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Theorem heicant 37642
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 5152 . . . . . . . . . . 11 (𝑑 = 𝑦 → (((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑 ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
21imbi2d 340 . . . . . . . . . 10 (𝑑 = 𝑦 → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
322ralbidv 3219 . . . . . . . . 9 (𝑑 = 𝑦 → (∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∀𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
43rexbidv 3177 . . . . . . . 8 (𝑑 = 𝑦 → (∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦)))
54cbvralvw 3235 . . . . . . 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 3312 . . . . . . . 8 (∃𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) → ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
76ralimi 3081 . . . . . . 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 217 . . . . . 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 13060 . . . . . . . . 9 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ+)
10 breq2 5152 . . . . . . . . . . . . . . . 16 (𝑦 = (𝑑 / 2) → (((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦 ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
1110imbi2d 340 . . . . . . . . . . . . . . 15 (𝑦 = (𝑑 / 2) → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1211ralbidv 3176 . . . . . . . . . . . . . 14 (𝑦 = (𝑑 / 2) → (∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1312rexbidv 3177 . . . . . . . . . . . . 13 (𝑦 = (𝑑 / 2) → (∃𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∃𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1413ralbidv 3176 . . . . . . . . . . . 12 (𝑦 = (𝑑 / 2) → (∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
1514rspcva 3620 . . . . . . . . . . 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 730 . . . . . . . . . . . . . 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 726 . . . . . . . . . . . . . . . . . . . . . 22 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → 𝐶 ∈ (∞Met‘𝑋))
2019anim1i 615 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) → (𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋))
21 rphalfcl 13060 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ+)
2221rpxrd 13076 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ℝ+ → (𝑧 / 2) ∈ ℝ*)
23 eqid 2735 . . . . . . . . . . . . . . . . . . . . . . 23 (MetOpen‘𝐶) = (MetOpen‘𝐶)
2423blopn 24529 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋 ∧ (𝑧 / 2) ∈ ℝ*) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
25243expa 1117 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋) ∧ (𝑧 / 2) ∈ ℝ*) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2620, 22, 25syl2an 596 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2726adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶))
2821rpgt0d 13078 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ ℝ+ → 0 < (𝑧 / 2))
2922, 28jca 511 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 ∈ ℝ+ → ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2)))
30 xblcntr 24437 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋 ∧ ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
31303expa 1117 . . . . . . . . . . . . . . . . . . . . 21 (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋) ∧ ((𝑧 / 2) ∈ ℝ* ∧ 0 < (𝑧 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
3220, 29, 31syl2an 596 . . . . . . . . . . . . . . . . . . . 20 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
3332adantr 480 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))
34 opelxpi 5726 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥𝑋 ∧ (𝑧 / 2) ∈ ℝ+) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
3521, 34sylan2 593 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑋𝑧 ∈ ℝ+) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
3635ad4ant23 753 . . . . . . . . . . . . . . . . . . . 20 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥𝑋) ∧ 𝑧 ∈ ℝ+) ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ⟨𝑥, (𝑧 / 2)⟩ ∈ (𝑋 × ℝ+))
37 rpcn 13043 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑧 ∈ ℝ+𝑧 ∈ ℂ)
38372halvesd 12510 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 ∈ ℝ+ → ((𝑧 / 2) + (𝑧 / 2)) = 𝑧)
3938breq2d 5160 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑧 ∈ ℝ+ → ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) ↔ (𝑥𝐶𝑐) < 𝑧))
4039imbi1d 341 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑧 ∈ ℝ+ → (((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
4140ralbidv 3176 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑧 ∈ ℝ+ → (∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑐𝑋 ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2))))
42 oveq2 7439 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑤 → (𝑥𝐶𝑐) = (𝑥𝐶𝑤))
4342breq1d 5158 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑤 → ((𝑥𝐶𝑐) < 𝑧 ↔ (𝑥𝐶𝑤) < 𝑧))
44 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑐 = 𝑤 → (𝑓𝑐) = (𝑓𝑤))
4544oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑐 = 𝑤 → ((𝑓𝑥)𝐷(𝑓𝑐)) = ((𝑓𝑥)𝐷(𝑓𝑤)))
