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Theorem heicant 34808
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 5061 . . . . . . . . . . 11 (𝑑 = 𝑦 → (((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑 ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
21imbi2d 342 . . . . . . . . . 10 (𝑑 = 𝑦 → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
322ralbidv 3196 . . . . . . . . 9 (𝑑 = 𝑦 → (∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
43rexbidv 3294 . . . . . . . 8 (𝑑 = 𝑦 → (∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
54cbvralvw 3447 . . . . . . 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 3321 . . . . . . . 8 (∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
76ralimi 3157 . . . . . . 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 218 . . . . . 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 12404 . . . . . . . . 9 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ+)
10 breq2 5061 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑑 / 2) → (((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦 ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
1110imbi2d 342 . . . . . . . . . . . . . . 15 (𝑦 = (𝑑 / 2) → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1211ralbidv 3194 . . . . . . . . . . . . . 14 (𝑦 = (𝑑 / 2) → (∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1312rexbidv 3294 . . . . . . . . . . . . 13 (𝑦 = (𝑑 / 2) → (∃𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∃𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1413ralbidv 3194 . . . . . . . . . . . 12 (𝑦 = (𝑑 / 2) → (∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1514rspcva 3618 . . . . . . . . . . 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 726 . . . . . . . . . . . . . 14 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → (MetOpen‘𝐶) ∈ Comp)
18 heicant.c . . . . . . . . . . . . . . . . . . . . . . 23 (𝜑𝐶 ∈ (∞Met‘𝑋))
1918ad2antrr 722 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → 𝐶 ∈ (∞Met‘𝑋))
2019anim1i 614 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) → (𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋))
21 rphalfcl 12404 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ+)
2221rpxrd 12420 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ*)
23 eqid 2818 . . . . . . . . . . . . . . . . . . . . . . 23 (MetOpen‘𝐶) = (MetOpen‘𝐶)
2423blopn 23037 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋 ∧ (𝑧 / 2) ∈ ℝ*) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
25243expa 1110 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋) ∧ (𝑧 / 2) ∈ ℝ*) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2620, 22, 25syl2an 595 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2726adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2821rpgt0d 12422 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ ℝ+ → 0 < (𝑧 / 2))
2922, 28jca 512 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ℝ+ → ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2)))
30 xblcntr 22948 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋 ∧ ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
31303expa 1110 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋) ∧ ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
3220, 29, 31syl2an 595 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
3332adantr 481 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
34 opelxpi 5585 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑋 ∧ (𝑧 / 2) ∈ ℝ+) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
3521, 34sylan2 592 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑋𝑧 ∈ ℝ+) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
3635ad4ant23 749 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
37 rpcn 12387 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ ℝ+𝑧 ∈ ℂ)
38372halvesd 11871 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ ℝ+ → ((𝑧 / 2) + (𝑧 / 2)) = 𝑧)
3938breq2d 5069 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ ℝ+ → ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) ↔ (𝑥𝐶𝑐) < 𝑧))
4039imbi1d 343 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℝ+ → (((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
4140ralbidv 3194 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℝ+ → (∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑐𝑋 ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
42 oveq2 7153 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑤 → (𝑥𝐶𝑐) = (𝑥𝐶𝑤))
4342breq1d 5067 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑤 → ((𝑥𝐶𝑐) < 𝑧 ↔ (𝑥𝐶𝑤) < 𝑧))
44 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐 = 𝑤 → (𝑓𝑐) = (𝑓𝑤))
4544oveq2d 7161 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑤 → ((𝑓𝑥)𝐷(𝑓𝑐)) = ((𝑓𝑥)𝐷(𝑓𝑤)))
4645breq1d 5067 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑤 → (((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
4743, 46imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = 𝑤 → (((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
4847cbvralvw 3447 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑐𝑋 ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
4941, 48syl6bb 288 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ ℝ+ → (∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
5049biimpar 478 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℝ+ ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)))
5150adantll 710 . . . . . . . . . . . . . . . . . . . 20 ((((((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 3495 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑥 ∈ V
53 ovex 7178 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 / 2) ∈ V
