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Theorem heicant 33757
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 4808 . . . . . . . . . . 11 (𝑑 = 𝑦 → (((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑 ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
21imbi2d 329 . . . . . . . . . 10 (𝑑 = 𝑦 → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
322ralbidv 3127 . . . . . . . . 9 (𝑑 = 𝑦 → (∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
43rexbidv 3190 . . . . . . . 8 (𝑑 = 𝑦 → (∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
54cbvralv 3310 . . . . . . 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 3201 . . . . . . . 8 (∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
76ralimi 3090 . . . . . . 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 12051 . . . . . . . . 9 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ+)
10 breq2 4808 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑑 / 2) → (((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦 ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
1110imbi2d 329 . . . . . . . . . . . . . . 15 (𝑦 = (𝑑 / 2) → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1211ralbidv 3124 . . . . . . . . . . . . . 14 (𝑦 = (𝑑 / 2) → (∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1312rexbidv 3190 . . . . . . . . . . . . 13 (𝑦 = (𝑑 / 2) → (∃𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∃𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1413ralbidv 3124 . . . . . . . . . . . 12 (𝑦 = (𝑑 / 2) → (∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1514rspcva 3447 . . . . . . . . . . 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 768 . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → 𝐶 ∈ (∞Met‘𝑋))
2019anim1i 593 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) → (𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋))
21 rphalfcl 12051 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ+)
2221rpxrd 12066 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ*)
23 eqid 2760 . . . . . . . . . . . . . . . . . . . . . . . 24 (MetOpen‘𝐶) = (MetOpen‘𝐶)
2423blopn 22506 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋 ∧ (𝑧 / 2) ∈ ℝ*) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
25243expa 1112 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋) ∧ (𝑧 / 2) ∈ ℝ*) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2620, 22, 25syl2an 495 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2726adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2821rpgt0d 12068 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℝ+ → 0 < (𝑧 / 2))
2922, 28jca 555 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ ℝ+ → ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2)))
30 xblcntr 22417 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋 ∧ ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
31303expa 1112 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋) ∧ ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
3220, 29, 31syl2an 495 . . . . . . . . . . . . . . . . . . . . 21 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
3332adantr 472 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
34 opelxpi 5305 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥𝑋 ∧ (𝑧 / 2) ∈ ℝ+) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
3521, 34sylan2 492 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑋𝑧 ∈ ℝ+) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
3635ad4ant23 1211 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
37 rpcn 12034 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑧 ∈ ℝ+𝑧 ∈ ℂ)
38372halvesd 11470 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ ℝ+ → ((𝑧 / 2) + (𝑧 / 2)) = 𝑧)
3938breq2d 4816 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ ℝ+ → ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) ↔ (𝑥𝐶𝑐) < 𝑧))
4039imbi1d 330 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ ℝ+ → (((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
4140ralbidv 3124 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℝ+ → (∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑐𝑋 ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
42 oveq2 6821 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐 = 𝑤 → (𝑥𝐶𝑐) = (𝑥𝐶𝑤))
4342breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑤 → ((𝑥𝐶𝑐) < 𝑧 ↔ (𝑥𝐶𝑤) < 𝑧))
44 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑐 = 𝑤 → (𝑓𝑐) = (𝑓𝑤))
4544oveq2d 6829 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐 = 𝑤 → ((𝑓𝑥)𝐷(𝑓𝑐)) = ((𝑓𝑥)𝐷(𝑓𝑤)))
4645breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑤 → (((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
4743, 46imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑤 → (((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
4847cbvralv 3310 . . . . . . . . . . . . . . . . . . . . . . . 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 503 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑧 ∈ ℝ+ ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)))
5150adantll 752 . . . . . . . . . . . . . . . . . . . . 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 3343 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 𝑥 ∈ V
53 ovex 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 / 2) ∈ V
5452, 53op1std 7343 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (1st𝑝) = 𝑥)
5552, 53op2ndd 7344 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (2nd𝑝) = (𝑧 / 2))
5654, 55oveq12d 6831 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((1st𝑝)(ball‘𝐶)(2nd𝑝)) = (𝑥(ball‘𝐶)(𝑧 / 2)))
5756eqcomd 2766 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)))
5857biantrurd 530 . . . . . . . . . . . . . . . . . . . . . . 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 6828 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((1st𝑝)𝐶𝑐) = (𝑥𝐶𝑐))
