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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)) ∧ 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 781	. . . 4 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ⁺) ∧ 𝐴 = (^◡𝐷 “ (0[,)𝑎))) → 𝐷 ∈ (PsMet‘𝑋))
2		simplr 766	. . . . . 6 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ⁺) ∧ 𝐴 = (^◡𝐷 “ (0[,)𝑎))) → 𝑎 ∈ ℝ⁺)
3	2	rphalfcld 12784	. . . . 5 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 𝑎 ∈ ℝ⁺) ∧ 𝐴 = (^◡𝐷 “ (0[,)𝑎))) → (𝑎 / 2) ∈ ℝ⁺)
4		eqidd 2739	. . . . 5 ⊢ (((((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) ∧ 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)))
6	5	imaeq2d 5969	. . . . . 6 ⊢ (𝑏 = (𝑎 / 2) → (^◡𝐷 “ (0[,)𝑏)) = (^◡𝐷 “ (0[,)(𝑎 / 2))))
7	6	rspceeqv 3575	. . . . 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 7283	. . . . . . . . . 10 ⊢ (𝑎 = 𝑏 → (0[,)𝑎) = (0[,)𝑏))
11	10	imaeq2d 5969	. . . . . . . . 9 ⊢ (𝑎 = 𝑏 → (^◡𝐷 “ (0[,)𝑎)) = (^◡𝐷 “ (0[,)𝑏)))
12	11	cbvmptv 5187	. . . . . . . 8 ⊢ (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎))) = (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))
13	12	rneqi 5846	. . . . . . 7 ⊢ ran (𝑎 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑎))) = ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))
14	9, 13	eqtri 2766	. . . . . 6 ⊢ 𝐹 = ran (𝑏 ∈ ℝ⁺ ↦ (^◡𝐷 “ (0[,)𝑏)))
15	14	metustel 23706	. . . . 5 ⊢ (𝐷 ∈ (PsMet‘𝑋) → ((^◡𝐷 “ (0[,)(𝑎 / 2))) ∈ 𝐹 ↔ ∃𝑏 ∈ ℝ⁺ (^◡𝐷 “ (0[,)(𝑎 / 2))) = (^◡𝐷 “ (0[,)𝑏))))
16	15	biimpar 478	. . . 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 6148	. . . . 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 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^*)
23	21, 22	fssdm 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)))
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 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)
32	30, 31	xpeq12d 5620	. . . . . . . . . . . 12 ⊢ (𝑋 ≠ ∅ → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋))
33	32	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (𝑋 × 𝑋) × ran (𝑋 × 𝑋)) = (𝑋 × 𝑋))
34	29, 33	sseqtrd 3961	. . . . . . . . . 10 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (dom (^◡𝐷 “ (0[,)(𝑎 / 2))) × ran (^◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
35	20, 34	sstrid 3932	. . . . . . . . 9 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((^◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (^◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ (𝑋 × 𝑋))
36	35	ad3antrrr 727	. . . . . . . 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 3921	. . . . . . 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 5625	. . . . . . 7 ⊢ (⟨𝑝, 𝑞⟩ ∈ (𝑋 × 𝑋) ↔ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋))
39	37, 38	sylib 217	. . . . . 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 764	. . . . . . 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 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 ∈ 𝑋)
42		simprr 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)) → 𝑞 ∈ 𝑋)
43		simplr 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)) → ⟨𝑝, 𝑞⟩ ∈ ((^◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (^◡𝐷 “ (0[,)(𝑎 / 2)))))
44		simplll 772	. . . . . . . . . . . . . . 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 1141	. . . . . . . . . . . . . 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 512	. . . . . . . . . . . 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 1142	. . . . . . . . . . . 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 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)) → 𝑞 ∈ 𝑋)
51	48, 49, 50	3jca 1127	. . . . . . . . . . 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 766	. . . . . . . . . . 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 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)) → 𝑝(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑟)
54		simprr 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)) → 𝑟(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)
55		simpll 764	. . . . . . . . . . . . . . 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 1141	. . . . . . . . . . . . . 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 495	. . . . . . . . . . . . 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 6604	. . . . . . . . . . . . 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 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)) → 𝑝 ∈ 𝑋)
61	55	simp3d 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)) → 𝑞 ∈ 𝑋)
62	60, 61	opelxpd 5627	. . . . . . . . . . . . 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 6611	. . . . . . . . . . . . . 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 2842	. . . . . . . . . . . 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 11022	. . . . . . . . . . . . . 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 496	. . . . . . . . . . . . . 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 12773	. . . . . . . . . . . . 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	ffvelrnd 6962	. . . . . . . . . . . . 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 23465	. . . . . . . . . . . . . . 15 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ 𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋) → 0 ≤ (𝑝𝐷𝑞))