5452, 53op1std 7688 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (1st𝑝) = 𝑥)
5552, 53op2ndd 7689 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (2nd𝑝) = (𝑧 / 2))
5654, 55oveq12d 7163 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((1st𝑝)(ball‘𝐶)(2nd𝑝)) = (𝑥(ball‘𝐶)(𝑧 / 2)))
5756eqcomd 2824 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)))
5857biantrurd 533 . . . . . . . . . . . . . . . . . . . . . 22 (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 7160 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((1st𝑝)𝐶𝑐) = (𝑥𝐶𝑐))
6055, 55oveq12d 7163 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((2nd𝑝) + (2nd𝑝)) = ((𝑧 / 2) + (𝑧 / 2)))
6159, 60breq12d 5070 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) ↔ (𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2))))
6254fveq2d 6667 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (𝑓‘(1st𝑝)) = (𝑓𝑥))
6362oveq1d 7160 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) = ((𝑓𝑥)𝐷(𝑓𝑐)))
6463breq1d 5067 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)))
6561, 64imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . 23 (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 3194 . . . . . . . . . . . . . . . . . . . . . 22 (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 282 . . . . . . . . . . . . . . . . . . . . 21 (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 3620 . . . . . . . . . . . . . . . . . . . 20 ((⟨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 584 . . . . . . . . . . . . . . . . . . 19 ((((((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 2898 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (𝑥𝑏𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2))))
71 eqeq1 2822 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ↔ (𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝))))
7271anbi1d 629 . . . . . . . . . . . . . . . . . . . . . 22 (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 3294 . . . . . . . . . . . . . . . . . . . . 21 (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 630 . . . . . . . . . . . . . . . . . . . 20 (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 3620 . . . . . . . . . . . . . . . . . . 19 (((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 832 . . . . . . . . . . . . . . . . . 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)))))
7776rexlimdva2 3284 . . . . . . . . . . . . . . . . 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))))))
7877ralimdva 3174 . . . . . . . . . . . . . . . 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))))))
7978imp 407 . . . . . . . . . . . . . . 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)))))
8023mopnuni 22978 . . . . . . . . . . . . . . . . . 18 (𝐶 ∈ (∞Met‘𝑋) → 𝑋 = (MetOpen‘𝐶))
8118, 80syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 = (MetOpen‘𝐶))
8281raleqdv 3413 . . . . . . . . . . . . . . . 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))))))
8382ad3antrrr 726 . . . . . . . . . . . . . . 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))))))
8479, 83mpbid 233 . . . . . . . . . . . . . 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)))))
85 eqid 2818 . . . . . . . . . . . . . . 15 (MetOpen‘𝐶) = (MetOpen‘𝐶)
86 fveq2 6663 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (1st𝑝) = (1st ‘(𝑔𝑏)))
87 fveq2 6663 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (2nd𝑝) = (2nd ‘(𝑔𝑏)))
8886, 87oveq12d 7163 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝑔𝑏) → ((1st𝑝)(ball‘𝐶)(2nd𝑝)) = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))))
8988eqeq2d 2829 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑔𝑏) → (𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ↔ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))))
9086oveq1d 7160 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((1st𝑝)𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑐))
9187, 87oveq12d 7163 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((2nd𝑝) + (2nd𝑝)) = ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
9290, 91breq12d 5070 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) ↔ ((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
9386fveq2d 6667 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = (𝑔𝑏) → (𝑓‘(1st𝑝)) = (𝑓‘(1st ‘(𝑔𝑏))))
9493oveq1d 7160 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)))
9594breq1d 5067 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))
9692, 95imbi12d 346 . . . . . . . . . . . . . . . . 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))))
9796ralbidv 3194 . . . . . . . . . . . . . . . 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))))
9889, 97anbi12d 630 . . . . . . . . . . . . . . 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)))))
9985, 98cmpcovf 21927 . . . . . . . . . . . . . 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))))))
10017, 84, 99syl2anc 584 . . . . . . . . . . . . 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))))))
101100ex 413 . . . . . . . . . . . 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)))))))
102 elinel2 4170 . . . . . . . . . . . . . 14 (𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin) → 𝑠 ∈ Fin)
103 simpll 763 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → 𝜑)
104103anim1i 614 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → (𝜑𝑠 ∈ Fin))
105 frn 6513 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran 𝑔 ⊆ (𝑋 × ℝ+))
106 rnss 5802 . . . . . . . . . . . . . . . . . . . . . . . 24 (ran 𝑔 ⊆ (𝑋 × ℝ+) → ran ran 𝑔 ⊆ ran (𝑋 × ℝ+))
107105, 106syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran 𝑔 ⊆ ran (𝑋 × ℝ+))