6055, 55oveq12d 6831 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((2nd𝑝) + (2nd𝑝)) = ((𝑧 / 2) + (𝑧 / 2)))
6159, 60breq12d 4817 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) ↔ (𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2))))
6254fveq2d 6356 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (𝑓‘(1st𝑝)) = (𝑓𝑥))
6362oveq1d 6828 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) = ((𝑓𝑥)𝐷(𝑓𝑐)))
6463breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)))
6561, 64imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . 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 3124 . . . . . . . . . . . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (𝑥𝑏𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2))))
71 eqeq1 2764 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ↔ (𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝))))
7271anbi1d 743 . . . . . . . . . . . . . . . . . . . . . . 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 3190 . . . . . . . . . . . . . . . . . . . . . 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 749 . . . . . . . . . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . . . . . . 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 1475 . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . 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 3169 . . . . . . . . . . . . . . . . 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 3100 . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . 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 22447 . . . . . . . . . . . . . . . . . 18 (𝐶 ∈ (∞Met‘𝑋) → 𝑋 = (MetOpen‘𝐶))
8218, 81syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 = (MetOpen‘𝐶))
8382raleqdv 3283 . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . 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 2760 . . . . . . . . . . . . . . 15 (MetOpen‘𝐶) = (MetOpen‘𝐶)
87 fveq2 6352 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (1st𝑝) = (1st ‘(𝑔𝑏)))
88 fveq2 6352 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (2nd𝑝) = (2nd ‘(𝑔𝑏)))
8987, 88oveq12d 6831 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝑔𝑏) → ((1st𝑝)(ball‘𝐶)(2nd𝑝)) = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))))
9089eqeq2d 2770 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑔𝑏) → (𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ↔ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))))
9187oveq1d 6828 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((1st𝑝)𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑐))
9288, 88oveq12d 6831 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((2nd𝑝) + (2nd𝑝)) = ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
9391, 92breq12d 4817 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) ↔ ((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
9487fveq2d 6356 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = (𝑔𝑏) → (𝑓‘(1st𝑝)) = (𝑓‘(1st ‘(𝑔𝑏))))
9594oveq1d 6828 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)))
9695breq1d 4814 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))
9793, 96imbi12d 333 . . . . . . . . . . . . . . . . 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 3124 . . . . . . . . . . . . . . . 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 749 . . . . . . . . . . . . . . 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 21396 . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . 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 449 . . . . . . . . . . . 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 3943 . . . . . . . . . . . . . 14 (𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin) → 𝑠 ∈ Fin)
104 simpll 807 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → 𝜑)
105104anim1i 593 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → (𝜑𝑠 ∈ Fin))
106 frn 6214 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran 𝑔 ⊆ (𝑋 × ℝ+))
107 rnss 5509 . . . . . . . . . . . . . . . . . . . . . . . 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 5724 . . . . . . . . . . . . . . . . . . . . . . 23 ran (𝑋 × ℝ+) ⊆ ℝ+
110108, 109syl6ss 3756 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran 𝑔 ⊆ ℝ+)
111110adantl 473 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ⊆ ℝ+)
112 simplr 809 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → 𝑠 ∈ Fin)
113 ffun 6209 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:𝑠⟶(𝑋 × ℝ+) → Fun 𝑔)
114 vex 3343 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑔 ∈ V
115114fundmen 8195 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun 𝑔 → dom 𝑔𝑔)
116115ensymd 8172 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Fun 𝑔𝑔 ≈ dom 𝑔)
117113, 116syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 ≈ dom 𝑔)
118 fdm 6212 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:𝑠⟶(𝑋 × ℝ+) → dom 𝑔 = 𝑠)
119117, 118breqtrd 4830 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔𝑠)
120 enfii 8342 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ Fin ∧ 𝑔𝑠) → 𝑔 ∈ Fin)
121119, 120sylan2 492 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ Fin ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑔 ∈ Fin)
122 rnfi 8414 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔 ∈ Fin → ran 𝑔 ∈ Fin)
123 rnfi 8414 . . . . . . . . . . . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ∈ Fin)
126118adantl 473 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → dom 𝑔 = 𝑠)
127 eqtr 2779 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → 𝑋 = 𝑠)
12882, 127sylan 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑋 = 𝑠)
129 heicant.x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝑋 ≠ ∅)
130129adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑋 ≠ ∅)
