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Theorem heicant 33115
Description: Heine-Cantor theorem: a continuous mapping between metric spaces whose domain is compact is uniformly continuous. Theorem on [Rosenlicht] p. 80. (Contributed by Brendan Leahy, 13-Aug-2018.) (Proof shortened by AV, 27-Sep-2020.)
Hypotheses
Ref Expression
heicant.c (𝜑𝐶 ∈ (∞Met‘𝑋))
heicant.d (𝜑𝐷 ∈ (∞Met‘𝑌))
heicant.j (𝜑 → (MetOpen‘𝐶) ∈ Comp)
heicant.x (𝜑𝑋 ≠ ∅)
heicant.y (𝜑𝑌 ≠ ∅)
Assertion
Ref Expression
heicant (𝜑 → ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) = ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)))

Proof of Theorem heicant
Dummy variables 𝑏 𝑐 𝑑 𝑓 𝑔 𝑝 𝑠 𝑤 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq2 4627 . . . . . . . . . . 11 (𝑑 = 𝑦 → (((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑 ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
21imbi2d 330 . . . . . . . . . 10 (𝑑 = 𝑦 → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
322ralbidv 2985 . . . . . . . . 9 (𝑑 = 𝑦 → (∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
43rexbidv 3047 . . . . . . . 8 (𝑑 = 𝑦 → (∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
54cbvralv 3163 . . . . . . 7 (∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
6 r19.12 3058 . . . . . . . 8 (∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
76ralimi 2948 . . . . . . 7 (∀𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
85, 7sylbi 207 . . . . . 6 (∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) → ∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
9 rphalfcl 11818 . . . . . . . . 9 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ+)
10 breq2 4627 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑑 / 2) → (((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦 ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
1110imbi2d 330 . . . . . . . . . . . . . . 15 (𝑦 = (𝑑 / 2) → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1211ralbidv 2982 . . . . . . . . . . . . . 14 (𝑦 = (𝑑 / 2) → (∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1312rexbidv 3047 . . . . . . . . . . . . 13 (𝑦 = (𝑑 / 2) → (∃𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∃𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1413ralbidv 2982 . . . . . . . . . . . 12 (𝑦 = (𝑑 / 2) → (∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1514rspcva 3297 . . . . . . . . . . 11 (((𝑑 / 2) ∈ ℝ+ ∧ ∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)) → ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
16 heicant.j . . . . . . . . . . . . . . 15 (𝜑 → (MetOpen‘𝐶) ∈ Comp)
1716ad3antrrr 765 . . . . . . . . . . . . . 14 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → (MetOpen‘𝐶) ∈ Comp)
18 heicant.c . . . . . . . . . . . . . . . . . . . . . . . 24 (𝜑𝐶 ∈ (∞Met‘𝑋))
1918ad2antrr 761 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → 𝐶 ∈ (∞Met‘𝑋))
2019anim1i 591 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) → (𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋))
21 rphalfcl 11818 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ+)
2221rpxrd 11833 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ*)
23 eqid 2621 . . . . . . . . . . . . . . . . . . . . . . . 24 (MetOpen‘𝐶) = (MetOpen‘𝐶)
2423blopn 22245 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋 ∧ (𝑧 / 2) ∈ ℝ*) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
25243expa 1262 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋) ∧ (𝑧 / 2) ∈ ℝ*) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2620, 22, 25syl2an 494 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2726adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2821rpgt0d 11835 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℝ+ → 0 < (𝑧 / 2))
2922, 28jca 554 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ ℝ+ → ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2)))
30 xblcntr 22156 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋 ∧ ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
31303expa 1262 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋) ∧ ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
3220, 29, 31syl2an 494 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
3332adantr 481 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
34 opelxpi 5118 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝑋 ∧ (𝑧 / 2) ∈ ℝ+) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
3521, 34sylan2 491 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑋𝑧 ∈ ℝ+) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
3635ad4ant23 1294 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
37 rpcn 11801 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℝ+𝑧 ∈ ℂ)
38372halvesd 11238 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ ℝ+ → ((𝑧 / 2) + (𝑧 / 2)) = 𝑧)
3938breq2d 4635 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ ℝ+ → ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) ↔ (𝑥𝐶𝑐) < 𝑧))
4039imbi1d 331 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ ℝ+ → (((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
4140ralbidv 2982 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℝ+ → (∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑐𝑋 ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
42 oveq2 6623 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐 = 𝑤 → (𝑥𝐶𝑐) = (𝑥𝐶𝑤))
4342breq1d 4633 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑤 → ((𝑥𝐶𝑐) < 𝑧 ↔ (𝑥𝐶𝑤) < 𝑧))
44 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = 𝑤 → (𝑓𝑐) = (𝑓𝑤))
4544oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐 = 𝑤 → ((𝑓𝑥)𝐷(𝑓𝑐)) = ((𝑓𝑥)𝐷(𝑓𝑤)))
4645breq1d 4633 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑤 → (((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
4743, 46imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑤 → (((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
4847cbvralv 3163 . . . . . . . . . . . . . . . . . . . . . . . 24 (∀𝑐𝑋 ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
4941, 48syl6bb 276 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℝ+ → (∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
5049biimpar 502 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℝ+ ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)))
5150adantll 749 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)))
52 vex 3193 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑥 ∈ V
53 ovex 6643 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 / 2) ∈ V
5452, 53op1std 7138 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (1st𝑝) = 𝑥)
5552, 53op2ndd 7139 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (2nd𝑝) = (𝑧 / 2))
5654, 55oveq12d 6633 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((1st𝑝)(ball‘𝐶)(2nd𝑝)) = (𝑥(ball‘𝐶)(𝑧 / 2)))
5756eqcomd 2627 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)))
5857biantrurd 529 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)))))