108 rnxpss 6022 . . . . . . . . . . . . . . . . . . . . . . 23 ran (𝑋 × ℝ+) ⊆ ℝ+
109107, 108sstrdi 3976 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran 𝑔 ⊆ ℝ+)
110109adantl 482 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ⊆ ℝ+)
111 simplr 765 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → 𝑠 ∈ Fin)
112 ffun 6510 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:𝑠⟶(𝑋 × ℝ+) → Fun 𝑔)
113 vex 3495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑔 ∈ V
114113fundmen 8571 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun 𝑔 → dom 𝑔𝑔)
115114ensymd 8548 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Fun 𝑔𝑔 ≈ dom 𝑔)
116112, 115syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 ≈ dom 𝑔)
117 fdm 6515 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:𝑠⟶(𝑋 × ℝ+) → dom 𝑔 = 𝑠)
118116, 117breqtrd 5083 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔𝑠)
119 enfii 8723 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ Fin ∧ 𝑔𝑠) → 𝑔 ∈ Fin)
120118, 119sylan2 592 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ Fin ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑔 ∈ Fin)
121 rnfi 8795 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔 ∈ Fin → ran 𝑔 ∈ Fin)
122 rnfi 8795 . . . . . . . . . . . . . . . . . . . . . . . 24 (ran 𝑔 ∈ Fin → ran ran 𝑔 ∈ Fin)
123120, 121, 1223syl 18 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑠 ∈ Fin ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ∈ Fin)
124111, 123sylan 580 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ∈ Fin)
125117adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → dom 𝑔 = 𝑠)
126 eqtr 2838 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → 𝑋 = 𝑠)
12781, 126sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑋 = 𝑠)
128 heicant.x . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝜑𝑋 ≠ ∅)
129128adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑋 ≠ ∅)
130127, 129eqnetrrd 3081 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑠 ≠ ∅)
131 unieq 4838 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑠 = ∅ → 𝑠 = ∅)
132 uni0 4857 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ∅ = ∅
133131, 132syl6eq 2869 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = ∅ → 𝑠 = ∅)
134133necon3i 3045 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( 𝑠 ≠ ∅ → 𝑠 ≠ ∅)
135130, 134syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑠 ≠ ∅)
136135adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑠 ≠ ∅)
137125, 136eqnetrd 3080 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → dom 𝑔 ≠ ∅)
138 dm0rn0 5788 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
139138necon3bii 3065 . . . . . . . . . . . . . . . . . . . . . . . . 25 (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
140137, 139sylib 219 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran 𝑔 ≠ ∅)
141 relxp 5566 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Rel (𝑋 × ℝ+)
142 relss 5649 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (ran 𝑔 ⊆ (𝑋 × ℝ+) → (Rel (𝑋 × ℝ+) → Rel ran 𝑔))
143105, 141, 142mpisyl 21 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:𝑠⟶(𝑋 × ℝ+) → Rel ran 𝑔)
144 relrn0 5833 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Rel ran 𝑔 → (ran 𝑔 = ∅ ↔ ran ran 𝑔 = ∅))
145144necon3bid 3057 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Rel ran 𝑔 → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
146143, 145syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑔:𝑠⟶(𝑋 × ℝ+) → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
147146adantl 482 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
148140, 147mpbid 233 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ≠ ∅)
149148adantllr 715 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ≠ ∅)
150 rpssre 12384 . . . . . . . . . . . . . . . . . . . . . . 23 + ⊆ ℝ
151110, 150sstrdi 3976 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ⊆ ℝ)
152 ltso 10709 . . . . . . . . . . . . . . . . . . . . . . 23 < Or ℝ
153 fiinfcl 8953 . . . . . . . . . . . . . . . . . . . . . . 23 (( < Or ℝ ∧ (ran ran 𝑔 ∈ Fin ∧ ran ran 𝑔 ≠ ∅ ∧ ran ran 𝑔 ⊆ ℝ)) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
154152, 153mpan 686 . . . . . . . . . . . . . . . . . . . . . 22 ((ran ran 𝑔 ∈ Fin ∧ ran ran 𝑔 ≠ ∅ ∧ ran ran 𝑔 ⊆ ℝ) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
155124, 149, 151, 154syl3anc 1363 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
156110, 155sseldd 3965 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ+)
157104, 156sylanl1 676 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ+)
158157adantr 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+)
15981ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → 𝑋 = (MetOpen‘𝐶))
160159anim1i 614 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠))
161160ad2antrr 722 . . . . . . . . . . . . . . . . . . . . 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))
162 simpl 483 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑋𝑤𝑋) → 𝑥𝑋)
163126eleq2d 2895 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑥𝑋𝑥 𝑠))
164 eluni2 4834 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 𝑠 ↔ ∃𝑏𝑠 𝑥𝑏)
165163, 164syl6bb 288 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑥𝑋 ↔ ∃𝑏𝑠 𝑥𝑏))
166165biimpa 477 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑥𝑋) → ∃𝑏𝑠 𝑥𝑏)
167161, 162, 166syl2an 595 . . . . . . . . . . . . . . . . . . . 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)
168 nfv 1906 . . . . . . . . . . . . . . . . . . . . . . 23 𝑏(((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+))