131128, 130eqnetrrd 3000 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑠 ≠ ∅)
132 unieq 4596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑠 = ∅ → 𝑠 = ∅)
133 uni0 4617 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ∅ = ∅
134132, 133syl6eq 2810 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = ∅ → 𝑠 = ∅)
135134necon3i 2964 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( 𝑠 ≠ ∅ → 𝑠 ≠ ∅)
136131, 135syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑠 ≠ ∅)
137136adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑠 ≠ ∅)
138126, 137eqnetrd 2999 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → dom 𝑔 ≠ ∅)
139 dm0rn0 5497 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
140139necon3bii 2984 . . . . . . . . . . . . . . . . . . . . . . . . 25 (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
141138, 140sylib 208 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran 𝑔 ≠ ∅)
142 relxp 5283 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Rel (𝑋 × ℝ+)
143 relss 5363 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5538 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Rel ran 𝑔 → (ran 𝑔 = ∅ ↔ ran ran 𝑔 = ∅))
146145necon3bid 2976 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Rel ran 𝑔 → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
147144, 146syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑔:𝑠⟶(𝑋 × ℝ+) → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
148147adantl 473 . . . . . . . . . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ≠ ∅)
151 rpssre 12036 . . . . . . . . . . . . . . . . . . . . . . 23 + ⊆ ℝ
152111, 151syl6ss 3756 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ⊆ ℝ)
153 ltso 10310 . . . . . . . . . . . . . . . . . . . . . . 23 < Or ℝ
154 fiinfcl 8572 . . . . . . . . . . . . . . . . . . . . . . 23 (( < Or ℝ ∧ (ran ran 𝑔 ∈ Fin ∧ ran ran 𝑔 ≠ ∅ ∧ ran ran 𝑔 ⊆ ℝ)) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
155153, 154mpan 708 . . . . . . . . . . . . . . . . . . . . . 22 ((ran ran 𝑔 ∈ Fin ∧ ran ran 𝑔 ≠ ∅ ∧ ran ran 𝑔 ⊆ ℝ) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
156125, 150, 152, 155syl3anc 1477 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
157111, 156sseldd 3745 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ+)
158105, 157sylanl1 685 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ+)
159158adantr 472 . . . . . . . . . . . . . . . . . 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 768 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → 𝑋 = (MetOpen‘𝐶))
161160anim1i 593 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠))
162161ad2antrr 764 . . . . . . . . . . . . . . . . . . . . 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 474 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑋𝑤𝑋) → 𝑥𝑋)
164127eleq2d 2825 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑥𝑋𝑥 𝑠))
165 eluni2 4592 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 𝑠 ↔ ∃𝑏𝑠 𝑥𝑏)
166164, 165syl6bb 276 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑥𝑋 ↔ ∃𝑏𝑠 𝑥𝑏))
167166biimpa 502 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑥𝑋) → ∃𝑏𝑠 𝑥𝑏)
168162, 163, 167syl2an 495 . . . . . . . . . . . . . . . . . . . 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 1992 . . . . . . . . . . . . . . . . . . . . . . 23 𝑏(((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+))
170 nfra1 3079 . . . . . . . . . . . . . . . . . . . . . . 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 1977 . . . . . . . . . . . . . . . . . . . . . 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 1992 . . . . . . . . . . . . . . . . . . . . . 22 𝑏(𝑥𝑋𝑤𝑋)
173171, 172nfan 1977 . . . . . . . . . . . . . . . . . . . . 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 1992 . . . . . . . . . . . . . . . . . . . . 21 𝑏((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)
175 rspa 3068 . . . . . . . . . . . . . . . . . . . . . . . . 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 6821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑥 → ((1st ‘(𝑔𝑏))𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑥))
177176breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑥 → (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ↔ ((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
178 fveq2 6352 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑐 = 𝑥 → (𝑓𝑐) = (𝑓𝑥))
179178oveq2d 6829 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑥 → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
180179breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑥 → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)))
181177, 180imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → ((1st ‘(𝑔𝑏))𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑤))
184183breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ↔ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
18544oveq2d 6829 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
186185breq1d 4814 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)))
187184, 186imbi12d 333 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 591 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 909 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 754 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1211 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 807 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → ((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+))
196195anim1i 593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)))
197196anim1i 593 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3756 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran 𝑔 ⊆ ℝ)
199198adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ran ran 𝑔 ⊆ ℝ)
200 0re 10232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 0 ∈ ℝ