5954oveq1d 6630 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((1st𝑝)𝐶𝑐) = (𝑥𝐶𝑐))
6055, 55oveq12d 6633 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((2nd𝑝) + (2nd𝑝)) = ((𝑧 / 2) + (𝑧 / 2)))
6159, 60breq12d 4636 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) ↔ (𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2))))
6254fveq2d 6162 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (𝑓‘(1st𝑝)) = (𝑓𝑥))
6362oveq1d 6630 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) = ((𝑓𝑥)𝐷(𝑓𝑐)))
6463breq1d 4633 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)))
6561, 64imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
6665ralbidv 2982 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
6758, 66bitr3d 270 . . . . . . . . . . . . . . . . . . . . . 22 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))) ↔ ∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
6867rspcev 3299 . . . . . . . . . . . . . . . . . . . . 21 ((⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+) ∧ ∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))) → ∃𝑝 ∈ (𝑋 × ℝ+)((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))))
6936, 51, 68syl2anc 692 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ∃𝑝 ∈ (𝑋 × ℝ+)((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))))
70 eleq2 2687 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (𝑥𝑏𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2))))
71 eqeq1 2625 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ↔ (𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝))))
7271anbi1d 740 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → ((𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))) ↔ ((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)))))
7372rexbidv 3047 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))) ↔ ∃𝑝 ∈ (𝑋 × ℝ+)((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)))))
7470, 73anbi12d 746 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → ((𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ↔ (𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)) ∧ ∃𝑝 ∈ (𝑋 × ℝ+)((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))))))
7574rspcev 3299 . . . . . . . . . . . . . . . . . . . 20 (((𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶) ∧ (𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)) ∧ ∃𝑝 ∈ (𝑋 × ℝ+)((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))))) → ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)))))
7627, 33, 69, 75syl12anc 1321 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)))))
7776ex 450 . . . . . . . . . . . . . . . . . 18 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) → (∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)) → ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))))))
7877rexlimdva 3026 . . . . . . . . . . . . . . . . 17 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) → (∃𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)) → ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))))))
7978ralimdva 2958 . . . . . . . . . . . . . . . 16 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → (∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)) → ∀𝑥𝑋𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))))))
8079imp 445 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ∀𝑥𝑋𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)))))
8123mopnuni 22186 . . . . . . . . . . . . . . . . . 18 (𝐶 ∈ (∞Met‘𝑋) → 𝑋 = (MetOpen‘𝐶))
8218, 81syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 = (MetOpen‘𝐶))
8382raleqdv 3137 . . . . . . . . . . . . . . . 16 (𝜑 → (∀𝑥𝑋𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ↔ ∀𝑥 (MetOpen‘𝐶)∃𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))))))
8483ad3antrrr 765 . . . . . . . . . . . . . . 15 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → (∀𝑥𝑋𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ↔ ∀𝑥 (MetOpen‘𝐶)∃𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))))))
8580, 84mpbid 222 . . . . . . . . . . . . . 14 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ∀𝑥 (MetOpen‘𝐶)∃𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)))))
86 eqid 2621 . . . . . . . . . . . . . . 15 (MetOpen‘𝐶) = (MetOpen‘𝐶)
87 fveq2 6158 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (1st𝑝) = (1st ‘(𝑔𝑏)))
88 fveq2 6158 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (2nd𝑝) = (2nd ‘(𝑔𝑏)))
8987, 88oveq12d 6633 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝑔𝑏) → ((1st𝑝)(ball‘𝐶)(2nd𝑝)) = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))))
9089eqeq2d 2631 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑔𝑏) → (𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ↔ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))))
9187oveq1d 6630 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((1st𝑝)𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑐))
9288, 88oveq12d 6633 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((2nd𝑝) + (2nd𝑝)) = ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
9391, 92breq12d 4636 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) ↔ ((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
9487fveq2d 6162 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = (𝑔𝑏) → (𝑓‘(1st𝑝)) = (𝑓‘(1st ‘(𝑔𝑏))))
9594oveq1d 6630 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)))
9695breq1d 4633 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))
9793, 96imbi12d 334 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝑔𝑏) → ((((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))))
9897ralbidv 2982 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑔𝑏) → (∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))))
9990, 98anbi12d 746 . . . . . . . . . . . . . . 15 (𝑝 = (𝑔𝑏) → ((𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))) ↔ (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))))
10086, 99cmpcovf 21134 . . . . . . . . . . . . . 14 (((MetOpen‘𝐶) ∈ Comp ∧ ∀𝑥 (MetOpen‘𝐶)∃𝑏 ∈ (MetOpen‘𝐶)(𝑥𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ∧ ∀𝑐𝑋 (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2))))) → ∃𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin)( (MetOpen‘𝐶) = 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))))))
10117, 85, 100syl2anc 692 . . . . . . . . . . . . 13 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ∃𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin)( (MetOpen‘𝐶) = 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))))))
102101ex 450 . . . . . . . . . . . 12 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → (∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)) → ∃𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin)( (MetOpen‘𝐶) = 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))))))
103 elinel2 3784 . . . . . . . . . . . . . 14 (𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin) → 𝑠 ∈ Fin)
104 simpll 789 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → 𝜑)
