Theorem metustexhalf 24451

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)) ∧ 𝐴 ∈ 𝐹) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 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.

Step	Hyp	Ref	Expression
1		simp-4r 783	. . . 4 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋))
2		simplr 768	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → 𝑎 ∈ ℝ+)
3	2	rphalfcld 13014	. . . . 5 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (𝑎 / 2) ∈ ℝ+)
4		eqidd 2731	. . . . 5 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)(𝑎 / 2))))
5		oveq2 7398	. . . . . . 7 ⊢ (𝑏 = (𝑎 / 2) → (0[,)𝑏) = (0[,)(𝑎 / 2)))
6	5	imaeq2d 6034	. . . . . 6 ⊢ (𝑏 = (𝑎 / 2) → (◡𝐷 “ (0[,)𝑏)) = (◡𝐷 “ (0[,)(𝑎 / 2))))
7	6	rspceeqv 3614	. . . . 5 ⊢ (((𝑎 / 2) ∈ ℝ+ ∧ (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)(𝑎 / 2)))) → ∃𝑏 ∈ ℝ+ (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)𝑏)))
8	3, 4, 7	syl2anc 584	. . . 4 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 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 7398	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
11	10	imaeq2d 6034	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (◡𝐷 “ (0[,)𝑎)) = (◡𝐷 “ (0[,)𝑏)))
12	11	cbvmptv 5214	. . . . . . . 8 ⊢ (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
13	12	rneqi 5904	. . . . . . 7 ⊢ ran (𝑎 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
14	9, 13	eqtri 2753	. . . . . 6 ⊢ 𝐹 = ran (𝑏 ∈ ℝ+ ↦ (◡𝐷 “ (0[,)𝑏)))
15	14	metustel 24445	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ↔ ∃𝑏 ∈ ℝ+ (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)𝑏))))
16	15	biimpar 477	. . . 4 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ ∃𝑏 ∈ ℝ+ (◡𝐷 “ (0[,)(𝑎 / 2))) = (◡𝐷 “ (0[,)𝑏))) → (◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹)
17	1, 8, 16	syl2anc 584	. . 3 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹)
18		relco 6082	. . . . 5 ⊢ Rel ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))
19	18	a1i 11	. . . 4 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → Rel ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))))
20		cossxp 6248	. . . . . . . . . 10 ⊢ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2))))
21		cnvimass 6056	. . . . . . . . . . . . . 14 ⊢ (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom 𝐷
22		psmetf 24201	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (PsMet‘𝑋) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
23	21, 22	fssdm 6710	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋))
24		dmss 5869	. . . . . . . . . . . . . 14 ⊢ ((◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → dom (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ dom (𝑋 × 𝑋))
25		rnss 5906	. . . . . . . . . . . . . 14 ⊢ ((◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋))
26		xpss12 5656	. . . . . . . . . . . . . 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)))
27	24, 25, 26	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋) → (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)))
28	23, 27	syl 17	. . . . . . . . . . . 12 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)))
29	28	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)))
30		dmxp 5895	. . . . . . . . . . . . 13 ⊢ (𝑋 ≠ ∅ → dom (𝑋 × 𝑋) = 𝑋)
31		rnxp 6146	. . . . . . . . . . . . 13 ⊢ (𝑋 ≠ ∅ → ran (𝑋 × 𝑋) = 𝑋)
32	30, 31	xpeq12d 5672	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ ∅ → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋))
33	32	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋))
34	29, 33	sseqtrd 3986	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (◡𝐷 “ (0[,)(𝑎 / 2))) × ran (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
35	20, 34	sstrid 3961	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
36	35	ad3antrrr 730	. . . . . . . 8 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
37	36	sselda 3949	. . . . . . 7 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
38		opelxp 5677	. . . . . . 7 ⊢ (⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋) ↔ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))