169 nfra1 3216 . . . . . . . . . . . . . . . . . . . . . . 23 𝑏𝑏𝑠 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ∧ ∀𝑐𝑋 (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))
170168, 169nfan 1891 . . . . . . . . . . . . . . . . . . . . . 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))))
171 nfv 1906 . . . . . . . . . . . . . . . . . . . . . 22 𝑏(𝑥𝑋𝑤𝑋)
172170, 171nfan 1891 . . . . . . . . . . . . . . . . . . . . 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))
173 nfv 1906 . . . . . . . . . . . . . . . . . . . . 21 𝑏((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)
174 rspa 3203 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
175 oveq2 7153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑥 → ((1st ‘(𝑔𝑏))𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑥))
176175breq1d 5067 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑥 → (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ↔ ((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
177 fveq2 6663 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑐 = 𝑥 → (𝑓𝑐) = (𝑓𝑥))
178177oveq2d 7161 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑥 → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
179178breq1d 5067 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑥 → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)))
180176, 179imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
181180rspcva 3618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
182 oveq2 7153 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → ((1st ‘(𝑔𝑏))𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑤))
183182breq1d 5067 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ↔ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
18444oveq2d 7161 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
185184breq1d 5067 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)))
186183, 185imbi12d 346 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
187186rspcva 3618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
188181, 187anim12i 612 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
189188anandirs 675 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
190 anim12 805 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
191189, 190syl 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))))
192191adantrl 712 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
193192ad4ant23 749 . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
194 simpll 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → ((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+))
195194anim1i 614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)))
196195anim1i 614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)))
197109, 150sstrdi 3976 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran 𝑔 ⊆ ℝ)
198197adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ran ran 𝑔 ⊆ ℝ)
199 0re 10631 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 0 ∈ ℝ
200 rpge0 12390 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
201200rgen 3145 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 𝑦 ∈ ℝ+ 0 ≤ 𝑦
202 ssralv 4030 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (ran ran 𝑔 ⊆ ℝ+ → (∀𝑦 ∈ ℝ+ 0 ≤ 𝑦 → ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦))
203109, 201, 202mpisyl 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑔:𝑠⟶(𝑋 × ℝ+) → ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦)
204 breq1 5060 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑥 = 0 → (𝑥𝑦 ↔ 0 ≤ 𝑦))
205204ralbidv 3194 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑥 = 0 → (∀𝑦 ∈ ran ran 𝑔 𝑥𝑦 ↔ ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦))
206205rspcev 3620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((0 ∈ ℝ ∧ ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦)
207199, 203, 206sylancr 587 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑔:𝑠⟶(𝑋 × ℝ+) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦)
208207adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦)
209143adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → Rel ran 𝑔)
210 ffn 6507 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 Fn 𝑠)
211 fnfvelrn 6840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔 Fn 𝑠𝑏𝑠) → (𝑔𝑏) ∈ ran 𝑔)
212210, 211sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ ran 𝑔)
213 2ndrn 7729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((Rel ran 𝑔 ∧ (𝑔𝑏) ∈ ran 𝑔) → (2nd ‘(𝑔𝑏)) ∈ ran ran 𝑔)
214209, 212, 213syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ran ran 𝑔)
215 infrelb 11614 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((ran ran 𝑔 ⊆ ℝ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦 ∧ (2nd ‘(𝑔𝑏)) ∈ ran ran 𝑔) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
216198, 208, 214, 215syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
217216adantll 710 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
218217ad2ant2r 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
21918ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝐶 ∈ (∞Met‘𝑋))
220 xmetcl 22868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑤𝑋) → (𝑥𝐶𝑤) ∈ ℝ*)
2212203expb 1112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐶𝑤) ∈ ℝ*)
222219, 221sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐶𝑤) ∈ ℝ*)
223222adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (𝑥𝐶𝑤) ∈ ℝ*)
224 simplr 765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → 𝑔:𝑠⟶(𝑋 × ℝ+))
225 simpl 483 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑏𝑠𝑥𝑏) → 𝑏𝑠)
226214ne0d 4298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ran ran 𝑔 ≠ ∅)