201 rpge0 12038 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
202201rgen 3060 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 𝑦 ∈ ℝ+ 0 ≤ 𝑦
203 ssralv 3807 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4807 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑥 = 0 → (𝑥𝑦 ↔ 0 ≤ 𝑦))
206205ralbidv 3124 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑥 = 0 → (∀𝑦 ∈ ran ran 𝑔 𝑥𝑦 ↔ ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦))
207206rspcev 3449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦)
208200, 204, 207sylancr 698 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑔:𝑠⟶(𝑋 × ℝ+) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦)
209208adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦)
210144adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → Rel ran 𝑔)
211 ffn 6206 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 Fn 𝑠)
212 fnfvelrn 6519 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔 Fn 𝑠𝑏𝑠) → (𝑔𝑏) ∈ ran 𝑔)
213211, 212sylan 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ ran 𝑔)
214 2ndrn 7383 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((Rel ran 𝑔 ∧ (𝑔𝑏) ∈ ran 𝑔) → (2nd ‘(𝑔𝑏)) ∈ ran ran 𝑔)
215210, 213, 214syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ran ran 𝑔)
216 infrelb 11200 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
218217adantll 752 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
219218ad2ant2r 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
22018ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝐶 ∈ (∞Met‘𝑋))
221 xmetcl 22337 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑤𝑋) → (𝑥𝐶𝑤) ∈ ℝ*)
2222213expb 1114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐶𝑤) ∈ ℝ*)
223220, 222sylan 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐶𝑤) ∈ ℝ*)
224223adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (𝑥𝐶𝑤) ∈ ℝ*)
225 simplr 809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → 𝑔:𝑠⟶(𝑋 × ℝ+))
226 simpl 474 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑏𝑠𝑥𝑏) → 𝑏𝑠)
227 ne0i 4064 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((2nd ‘(𝑔𝑏)) ∈ ran ran 𝑔 → ran ran 𝑔 ≠ ∅)
228215, 227syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ran ran 𝑔 ≠ ∅)
229 infrecl 11197 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ)
231230rexrd 10281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ*)
232225, 226, 231syl2an 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ*)
233 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑔:𝑠⟶(𝑋 × ℝ+))
234233ffvelrnda 6522 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ (𝑋 × ℝ+))
235 xp2nd 7366 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔𝑏) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
236234, 235syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
237236rpxrd 12066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
238237ad2ant2r 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
239 xrltletr 12181 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))))
242241adantlr 753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 779 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝐶 ∈ (∞Met‘𝑋))
244 simpllr 817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) → 𝑔:𝑠⟶(𝑋 × ℝ+))
245 ffvelrn 6520 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ (𝑋 × ℝ+))
246 xp1st 7365 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔𝑏) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
247245, 246syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
248244, 226, 247syl2an 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
249 simpr 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑥𝑋𝑤𝑋) → 𝑤𝑋)
250249ad3antlr 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝑤𝑋)
251 xmetcl 22337 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑤𝑋) → ((1st ‘(𝑔𝑏))𝐶𝑤) ∈ ℝ*)
252243, 248, 250, 251syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑤) ∈ ℝ*)
253252adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
256255ad2ant2r 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
257256rpred 12065 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
258 eleq2 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) → (𝑥𝑏𝑥 ∈ ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))))
25918ad5antr 775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝐶 ∈ (∞Met‘𝑋))
260225, 247sylan 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
261255rpxrd 12066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
262 elbl 22394 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 739 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 650 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))
269163ad3antlr 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝑥𝑋)
270 xmetcl 22337 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑥𝑋) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
271243, 248, 269, 270syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
272254rpxrd 12066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
273244, 226, 272syl2an 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
274 xrltle 12175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ*) → (((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏))))
275271, 273, 274syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6522 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 819 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝑥𝑋)
280259, 278, 279, 270syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
281 xmetge0 22350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑥𝑋) → 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥))
282259, 278, 279, 281syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥))