39	37, 38	sylib 218	. . . . . 6 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))
40		simpll 766	. . . . . . 7 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
41		simprl 770	. . . . . . 7 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → 𝑝 ∈ 𝑋)
42		simprr 772	. . . . . . 7 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → 𝑞 ∈ 𝑋)
43		simplr 768	. . . . . . 7 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))))
44		simplll 774	. . . . . . . . . . . . . . 15 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))
45	44	simp1d 1142	. . . . . . . . . . . . . 14 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
46	45, 1	syl 17	. . . . . . . . . . . . 13 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝐷 ∈ (PsMet‘𝑋))
47	45, 2	syl 17	. . . . . . . . . . . . 13 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ+)
48	46, 47	jca 511	. . . . . . . . . . . 12 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+))
49	44	simp2d 1143	. . . . . . . . . . . 12 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝 ∈ 𝑋)
50	44	simp3d 1144	. . . . . . . . . . . 12 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑞 ∈ 𝑋)
51	48, 49, 50	3jca 1128	. . . . . . . . . . 11 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))
52		simplr 768	. . . . . . . . . . 11 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟 ∈ 𝑋)
53		simprl 770	. . . . . . . . . . 11 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟)
54		simprr 772	. . . . . . . . . . 11 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)
55		simpll 766	. . . . . . . . . . . . . . 15 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))
56	55	simp1d 1142	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+))
57	56	simpld 494	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝐷 ∈ (PsMet‘𝑋))
58	22	ffund 6695	. . . . . . . . . . . . 13 ⊢ (𝐷 ∈ (PsMet‘𝑋) → Fun 𝐷)
59	57, 58	syl 17	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → Fun 𝐷)
60	55	simp2d 1143	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝 ∈ 𝑋)
61	55	simp3d 1144	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑞 ∈ 𝑋)
62	60, 61	opelxpd 5680	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋))
63	22	fdmd 6701	. . . . . . . . . . . . . 14 ⊢ (𝐷 ∈ (PsMet‘𝑋) → dom 𝐷 = (𝑋 × 𝑋))
64	57, 63	syl 17	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → dom 𝐷 = (𝑋 × 𝑋))
65	62, 64	eleqtrrd 2832	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑞⟩ ∈ dom 𝐷)
66		0xr 11228	. . . . . . . . . . . . . 14 ⊢ 0 ∈ ℝ*
67	66	a1i 11	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ∈ ℝ*)
68	56	simprd 495	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ+)
69	68	rpxrd 13003	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ*)
70	57, 22	syl 17	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝐷:(𝑋 × 𝑋)⟶ℝ*)
71	70, 62	ffvelcdmd 7060	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ ℝ*)
72		psmetge0 24207	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → 0 ≤ (𝑝𝐷𝑞))
73	57, 60, 61, 72	syl3anc 1373	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ≤ (𝑝𝐷𝑞))
74		df-ov 7393	. . . . . . . . . . . . . 14 ⊢ (𝑝𝐷𝑞) = (𝐷‘⟨𝑝, 𝑞⟩)
75	73, 74	breqtrdi 5151	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ≤ (𝐷‘⟨𝑝, 𝑞⟩))
76	74, 71	eqeltrid 2833	. . . . . . . . . . . . . . 15 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) ∈ ℝ*)
77		0red 11184	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 0 ∈ ℝ)
78	68	rpred 13002	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑎 ∈ ℝ)
79	78	rehalfcld 12436	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑎 / 2) ∈ ℝ)
80	79	rexrd 11231	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑎 / 2) ∈ ℝ*)
81		df-ov 7393	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝𝐷𝑟) = (𝐷‘⟨𝑝, 𝑟⟩)
82		simplr 768	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟 ∈ 𝑋)
83	60, 82	opelxpd 5680	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑟⟩ ∈ (𝑋 × 𝑋))