4645breq1d 5158 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑐 = 𝑤 → (((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
4743, 46imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑐 = 𝑤 → (((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
4847cbvralvw 3235 . . . . . . . . . . . . . . . . . . . . . . 23 (∀𝑐𝑋 ((𝑥𝐶𝑐) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2)))
4941, 48bitrdi 287 . . . . . . . . . . . . . . . . . . . . . 22 (𝑧 ∈ ℝ+ → (∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)) ↔ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))))
5049biimpar 477 . . . . . . . . . . . . . . . . . . . . 21 ((𝑧 ∈ ℝ+ ∧ ∀𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < (𝑑 / 2))) → ∀𝑐𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)))
5150adantll 714 . . . . . . . . . . . . . . . . . . . 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 3482 . . . . . . . . . . . . . . . . . . . . . . . . . 26 𝑥 ∈ V
53 ovex 7464 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑧 / 2) ∈ V
5452, 53op1std 8023 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (1st𝑝) = 𝑥)
5552, 53op2ndd 8024 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (2nd𝑝) = (𝑧 / 2))
5654, 55oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((1st𝑝)(ball‘𝐶)(2nd𝑝)) = (𝑥(ball‘𝐶)(𝑧 / 2)))
5756eqcomd 2741 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝)))
5857biantrurd 532 . . . . . . . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((1st𝑝)𝐶𝑐) = (𝑥𝐶𝑐))
6055, 55oveq12d 7449 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((2nd𝑝) + (2nd𝑝)) = ((𝑧 / 2) + (𝑧 / 2)))
6159, 60breq12d 5161 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) ↔ (𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2))))
6254fveq2d 6911 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (𝑓‘(1st𝑝)) = (𝑓𝑥))
6362oveq1d 7446 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) = ((𝑓𝑥)𝐷(𝑓𝑐)))
6463breq1d 5158 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑝 = ⟨𝑥, (𝑧 / 2)⟩ → (((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓𝑥)𝐷(𝑓𝑐)) < (𝑑 / 2)))
6561, 64imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . . . . . . . . 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 281 . . . . . . . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . . . . . . 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 2828 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (𝑥𝑏𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2))))
71 eqeq1 2739 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ↔ (𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st𝑝)(ball‘𝐶)(2nd𝑝))))
7271anbi1d 631 . . . . . . . . . . . . . . . . . . . . . 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 3177 . . . . . . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . . . . . 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 837 . . . . . . . . . . . . . . . . . 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 3155 . . . . . . . . . . . . . . . . 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 3165 . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . 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 24467 . . . . . . . . . . . . . . . . . 18 (𝐶 ∈ (∞Met‘𝑋) → 𝑋 = (MetOpen‘𝐶))
8118, 80syl 17 . . . . . . . . . . . . . . . . 17 (𝜑𝑋 = (MetOpen‘𝐶))
8281raleqdv 3324 . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . 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 232 . . . . . . . . . . . . . 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 2735 . . . . . . . . . . . . . . 15 (MetOpen‘𝐶) = (MetOpen‘𝐶)
86 fveq2 6907 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (1st𝑝) = (1st ‘(𝑔𝑏)))
87 fveq2 6907 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (2nd𝑝) = (2nd ‘(𝑔𝑏)))
8886, 87oveq12d 7449 . . . . . . . . . . . . . . . . 17 (𝑝 = (𝑔𝑏) → ((1st𝑝)(ball‘𝐶)(2nd𝑝)) = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))))
8988eqeq2d 2746 . . . . . . . . . . . . . . . 16 (𝑝 = (𝑔𝑏) → (𝑏 = ((1st𝑝)(ball‘𝐶)(2nd𝑝)) ↔ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))))
9086oveq1d 7446 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((1st𝑝)𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑐))
9187, 87oveq12d 7449 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((2nd𝑝) + (2nd𝑝)) = ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
9290, 91breq12d 5161 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (((1st𝑝)𝐶𝑐) < ((2nd𝑝) + (2nd𝑝)) ↔ ((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
9386fveq2d 6911 . . . . . . . . . . . . . . . . . . . 20 (𝑝 = (𝑔𝑏) → (𝑓‘(1st𝑝)) = (𝑓‘(1st ‘(𝑔𝑏))))
9493oveq1d 7446 . . . . . . . . . . . . . . . . . . 19 (𝑝 = (𝑔𝑏) → ((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)))
9594breq1d 5158 . . . . . . . . . . . . . . . . . 18 (𝑝 = (𝑔𝑏) → (((𝑓‘(1st𝑝))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2)))