283 xrrege0 12198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) ∧ (0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥) ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
284283an4s 904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥)) ∧ ((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
285284ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥)) → (((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ))
286280, 282, 285syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ))
287286ad2ant2r 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
289288adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑥𝐶𝑤) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ*) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))))
291224, 238, 290syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))))
292 xmetge0 22350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑤𝑋) → 0 ≤ (𝑥𝐶𝑤))
2932923expb 1114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥𝑋𝑤𝑋)) → 0 ≤ (𝑥𝐶𝑤))
294220, 293sylan 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → 0 ≤ (𝑥𝐶𝑤))
295294adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → 0 ≤ (𝑥𝐶𝑤))
296236rpred 12065 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
297296ad2ant2r 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
298 xrrege0 12198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((𝑥𝐶𝑤) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) ∧ (0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏)))) → (𝑥𝐶𝑤) ∈ ℝ)
299298ex 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝑥𝐶𝑤) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) → ((0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) ∈ ℝ))
300224, 297, 299syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) ∈ ℝ))
301295, 300mpand 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ∈ ℝ))
304303imp 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12080 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∈ ℝ+)
308307rpxrd 12066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∈ ℝ*)
309308adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐶 ∈ (∞Met‘𝑋) ∧ ((1st ‘(𝑔𝑏)) ∈ 𝑋𝑤𝑋𝑥𝑋)) → ((1st ‘(𝑔𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)))
311243, 248, 250, 269, 310syl13anc 1479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ ∧ (𝑥𝐶𝑤) ∈ ℝ) → (((1st ‘(𝑔𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)) = (((1st ‘(𝑔𝑏))𝐶𝑥) + (𝑥𝐶𝑤)))
314289, 304, 313syl2anc 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12065 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
323322, 254ltaddrpd 12098 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
324244, 226, 323syl2an 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12183 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 567 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 807 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (𝜑𝑓:𝑋𝑌))
329 heicant.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝜑𝐷 ∈ (∞Met‘𝑌))
330 ffvelrn 6520 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓:𝑋𝑌𝑥𝑋) → (𝑓𝑥) ∈ 𝑌)
331 ffvelrn 6520 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓:𝑋𝑌𝑤𝑋) → (𝑓𝑤) ∈ 𝑌)
332330, 331anim12dan 918 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓:𝑋𝑌 ∧ (𝑥𝑋𝑤𝑋)) → ((𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌))
333 xmetcl 22337 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
3343333expb 1114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐷 ∈ (∞Met‘𝑌) ∧ ((𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
335329, 332, 334syl2an 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑓:𝑋𝑌 ∧ (𝑥𝑋𝑤𝑋))) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
336335anassrs 683 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑓:𝑋𝑌) ∧ (𝑥𝑋𝑤𝑋)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
337328, 336sylan 489 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
338337ad3antrrr 768 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝐷 ∈ (∞Met‘𝑌))
340 simp-5r 831 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝑓:𝑋𝑌)
341340, 278ffvelrnd 6523 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌)
342 simpllr 817 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑓:𝑋𝑌)
343342ffvelrnda 6522 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ 𝑌)
344343adantrr 755 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑓𝑥) ∈ 𝑌)
345344adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓𝑥) ∈ 𝑌)
346 xmetcl 22337 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑥) ∈ 𝑌) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ*)
347339, 341, 345, 346syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ*)
3489rpxrd 12066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ*)
349348ad4antlr 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑑 / 2) ∈ ℝ*)
350 xrltle 12175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑥) ∈ 𝑌) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
353339, 341, 345, 352syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
3549rpred 12065 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ)
355354ad4antlr 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑑 / 2) ∈ ℝ)
356 xrrege0 12198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6522 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑤𝑋) → (𝑓𝑤) ∈ 𝑌)
364363adantrl 754 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑓𝑤) ∈ 𝑌)
365364adantr 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓𝑤) ∈ 𝑌)
366 xmetcl 22337 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ*)
367339, 341, 365, 366syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ*)
368 xrltle 12175 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22350 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
371339, 341, 365, 370syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