227 infrecl 11611 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((ran ran 𝑔 ⊆ ℝ ∧ ran ran 𝑔 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ)
228198, 226, 208, 227syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ)
229228rexrd 10679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ*)
230224, 225, 229syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ*)
231 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑔:𝑠⟶(𝑋 × ℝ+))
232231ffvelrnda 6843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ (𝑋 × ℝ+))
233 xp2nd 7711 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔𝑏) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
234232, 233syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
235234rpxrd 12420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
236235ad2ant2r 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
237 xrltletr 12538 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
238223, 230, 236, 237syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
239218, 238mpan2d 690 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))))
240239adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
24118ad6antr 732 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝐶 ∈ (∞Met‘𝑋))
242 simpllr 772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) → 𝑔:𝑠⟶(𝑋 × ℝ+))
243 ffvelrn 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ (𝑋 × ℝ+))
244 xp1st 7710 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔𝑏) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
245243, 244syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
246242, 225, 245syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
247 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑥𝑋𝑤𝑋) → 𝑤𝑋)
248247ad3antlr 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝑤𝑋)
249 xmetcl 22868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑤𝑋) → ((1st ‘(𝑔𝑏))𝐶𝑤) ∈ ℝ*)
250241, 246, 248, 249syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑤) ∈ ℝ*)
251250adantr 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*)
252243, 233syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
253224, 252sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
254253ad2ant2r 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
255254rpred 12419 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
256162ad3antlr 727 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝑥𝑋)
257 xmetcl 22868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑥𝑋) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
258241, 246, 256, 257syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
259252rpxrd 12420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
260242, 225, 259syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
261 eleq2 2898 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) → (𝑥𝑏𝑥 ∈ ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))))
26218ad5antr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝐶 ∈ (∞Met‘𝑋))
263224, 245sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
264253rpxrd 12420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
265 elbl 22925 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋 ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ*) → (𝑥 ∈ ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ↔ (𝑥𝑋 ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))))
266262, 263, 264, 265syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑥 ∈ ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) ↔ (𝑥𝑋 ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))))
267261, 266sylan9bbr 511 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
268267biimpd 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
269268an32s 648 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
270269impr 455 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (𝑥𝑋 ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏))))
271270simprd 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) < (2nd ‘(𝑔𝑏)))
272258, 260, 271xrltled 12531 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))
273224ffvelrnda 6843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ (𝑋 × ℝ+))
274273, 244syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
275 simplrl 773 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝑥𝑋)
276262, 274, 275, 257syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
277 xmetge0 22881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑥𝑋) → 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥))
278262, 274, 275, 277syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥))
279 xrrege0 12555 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) ∧ (0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥) ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
280279an4s 656 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥)) ∧ ((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
281280ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥)) → (((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ))
282276, 278, 281syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ))
283282ad2ant2r 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
284255, 272, 283mp2and 695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
285284adantr 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)
286 xrltle 12530 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((𝑥𝐶𝑤) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ*) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))))
287223, 236, 286syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))))
288 xmetge0 22881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑤𝑋) → 0 ≤ (𝑥𝐶𝑤))