84	83, 64	eleqtrrd 2832	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑟⟩ ∈ dom 𝐷)
85		simprl 770	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟)
86		df-br 5111	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ↔ ⟨𝑝, 𝑟⟩ ∈ (◡𝐷 “ (0[,)(𝑎 / 2))))
87	85, 86	sylib 218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑝, 𝑟⟩ ∈ (◡𝐷 “ (0[,)(𝑎 / 2))))
88		fvimacnv 7028	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝐷 ∧ ⟨𝑝, 𝑟⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑟⟩) ∈ (0[,)(𝑎 / 2)) ↔ ⟨𝑝, 𝑟⟩ ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))))
89	88	biimpar 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((Fun 𝐷 ∧ ⟨𝑝, 𝑟⟩ ∈ dom 𝐷) ∧ ⟨𝑝, 𝑟⟩ ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))) → (𝐷‘⟨𝑝, 𝑟⟩) ∈ (0[,)(𝑎 / 2)))
90	59, 84, 87, 89	syl21anc 837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷‘⟨𝑝, 𝑟⟩) ∈ (0[,)(𝑎 / 2)))
91	81, 90	eqeltrid 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2)))
92		elico2 13378	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) → ((𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2))))
93	92	biimpa 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2)))
94	93	simp1d 1142	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) ∈ ℝ)
95	77, 80, 91, 94	syl21anc 837	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) ∈ ℝ)
96		df-ov 7393	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟𝐷𝑞) = (𝐷‘⟨𝑟, 𝑞⟩)
97	82, 61	opelxpd 5680	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑟, 𝑞⟩ ∈ (𝑋 × 𝑋))
98	97, 64	eleqtrrd 2832	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑟, 𝑞⟩ ∈ dom 𝐷)
99		simprr 772	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)
100		df-br 5111	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞 ↔ ⟨𝑟, 𝑞⟩ ∈ (◡𝐷 “ (0[,)(𝑎 / 2))))
101	99, 100	sylib 218	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ⟨𝑟, 𝑞⟩ ∈ (◡𝐷 “ (0[,)(𝑎 / 2))))
102		fvimacnv 7028	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝐷 ∧ ⟨𝑟, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑟, 𝑞⟩) ∈ (0[,)(𝑎 / 2)) ↔ ⟨𝑟, 𝑞⟩ ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))))
103	102	biimpar 477	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((Fun 𝐷 ∧ ⟨𝑟, 𝑞⟩ ∈ dom 𝐷) ∧ ⟨𝑟, 𝑞⟩ ∈ (◡𝐷 “ (0[,)(𝑎 / 2)))) → (𝐷‘⟨𝑟, 𝑞⟩) ∈ (0[,)(𝑎 / 2)))
104	59, 98, 101, 103	syl21anc 837	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷‘⟨𝑟, 𝑞⟩) ∈ (0[,)(𝑎 / 2)))
105	96, 104	eqeltrid 2833	. . . . . . . . . . . . . . . . . . 19 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2)))
106		elico2 13378	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) → ((𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2))))
107	106	biimpa 476	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2)))
108	107	simp1d 1142	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) ∈ ℝ)
109	77, 80, 105, 108	syl21anc 837	. . . . . . . . . . . . . . . . . 18 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑟𝐷𝑞) ∈ ℝ)
110	95, 109	rexaddd 13201	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) = ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)))
111	95, 109	readdcld 11210	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)) ∈ ℝ)
112	110, 111	eqeltrd 2829	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) ∈ ℝ)
113	112	rexrd 11231	. . . . . . . . . . . . . . 15 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) ∈ ℝ*)
114		psmettri 24206	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)))
115	57, 60, 61, 82, 114	syl13anc 1374	. . . . . . . . . . . . . . 15 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)))
116	93	simp3d 1144	. . . . . . . . . . . . . . . . . 18 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) < (𝑎 / 2))
117	77, 80, 91, 116	syl21anc 837	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑟) < (𝑎 / 2))
118	107	simp3d 1144	. . . . . . . . . . . . . . . . . 18 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) < (𝑎 / 2))
119	77, 80, 105, 118	syl21anc 837	. . . . . . . . . . . . . . . . 17 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑟𝐷𝑞) < (𝑎 / 2))