73	57, 60, 61, 72	syl3anc 1370	. . . . . . . . . . . . . 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 7278	. . . . . . . . . . . . . 14 ⊢ (𝑝𝐷𝑞) = (𝐷‘⟨𝑝, 𝑞⟩)
75	73, 74	breqtrdi 5115	. . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . 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 10978	. . . . . . . . . . . . . . . . . . 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 12772	. . . . . . . . . . . . . . . . . . . . 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 12220	. . . . . . . . . . . . . . . . . . . 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 11025	. . . . . . . . . . . . . . . . . . 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 7278	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝𝐷𝑟) = (𝐷‘⟨𝑝, 𝑟⟩)
82		simplr 766	. . . . . . . . . . . . . . . . . . . . . . 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 5627	. . . . . . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . . . . . . 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 768	. . . . . . . . . . . . . . . . . . . . . 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 5075	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑝(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ↔ ⟨𝑝, 𝑟⟩ ∈ (^◡𝐷 “ (0[,)(𝑎 / 2))))
87	85, 86	sylib 217	. . . . . . . . . . . . . . . . . . . . 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 6930	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝐷 ∧ ⟨𝑝, 𝑟⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑟⟩) ∈ (0[,)(𝑎 / 2)) ↔ ⟨𝑝, 𝑟⟩ ∈ (^◡𝐷 “ (0[,)(𝑎 / 2)))))
89	88	biimpar 478	. . . . . . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . . . . . 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 13143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ^*) → ((𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2))))
93	92	biimpa 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ^*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → ((𝑝𝐷𝑟) ∈ ℝ ∧ 0 ≤ (𝑝𝐷𝑟) ∧ (𝑝𝐷𝑟) < (𝑎 / 2)))
94	93	simp1d 1141	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ^*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) ∈ ℝ)
95	77, 80, 91, 94	syl21anc 835	. . . . . . . . . . . . . . . . . 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 7278	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑟𝐷𝑞) = (𝐷‘⟨𝑟, 𝑞⟩)
97	82, 61	opelxpd 5627	. . . . . . . . . . . . . . . . . . . . . 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 2842	. . . . . . . . . . . . . . . . . . . . 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 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)) → 𝑟(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)
100		df-br 5075	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑟(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑞 ↔ ⟨𝑟, 𝑞⟩ ∈ (^◡𝐷 “ (0[,)(𝑎 / 2))))
101	99, 100	sylib 217	. . . . . . . . . . . . . . . . . . . . 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 6930	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((Fun 𝐷 ∧ ⟨𝑟, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑟, 𝑞⟩) ∈ (0[,)(𝑎 / 2)) ↔ ⟨𝑟, 𝑞⟩ ∈ (^◡𝐷 “ (0[,)(𝑎 / 2)))))
103	102	biimpar 478	. . . . . . . . . . . . . . . . . . . . 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 835	. . . . . . . . . . . . . . . . . . . 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 2843	. . . . . . . . . . . . . . . . . . 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 13143	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ^*) → ((𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2)) ↔ ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2))))
107	106	biimpa 477	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ^*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → ((𝑟𝐷𝑞) ∈ ℝ ∧ 0 ≤ (𝑟𝐷𝑞) ∧ (𝑟𝐷𝑞) < (𝑎 / 2)))
108	107	simp1d 1141	. . . . . . . . . . . . . . . . . . 19 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ^*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) ∈ ℝ)
109	77, 80, 105, 108	syl21anc 835	. . . . . . . . . . . . . . . . . 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 12968	. . . . . . . . . . . . . . . . 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 11004	. . . . . . . . . . . . . . . . 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 2839	. . . . . . . . . . . . . . . 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 11025	. . . . . . . . . . . . . . 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 23464	. . . . . . . . . . . . . . . 16 ⊢ ((𝐷 ∈ (PsMet‘𝑋) ∧ (𝑝 ∈ 𝑋 ∧ 𝑞 ∈ 𝑋 ∧ 𝑟 ∈ 𝑋)) → (𝑝𝐷𝑞) ≤ ((𝑝𝐷𝑟) +_𝑒 (𝑟𝐷𝑞)))
115	57, 60, 61, 82, 114	syl13anc 1371	. . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . . . . . . 18 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ^*) ∧ (𝑝𝐷𝑟) ∈ (0[,)(𝑎 / 2))) → (𝑝𝐷𝑟) < (𝑎 / 2))
117	77, 80, 91, 116	syl21anc 835	. . . . . . . . . . . . . . . . 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 1143	. . . . . . . . . . . . . . . . . 18 ⊢ (((0 ∈ ℝ ∧ (𝑎 / 2) ∈ ℝ^*) ∧ (𝑟𝐷𝑞) ∈ (0[,)(𝑎 / 2))) → (𝑟𝐷𝑞) < (𝑎 / 2))