9692, 95imbi12d 344 . . . . . . . . . . . . . . . . 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 3176 . . . . . . . . . . . . . . . 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 632 . . . . . . . . . . . . . . 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 23415 . . . . . . . . . . . . . 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 412 . . . . . . . . . . . 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 4212 . . . . . . . . . . . . . 14 (𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin) → 𝑠 ∈ Fin)
103 simpll 767 . . . . . . . . . . . . . . . . . . . . 21 (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) → 𝜑)
104103anim1i 615 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → (𝜑𝑠 ∈ Fin))
105 frn 6744 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran 𝑔 ⊆ (𝑋 × ℝ+))
106 rnss 5953 . . . . . . . . . . . . . . . . . . . . . . . 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 6194 . . . . . . . . . . . . . . . . . . . . . . 23 ran (𝑋 × ℝ+) ⊆ ℝ+
109107, 108sstrdi 4008 . . . . . . . . . . . . . . . . . . . . . 22 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran 𝑔 ⊆ ℝ+)
110109adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ⊆ ℝ+)
111 simplr 769 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → 𝑠 ∈ Fin)
112 ffun 6740 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑔:𝑠⟶(𝑋 × ℝ+) → Fun 𝑔)
113 vex 3482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 𝑔 ∈ V
114113fundmen 9070 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Fun 𝑔 → dom 𝑔𝑔)
115114ensymd 9044 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Fun 𝑔𝑔 ≈ dom 𝑔)
116112, 115syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 ≈ dom 𝑔)
117 fdm 6746 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑔:𝑠⟶(𝑋 × ℝ+) → dom 𝑔 = 𝑠)
118116, 117breqtrd 5174 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔𝑠)
119 enfii 9224 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 ∈ Fin ∧ 𝑔𝑠) → 𝑔 ∈ Fin)
120118, 119sylan2 593 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 ∈ Fin ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑔 ∈ Fin)
121 rnfi 9378 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑔 ∈ Fin → ran 𝑔 ∈ Fin)
122 rnfi 9378 . . . . . . . . . . . . . . . . . . . . . . . 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 481 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → dom 𝑔 = 𝑠)
126 eqtr 2758 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑋 ≠ ∅)
130127, 129eqnetrrd 3007 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑠 ≠ ∅)
131 unieq 4923 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑠 = ∅ → 𝑠 = ∅)
132 uni0 4940 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ∅ = ∅
133131, 132eqtrdi 2791 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑠 = ∅ → 𝑠 = ∅)
134133necon3i 2971 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ( 𝑠 ≠ ∅ → 𝑠 ≠ ∅)
135130, 134syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝜑 (MetOpen‘𝐶) = 𝑠) → 𝑠 ≠ ∅)
136135adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑠 ≠ ∅)
137125, 136eqnetrd 3006 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → dom 𝑔 ≠ ∅)
138 dm0rn0 5938 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (dom 𝑔 = ∅ ↔ ran 𝑔 = ∅)
139138necon3bii 2991 . . . . . . . . . . . . . . . . . . . . . . . . 25 (dom 𝑔 ≠ ∅ ↔ ran 𝑔 ≠ ∅)
140137, 139sylib 218 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran 𝑔 ≠ ∅)
141 relxp 5707 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 Rel (𝑋 × ℝ+)
142 relss 5794 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5986 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (Rel ran 𝑔 → (ran 𝑔 = ∅ ↔ ran ran 𝑔 = ∅))
145144necon3bid 2983 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (Rel ran 𝑔 → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
146143, 145syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑔:𝑠⟶(𝑋 × ℝ+) → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
147146adantl 481 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (ran 𝑔 ≠ ∅ ↔ ran ran 𝑔 ≠ ∅))
148140, 147mpbid 232 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝜑 (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ≠ ∅)
149148adantllr 719 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ≠ ∅)
150 rpssre 13040 . . . . . . . . . . . . . . . . . . . . . . 23 + ⊆ ℝ
151110, 150sstrdi 4008 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran ran 𝑔 ⊆ ℝ)
152 ltso 11339 . . . . . . . . . . . . . . . . . . . . . . 23 < Or ℝ
153 fiinfcl 9539 . . . . . . . . . . . . . . . . . . . . . . 23 (( < Or ℝ ∧ (ran ran 𝑔 ∈ Fin ∧ ran ran 𝑔 ≠ ∅ ∧ ran ran 𝑔 ⊆ ℝ)) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
154152, 153mpan 690 . . . . . . . . . . . . . . . . . . . . . 22 ((ran ran 𝑔 ∈ Fin ∧ ran ran 𝑔 ≠ ∅ ∧ ran ran 𝑔 ⊆ ℝ) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
155124, 149, 151, 154syl3anc 1370 . . . . . . . . . . . . . . . . . . . . 21 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran 𝑔)
156110, 155sseldd 3996 . . . . . . . . . . . . . . . . . . . 20 ((((𝜑𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ+)
157104, 156sylanl1 680 . . . . . . . . . . . . . . . . . . 19 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ+)
158157adantr 480 . . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → 𝑋 = (MetOpen‘𝐶))