120	95, 109, 78, 117, 119	lt2halvesd 12437	. . . . . . . . . . . . . . . 16 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) + (𝑟𝐷𝑞)) < 𝑎)
121	110, 120	eqbrtrd 5132	. . . . . . . . . . . . . . 15 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → ((𝑝𝐷𝑟) +𝑒 (𝑟𝐷𝑞)) < 𝑎)
122	76, 113, 69, 115, 121	xrlelttrd 13127	. . . . . . . . . . . . . 14 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐷𝑞) < 𝑎)
123	74, 122	eqbrtrrid 5146	. . . . . . . . . . . . 13 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷‘⟨𝑝, 𝑞⟩) < 𝑎)
124	67, 69, 71, 75, 123	elicod 13363	. . . . . . . . . . . 12 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎))
125		fvimacnv 7028	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎))))
126	125	biimpa 476	. . . . . . . . . . . . 13 ⊢ (((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) ∧ (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎)) → ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎)))
127		df-br 5111	. . . . . . . . . . . . 13 ⊢ (𝑝(◡𝐷 “ (0[,)𝑎))𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ (◡𝐷 “ (0[,)𝑎)))
128	126, 127	sylibr 234	. . . . . . . . . . . 12 ⊢ (((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) ∧ (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎)) → 𝑝(◡𝐷 “ (0[,)𝑎))𝑞)
129	59, 65, 124, 128	syl21anc 837	. . . . . . . . . . 11 ⊢ (((((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑎 ∈ ℝ+) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝(◡𝐷 “ (0[,)𝑎))𝑞)
130	51, 52, 53, 54, 129	syl22anc 838	. . . . . . . . . 10 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝(◡𝐷 “ (0[,)𝑎))𝑞)
131	45	simprd 495	. . . . . . . . . . 11 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝐴 = (◡𝐷 “ (0[,)𝑎)))
132	131	breqd 5121	. . . . . . . . . 10 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → (𝑝𝐴𝑞 ↔ 𝑝(◡𝐷 “ (0[,)𝑎))𝑞))
133	130, 132	mpbird 257	. . . . . . . . 9 ⊢ (((((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ 𝑟 ∈ 𝑋) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑝𝐴𝑞)
134		simpr 484	. . . . . . . . . . . . 13 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))))
135		df-br 5111	. . . . . . . . . . . . 13 ⊢ (𝑝((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))))
136	134, 135	sylibr 234	. . . . . . . . . . . 12 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → 𝑝((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))𝑞)
137		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑝 ∈ V
138		vex 3454	. . . . . . . . . . . . 13 ⊢ 𝑞 ∈ V
139	137, 138	brco 5837	. . . . . . . . . . . 12 ⊢ (𝑝((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))𝑞 ↔ ∃𝑟(𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞))
140	136, 139	sylib 218	. . . . . . . . . . 11 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → ∃𝑟(𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞))
141	23	adantl 481	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ (𝑋 × 𝑋))
142	141, 25	syl 17	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ ran (𝑋 × 𝑋))
143	31	adantr 480	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑋 × 𝑋) = 𝑋)
144	142, 143	sseqtrd 3986	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋)
145	144	adantr 480	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → ran (◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋)
146		vex 3454	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑟 ∈ V
147	137, 146	brelrn 5909	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 → 𝑟 ∈ ran (◡𝐷 “ (0[,)(𝑎 / 2))))
148	147	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟 ∈ ran (◡𝐷 “ (0[,)(𝑎 / 2))))
149	145, 148	sseldd 3950	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟 ∈ 𝑋)
150	149	adantrr 717	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟 ∈ 𝑋)
151	150	ex 412	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞) → 𝑟 ∈ 𝑋))
152	151	ancrd 551	. . . . . . . . . . . . . . 15 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞) → (𝑟 ∈ 𝑋 ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞))))
