| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | breq2 5147 | . . . . . . . . . . 11
⊢ (𝑑 = 𝑦 → (((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑 ↔ ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)) | 
| 2 | 1 | imbi2d 340 | . . . . . . . . . 10
⊢ (𝑑 = 𝑦 → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦))) | 
| 3 | 2 | 2ralbidv 3221 | . . . . . . . . 9
⊢ (𝑑 = 𝑦 → (∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) ↔ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦))) | 
| 4 | 3 | rexbidv 3179 | . . . . . . . 8
⊢ (𝑑 = 𝑦 → (∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) ↔ ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦))) | 
| 5 | 4 | cbvralvw 3237 | . . . . . . 7
⊢
(∀𝑑 ∈
ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) ↔ ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)) | 
| 6 |  | r19.12 3314 | . . . . . . . 8
⊢
(∃𝑧 ∈
ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) → ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)) | 
| 7 | 6 | ralimi 3083 | . . . . . . 7
⊢
(∀𝑦 ∈
ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) → ∀𝑦 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)) | 
| 8 | 5, 7 | sylbi 217 | . . . . . 6
⊢
(∀𝑑 ∈
ℝ+ ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) → ∀𝑦 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)) | 
| 9 |  | rphalfcl 13062 | . . . . . . . . 9
⊢ (𝑑 ∈ ℝ+
→ (𝑑 / 2) ∈
ℝ+) | 
| 10 |  | breq2 5147 | . . . . . . . . . . . . . . . 16
⊢ (𝑦 = (𝑑 / 2) → (((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦 ↔ ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) | 
| 11 | 10 | imbi2d 340 | . . . . . . . . . . . . . . 15
⊢ (𝑦 = (𝑑 / 2) → (((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 12 | 11 | ralbidv 3178 | . . . . . . . . . . . . . 14
⊢ (𝑦 = (𝑑 / 2) → (∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) ↔ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 13 | 12 | rexbidv 3179 | . . . . . . . . . . . . 13
⊢ (𝑦 = (𝑑 / 2) → (∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) ↔ ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 14 | 13 | ralbidv 3178 | . . . . . . . . . . . 12
⊢ (𝑦 = (𝑑 / 2) → (∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) ↔ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 15 | 14 | rspcva 3620 | . . . . . . . . . . 11
⊢ (((𝑑 / 2) ∈ ℝ+
∧ ∀𝑦 ∈
ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)) → ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) | 
| 16 |  | heicant.j | . . . . . . . . . . . . . . 15
⊢ (𝜑 → (MetOpen‘𝐶) ∈ Comp) | 
| 17 | 16 | ad3antrrr 730 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧
∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (MetOpen‘𝐶) ∈ Comp) | 
| 18 |  | heicant.c | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝜑 → 𝐶 ∈ (∞Met‘𝑋)) | 
| 19 | 18 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) → 𝐶 ∈ (∞Met‘𝑋)) | 
| 20 | 19 | anim1i 615 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) → (𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋)) | 
| 21 |  | rphalfcl 13062 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 ∈ ℝ+
→ (𝑧 / 2) ∈
ℝ+) | 
| 22 | 21 | rpxrd 13078 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 ∈ ℝ+
→ (𝑧 / 2) ∈
ℝ*) | 
| 23 |  | eqid 2737 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
(MetOpen‘𝐶) =
(MetOpen‘𝐶) | 
| 24 | 23 | blopn 24513 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ (𝑧 / 2) ∈ ℝ*) →
(𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶)) | 
| 25 | 24 | 3expa 1119 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ (𝑧 / 2) ∈ ℝ*) →
(𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶)) | 
| 26 | 20, 22, 25 | syl2an 596 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ ℝ+) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶)) | 
| 27 | 26 | adantr 480 | . . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ ℝ+) ∧
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶)) | 
| 28 | 21 | rpgt0d 13080 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 ∈ ℝ+
→ 0 < (𝑧 /
2)) | 
| 29 | 22, 28 | jca 511 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 ∈ ℝ+
→ ((𝑧 / 2) ∈
ℝ* ∧ 0 < (𝑧 / 2))) | 
| 30 |  | xblcntr 24421 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ ((𝑧 / 2) ∈ ℝ* ∧ 0 <
(𝑧 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2))) | 
| 31 | 30 | 3expa 1119 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋) ∧ ((𝑧 / 2) ∈ ℝ* ∧ 0 <
