1141 . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝐷 ∈ (PsMet‘𝑋))
5822ffund 6604 . . . . . . . . . . . . 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 1142 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝𝑋)
6155simp3d 1143 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑞𝑋)
6260, 61opelxpd 5627 . . . . . . . . . . . . 13 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
6322fdmd 6611 . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . 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 11022 . . . . . . . . . . . . . 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 496 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ+)
6968rpxrd 12773 . . . . . . . . . . . . 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, 62ffvelrnd 6962 . . . . . . . . . . . . 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 23465 . . . . . . . . . . . . . . 15 ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝𝑋𝑞𝑋) → 0 ≤ (𝑝𝐷𝑞))
7357, 60, 61, 72syl3anc 1370 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ≤ (𝑝𝐷𝑞))
74 df-ov 7278 . . . . . . . . . . . . . 14 (𝑝𝐷𝑞) = (𝐷‘⟨𝑝, 𝑞⟩)
7573, 74breqtrdi 5115 . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) ∈ ℝ*)
77 0red 10978 . . . . . . . . . . . . . . . . . . 19 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ∈ ℝ)
7868rpred 12772 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ)
7978rehalfcld 12220 . . . . . . . . . . . . . . . . . . . 20 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑎 / 2) ∈ ℝ)
8079rexrd 11025 . . . . . . . . . . . . . . . . . . 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 7278 . . . . . . . . . . . . . . . . . . . 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 5627 . . . . . . . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . . . . . . 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 5075 . . . . . . . . . . . . . . . . . . . . . 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 6930 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝐷 ∧ ⟨𝑝, 𝑟⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑟⟩) ∈ (0[,)(𝑎 / 2)) ↔ ⟨𝑝, 𝑟⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2)))))
8988biimpar 478 . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . 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 13143 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) → ((𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2))))
9392biimpa 477 . . . . . . . . . . . . . . . . . . . 20 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2)))
9493simp1d 1141 . . . . . . . . . . . . . . . . . . 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 7278 . . . . . . . . . . . . . . . . . . . 20 (𝑟𝐷𝑞) = (𝐷‘⟨𝑟, 𝑞⟩)
9782, 61opelxpd 5627 . . . . . . . . . . . . . . . . . . . . . 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 2842 . . . . . . . . . . . . . . . . . . . . 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 5075 . . . . . . . . . . . . . . . . . . . . . 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 6930 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝐷 ∧ ⟨𝑟, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑟, 𝑞⟩) ∈ (0[,)(𝑎 / 2)) ↔ ⟨𝑟, 𝑞⟩ ∈ (𝐷 “ (0[,)(𝑎 / 2)))))
103102biimpar 478 . . . . . . . . . . . . . . . . . . . . 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 2843 . . . . . . . . . . . . . . . . . . 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 13143 . . . . . . . . . . . . . . . . . . . . 21 ((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) → ((𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2))))
107106biimpa 477 . . . . . . . . . . . . . . . . . . . 20 (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2)))
108107simp1d 1141 . . . . . . . . . . . . . . . . . . 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 12968 . . . . . . . . . . . . . . . . 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 11004 . . . . . . . . . . . . . . . . 17 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)) ∈ ℝ)
112110, 111eqeltrd 2839 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) ∈ ℝ)
113112rexrd 11025 . . . . . . . . . . . . . . 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 23464 . . . . . . . . . . . . . . . 16 ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑝𝑋𝑞𝑋𝑟𝑋)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)))
11557, 60, 61, 82, 114syl13anc 1371 . . . . . . . . . . . . . . 15 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)))
11693simp3d 1143 . . . . . . . . . . . . . . . . . 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 1143 . . . . . . . . . . . . . . . . . 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 12221 . . . . . . . . . . . . . . . 16 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)) < 𝑎)