2892883expb 1112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥𝑋𝑤𝑋)) → 0 ≤ (𝑥𝐶𝑤))
290219, 289sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → 0 ≤ (𝑥𝐶𝑤))
291290adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → 0 ≤ (𝑥𝐶𝑤))
292234rpred 12419 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
293292ad2ant2r 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
294 xrrege0 12555 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((𝑥𝐶𝑤) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) ∧ (0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏)))) → (𝑥𝐶𝑤) ∈ ℝ)
295294ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((𝑥𝐶𝑤) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) → ((0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) ∈ ℝ))
296223, 293, 295syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) ∈ ℝ))
297291, 296mpand 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ∈ ℝ))
298287, 297syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ∈ ℝ))
299298adantlr 711 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ∈ ℝ))
300299imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) ∈ ℝ)
301285, 300readdcld 10658 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
302301rexrd 10679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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*)
303254, 254rpaddcld 12434 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∈ ℝ+)
304303rpxrd 12420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∈ ℝ*)
305304adantr 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*)
306 xmettri 22888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐶 ∈ (∞Met‘𝑋) ∧ ((1st ‘(𝑔𝑏)) ∈ 𝑋𝑤𝑋𝑥𝑋)) → ((1st ‘(𝑔𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)))
307241, 246, 248, 256, 306syl13anc 1364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
308307adantr 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)))
309 rexadd 12613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ ∧ (𝑥𝐶𝑤) ∈ ℝ) → (((1st ‘(𝑔𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)) = (((1st ‘(𝑔𝑏))𝐶𝑥) + (𝑥𝐶𝑤)))
310285, 300, 309syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
311308, 310breqtrd 5083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
312255adantr 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)
313271adantr 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)))
314 simpr 485 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))) → (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)))
315285, 300, 312, 312, 313, 314lt2addd 11251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
316251, 302, 305, 311, 315xrlelttrd 12541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
317316ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
318252rpred 12419 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
319318, 252ltaddrpd 12452 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
320242, 225, 319syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
321258, 260, 304, 271, 320xrlttrd 12540 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
322317, 321jctild 526 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))))
323240, 322syld 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))))))
324 simpll 763 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (𝜑𝑓:𝑋𝑌))
325 heicant.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝜑𝐷 ∈ (∞Met‘𝑌))
326 ffvelrn 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓:𝑋𝑌𝑥𝑋) → (𝑓𝑥) ∈ 𝑌)
327 ffvelrn 6841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓:𝑋𝑌𝑤𝑋) → (𝑓𝑤) ∈ 𝑌)
328326, 327anim12dan 618 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓:𝑋𝑌 ∧ (𝑥𝑋𝑤𝑋)) → ((𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌))
329 xmetcl 22868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
3303293expb 1112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐷 ∈ (∞Met‘𝑌) ∧ ((𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
331325, 328, 330syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑓:𝑋𝑌 ∧ (𝑥𝑋𝑤𝑋))) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
332331anassrs 468 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((𝜑𝑓:𝑋𝑌) ∧ (𝑥𝑋𝑤𝑋)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
333324, 332sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
334333ad3antrrr 726 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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*)
335325ad5antr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝐷 ∈ (∞Met‘𝑌))
336 simp-5r 782 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝑓:𝑋𝑌)
337336, 274ffvelrnd 6844 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌)
338 simpllr 772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑓:𝑋𝑌)
339338ffvelrnda 6843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ 𝑌)
340339adantrr 713 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑓𝑥) ∈ 𝑌)
341340adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓𝑥) ∈ 𝑌)
342 xmetcl 22868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑥) ∈ 𝑌) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ*)
343335, 337, 341, 342syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ*)
3449rpxrd 12420 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ*)
345344ad4antlr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑑 / 2) ∈ ℝ*)
346 xrltle 12530 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ*) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ≤ (𝑑 / 2)))
347343, 345, 346syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
348 xmetge0 22881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑥) ∈ 𝑌) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