(𝑧 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2))) | 
| 32 | 20, 29, 31 | syl2an 596 | . . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ ℝ+) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2))) | 
| 33 | 32 | adantr 480 | . . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ ℝ+) ∧
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2))) | 
| 34 |  | opelxpi 5722 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑥 ∈ 𝑋 ∧ (𝑧 / 2) ∈ ℝ+) →
〈𝑥, (𝑧 / 2)〉 ∈ (𝑋 ×
ℝ+)) | 
| 35 | 21, 34 | sylan2 593 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑧 ∈ ℝ+) →
〈𝑥, (𝑧 / 2)〉 ∈ (𝑋 ×
ℝ+)) | 
| 36 | 35 | ad4ant23 753 | . . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ ℝ+) ∧
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → 〈𝑥, (𝑧 / 2)〉 ∈ (𝑋 ×
ℝ+)) | 
| 37 |  | rpcn 13045 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑧 ∈ ℝ+
→ 𝑧 ∈
ℂ) | 
| 38 | 37 | 2halvesd 12512 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 ∈ ℝ+
→ ((𝑧 / 2) + (𝑧 / 2)) = 𝑧) | 
| 39 | 38 | breq2d 5155 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑧 ∈ ℝ+
→ ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) ↔ (𝑥𝐶𝑐) < 𝑧)) | 
| 40 | 39 | imbi1d 341 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑧 ∈ ℝ+
→ (((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑐) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 41 | 40 | ralbidv 3178 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑧 ∈ ℝ+
→ (∀𝑐 ∈
𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ ∀𝑐 ∈ 𝑋 ((𝑥𝐶𝑐) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 42 |  | oveq2 7439 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑐 = 𝑤 → (𝑥𝐶𝑐) = (𝑥𝐶𝑤)) | 
| 43 | 42 | breq1d 5153 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑐 = 𝑤 → ((𝑥𝐶𝑐) < 𝑧 ↔ (𝑥𝐶𝑤) < 𝑧)) | 
| 44 |  | fveq2 6906 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑐 = 𝑤 → (𝑓‘𝑐) = (𝑓‘𝑤)) | 
| 45 | 44 | oveq2d 7447 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑐 = 𝑤 → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) = ((𝑓‘𝑥)𝐷(𝑓‘𝑤))) | 
| 46 | 45 | breq1d 5153 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑐 = 𝑤 → (((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2) ↔ ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) | 
| 47 | 43, 46 | imbi12d 344 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑐 = 𝑤 → (((𝑥𝐶𝑐) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 48 | 47 | cbvralvw 3237 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
(∀𝑐 ∈
𝑋 ((𝑥𝐶𝑐) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) | 
| 49 | 41, 48 | bitrdi 287 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑧 ∈ ℝ+
→ (∀𝑐 ∈
𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 50 | 49 | biimpar 477 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑧 ∈ ℝ+
∧ ∀𝑤 ∈
𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ∀𝑐 ∈ 𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2))) | 
| 51 | 50 | adantll 714 | . . . . . . . . . . . . . . . . . . . 20
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ ℝ+) ∧
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ∀𝑐 ∈ 𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2))) | 
| 52 |  | vex 3484 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ 𝑥 ∈ V | 
| 53 |  | ovex 7464 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑧 / 2) ∈ V | 
| 54 | 52, 53 | op1std 8024 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → (1st
‘𝑝) = 𝑥) | 
| 55 | 52, 53 | op2ndd 8025 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → (2nd
‘𝑝) = (𝑧 / 2)) | 
| 56 | 54, 55 | oveq12d 7449 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → ((1st
‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) = (𝑥(ball‘𝐶)(𝑧 / 2))) | 
| 57 | 56 | eqcomd 2743 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → (𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝))) | 
| 58 | 57 | biantrurd 532 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → (∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ ((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) | 
| 59 | 54 | oveq1d 7446 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → ((1st
‘𝑝)𝐶𝑐) = (𝑥𝐶𝑐)) | 
| 60 | 55, 55 | oveq12d 7449 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → ((2nd
‘𝑝) + (2nd
‘𝑝)) = ((𝑧 / 2) + (𝑧 / 2))) | 
| 61 | 59, 60 | breq12d 5156 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → (((1st
‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) ↔ (𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)))) | 
| 62 | 54 | fveq2d 6910 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → (𝑓‘(1st ‘𝑝)) = (𝑓‘𝑥)) | 
| 63 | 62 | oveq1d 7446 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) = ((𝑓‘𝑥)𝐷(𝑓‘𝑐))) | 
| 64 | 63 | breq1d 5153 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → (((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2) ↔ ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2))) | 
| 65 | 61, 64 | imbi12d 344 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → ((((1st
‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 66 | 65 | ralbidv 3178 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → (∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ ∀𝑐 ∈ 𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 67 | 58, 66 | bitr3d 281 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑝 = 〈𝑥, (𝑧 / 2)〉 → (((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) ↔ ∀𝑐 ∈ 𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 68 | 67 | rspcev 3622 | . . . . . . . . . . . . . . . . . . . 20
⊢
((〈𝑥, (𝑧 / 2)〉 ∈ (𝑋 × ℝ+)
∧ ∀𝑐 ∈
𝑋 ((𝑥𝐶𝑐) < ((𝑧 / 2) + (𝑧 / 2)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑐)) < (𝑑 / 2))) → ∃𝑝 ∈ (𝑋 × ℝ+)((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 69 | 36, 51, 68 | syl2anc 584 | . . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ ℝ+) ∧
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ∃𝑝 ∈ (𝑋 × ℝ+)((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 70 |  | eleq2 2830 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (𝑥 ∈ 𝑏 ↔ 𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)))) | 
| 71 |  | eqeq1 2741 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ↔ (𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)))) | 
| 72 | 71 | anbi1d 631 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → ((𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) ↔ ((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) | 
| 73 | 72 | rexbidv 3179 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → (∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) ↔ ∃𝑝 ∈ (𝑋 × ℝ+)((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) | 
| 74 | 70, 73 | anbi12d 632 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑏 = (𝑥(ball‘𝐶)(𝑧 / 2)) → ((𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ↔ (𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)) ∧ ∃𝑝 ∈ (𝑋 × ℝ+)((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))))) | 
| 75 | 74 | rspcev 3622 | . . . . . . . . . . . . . . . . . . 19
⊢ (((𝑥(ball‘𝐶)(𝑧 / 2)) ∈ (MetOpen‘𝐶) ∧ (𝑥 ∈ (𝑥(ball‘𝐶)(𝑧 / 2)) ∧ ∃𝑝 ∈ (𝑋 × ℝ+)((𝑥(ball‘𝐶)(𝑧 / 2)) = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) → ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) | 
| 76 | 27, 33, 69, 75 | syl12anc 837 | . . . . . . . . . . . . . . . . . 18
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) ∧ 𝑧 ∈ ℝ+) ∧
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) | 
| 77 | 76 | rexlimdva2 3157 | . . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑥 ∈ 𝑋) → (∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))))) | 
| 78 | 77 | ralimdva 3167 | . . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) →
(∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → ∀𝑥 ∈ 𝑋 ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))))) | 
| 79 | 78 | imp 406 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧
∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ∀𝑥 ∈ 𝑋 ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) | 
| 80 | 23 | mopnuni 24451 | . . . . . . . . . . . . . . . . . 18
⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝑋 = ∪
(MetOpen‘𝐶)) | 
| 81 | 18, 80 | syl 17 | . . . . . . . . . . . . . . . . 17
⊢ (𝜑 → 𝑋 = ∪
(MetOpen‘𝐶)) | 
| 82 | 81 | raleqdv 3326 | . . . . . . . . . . . . . . . 16
⊢ (𝜑 → (∀𝑥 ∈ 𝑋 ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ↔ ∀𝑥 ∈ ∪
(MetOpen‘𝐶)∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))))) | 
| 83 | 82 | ad3antrrr 730 | . . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧
∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (∀𝑥 ∈ 𝑋 ∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ↔ ∀𝑥 ∈ ∪
(MetOpen‘𝐶)∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))))) | 