105104anim1i 591 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → (𝜑𝑠 ∈ Fin))
106 frn 6020 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran 𝑔 ⊆ (𝑋 × ℝ+))
107 rnss 5324 . . . . . . . . . . . . . . . . . . . . . . . 24 (ran 𝑔 ⊆ (𝑋 × ℝ+) → ran ran 𝑔 ⊆ ran (𝑋 × ℝ+))
108106, 107syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran 𝑔 ⊆ ran (𝑋 × ℝ+))
109 rnxpss 5535 . . . . . . . . . . . . . . . . . . . . . . 23 ran (𝑋 × ℝ+) ⊆ ℝ+
110108, 109syl6ss 3600 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran 𝑔 ⊆ ℝ+)
111110adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ⊆ ℝ+)
112 simplr 791 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → 𝑠 ∈ Fin)
113 ffun 6015 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:𝑠⟶(𝑋 × ℝ+) → Fun 𝑔)
114 vex 3193 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑔 ∈ V
115114fundmen 7990 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun 𝑔 → dom 𝑔𝑔)
116115ensymd 7967 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Fun 𝑔𝑔 ≈ dom 𝑔)
117113, 116syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 ≈ dom 𝑔)
118 fdm 6018 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:𝑠⟶(𝑋 × ℝ+) → dom 𝑔 = 𝑠)
119117, 118breqtrd 4649 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔𝑠)
120 enfii 8137 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ Fin ∧ 𝑔𝑠) → 𝑔 ∈ Fin)
121119, 120sylan2 491 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ Fin ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑔 ∈ Fin)
122 rnfi 8209 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔 ∈ Fin → ran 𝑔 ∈ Fin)
123 rnfi 8209 . . . . . . . . . . . . . . . . . . . . . . . 24 (ran 𝑔 ∈ Fin → ran ran 𝑔 ∈ Fin)
124121, 122, 1233syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ Fin ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ∈ Fin)
125112, 124sylan 488 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ∈ Fin)
126118adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → dom 𝑔 = 𝑠)
127 eqtr 2640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → 𝑋 = 𝑠)
12882, 127sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑋 = 𝑠)
129 heicant.x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝑋 ≠ ∅)
130129adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑋 ≠ ∅)
131128, 130eqnetrrd 2858 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑠 ≠ ∅)
132 unieq 4417 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑠 = ∅ → 𝑠 = ∅)
133 uni0 4438 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ∅ = ∅
134132, 133syl6eq 2671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = ∅ → 𝑠 = ∅)
135134necon3i 2822 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( 𝑠 ≠ ∅ → 𝑠 ≠ ∅)
136131, 135syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑠 ≠ ∅)
137136adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑠 ≠ ∅)
138126, 137eqnetrd 2857 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → dom 𝑔 ≠ ∅)
139 dm0rn0 5312 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
140139necon3bii 2842 . . . . . . . . . . . . . . . . . . . . . . . . 25 (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
141138, 140sylib 208 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran 𝑔 ≠ ∅)
142 relxp 5198 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Rel (𝑋 × ℝ+)
143 relss 5177 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (ran 𝑔 ⊆ (𝑋 × ℝ+) → (Rel (𝑋 × ℝ+) → Rel ran 𝑔))
144106, 142, 143mpisyl 21 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:𝑠⟶(𝑋 × ℝ+) → Rel ran 𝑔)
145 relrn0 5353 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Rel ran 𝑔 → (ran 𝑔 = ∅ ↔ ran ran 𝑔 = ∅))
146145necon3bid 2834 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Rel ran 𝑔 → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
147144, 146syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑔:𝑠⟶(𝑋 × ℝ+) → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
148147adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
149141, 148mpbid 222 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ≠ ∅)
150149adantllr 754 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ≠ ∅)
151 rpssre 11803 . . . . . . . . . . . . . . . . . . . . . . 23 + ⊆ ℝ
152111, 151syl6ss 3600 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ⊆ ℝ)
153 ltso 10078 . . . . . . . . . . . . . . . . . . . . . . 23 < Or ℝ
154 fiinfcl 8367 . . . . . . . . . . . . . . . . . . . . . . 23 (( < Or ℝ ∧ (ran ran 𝑔 ∈ Fin ∧ ran ran 𝑔 ≠ ∅ ∧ ran ran 𝑔 ⊆ ℝ)) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
155153, 154mpan 705 . . . . . . . . . . . . . . . . . . . . . 22 ((ran ran 𝑔 ∈ Fin ∧ ran ran 𝑔 ≠ ∅ ∧ ran ran 𝑔 ⊆ ℝ) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
156125, 150, 152, 155syl3anc 1323 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
157111, 156sseldd 3589 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ+)
158105, 157sylanl1 681 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ+)
159158adantr 481 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ+)
16082ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → 𝑋 = (MetOpen‘𝐶))
161160anim1i 591 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠))
162161ad2antrr 761 . . . . . . . . . . . . . . . . . . . . 21 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → (𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠))
163 simpl 473 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑋𝑤𝑋) → 𝑥𝑋)
164127eleq2d 2684 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑥𝑋𝑥 𝑠))
165 eluni2 4413 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 𝑠 ↔ ∃𝑏𝑠 𝑥𝑏)
166164, 165syl6bb 276 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑥𝑋 ↔ ∃𝑏𝑠 𝑥𝑏))
167166biimpa 501 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑥𝑋) → ∃𝑏𝑠 𝑥𝑏)
168162, 163, 167syl2an 494 . . . . . . . . . . . . . . . . . . . 20 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ∧ (𝑥𝑋𝑤𝑋)) → ∃𝑏𝑠 𝑥𝑏)
169 nfv 1840 . . . . . . . . . . . . . . . . . . . . . . 23 𝑏(((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+))
170 nfra1 2937 . . . . . . . . . . . . . . . . . . . . . . 23 𝑏𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))
171169, 170nfan 1825 . . . . . . . . . . . . . . . . . . . . . 22 𝑏((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))))
172 nfv 1840 . . . . . . . . . . . . . . . . . . . . . 22 𝑏(𝑥𝑋𝑤𝑋)
173171, 172nfan 1825 . . . . . . . . . . . . . . . . . . . . 21 𝑏(((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ∧ (𝑥𝑋𝑤𝑋))
174 nfv 1840 . . . . . . . . . . . . . . . . . . . . 21 𝑏((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)
175 rspa 2926 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))) ∧ 𝑏𝑠) → (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))))
176 oveq2 6623 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑥 → ((1st ‘(𝑔𝑏))𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑥))