160159anim1i 615 . . . . . . . . . . . . . . . . . . . . . 22 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠))
161160ad2antrr 726 . . . . . . . . . . . . . . . . . . . . 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 482 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥𝑋𝑤𝑋) → 𝑥𝑋)
163126eleq2d 2825 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑥𝑋𝑥 𝑠))
164 eluni2 4916 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 𝑠 ↔ ∃𝑏𝑠 𝑥𝑏)
165163, 164bitrdi 287 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) → (𝑥𝑋 ↔ ∃𝑏𝑠 𝑥𝑏))
166165biimpa 476 . . . . . . . . . . . . . . . . . . . . 21 (((𝑋 = (MetOpen‘𝐶) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑥𝑋) → ∃𝑏𝑠 𝑥𝑏)
167161, 162, 166syl2an 596 . . . . . . . . . . . . . . . . . . . 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 1912 . . . . . . . . . . . . . . . . . . . . . . 23 𝑏(((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+))
169 nfra1 3282 . . . . . . . . . . . . . . . . . . . . . . 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 1897 . . . . . . . . . . . . . . . . . . . . . 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 1912 . . . . . . . . . . . . . . . . . . . . . 22 𝑏(𝑥𝑋𝑤𝑋)
172170, 171nfan 1897 . . . . . . . . . . . . . . . . . . . . 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 1912 . . . . . . . . . . . . . . . . . . . . 21 𝑏((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)
174 rspa 3246 . . . . . . . . . . . . . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑥 → ((1st ‘(𝑔𝑏))𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑥))
176175breq1d 5158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑥 → (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ↔ ((1st ‘(𝑔𝑏))𝐶𝑥) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
177 fveq2 6907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑐 = 𝑥 → (𝑓𝑐) = (𝑓𝑥))
178177oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑥 → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
179178breq1d 5158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑥 → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) < (𝑑 / 2)))
180176, 179imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → ((1st ‘(𝑔𝑏))𝐶𝑐) = ((1st ‘(𝑔𝑏))𝐶𝑤))
183182breq1d 5158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (((1st ‘(𝑔𝑏))𝐶𝑐) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ↔ ((1st ‘(𝑔𝑏))𝐶𝑤) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏)))))
18444oveq2d 7447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑐 = 𝑤 → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
185184breq1d 5158 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑐 = 𝑤 → (((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) < (𝑑 / 2)))
186183, 185imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 679 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 809 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 716 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 753 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) → ((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+))
195194anim1i 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ (MetOpen‘𝐶) = 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)))
196195anim1i 615 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran 𝑔 ⊆ ℝ)
198197adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ran ran 𝑔 ⊆ ℝ)
199 0re 11261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 0 ∈ ℝ
200 rpge0 13046 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦 ∈ ℝ+ → 0 ≤ 𝑦)
201200rgen 3061 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 𝑦 ∈ ℝ+ 0 ≤ 𝑦
202 ssralv 4064 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5151 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑥 = 0 → (𝑥𝑦 ↔ 0 ≤ 𝑦))
205204ralbidv 3176 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑥 = 0 → (∀𝑦 ∈ ran ran 𝑔 𝑥𝑦 ↔ ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦))
206205rspcev 3622 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥𝑦)
209143adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → Rel ran 𝑔)
210 ffn 6737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 Fn 𝑠)
211 fnfvelrn 7100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔 Fn 𝑠𝑏𝑠) → (𝑔𝑏) ∈ ran 𝑔)
212210, 211sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ ran 𝑔)
213 2ndrn 8065 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 12251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
217216adantll 714 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
218217ad2ant2r 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd ‘(𝑔𝑏)))
21918ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝐶 ∈ (∞Met‘𝑋))
220 xmetcl 24357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑤𝑋) → (𝑥𝐶𝑤) ∈ ℝ*)
2212203expb 1119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐶𝑤) ∈ ℝ*)
222219, 221sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑥𝐶𝑤) ∈ ℝ*)
223222adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (𝑥𝐶𝑤) ∈ ℝ*)
224 simplr 769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → 𝑔:𝑠⟶(𝑋 × ℝ+))