| 84 | 79, 83 | mpbid 232 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧
∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ∀𝑥 ∈ ∪
(MetOpen‘𝐶)∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) | 
| 85 |  | eqid 2737 | . . . . . . . . . . . . . . 15
⊢ ∪ (MetOpen‘𝐶) = ∪
(MetOpen‘𝐶) | 
| 86 |  | fveq2 6906 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = (𝑔‘𝑏) → (1st ‘𝑝) = (1st
‘(𝑔‘𝑏))) | 
| 87 |  | fveq2 6906 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = (𝑔‘𝑏) → (2nd ‘𝑝) = (2nd
‘(𝑔‘𝑏))) | 
| 88 | 86, 87 | oveq12d 7449 | . . . . . . . . . . . . . . . . 17
⊢ (𝑝 = (𝑔‘𝑏) → ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) | 
| 89 | 88 | eqeq2d 2748 | . . . . . . . . . . . . . . . 16
⊢ (𝑝 = (𝑔‘𝑏) → (𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ↔ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))))) | 
| 90 | 86 | oveq1d 7446 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑔‘𝑏) → ((1st ‘𝑝)𝐶𝑐) = ((1st ‘(𝑔‘𝑏))𝐶𝑐)) | 
| 91 | 87, 87 | oveq12d 7449 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑔‘𝑏) → ((2nd ‘𝑝) + (2nd ‘𝑝)) = ((2nd
‘(𝑔‘𝑏)) + (2nd
‘(𝑔‘𝑏)))) | 
| 92 | 90, 91 | breq12d 5156 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = (𝑔‘𝑏) → (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) ↔ ((1st
‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))))) | 
| 93 | 86 | fveq2d 6910 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑝 = (𝑔‘𝑏) → (𝑓‘(1st ‘𝑝)) = (𝑓‘(1st ‘(𝑔‘𝑏)))) | 
| 94 | 93 | oveq1d 7446 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑝 = (𝑔‘𝑏) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) = ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐))) | 
| 95 | 94 | breq1d 5153 | . . . . . . . . . . . . . . . . . 18
⊢ (𝑝 = (𝑔‘𝑏) → (((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) | 
| 96 | 92, 95 | imbi12d 344 | . . . . . . . . . . . . . . . . 17
⊢ (𝑝 = (𝑔‘𝑏) → ((((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 97 | 96 | ralbidv 3178 | . . . . . . . . . . . . . . . 16
⊢ (𝑝 = (𝑔‘𝑏) → (∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 98 | 89, 97 | anbi12d 632 | . . . . . . . . . . . . . . 15
⊢ (𝑝 = (𝑔‘𝑏) → ((𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) ↔ (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) | 
| 99 | 85, 98 | cmpcovf 23399 | . . . . . . . . . . . . . 14
⊢
(((MetOpen‘𝐶)
∈ Comp ∧ ∀𝑥
∈ ∪ (MetOpen‘𝐶)∃𝑏 ∈ (MetOpen‘𝐶)(𝑥 ∈ 𝑏 ∧ ∃𝑝 ∈ (𝑋 × ℝ+)(𝑏 = ((1st ‘𝑝)(ball‘𝐶)(2nd ‘𝑝)) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘𝑝)𝐶𝑐) < ((2nd ‘𝑝) + (2nd ‘𝑝)) → ((𝑓‘(1st ‘𝑝))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) → ∃𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin)(∪ (MetOpen‘𝐶) = ∪ 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))))) | 
| 100 | 17, 84, 99 | syl2anc 584 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧
∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ∃𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin)(∪ (MetOpen‘𝐶) = ∪ 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))))) | 
| 101 | 100 | ex 412 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) →
(∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → ∃𝑠 ∈ (𝒫 (MetOpen‘𝐶) ∩ Fin)(∪ (MetOpen‘𝐶) = ∪ 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))))) | 
| 102 |  | elinel2 4202 | . . . . . . . . . . . . . 14
⊢ (𝑠 ∈ (𝒫
(MetOpen‘𝐶) ∩
Fin) → 𝑠 ∈
Fin) | 
| 103 |  | simpll 767 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) → 𝜑) | 
| 104 | 103 | anim1i 615 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → (𝜑 ∧ 𝑠 ∈ Fin)) | 
| 105 |  | frn 6743 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → ran
𝑔 ⊆ (𝑋 ×
ℝ+)) | 
| 106 |  | rnss 5950 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ran
𝑔 ⊆ (𝑋 × ℝ+)
→ ran ran 𝑔 ⊆
ran (𝑋 ×
ℝ+)) | 
| 107 | 105, 106 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran
𝑔 ⊆ ran (𝑋 ×
ℝ+)) | 
| 108 |  | rnxpss 6192 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ran
(𝑋 ×
ℝ+) ⊆ ℝ+ | 
| 109 | 107, 108 | sstrdi 3996 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran
𝑔 ⊆
ℝ+) | 
| 110 | 109 | adantl 481 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran
ran 𝑔 ⊆
ℝ+) | 
| 111 |  | simplr 769 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) → 𝑠 ∈ Fin) | 
| 112 |  | ffun 6739 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → Fun
𝑔) | 
| 113 |  | vex 3484 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ 𝑔 ∈ V | 