177176breq1d 4633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑥 → (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ↔ ((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
178 fveq2 6158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑐 = 𝑥 → (𝑓𝑐) = (𝑓𝑥))
179178oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑥 → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
180179breq1d 4633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑥 → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)))
181177, 180imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = 𝑥 → ((((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ (((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2))))
182181rspcva 3297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑥𝑋 ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))) → (((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)))
183 oveq2 6623 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → ((1st ‘(𝑔𝑏))𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑤))
184183breq1d 4633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ↔ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
18544oveq2d 6631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
186185breq1d 4633 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)))
187184, 186imbi12d 334 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑐 = 𝑤 → ((((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ (((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))))
188187rspcva 3297 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑤𝑋 ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))) → (((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)))
189182, 188anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝑥𝑋 ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))) ∧ (𝑤𝑋 ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → ((((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)) ∧ (((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))))
190189anandirs 873 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝑥𝑋𝑤𝑋) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))) → ((((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)) ∧ (((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))))
191 prth 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)) ∧ (((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → ((((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∧ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))))
192190, 191syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝑥𝑋𝑤𝑋) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))) → ((((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∧ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))))
193192adantrl 751 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝑥𝑋𝑤𝑋) ∧ (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → ((((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∧ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))))
194193ad4ant23 1294 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ∧ (𝑏𝑠𝑥𝑏)) → ((((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∧ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))))
195 simpll 789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → ((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+))
196195anim1i 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)))
197196anim1i 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)))
198110, 151syl6ss 3600 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran 𝑔 ⊆ ℝ)
199198adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ran ran 𝑔 ⊆ ℝ)
200 0re 10000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 0 ∈ ℝ
201 rpge0 11805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
202201rgen 2918 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 𝑦 ∈ ℝ+ 0 ≤ 𝑦
203 ssralv 3651 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (ran ran 𝑔 ⊆ ℝ+ → (∀𝑦 ∈ ℝ+ 0 ≤ 𝑦 → ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦))
204110, 202, 203mpisyl 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑔:𝑠⟶(𝑋 × ℝ+) → ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦)
205 breq1 4626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑥 = 0 → (𝑥𝑦 ↔ 0 ≤ 𝑦))
206205ralbidv 2982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑥 = 0 → (∀𝑦 ∈ ran ran 𝑔 𝑥𝑦 ↔ ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦))
207206rspcev 3299 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦)
208200, 204, 207sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑔:𝑠⟶(𝑋 × ℝ+) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦)
209208adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦)
210144adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → Rel ran 𝑔)
211 ffn 6012 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 Fn 𝑠)
212 fnfvelrn 6322 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔 Fn 𝑠𝑏𝑠) → (𝑔𝑏) ∈ ran 𝑔)
213211, 212sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ ran 𝑔)
214 2ndrn 7176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((Rel ran 𝑔 ∧ (𝑔𝑏) ∈ ran 𝑔) → (2nd ‘(𝑔𝑏)) ∈ ran ran 𝑔)
215210, 213, 214syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ran ran 𝑔)
216 infrelb 10968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((ran ran 𝑔 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦 ∧ (2nd ‘(𝑔𝑏)) ∈ ran ran 𝑔) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
217199, 209, 215, 216syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
218217adantll 749 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
219218ad2ant2r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
22018ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝐶 ∈ (∞Met‘𝑋))
221 xmetcl 22076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑤𝑋) → (𝑥𝐶𝑤) ∈ ℝ*)
2222213expb 1263 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐶𝑤) ∈ ℝ*)
223220, 222sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐶𝑤) ∈ ℝ*)
224223adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (𝑥𝐶𝑤) ∈ ℝ*)
225 simplr 791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → 𝑔:𝑠⟶(𝑋 × ℝ+))
226 simpl 473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑏𝑠𝑥𝑏) → 𝑏𝑠)
227 ne0i 3903 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((2nd ‘(𝑔𝑏)) ∈ ran ran 𝑔 → ran ran 𝑔 ≠ ∅)
228215, 227syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ran ran 𝑔 ≠ ∅)
229 infrecl 10965 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((ran ran 𝑔 ⊆ ℝ ∧ ran ran 𝑔 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ)
230199, 228, 209, 229syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ)
231230rexrd 10049 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ*)
232225, 226, 231syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ*)
233 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑔:𝑠⟶(𝑋 × ℝ+))
234233ffvelrnda 6325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ (𝑋 × ℝ+))
235 xp2nd 7159 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔𝑏) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