372 xrrege0 12198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10211 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 908 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10281 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12033 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑑 ∈ ℝ+𝑑 ∈ ℝ*)
384383ad6antlr 781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 22353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓𝑥) ∈ 𝑌 ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌) → ((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
390339, 345, 341, 389syl3anc 1477 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
391390ad2ant2r 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12256 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 908 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2794 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 10705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 501 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 444 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12034 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑑 ∈ ℝ+𝑑 ∈ ℂ)
4074062halvesd 11470 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑑 ∈ ℝ+ → ((𝑑 / 2) + (𝑑 / 2)) = 𝑑)
408407ad6antlr 781 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12184 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 685 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 759 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . . . . . . . . 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 881 . . . . . . . . . . . . . . . . . . . . 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 3164 . . . . . . . . . . . . . . . . . . . 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 3109 . . . . . . . . . . . . . . . . . 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 4808 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = inf(ran ran 𝑔, ℝ, < ) → ((𝑥𝐶𝑤) < 𝑧 ↔ (𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < )))
425424imbi1d 330 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = inf(ran ran 𝑔, ℝ, < ) → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
4264252ralbidv 3127 . . . . . . . . . . . . . . . . . . 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 3449 . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . 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 649 . . . . . . . . . . . . . . . 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 2010 . . . . . . . . . . . . . . 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 630 . . . . . . . . . . . . . 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 492 . . . . . . . . . . . . 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 3169 . . . . . . . . . . . 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 633 . . . . . . . . 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 3103 . . . . . . 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 3098 . . . . . . 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 3236 . . . . 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 676 . . 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 2760 . . . 4 (metUnif‘𝐶) = (metUnif‘𝐶)
446 eqid 2760 . . . 4 (metUnif‘𝐷) = (metUnif‘𝐷)
447 heicant.y . . . 4 (𝜑𝑌 ≠ ∅)
448 xmetpsmet 22354 . . . . 5 (𝐶 ∈ (∞Met‘𝑋) → 𝐶 ∈ (PsMet‘𝑋))
44918, 448syl 17 . . . 4 (𝜑𝐶 ∈ (PsMet‘𝑋))
450 xmetpsmet 22354 . . . . 5 (𝐷 ∈ (∞Met‘𝑌) → 𝐷 ∈ (PsMet‘𝑌))
451329, 450syl 17 . . . 4 (𝜑𝐷 ∈ (PsMet‘𝑌))
452445, 446, 129, 447, 449, 451metucn 22577 . . 3 (𝜑 → (𝑓 ∈ ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))))
453 eqid 2760 . . . . 5 (MetOpen‘𝐷) = (MetOpen‘𝐷)
45423, 453metcn 22549 . . . 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 696 . . 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 2758 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 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  wne 2932  wral 3050  wrex 3051  cin 3714  wss 3715  c0 4058  𝒫 cpw 4302  cop 4327   cuni 4588   class class class wbr 4804   Or wor 5186   × cxp 5264  dom cdm 5266  ran crn 5267  Rel wrel 5271  Fun wfun 6043   Fn wfn 6044  wf 6045  cfv 6049  (class class class)co 6813  1st c1st 7331  2nd c2nd 7332  cen 8118  Fincfn 8121  infcinf 8512  cr 10127  0cc0 10128   + caddc 10131  *cxr 10265   < clt 10266  cle 10267   / cdiv 10876  2c2 11262  +crp 12025   +𝑒 cxad 12137  PsMetcpsmet 19932  ∞Metcxmt 19933  ballcbl 19935  MetOpencmopn 19938  metUnifcmetu 19939   Cn ccn 21230  Compccmp 21391   Cnucucn 22280
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114  ax-cnex 10184  ax-resscn 10185  ax-1cn 10186  ax-icn 10187  ax-addcl 10188  ax-addrcl 10189  ax-mulcl 10190  ax-mulrcl 10191  ax-mulcom 10192  ax-addass 10193  ax-mulass 10194  ax-distr 10195  ax-i2m1 10196  ax-1ne0 10197  ax-1rid 10198  ax-rnegex 10199  ax-rrecex 10200  ax-cnre 10201  ax-pre-lttri 10202  ax-pre-lttrn 10203  ax-pre-ltadd 10204  ax-pre-mulgt0 10205  ax-pre-sup 10206
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6774  df-ov 6816  df-oprab 6817  df-mpt2 6818  df-om 7231  df-1st 7333  df-2nd 7334  df-wrecs 7576  df-recs 7637  df-rdg 7675  df-1o 7729  df-er 7911  df-map 8025  df-en 8122  df-dom 8123  df-sdom 8124  df-fin 8125  df-sup 8513  df-inf 8514  df-pnf 10268  df-mnf 10269  df-xr 10270  df-ltxr 10271  df-le 10272  df-sub 10460  df-neg 10461  df-div 10877  df-nn 11213  df-2 11271  df-n0 11485  df-z 11570  df-uz 11880  df-q 11982  df-rp 12026  df-xneg 12139  df-xadd 12140  df-xmul 12141  df-ico 12374  df-topgen 16306  df-psmet 19940  df-xmet 19941  df-bl 19943  df-mopn 19944  df-fbas 19945  df-fg 19946  df-metu 19947  df-top 20901  df-topon 20918  df-bases 20952  df-cn 21233  df-cnp 21234  df-cmp 21392  df-fil 21851  df-ust 22205  df-ucn 22281
This theorem is referenced by: (None)
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