153	152	eximdv 1917	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (∃𝑟(𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞) → ∃𝑟(𝑟 ∈ 𝑋 ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞))))
154	153	ad3antrrr 730	. . . . . . . . . . . . 13 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (∃𝑟(𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞) → ∃𝑟(𝑟 ∈ 𝑋 ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞))))
155	154	3ad2ant1 1133	. . . . . . . . . . . 12 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → (∃𝑟(𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞) → ∃𝑟(𝑟 ∈ 𝑋 ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞))))
156	155	adantr 480	. . . . . . . . . . 11 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → (∃𝑟(𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞) → ∃𝑟(𝑟 ∈ 𝑋 ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞))))
157	140, 156	mpd 15	. . . . . . . . . 10 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → ∃𝑟(𝑟 ∈ 𝑋 ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)))
158		df-rex 3055	. . . . . . . . . 10 ⊢ (∃𝑟 ∈ 𝑋 (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞) ↔ ∃𝑟(𝑟 ∈ 𝑋 ∧ (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)))
159	157, 158	sylibr 234	. . . . . . . . 9 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → ∃𝑟 ∈ 𝑋 (𝑝(◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(◡𝐷 “ (0[,)(𝑎 / 2)))𝑞))
160	133, 159	r19.29a 3142	. . . . . . . 8 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → 𝑝𝐴𝑞)
161		df-br 5111	. . . . . . . 8 ⊢ (𝑝𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴)
162	160, 161	sylib 218	. . . . . . 7 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
163	40, 41, 42, 43, 162	syl31anc 1375	. . . . . 6 ⊢ (((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋)) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
164	39, 163	mpdan 687	. . . . 5 ⊢ ((((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) ∧ ⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2))))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴)
165	164	ex 412	. . . 4 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → (⟨𝑝, 𝑞⟩ ∈ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) → ⟨𝑝, 𝑞⟩ ∈ 𝐴))
166	19, 165	relssdv 5754	. . 3 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 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))))
168	167, 167	coeq12d 5831	. . . . 5 ⊢ (𝑣 = (◡𝐷 “ (0[,)(𝑎 / 2))) → (𝑣 ∘ 𝑣) = ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))))
169	168	sseq1d 3981	. . . 4 ⊢ (𝑣 = (◡𝐷 “ (0[,)(𝑎 / 2))) → ((𝑣 ∘ 𝑣) ⊆ 𝐴 ↔ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴))
170	169	rspcev 3591	. . 3 ⊢ (((◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ∧ ((◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 𝐴)
171	17, 166, 170	syl2anc 584	. 2 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ+) ∧ 𝐴 = (◡𝐷 “ (0[,)𝑎))) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 𝐴)
172	9	metustel 24445	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
173	172	adantl 481	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎))))
174	173	biimpa 476	. 2 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) → ∃𝑎 ∈ ℝ+ 𝐴 = (◡𝐷 “ (0[,)𝑎)))
175	171, 174	r19.29a 3142	1 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540  ∃wex 1779   ∈ wcel 2109   ≠ wne 2926  ∃wrex 3054   ⊆ wss 3917  ∅c0 4299  ⟨cop 4598   class class class wbr 5110   ↦ cmpt 5191   × cxp 5639  ^◡ccnv 5640  dom cdm 5641  ran crn 5642   “ cima 5644   ∘ ccom 5645  Rel wrel 5646  Fun wfun 6508  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390  ℝcr 11074  0cc0 11075   + caddc 11078  ℝ^*cxr 11214   < clt 11215   ≤ cle 11216   / cdiv 11842  2c2 12248  ℝ⁺crp 12958   +_𝑒 cxad 13077  [,)cico 13315  PsMetcpsmet 21255

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-map 8804  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-rp 12959  df-xneg 13079  df-xadd 13080  df-xmul 13081  df-ico 13319  df-psmet 21263

This theorem is referenced by:  metust  24453