236234, 235syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
237236rpxrd 11833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
238237ad2ant2r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
239 xrltletr 11948 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝑥𝐶𝑤) ∈ ℝ* ∧ inf(ran ran 𝑔, ℝ, < ) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ*) → (((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) ∧ inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))))
240224, 232, 238, 239syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) ∧ inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))))
241219, 240mpan2d 709 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))))
242241adantlr 750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))))
24318ad6antr 771 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝐶 ∈ (∞Met‘𝑋))
244 simpllr 798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) → 𝑔:𝑠⟶(𝑋 × ℝ+))
245 ffvelrn 6323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ (𝑋 × ℝ+))
246 xp1st 7158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔𝑏) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
247245, 246syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
248244, 226, 247syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
249 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑥𝑋𝑤𝑋) → 𝑤𝑋)
250249ad3antlr 766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝑤𝑋)
251 xmetcl 22076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑤𝑋) → ((1st ‘(𝑔𝑏))𝐶𝑤) ∈ ℝ*)
252243, 248, 250, 251syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑤) ∈ ℝ*)
253252adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑤) ∈ ℝ*)
254245, 235syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
255225, 254sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
256255ad2ant2r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
257256rpred 11832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
258 eleq2 2687 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) → (𝑥𝑏𝑥 ∈ ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))))
25918ad5antr 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝐶 ∈ (∞Met‘𝑋))
260225, 247sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
261255rpxrd 11833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
262 elbl 22133 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋 ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ*) → (𝑥 ∈ ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ↔ (𝑥𝑋 ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))))
263259, 260, 261, 262syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑥 ∈ ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ↔ (𝑥𝑋 ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))))
264258, 263sylan9bbr 736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) → (𝑥𝑏 ↔ (𝑥𝑋 ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))))
265264biimpd 219 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) → (𝑥𝑏 → (𝑥𝑋 ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))))
266265an32s 845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ 𝑏𝑠) → (𝑥𝑏 → (𝑥𝑋 ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))))
267266impr 648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (𝑥𝑋 ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏))))
268267simprd 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))
269163ad3antlr 766 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝑥𝑋)
270 xmetcl 22076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑥𝑋) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
271243, 248, 269, 270syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
272254rpxrd 11833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
273244, 226, 272syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
274 xrltle 11942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ*) → (((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏))))
275271, 273, 274syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏))))
276268, 275mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))
277225ffvelrnda 6325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ (𝑋 × ℝ+))
278277, 246syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
279 simplrl 799 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝑥𝑋)
280259, 278, 279, 270syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
281 xmetge0 22089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑥𝑋) → 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥))
282259, 278, 279, 281syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥))
283 xrrege0 11964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) ∧ (0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥) ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
284283an4s 868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥)) ∧ ((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
285284ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥)) → (((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ))
286280, 282, 285syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ))
287286ad2ant2r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ))
288257, 276, 287mp2and 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
289288adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
290 xrltle 11942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑥𝐶𝑤) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ*) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))))
291224, 238, 290syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))))
292 xmetge0 22089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑤𝑋) → 0 ≤ (𝑥𝐶𝑤))
2932923expb 1263 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥𝑋𝑤𝑋)) → 0 ≤ (𝑥𝐶𝑤))
294220, 293sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → 0 ≤ (𝑥𝐶𝑤))
295294adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → 0 ≤ (𝑥𝐶𝑤))
296236rpred 11832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
297296ad2ant2r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
298 xrrege0 11964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((𝑥𝐶𝑤) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) ∧ (0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏)))) → (𝑥𝐶𝑤) ∈ ℝ)
299298ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝑥𝐶𝑤) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) → ((0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) ∈ ℝ))
300224, 297, 299syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) ∈ ℝ))
301295, 300mpand 710 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ∈ ℝ))
302291, 301syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ∈ ℝ))
303302adantlr 750 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ∈ ℝ))
304303imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) ∈ ℝ)