349335, 337, 341, 348syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
3509rpred 12419 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ)
351350ad4antlr 729 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑑 / 2) ∈ ℝ)
352 xrrege0 12555 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
353352ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
354343, 351, 353syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
355349, 354mpand 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ≤ (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ))
356347, 355syld 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))
357356ad2ant2r 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
358357imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
359338ffvelrnda 6843 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑤𝑋) → (𝑓𝑤) ∈ 𝑌)
360359adantrl 712 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑓𝑤) ∈ 𝑌)
361360adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓𝑤) ∈ 𝑌)
362 xmetcl 22868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ*)
363335, 337, 361, 362syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ*)
364 xrltle 12530 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ*) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ≤ (𝑑 / 2)))
365363, 345, 364syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
366 xmetge0 22881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
367335, 337, 361, 366syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
368 xrrege0 12555 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
369368ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
370363, 351, 369syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
371367, 370mpand 691 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ≤ (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ))
372365, 371syld 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))
373372ad2ant2r 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
374373imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
375 readdcl 10608 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ) → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) + ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤))) ∈ ℝ)
376358, 374, 375syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
377376anandis 674 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
378377rexrd 10679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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*)
379 rpxr 12386 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑑 ∈ ℝ+𝑑 ∈ ℝ*)
380379ad6antlr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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*)
381 xmettri 22888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
382335, 341, 361, 337, 381syl13anc 1364 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
383382ad2ant2r 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
384383adantr 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))))
385 xmetsym 22884 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓𝑥) ∈ 𝑌 ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌) → ((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
386335, 341, 337, 385syl3anc 1363 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
387386ad2ant2r 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
388387adantr 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)))
389388oveq1d 7160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
390 rexadd 12613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
391358, 374, 390syl2an 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
392391anandis 674 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
393389, 392eqtrd 2853 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
394384, 393breqtrd 5083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
395 lt2add 11113 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
396395expcom 414 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
397351, 351, 396syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
398356, 372, 397syl2and 607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))))
399398pm2.43d 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))))
400399ad2ant2r 743 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
401400imp 407 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
402 rpcn 12387 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑑 ∈ ℝ+𝑑 ∈ ℂ)
4034022halvesd 11871 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑑 ∈ ℝ+ → ((𝑑 / 2) + (𝑑 / 2)) = 𝑑)
404403ad6antlr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
405401, 404breqtrd 5083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
406334, 378, 380, 394, 405xrlelttrd 12541 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
407406ex 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))
408323, 407imim12d 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)))
409196, 408sylanl1 676 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
410409adantlrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . 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)))
411193, 410mpd 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))
412411exp32 421 . . . . . . . . . . . . . . . . . . . . . . . . 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))))
413174, 412sylan2 592 . . . . . . . . . . . . . . . . . . . . . . . 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))))
414413expr 457 . . . . . . . . . . . . . . . . . . . . . . 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)))))