225 simpl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑏𝑠𝑥𝑏) → 𝑏𝑠)
226214ne0d 4348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → ran ran 𝑔 ≠ ∅)
227 infrecl 12248 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ)
229228rexrd 11309 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ*)
230224, 225, 229syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → inf(ran ran 𝑔, ℝ, < ) ∈ ℝ*)
231 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑔:𝑠⟶(𝑋 × ℝ+))
232231ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ (𝑋 × ℝ+))
233 xp2nd 8046 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔𝑏) ∈ (𝑋 × ℝ+) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
234232, 233syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
235234rpxrd 13076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
236235ad2ant2r 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
237 xrltletr 13196 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → (𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏))))
240239adantlr 715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝐶 ∈ (∞Met‘𝑋))
242 simpllr 776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) → 𝑔:𝑠⟶(𝑋 × ℝ+))
243 ffvelcdm 7101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (𝑔𝑏) ∈ (𝑋 × ℝ+))
244 xp1st 8045 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑔𝑏) ∈ (𝑋 × ℝ+) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
245243, 244syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
246242, 225, 245syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (1st ‘(𝑔𝑏)) ∈ 𝑋)
247 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑥𝑋𝑤𝑋) → 𝑤𝑋)
248247ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝑤𝑋)
249 xmetcl 24357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑤𝑋) → ((1st ‘(𝑔𝑏))𝐶𝑤) ∈ ℝ*)
250241, 246, 248, 249syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑤) ∈ ℝ*)
251250adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ+)
255254rpred 13075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
256162ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → 𝑥𝑋)
257 xmetcl 24357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑥𝑋) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
258241, 246, 256, 257syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
259252rpxrd 13076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
260242, 225, 259syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
261 eleq2 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 (𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏))) → (𝑥𝑏𝑥 ∈ ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))))
26218ad5antr 734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ*)
265 elbl 24414 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 510 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 229 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 652 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13189 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))
273224ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 777 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝑥𝑋)
276262, 274, 275, 257syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ*)
277 xmetge0 24370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st ‘(𝑔𝑏)) ∈ 𝑋𝑥𝑋) → 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥))
278262, 274, 275, 277syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥))
279 xrrege0 13213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) ∧ (0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥) ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
280279an4s 660 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ* ∧ 0 ≤ ((1st ‘(𝑔𝑏))𝐶𝑥)) ∧ ((2nd ‘(𝑔𝑏)) ∈ ℝ ∧ ((1st ‘(𝑔𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔𝑏)))) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
281280ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 699 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((1st ‘(𝑔𝑏))𝐶𝑥) ∈ ℝ)
285284adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13188 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥𝑋𝑤𝑋) → 0 ≤ (𝑥𝐶𝑤))
2892883expb 1119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥𝑋𝑤𝑋)) → 0 ≤ (𝑥𝐶𝑤))
290219, 289sylan 580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → 0 ≤ (𝑥𝐶𝑤))
291290adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → 0 ≤ (𝑥𝐶𝑤))
292234rpred 13075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
293292ad2ant2r 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ (𝑏𝑠𝑥𝑏)) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
294 xrrege0 13213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 ((((𝑥𝐶𝑤) ∈ ℝ* ∧ (2nd ‘(𝑔𝑏)) ∈ ℝ) ∧ (0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔𝑏)))) → (𝑥𝐶𝑤) ∈ ℝ)
295294ex 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 715 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔𝑏)) → (𝑥𝐶𝑤) ∈ ℝ))
300299imp 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11288 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11309 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13090 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∈ ℝ+)