305289, 304readdcld 10029 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → (((1st ‘(𝑔𝑏))𝐶𝑥) + (𝑥𝐶𝑤)) ∈ ℝ)
306305rexrd 10049 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → (((1st ‘(𝑔𝑏))𝐶𝑥) + (𝑥𝐶𝑤)) ∈ ℝ*)
307256, 256rpaddcld 11847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∈ ℝ+)
308307rpxrd 11833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∈ ℝ*)
309308adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∈ ℝ*)
310 xmettri 22096 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐶 ∈ (∞Met‘𝑋) ∧ ((1st ‘(𝑔𝑏)) ∈ 𝑋𝑤𝑋𝑥𝑋)) → ((1st ‘(𝑔𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)))
311243, 248, 250, 269, 310syl13anc 1325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)))
312311adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)))
313 rexadd 12022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ ∧ (𝑥𝐶𝑤) ∈ ℝ) → (((1st ‘(𝑔𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)) = (((1st ‘(𝑔𝑏))𝐶𝑥) + (𝑥𝐶𝑤)))
314289, 304, 313syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → (((1st ‘(𝑔𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)) = (((1st ‘(𝑔𝑏))𝐶𝑥) + (𝑥𝐶𝑤)))
315312, 314breqtrd 4649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔𝑏))𝐶𝑥) + (𝑥𝐶𝑤)))
316257adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
317268adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))
318 simpr 477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)))
319289, 304, 316, 316, 317, 318lt2addd 10610 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → (((1st ‘(𝑔𝑏))𝐶𝑥) + (𝑥𝐶𝑤)) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
320253, 306, 309, 315, 319xrlelttrd 11951 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
321320ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
322254rpred 11832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
323322, 254ltaddrpd 11865 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
324244, 226, 323syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
325271, 273, 308, 268, 324xrlttrd 11950 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
326321, 325jctild 565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∧ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))))
327242, 326syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → (((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∧ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))))
328 simpll 789 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (𝜑𝑓:𝑋𝑌))
329 heicant.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝜑𝐷 ∈ (∞Met‘𝑌))
330 ffvelrn 6323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓:𝑋𝑌𝑥𝑋) → (𝑓𝑥) ∈ 𝑌)
331 ffvelrn 6323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓:𝑋𝑌𝑤𝑋) → (𝑓𝑤) ∈ 𝑌)
332330, 331anim12dan 881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓:𝑋𝑌 ∧ (𝑥𝑋𝑤𝑋)) → ((𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌))
333 xmetcl 22076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
3343333expb 1263 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐷 ∈ (∞Met‘𝑌) ∧ ((𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
335329, 332, 334syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑓:𝑋𝑌 ∧ (𝑥𝑋𝑤𝑋))) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
336335anassrs 679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑓:𝑋𝑌) ∧ (𝑥𝑋𝑤𝑋)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
337328, 336sylan 488 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
338337ad3antrrr 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
339329ad5antr 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝐷 ∈ (∞Met‘𝑌))
340 simp-5r 808 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝑓:𝑋𝑌)
341340, 278ffvelrnd 6326 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌)
342 simpllr 798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑓:𝑋𝑌)
343342ffvelrnda 6325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ 𝑌)
344343adantrr 752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑓𝑥) ∈ 𝑌)
345344adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓𝑥) ∈ 𝑌)
346 xmetcl 22076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑥) ∈ 𝑌) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ*)
347339, 341, 345, 346syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ*)
3489rpxrd 11833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ*)
349348ad4antlr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑑 / 2) ∈ ℝ*)
350 xrltle 11942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ*) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ≤ (𝑑 / 2)))
351347, 349, 350syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ≤ (𝑑 / 2)))
352 xmetge0 22089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑥) ∈ 𝑌) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
353339, 341, 345, 352syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
3549rpred 11832 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ)
355354ad4antlr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑑 / 2) ∈ ℝ)
356 xrrege0 11964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ) ∧ (0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ≤ (𝑑 / 2))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ)
357356ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ) → ((0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ≤ (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ))
358347, 355, 357syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ≤ (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ))
359353, 358mpand 710 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ≤ (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ))
360351, 359syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ))
361360ad2ant2r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ))
362361imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ)
363342ffvelrnda 6325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑤𝑋) → (𝑓𝑤) ∈ 𝑌)
364363adantrl 751 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑓𝑤) ∈ 𝑌)
365364adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓𝑤) ∈ 𝑌)
366 xmetcl 22076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ*)
367339, 341, 365, 366syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ*)
368 xrltle 11942 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ*) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ≤ (𝑑 / 2)))
369367, 349, 368syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ≤ (𝑑 / 2)))
370 xmetge0 22089 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
371339, 341, 365, 370syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
372 xrrege0 11964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ) ∧ (0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ≤ (𝑑 / 2))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ)
373372ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ) → ((0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ≤ (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ))