415414pm2.43d 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))))
416415an32s 648 . . . . . . . . . . . . . . . . . . . . 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))))
417172, 173, 416rexlimd 3314 . . . . . . . . . . . . . . . . . . . 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)))
418167, 417mpd 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))
419418ralrimivva 3188 . . . . . . . . . . . . . . . . . 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))
420 breq2 5061 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = inf(ran ran 𝑔, ℝ, < ) → ((𝑥𝐶𝑤) < 𝑧 ↔ (𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < )))
421420imbi1d 343 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = inf(ran ran 𝑔, ℝ, < ) → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
4224212ralbidv 3196 . . . . . . . . . . . . . . . . . . 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)))
423422rspcev 3620 . . . . . . . . . . . . . . . . . 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))
424158, 419, 423syl2anc 584 . . . . . . . . . . . . . . . . 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))
425424expl 458 . . . . . . . . . . . . . . . 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)))
426425exlimdv 1925 . . . . . . . . . . . . . . 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)))
427426expimpd 454 . . . . . . . . . . . . . 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)))
428102, 427sylan2 592 . . . . . . . . . . . . 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)))
429428rexlimdva 3281 . . . . . . . . . . . 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)))
430101, 429syld 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)))
43115, 430syl5 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)))
432431exp4b 431 . . . . . . . . 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)))))
4339, 432mpdi 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))))
434433ralrimiv 3178 . . . . . . 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)))
435 r19.21v 3172 . . . . . . 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)))
436434, 435sylib 219 . . . . . 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)))
4378, 436impbid2 227 . . . . 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)))
438 ralcom 3351 . . . . 5 (∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
439437, 438syl6bb 288 . . . 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)))
440439pm5.32da 579 . . 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))))
441 eqid 2818 . . . 4 (metUnif‘𝐶) = (metUnif‘𝐶)
442 eqid 2818 . . . 4 (metUnif‘𝐷) = (metUnif‘𝐷)
443 heicant.y . . . 4 (𝜑𝑌 ≠ ∅)
444 xmetpsmet 22885 . . . . 5 (𝐶 ∈ (∞Met‘𝑋) → 𝐶 ∈ (PsMet‘𝑋))
44518, 444syl 17 . . . 4 (𝜑𝐶 ∈ (PsMet‘𝑋))
446 xmetpsmet 22885 . . . . 5 (𝐷 ∈ (∞Met‘𝑌) → 𝐷 ∈ (PsMet‘𝑌))
447325, 446syl 17 . . . 4 (𝜑𝐷 ∈ (PsMet‘𝑌))
448441, 442, 128, 443, 445, 447metucn 23108 . . 3 (𝜑 → (𝑓 ∈ ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))))
449 eqid 2818 . . . . 5 (MetOpen‘𝐷) = (MetOpen‘𝐷)
45023, 449metcn 23080 . . . 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))))
45118, 325, 450syl2anc 584 . . 3 (𝜑 → (𝑓 ∈ ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))))
452440, 448, 4513bitr4d 312 . 2 (𝜑 → (𝑓 ∈ ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) ↔ 𝑓 ∈ ((MetOpen‘𝐶) Cn (MetOpen‘𝐷))))
453452eqrdv 2816 1 (𝜑 → ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) = ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1079   = wceq 1528  wex 1771  wcel 2105  wne 3013  wral 3135  wrex 3136  cin 3932  wss 3933  c0 4288  𝒫 cpw 4535  cop 4563   cuni 4830   class class class wbr 5057   Or wor 5466   × cxp 5546  dom cdm 5548  ran crn 5549  Rel wrel 5553  Fun wfun 6342   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  1st c1st 7676  2nd c2nd 7677  cen 8494  Fincfn 8497  infcinf 8893  cr 10524  0cc0 10525   + caddc 10528  *cxr 10662   < clt 10663  cle 10664   / cdiv 11285  2c2 11680  +crp 12377   +𝑒 cxad 12493  PsMetcpsmet 20457  ∞Metcxmet 20458  ballcbl 20460  MetOpencmopn 20463  metUnifcmetu 20464   Cn ccn 21760  Compccmp 21922   Cnucucn 22811
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450  ax-cnex 10581  ax-resscn 10582  ax-1cn 10583  ax-icn 10584  ax-addcl 10585  ax-addrcl 10586  ax-mulcl 10587  ax-mulrcl 10588  ax-mulcom 10589  ax-addass 10590  ax-mulass 10591  ax-distr 10592  ax-i2m1 10593  ax-1ne0 10594  ax-1rid 10595  ax-rnegex 10596  ax-rrecex 10597  ax-cnre 10598  ax-pre-lttri 10599  ax-pre-lttrn 10600  ax-pre-ltadd 10601  ax-pre-mulgt0 10602  ax-pre-sup 10603
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-nel 3121  df-ral 3140  df-rex 3141  df-reu 3142  df-rmo 3143  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7103  df-ov 7148  df-oprab 7149  df-mpo 7150  df-om 7570  df-1st 7678  df-2nd 7679  df-wrecs 7936  df-recs 7997  df-rdg 8035  df-1o 8091  df-er 8278  df-map 8397  df-en 8498  df-dom 8499  df-sdom 8500  df-fin 8501  df-sup 8894  df-inf 8895  df-pnf 10665  df-mnf 10666  df-xr 10667  df-ltxr 10668  df-le 10669  df-sub 10860  df-neg 10861  df-div 11286  df-nn 11627  df-2 11688  df-n0 11886  df-z 11970  df-uz 12232  df-q 12337  df-rp 12378  df-xneg 12495  df-xadd 12496  df-xmul 12497  df-ico 12732  df-topgen 16705  df-psmet 20465  df-xmet 20466  df-bl 20468  df-mopn 20469  df-fbas 20470  df-fg 20471  df-metu 20472  df-top 21430  df-topon 21447  df-bases 21482  df-cn 21763  df-cnp 21764  df-cmp 21923  df-fil 22382  df-ust 22736  df-ucn 22812
This theorem is referenced by: (None)
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