| 114 | 113 | fundmen 9071 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (Fun
𝑔 → dom 𝑔 ≈ 𝑔) | 
| 115 | 114 | ensymd 9045 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (Fun
𝑔 → 𝑔 ≈ dom 𝑔) | 
| 116 | 112, 115 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 ≈ dom 𝑔) | 
| 117 |  | fdm 6745 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → dom
𝑔 = 𝑠) | 
| 118 | 116, 117 | breqtrd 5169 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 ≈ 𝑠) | 
| 119 |  | enfii 9226 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ ((𝑠 ∈ Fin ∧ 𝑔 ≈ 𝑠) → 𝑔 ∈ Fin) | 
| 120 | 118, 119 | sylan2 593 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ ((𝑠 ∈ Fin ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑔 ∈ Fin) | 
| 121 |  | rnfi 9380 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑔 ∈ Fin → ran 𝑔 ∈ Fin) | 
| 122 |  | rnfi 9380 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (ran
𝑔 ∈ Fin → ran ran
𝑔 ∈
Fin) | 
| 123 | 120, 121,
122 | 3syl 18 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑠 ∈ Fin ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran
ran 𝑔 ∈
Fin) | 
| 124 | 111, 123 | sylan 580 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran
ran 𝑔 ∈
Fin) | 
| 125 | 117 | adantl 481 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → dom
𝑔 = 𝑠) | 
| 126 |  | eqtr 2760 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ((𝑋 = ∪
(MetOpen‘𝐶) ∧
∪ (MetOpen‘𝐶) = ∪ 𝑠) → 𝑋 = ∪ 𝑠) | 
| 127 | 81, 126 | sylan 580 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) → 𝑋 = ∪ 𝑠) | 
| 128 |  | heicant.x | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝜑 → 𝑋 ≠ ∅) | 
| 129 | 128 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ ((𝜑 ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) → 𝑋 ≠ ∅) | 
| 130 | 127, 129 | eqnetrrd 3009 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ ((𝜑 ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) → ∪ 𝑠
≠ ∅) | 
| 131 |  | unieq 4918 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (𝑠 = ∅ → ∪ 𝑠 =
∪ ∅) | 
| 132 |  | uni0 4935 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ ∪ ∅ = ∅ | 
| 133 | 131, 132 | eqtrdi 2793 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (𝑠 = ∅ → ∪ 𝑠 =
∅) | 
| 134 | 133 | necon3i 2973 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (∪ 𝑠
≠ ∅ → 𝑠 ≠
∅) | 
| 135 | 130, 134 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ ((𝜑 ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) → 𝑠 ≠ ∅) | 
| 136 | 135 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (((𝜑 ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑠 ≠ ∅) | 
| 137 | 125, 136 | eqnetrd 3008 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (((𝜑 ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → dom
𝑔 ≠
∅) | 
| 138 |  | dm0rn0 5935 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (dom
𝑔 = ∅ ↔ ran
𝑔 =
∅) | 
| 139 | 138 | necon3bii 2993 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (dom
𝑔 ≠ ∅ ↔ ran
𝑔 ≠
∅) | 
| 140 | 137, 139 | sylib 218 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran
𝑔 ≠
∅) | 
| 141 |  | relxp 5703 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ Rel
(𝑋 ×
ℝ+) | 
| 142 |  | relss 5791 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (ran
𝑔 ⊆ (𝑋 × ℝ+)
→ (Rel (𝑋 ×
ℝ+) → Rel ran 𝑔)) | 
| 143 | 105, 141,
142 | mpisyl 21 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → Rel ran
𝑔) | 
| 144 |  | relrn0 5983 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢ (Rel ran
𝑔 → (ran 𝑔 = ∅ ↔ ran ran 𝑔 = ∅)) | 
| 145 | 144 | necon3bid 2985 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (Rel ran
𝑔 → (ran 𝑔 ≠ ∅ ↔ ran ran
𝑔 ≠
∅)) | 
| 146 | 143, 145 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → (ran
𝑔 ≠ ∅ ↔ ran
ran 𝑔 ≠
∅)) | 
| 147 | 146 | adantl 481 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (((𝜑 ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (ran
𝑔 ≠ ∅ ↔ ran
ran 𝑔 ≠
∅)) | 
| 148 | 140, 147 | mpbid 232 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (((𝜑 ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran
ran 𝑔 ≠
∅) | 
| 149 | 148 | adantllr 719 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran
ran 𝑔 ≠
∅) | 
| 150 |  | rpssre 13042 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
ℝ+ ⊆ ℝ | 
| 151 | 110, 150 | sstrdi 3996 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((((𝜑 ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → ran