304303rpxrd 13076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏 = ((1st ‘(𝑔𝑏))(ball‘𝐶)(2nd ‘(𝑔𝑏)))) ∧ (𝑏𝑠𝑥𝑏)) → ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))) ∈ ℝ*)
305304adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24377 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐶 ∈ (∞Met‘𝑋) ∧ ((1st ‘(𝑔𝑏)) ∈ 𝑋𝑤𝑋𝑥𝑋)) → ((1st ‘(𝑔𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)))
307241, 246, 248, 256, 306syl13anc 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13271 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11884 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) ∈ ℝ)
319318, 252ltaddrpd 13108 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏𝑠) → (2nd ‘(𝑔𝑏)) < ((2nd ‘(𝑔𝑏)) + (2nd ‘(𝑔𝑏))))
320242, 225, 319syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 525 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (𝜑𝑓:𝑋𝑌))
325 heicant.d . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝜑𝐷 ∈ (∞Met‘𝑌))
326 ffvelcdm 7101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓:𝑋𝑌𝑥𝑋) → (𝑓𝑥) ∈ 𝑌)
327 ffvelcdm 7101 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝑓:𝑋𝑌𝑤𝑋) → (𝑓𝑤) ∈ 𝑌)
328326, 327anim12dan 619 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝑓:𝑋𝑌 ∧ (𝑥𝑋𝑤𝑋)) → ((𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌))
329 xmetcl 24357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
3303293expb 1119 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ((𝐷 ∈ (∞Met‘𝑌) ∧ ((𝑓𝑥) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌)) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
331325, 328, 330syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((𝜑 ∧ (𝑓:𝑋𝑌 ∧ (𝑥𝑋𝑤𝑋))) → ((𝑓𝑥)𝐷(𝑓𝑤)) ∈ ℝ*)
332331anassrs 467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 734 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝐷 ∈ (∞Met‘𝑌))
336 simp-5r 786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 𝑓:𝑋𝑌)
337336, 274ffvelcdmd 7105 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌)
338 simpllr 776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 ((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑓:𝑋𝑌)
339338ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑥𝑋) → (𝑓𝑥) ∈ 𝑌)
340339adantrr 717 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑓𝑥) ∈ 𝑌)
341340adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓𝑥) ∈ 𝑌)
342 xmetcl 24357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑥) ∈ 𝑌) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ*)
343335, 337, 341, 342syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)) ∈ ℝ*)
3449rpxrd 13076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ*)
345344ad4antlr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑑 / 2) ∈ ℝ*)
346 xrltle 13188 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑥) ∈ 𝑌) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
349335, 337, 341, 348syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
3509rpred 13075 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑑 ∈ ℝ+ → (𝑑 / 2) ∈ ℝ)
351350ad4antlr 733 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑑 / 2) ∈ ℝ)
352 xrrege0 13213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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)
359338ffvelcdmda 7104 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑤𝑋) → (𝑓𝑤) ∈ 𝑌)
360359adantrl 716 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) → (𝑓𝑤) ∈ 𝑌)
361360adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → (𝑓𝑤) ∈ 𝑌)
362 xmetcl 24357 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ*)
363335, 337, 361, 362syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)) ∈ ℝ*)
364 xrltle 13188 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌 ∧ (𝑓𝑤) ∈ 𝑌) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
367335, 337, 361, 366syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → 0 ≤ ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑤)))
368 xrrege0 13213 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 695 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11236 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11309 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13042 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑑 ∈ ℝ+𝑑 ∈ ℝ*)
380379ad6antlr 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24377 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1371 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 24373 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓𝑥) ∈ 𝑌 ∧ (𝑓‘(1st ‘(𝑔𝑏))) ∈ 𝑌) → ((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
386335, 341, 337, 385syl3anc 1370 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ((((((𝜑𝑓:𝑋𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥𝑋𝑤𝑋)) ∧ 𝑏𝑠) → ((𝑓𝑥)𝐷(𝑓‘(1st ‘(𝑔𝑏)))) = ((𝑓‘(1st ‘(𝑔𝑏)))𝐷(𝑓𝑥)))