121110, 120eqbrtrd 5096 . . . . . . . . . . . . . . 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 12894 . . . . . . . . . . . . . 14 (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝𝑋𝑞𝑋) ∧ 𝑟𝑋) ∧ (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) < 𝑎)
12374, 122eqbrtrrid 5110 . . . . . . . . . . . . 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 13129 . . . . . . . . . . . 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 6930 . . . . . . . . . . . . . 14 ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎))))
126125biimpa 477 . . . . . . . . . . . . 13 (((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) ∧ (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎)) → ⟨𝑝, 𝑞⟩ ∈ (𝐷 “ (0[,)𝑎)))
127 df-br 5075 . . . . . . . . . . . . 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 496 . . . . . . . . . . 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 5085 . . . . . . . . . 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 256 . . . . . . . . 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 485 . . . . . . . . . . . . 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 5075 . . . . . . . . . . . . 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 3436 . . . . . . . . . . . . 13 𝑝 ∈ V
138 vex 3436 . . . . . . . . . . . . 13 𝑞 ∈ V
139137, 138brco 5779 . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑋 × 𝑋) = 𝑋)
144142, 143sseqtrd 3961 . . . . . . . . . . . . . . . . . . . 20 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋)
145144adantr 481 . . . . . . . . . . . . . . . . . . 19 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟) → ran (𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋)
146 vex 3436 . . . . . . . . . . . . . . . . . . . . 21 𝑟 ∈ V
147137, 146brelrn 5851 . . . . . . . . . . . . . . . . . . . 20 (𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟 ∈ ran (𝐷 “ (0[,)(𝑎 / 2))))
148147adantl 482 . . . . . . . . . . . . . . . . . . 19 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟 ∈ ran (𝐷 “ (0[,)(𝑎 / 2))))
149145, 148sseldd 3922 . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . 16 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑝(𝐷 “ (0[,)(𝑎 / 2)))𝑟𝑟(𝐷 “ (0[,)(𝑎 / 2)))𝑞) → 𝑟𝑋))
152151ancrd 552 . . . . . . . . . . . . . . 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 1920 . . . . . . . . . . . . . 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 1132 . . . . . . . . . . . 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 481 . . . . . . . . . . 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 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 3218 . . . . . . . 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 5075 . . . . . . . 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 1372 . . . . . 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 413 . . . 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 5698 . . 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 5773 . . . . 5 (𝑣 = (𝐷 “ (0[,)(𝑎 / 2))) → (𝑣𝑣) = ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))))
169168sseq1d 3952 . . . 4 (𝑣 = (𝐷 “ (0[,)(𝑎 / 2))) → ((𝑣𝑣) ⊆ 𝐴 ↔ ((𝐷 “ (0[,)(𝑎 / 2))) ∘ (𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴))
170169rspcev 3561 . . 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 23706 . . . 4 (𝐷 ∈ (PsMet‘𝑋) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
173172adantl 482 . . 3 ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐴𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎))))
174173biimpa 477 . 2 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (𝐷 “ (0[,)𝑎)))
175171, 174r19.29a 3218 1 (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴𝐹) → ∃𝑣𝐹 (𝑣𝑣) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  wne 2943  wrex 3065  wss 3887  c0 4256  cop 4567   class class class wbr 5074  cmpt 5157   × cxp 5587  ccnv 5588  dom cdm 5589  ran crn 5590  cima 5592  ccom 5593  Rel wrel 5594  Fun wfun 6427  wf 6429  cfv 6433  (class class class)co 7275  cr 10870  0cc0 10871   + caddc 10874  *cxr 11008   < clt 11009  cle 11010   / cdiv 11632  2c2 12028  +crp 12730   +𝑒 cxad 12846  [,)cico 13081  PsMetcpsmet 20581
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-1cn 10929  ax-icn 10930  ax-addcl 10931  ax-addrcl 10932  ax-mulcl 10933  ax-mulrcl 10934  ax-mulcom 10935  ax-addass 10936  ax-mulass 10937  ax-distr 10938  ax-i2m1 10939  ax-1ne0 10940  ax-1rid 10941  ax-rnegex 10942  ax-rrecex 10943  ax-cnre 10944  ax-pre-lttri 10945  ax-pre-lttrn 10946  ax-pre-ltadd 10947  ax-pre-mulgt0 10948
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rmo 3071  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-riota 7232  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-map 8617  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-sub 11207  df-neg 11208  df-div 11633  df-2 12036  df-rp 12731  df-xneg 12848  df-xadd 12849  df-xmul 12850  df-ico 13085  df-psmet 20589
This theorem is referenced by:  metust  23714
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