ran 𝑔 ⊆
ℝ) | 
| 152 |  | ltso 11341 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢  < Or
ℝ | 
| 153 |  | fiinfcl 9541 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (( <
Or ℝ ∧ (ran ran 𝑔
∈ Fin ∧ ran ran 𝑔
≠ ∅ ∧ ran ran 𝑔 ⊆ ℝ)) → inf(ran ran 𝑔, ℝ, < ) ∈ ran ran
𝑔) | 
| 154 | 152, 153 | mpan 690 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((ran ran
𝑔 ∈ Fin ∧ ran ran
𝑔 ≠ ∅ ∧ ran
ran 𝑔 ⊆ ℝ)
→ inf(ran ran 𝑔,
ℝ, < ) ∈ ran ran 𝑔) | 
| 155 | 124, 149,
151, 154 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((((𝜑 ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) →
inf(ran ran 𝑔, ℝ,
< ) ∈ ran ran 𝑔) | 
| 156 | 110, 155 | sseldd 3984 | . . . . . . . . . . . . . . . . . . . 20
⊢ ((((𝜑 ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) →
inf(ran ran 𝑔, ℝ,
< ) ∈ ℝ+) | 
| 157 | 104, 156 | sylanl1 680 | . . . . . . . . . . . . . . . . . . 19
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) →
inf(ran ran 𝑔, ℝ,
< ) ∈ ℝ+) | 
| 158 | 157 | adantr 480 | . . . . . . . . . . . . . . . . . 18
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → inf(ran ran 𝑔, ℝ, < ) ∈
ℝ+) | 
| 159 | 81 | ad3antrrr 730 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → 𝑋 = ∪
(MetOpen‘𝐶)) | 
| 160 | 159 | anim1i 615 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) → (𝑋 = ∪
(MetOpen‘𝐶) ∧
∪ (MetOpen‘𝐶) = ∪ 𝑠)) | 
| 161 | 160 | ad2antrr 726 | . . . . . . . . . . . . . . . . . . . . 21
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → (𝑋 = ∪
(MetOpen‘𝐶) ∧
∪ (MetOpen‘𝐶) = ∪ 𝑠)) | 
| 162 |  | simpl 482 | . . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → 𝑥 ∈ 𝑋) | 
| 163 | 126 | eleq2d 2827 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ ((𝑋 = ∪
(MetOpen‘𝐶) ∧
∪ (MetOpen‘𝐶) = ∪ 𝑠) → (𝑥 ∈ 𝑋 ↔ 𝑥 ∈ ∪ 𝑠)) | 
| 164 |  | eluni2 4911 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 ∈ ∪ 𝑠
↔ ∃𝑏 ∈
𝑠 𝑥 ∈ 𝑏) | 
| 165 | 163, 164 | bitrdi 287 | . . . . . . . . . . . . . . . . . . . . . 22
⊢ ((𝑋 = ∪
(MetOpen‘𝐶) ∧
∪ (MetOpen‘𝐶) = ∪ 𝑠) → (𝑥 ∈ 𝑋 ↔ ∃𝑏 ∈ 𝑠 𝑥 ∈ 𝑏)) | 
| 166 | 165 | biimpa 476 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (((𝑋 = ∪
(MetOpen‘𝐶) ∧
∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑥 ∈ 𝑋) → ∃𝑏 ∈ 𝑠 𝑥 ∈ 𝑏) | 
| 167 | 161, 162,
166 | syl2an 596 | . . . . . . . . . . . . . . . . . . . 20
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ∃𝑏 ∈ 𝑠 𝑥 ∈ 𝑏) | 
| 168 |  | nfv 1914 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑏(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 ×
ℝ+)) | 
| 169 |  | nfra1 3284 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
Ⅎ𝑏∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) | 
| 170 | 168, 169 | nfan 1899 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑏((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 171 |  | nfv 1914 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
Ⅎ𝑏(𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) | 
| 172 | 170, 171 | nfan 1899 | . . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑏(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) | 
| 173 |  | nfv 1914 | . . . . . . . . . . . . . . . . . . . . 21
⊢
Ⅎ𝑏((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) | 
| 174 |  | rspa 3248 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((∀𝑏 ∈
𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) ∧ 𝑏 ∈ 𝑠) → (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) | 
| 175 |  | oveq2 7439 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑐 = 𝑥 → ((1st ‘(𝑔‘𝑏))𝐶𝑐) = ((1st ‘(𝑔‘𝑏))𝐶𝑥)) | 
| 176 | 175 | breq1d 5153 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑐 = 𝑥 → (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ↔ ((1st ‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))))) | 
| 177 |  | fveq2 6906 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝑐 = 𝑥 → (𝑓‘𝑐) = (𝑓‘𝑥)) | 
| 178 | 177 | oveq2d 7447 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑐 = 𝑥 → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) = ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥))) | 
| 179 | 178 | breq1d 5153 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑐 = 𝑥 → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2))) | 
| 180 | 176, 179 | imbi12d 344 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑐 = 𝑥 → ((((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ (((1st ‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2)))) | 