119	77, 80, 105, 118	syl21anc 835	. . . . . . . . . . . . . . . . 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 12221	. . . . . . . . . . . . . . . 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 5096	. . . . . . . . . . . . . . 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 12894	. . . . . . . . . . . . . 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 5110	. . . . . . . . . . . . 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 13129	. . . . . . . . . . . 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 6930	. . . . . . . . . . . . . 14 ⊢ ((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) → ((𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎) ↔ ⟨𝑝, 𝑞⟩ ∈ (^◡𝐷 “ (0[,)𝑎))))
126	125	biimpa 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)))
128	126, 127	sylibr 233	. . . . . . . . . . . 12 ⊢ (((Fun 𝐷 ∧ ⟨𝑝, 𝑞⟩ ∈ dom 𝐷) ∧ (𝐷‘⟨𝑝, 𝑞⟩) ∈ (0[,)𝑎)) → 𝑝(^◡𝐷 “ (0[,)𝑎))𝑞)
129	59, 65, 124, 128	syl21anc 835	. . . . . . . . . . 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 836	. . . . . . . . . 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 496	. . . . . . . . . . 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 5085	. . . . . . . . . 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 256	. . . . . . . . 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 485	. . . . . . . . . . . . 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 5075	. . . . . . . . . . . . 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 233	. . . . . . . . . . . 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 3436	. . . . . . . . . . . . 13 ⊢ 𝑝 ∈ V
138		vex 3436	. . . . . . . . . . . . 13 ⊢ 𝑞 ∈ V
139	137, 138	brco 5779	. . . . . . . . . . . 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 217	. . . . . . . . . . 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 482	. . . . . . . . . . . . . . . . . . . . . 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 481	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (𝑋 × 𝑋) = 𝑋)
144	142, 143	sseqtrd 3961	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ran (^◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋)
145	144	adantr 481	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → ran (^◡𝐷 “ (0[,)(𝑎 / 2))) ⊆ 𝑋)
146		vex 3436	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑟 ∈ V
147	137, 146	brelrn 5851	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑝(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 → 𝑟 ∈ ran (^◡𝐷 “ (0[,)(𝑎 / 2))))
148	147	adantl 482	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟 ∈ ran (^◡𝐷 “ (0[,)(𝑎 / 2))))
149	145, 148	sseldd 3922	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝑝(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑟) → 𝑟 ∈ 𝑋)
150	149	adantrr 714	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ (𝑝(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑞)) → 𝑟 ∈ 𝑋)
151	150	ex 413	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → ((𝑝(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑟 ∧ 𝑟(^◡𝐷 “ (0[,)(𝑎 / 2)))𝑞) → 𝑟 ∈ 𝑋))
152	151	ancrd 552	. . . . . . . . . . . . . . 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 1920	. . . . . . . . . . . . . 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 727	. . . . . . . . . . . . 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 1132	. . . . . . . . . . . 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 481	. . . . . . . . . . 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 3070	. . . . . . . . . 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 233	. . . . . . . . 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 3218	. . . . . . . 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 5075	. . . . . . . 8 ⊢ (𝑝𝐴𝑞 ↔ ⟨𝑝, 𝑞⟩ ∈ 𝐴)
162	160, 161	sylib 217	. . . . . . 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 1372	. . . . . 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 684	. . . . 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 413	. . . 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 5698	. . 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 5773	. . . . 5 ⊢ (𝑣 = (^◡𝐷 “ (0[,)(𝑎 / 2))) → (𝑣 ∘ 𝑣) = ((^◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (^◡𝐷 “ (0[,)(𝑎 / 2)))))
169	168	sseq1d 3952	. . . 4 ⊢ (𝑣 = (^◡𝐷 “ (0[,)(𝑎 / 2))) → ((𝑣 ∘ 𝑣) ⊆ 𝐴 ↔ ((^◡𝐷 “ (0[,)(𝑎 / 2))) ∘ (^◡𝐷 “ (0[,)(𝑎 / 2)))) ⊆ 𝐴))
170	169	rspcev 3561	. . 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 23706	. . . 4 ⊢ (𝐷 ∈ (PsMet‘𝑋) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ⁺ 𝐴 = (^◡𝐷 “ (0[,)𝑎))))
173	172	adantl 482	. . 3 ⊢ ((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) → (𝐴 ∈ 𝐹 ↔ ∃𝑎 ∈ ℝ⁺ 𝐴 = (^◡𝐷 “ (0[,)𝑎))))
174	173	biimpa 477	. 2 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) → ∃𝑎 ∈ ℝ⁺ 𝐴 = (^◡𝐷 “ (0[,)𝑎)))
175	171, 174	r19.29a 3218	1 ⊢ (((𝑋 ≠ ∅ ∧ 𝐷 ∈ (PsMet‘𝑋)) ∧ 𝐴 ∈ 𝐹) → ∃𝑣 ∈ 𝐹 (𝑣 ∘ 𝑣) ⊆ 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 ℝcr 10870 0cc0 10871 + caddc 10874 ℝ^*cxr 11008 < clt 11009 ≤ cle 11010 / cdiv 11632 2c2 12028 ℝ⁺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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