374367, 355, 373syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ≤ (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ))
375371, 374mpand 710 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ≤ (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ))
376369, 375syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ))
377376ad2ant2r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ))
378377imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ)
379 readdcl 9979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) ∈ ℝ)
380362, 378, 379syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)) ∧ (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) ∈ ℝ)
381380anandis 872 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) ∈ ℝ)
382381rexrd 10049 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) ∈ ℝ*)
383 rpxr 11800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑑 ∈ ℝ+𝑑 ∈ ℝ*)
384383ad6antlr 772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → 𝑑 ∈ ℝ*)
385 xmettri 22096 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐷 ∈ (∞Met‘𝑌) ∧ ((𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌 ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ≤ (((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) +𝑒 ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))))
386339, 345, 365, 341, 385syl13anc 1325 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓𝑥)𝐷(𝑓𝑤)) ≤ (((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) +𝑒 ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))))
387386ad2ant2r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ≤ (((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) +𝑒 ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))))
388387adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → ((𝑓𝑥)𝐷(𝑓𝑤)) ≤ (((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) +𝑒 ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))))
389 xmetsym 22092 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓𝑥) ∈ 𝑌 ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌) → ((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
390339, 345, 341, 389syl3anc 1323 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
391390ad2ant2r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
392391adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → ((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
393392oveq1d 6630 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → (((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) +𝑒 ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) = (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) +𝑒 ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))))
394 rexadd 12022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) +𝑒 ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) = (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))))
395362, 378, 394syl2an 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)) ∧ (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) +𝑒 ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) = (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))))
396395anandis 872 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) +𝑒 ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) = (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))))
397393, 396eqtrd 2655 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → (((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) +𝑒 ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) = (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))))
398388, 397breqtrd 4649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → ((𝑓𝑥)𝐷(𝑓𝑤)) ≤ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))))
399 lt2add 10473 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ) ∧ ((𝑑 / 2) ∈ ℝ ∧ (𝑑 / 2) ∈ ℝ)) → ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) < ((𝑑 / 2) + (𝑑 / 2))))
400399expcom 451 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((𝑑 / 2) ∈ ℝ ∧ (𝑑 / 2) ∈ ℝ) → ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ) → ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) < ((𝑑 / 2) + (𝑑 / 2)))))
401355, 355, 400syl2anc 692 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ) → ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) < ((𝑑 / 2) + (𝑑 / 2)))))
402360, 376, 401syl2and 500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)) → ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) < ((𝑑 / 2) + (𝑑 / 2)))))
403402pm2.43d 53 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) < ((𝑑 / 2) + (𝑑 / 2))))
404403ad2ant2r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) < ((𝑑 / 2) + (𝑑 / 2))))
405404imp 445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) < ((𝑑 / 2) + (𝑑 / 2)))
406 rpcn 11801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑑 ∈ ℝ+𝑑 ∈ ℂ)
4074062halvesd 11238 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑑 ∈ ℝ+ → ((𝑑 / 2) + (𝑑 / 2)) = 𝑑)
408407ad6antlr 772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → ((𝑑 / 2) + (𝑑 / 2)) = 𝑑)
409405, 408breqtrd 4649 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) < 𝑑)
410338, 382, 384, 398, 409xrlelttrd 11951 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)
411410ex 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))
412327, 411imim12d 81 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (((((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∧ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
413197, 412sylanl1 681 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (((((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∧ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
414413adantlrr 756 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ∧ (𝑏𝑠𝑥𝑏)) → (((((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∧ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2))) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
415194, 414mpd 15 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))
416415exp32 630 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → (𝑏𝑠 → (𝑥𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))))
417175, 416sylan2 491 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))) ∧ 𝑏𝑠)) → (𝑏𝑠 → (𝑥𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))))
418417expr 642 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → (𝑏𝑠 → (𝑏𝑠 → (𝑥𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))))
419418pm2.43d 53 . . . . . . . . . . . . . . . . . . . . . 22 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → (𝑏𝑠 → (𝑥𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))))
420419an32s 845 . . . . . . . . . . . . . . . . . . . . 21 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ∧ (𝑥𝑋𝑤𝑋)) → (𝑏𝑠 → (𝑥𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))))
421173, 174, 420rexlimd 3021 . . . . . . . . . . . . . . . . . . . 20 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ∧ (𝑥𝑋𝑤𝑋)) → (∃𝑏𝑠 𝑥𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