387386ad2ant2r 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13271 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 678 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2775 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 11746 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 413 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 608 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 406 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13043 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑑 ∈ ℝ+𝑑 ∈ ℂ)
4034022halvesd 12510 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑑 ∈ ℝ+ → ((𝑑 / 2) + (𝑑 / 2)) = 𝑑)
404403ad6antlr 737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5174 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 13199 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 412 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 680 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 721 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 420 . . . . . . . . . . . . . . . . . . . . . . . . 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 593 . . . . . . . . . . . . . . . . . . . . . . . 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 456 . . . . . . . . . . . . . . . . . . . . . . 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 652 . . . . . . . . . . . . . . . . . . . . 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 3264 . . . . . . . . . . . . . . . . . . . 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 3200 . . . . . . . . . . . . . . . . . 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 5152 . . . . . . . . . . . . . . . . . . . . 21 (𝑧 = inf(ran ran 𝑔, ℝ, < ) → ((𝑥𝐶𝑤) < 𝑧 ↔ (𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < )))
421420imbi1d 341 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = inf(ran ran 𝑔, ℝ, < ) → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑) ↔ ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑)))
4224212ralbidv 3219 . . . . . . . . . . . . . . . . . . 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 3622 . . . . . . . . . . . . . . . . . 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 457 . . . . . . . . . . . . . . . 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 1931 . . . . . . . . . . . . . . 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 453 . . . . . . . . . . . . . 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 593 . . . . . . . . . . . . 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 3153 . . . . . . . . . . . 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 430 . . . . . . . . 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 3143 . . . . . . 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 3178 . . . . . . 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 218 . . . . . 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 226 . . . . 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 3287 . . . . 5 (∀𝑦 ∈ ℝ+𝑥𝑋𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦) ↔ ∀𝑥𝑋𝑦 ∈ ℝ+𝑧 ∈ ℝ+𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑦))
439437, 438bitrdi 287 . . . 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 2735 . . . 4 (metUnif‘𝐶) = (metUnif‘𝐶)
442 eqid 2735 . . . 4 (metUnif‘𝐷) = (metUnif‘𝐷)
443 heicant.y . . . 4 (𝜑𝑌 ≠ ∅)
444 xmetpsmet 24374 . . . . 5 (𝐶 ∈ (∞Met‘𝑋) → 𝐶 ∈ (PsMet‘𝑋))
44518, 444syl 17 . . . 4 (𝜑𝐶 ∈ (PsMet‘𝑋))
446 xmetpsmet 24374 . . . . 5 (𝐷 ∈ (∞Met‘𝑌) → 𝐷 ∈ (PsMet‘𝑌))
447325, 446syl 17 . . . 4 (𝜑𝐷 ∈ (PsMet‘𝑌))
448441, 442, 128, 443, 445, 447metucn 24600 . . 3 (𝜑 → (𝑓 ∈ ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) ↔ (𝑓:𝑋𝑌 ∧ ∀𝑑 ∈ ℝ+𝑧 ∈ ℝ+𝑥𝑋𝑤𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓𝑥)𝐷(𝑓𝑤)) < 𝑑))))
449 eqid 2735 . . . . 5 (MetOpen‘𝐷) = (MetOpen‘𝐷)
45023, 449metcn 24572 . . . 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 311 . 2 (𝜑 → (𝑓 ∈ ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) ↔ 𝑓 ∈ ((MetOpen‘𝐶) Cn (MetOpen‘𝐷))))
453452eqrdv 2733 1 (𝜑 → ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) = ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  cin 3962  wss 3963  c0 4339  𝒫 cpw 4605  cop 4637   cuni 4912   class class class wbr 5148   Or wor 5596   × cxp 5687  dom cdm 5689  ran crn 5690  Rel wrel 5694  Fun wfun 6557   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  cen 8981  Fincfn 8984  infcinf 9479  cr 11152  0cc0 11153   + caddc 11156  *cxr 11292   < clt 11293  cle 11294   / cdiv 11918  2c2 12319  +crp 13032   +𝑒 cxad 13150  PsMetcpsmet 21366  ∞Metcxmet 21367  ballcbl 21369  MetOpencmopn 21372  metUnifcmetu 21373   Cn ccn 23248  Compccmp 23410   Cnucucn 24300
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8013  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-er 8744  df-map 8867  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-sup 9480  df-inf 9481  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-n0 12525  df-z 12612  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ico 13390  df-topgen 17490  df-psmet 21374  df-xmet 21375  df-bl 21377  df-mopn 21378  df-fbas 21379  df-fg 21380  df-metu 21381  df-top 22916  df-topon 22933  df-bases 22969  df-cn 23251  df-cnp 23252  df-cmp 23411  df-fil 23870  df-ust 24225  df-ucn 24301
This theorem is referenced by: (None)
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