422168, 421mpd 15 . . . . . . . . . . . . . . . . . . 19 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) ∧ (𝑥𝑋𝑤𝑋)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))
423422ralrimivva 2967 . . . . . . . . . . . . . . . . . 18 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → ∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))
424 breq2 4627 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = inf(ran ran 𝑔, ℝ, < ) → ((𝑥𝐶𝑤) < 𝑧 ↔ (𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < )))
425424imbi1d 331 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = inf(ran ran 𝑔, ℝ, < ) → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
4264252ralbidv 2985 . . . . . . . . . . . . . . . . . . 19 (𝑧 = inf(ran ran 𝑔, ℝ, < ) → (∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
427426rspcev 3299 . . . . . . . . . . . . . . . . . 18 ((inf(ran ran 𝑔, ℝ, < ) ∈ ℝ+ ∧ ∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))
428159, 423, 427syl2anc 692 . . . . . . . . . . . . . . . . 17 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))
429428expl 647 . . . . . . . . . . . . . . . 16 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
430429exlimdv 1858 . . . . . . . . . . . . . . 15 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → (∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
431430expimpd 628 . . . . . . . . . . . . . 14 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → (( (MetOpen‘𝐶) = 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))))) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
432103, 431sylan2 491 . . . . . . . . . . . . 13 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin)) → (( (MetOpen‘𝐶) = 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))))) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
433432rexlimdva 3026 . . . . . . . . . . . 12 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → (∃𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin)( (MetOpen‘𝐶) = 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧ ∀𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2))))) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
434102, 433syld 47 . . . . . . . . . . 11 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → (∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
43515, 434syl5 34 . . . . . . . . . 10 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → (((𝑑 / 2) ∈ ℝ+ ∧ ∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
436435exp4b 631 . . . . . . . . 9 ((𝜑𝑓:𝑋𝑌) → (𝑑 ∈ ℝ+ → ((𝑑 / 2) ∈ ℝ+ → (∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))))
4379, 436mpdi 45 . . . . . . . 8 ((𝜑𝑓:𝑋𝑌) → (𝑑 ∈ ℝ+ → (∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))))
438437ralrimiv 2961 . . . . . . 7 ((𝜑𝑓:𝑋𝑌) → ∀𝑑 ∈ ℝ+ (∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
439 r19.21v 2956 . . . . . . 7 (∀𝑑 ∈ ℝ+ (∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)) ↔ (∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
440438, 439sylib 208 . . . . . 6 ((𝜑𝑓:𝑋𝑌) → (∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
4418, 440impbid2 216 . . . . 5 ((𝜑𝑓:𝑋𝑌) → (∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
442 ralcom 3092 . . . . 5 (∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
443441, 442syl6bb 276 . . . 4 ((𝜑𝑓:𝑋𝑌) → (∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
444443pm5.32da 672 . . 3 (𝜑 → ((𝑓:𝑋𝑌 ∧ ∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))))
445 eqid 2621 . . . 4 (metUnif‘𝐶) = (metUnif‘𝐶)
446 eqid 2621 . . . 4 (metUnif‘𝐷) = (metUnif‘𝐷)
447 heicant.y . . . 4 (𝜑𝑌 ≠ ∅)
448 xmetpsmet 22093 . . . . 5 (𝐶 ∈ (∞Met‘𝑋) → 𝐶 ∈ (PsMet‘𝑋))
44918, 448syl 17 . . . 4 (𝜑𝐶 ∈ (PsMet‘𝑋))
450 xmetpsmet 22093 . . . . 5 (𝐷 ∈ (∞Met‘𝑌) → 𝐷 ∈ (PsMet‘𝑌))
451329, 450syl 17 . . . 4 (𝜑𝐷 ∈ (PsMet‘𝑌))
452445, 446, 129, 447, 449, 451metucn 22316 . . 3 (𝜑 → (𝑓 ∈ ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))))
453 eqid 2621 . . . . 5 (MetOpen‘𝐷) = (MetOpen‘𝐷)
45423, 453metcn 22288 . . . 4 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) → (𝑓 ∈ ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))))
45518, 329, 454syl2anc 692 . . 3 (𝜑 → (𝑓 ∈ ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))))
456444, 452, 4553bitr4d 300 . 2 (𝜑 → (𝑓 ∈ ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) ↔ 𝑓 ∈ ((MetOpen‘𝐶) Cn (MetOpen‘𝐷))))
457456eqrdv 2619 1 (𝜑 → ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) = ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2908  wrex 2909  cin 3559  wss 3560  c0 3897  𝒫 cpw 4136  cop 4161   cuni 4409   class class class wbr 4623   Or wor 5004   × cxp 5082  dom cdm 5084  ran crn 5085  Rel wrel 5089  Fun wfun 5851   Fn wfn 5852  wf 5853  cfv 5857  (class class class)co 6615  1st c1st 7126  2nd c2nd 7127  cen 7912  Fincfn 7915  infcinf 8307  cr 9895  0cc0 9896   + caddc 9899  *cxr 10033   < clt 10034  cle 10035   / cdiv 10644  2c2 11030  +crp 11792   +𝑒 cxad 11904  PsMetcpsmet 19670  ∞Metcxmt 19671  ballcbl 19673  MetOpencmopn 19676  metUnifcmetu 19677   Cn ccn 20968  Compccmp 21129   Cnucucn 22019
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914  ax-cnex 9952  ax-resscn 9953  ax-1cn 9954  ax-icn 9955  ax-addcl 9956  ax-addrcl 9957  ax-mulcl 9958  ax-mulrcl 9959  ax-mulcom 9960  ax-addass 9961  ax-mulass 9962  ax-distr 9963  ax-i2m1 9964  ax-1ne0 9965  ax-1rid 9966  ax-rnegex 9967  ax-rrecex 9968  ax-cnre 9969  ax-pre-lttri 9970  ax-pre-lttrn 9971  ax-pre-ltadd 9972  ax-pre-mulgt0 9973  ax-pre-sup 9974
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-reu 2915  df-rmo 2916  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-pss 3576  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-tp 4160  df-op 4162  df-uni 4410  df-int 4448  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-tr 4723  df-eprel 4995  df-id 4999  df-po 5005  df-so 5006  df-fr 5043  df-we 5045  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-pred 5649  df-ord 5695  df-on 5696  df-lim 5697  df-suc 5698  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-riota 6576  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-om 7028  df-1st 7128  df-2nd 7129  df-wrecs 7367  df-recs 7428  df-rdg 7466  df-1o 7520  df-er 7702  df-map 7819  df-en 7916  df-dom 7917  df-sdom 7918  df-fin 7919  df-sup 8308  df-inf 8309  df-pnf 10036  df-mnf 10037  df-xr 10038  df-ltxr 10039  df-le 10040  df-sub 10228  df-neg 10229  df-div 10645  df-nn 10981  df-2 11039  df-n0 11253  df-z 11338  df-uz 11648  df-q 11749  df-rp 11793  df-xneg 11906  df-xadd 11907  df-xmul 11908  df-ico 12139  df-topgen 16044  df-psmet 19678  df-xmet 19679  df-bl 19681  df-mopn 19682  df-fbas 19683  df-fg 19684  df-metu 19685  df-top 20639  df-topon 20656  df-bases 20690  df-cn 20971  df-cnp 20972  df-cmp 21130  df-fil 21590  df-ust 21944  df-ucn 22020
This theorem is referenced by: (None)
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