| 181 | 180 | rspcva 3620 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑥 ∈ 𝑋 ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) → (((1st
‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2))) | 
| 182 |  | oveq2 7439 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑐 = 𝑤 → ((1st ‘(𝑔‘𝑏))𝐶𝑐) = ((1st ‘(𝑔‘𝑏))𝐶𝑤)) | 
| 183 | 182 | breq1d 5153 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑐 = 𝑤 → (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ↔ ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))))) | 
| 184 | 44 | oveq2d 7447 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑐 = 𝑤 → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) = ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) | 
| 185 | 184 | breq1d 5153 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑐 = 𝑤 → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2) ↔ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) | 
| 186 | 183, 185 | imbi12d 344 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑐 = 𝑤 → ((((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)) ↔ (((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 187 | 186 | rspcva 3620 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢ ((𝑤 ∈ 𝑋 ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) → (((1st
‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) | 
| 188 | 181, 187 | anim12i 613 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢ (((𝑥 ∈ 𝑋 ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) ∧ (𝑤 ∈ 𝑋 ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → ((((1st
‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2)) ∧ (((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 189 | 188 | anandirs 679 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) → ((((1st
‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2)) ∧ (((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 190 |  | anim12 809 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((1st ‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2)) ∧ (((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ((((1st
‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 191 | 189, 190 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) → ((((1st
‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 192 | 191 | adantrl 716 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢ (((𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → ((((1st
‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 193 | 192 | ad4ant23 753 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((((1st ‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)))) | 
| 194 |  | simpll 767 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) → ((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈
ℝ+)) | 
| 195 | 194 | anim1i 615 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) →
(((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 ×
ℝ+))) | 
| 196 | 195 | anim1i 615 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋))) | 
| 197 | 109, 150 | sstrdi 3996 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → ran ran
𝑔 ⊆
ℝ) | 
| 198 | 197 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → ran ran 𝑔 ⊆ ℝ) | 
| 199 |  | 0re 11263 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ 0 ∈
ℝ | 
| 200 |  | rpge0 13048 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (𝑦 ∈ ℝ+
→ 0 ≤ 𝑦) | 
| 201 | 200 | rgen 3063 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
∀𝑦 ∈
ℝ+ 0 ≤ 𝑦 | 
| 202 |  | ssralv 4052 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ (ran ran
𝑔 ⊆
ℝ+ → (∀𝑦 ∈ ℝ+ 0 ≤ 𝑦 → ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦)) | 
| 203 | 109, 201,
202 | mpisyl 21 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) →
∀𝑦 ∈ ran ran
𝑔0 ≤ 𝑦) | 
| 204 |  | breq1 5146 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (𝑥 = 0 → (𝑥 ≤ 𝑦 ↔ 0 ≤ 𝑦)) | 
| 205 | 204 | ralbidv 3178 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ (𝑥 = 0 → (∀𝑦 ∈ ran ran 𝑔 𝑥 ≤ 𝑦 ↔ ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦)) | 
| 206 | 205 | rspcev 3622 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((0
∈ ℝ ∧ ∀𝑦 ∈ ran ran 𝑔0 ≤ 𝑦) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥 ≤ 𝑦) | 
| 207 | 199, 203,
206 | sylancr 587 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) →
∃𝑥 ∈ ℝ
∀𝑦 ∈ ran ran
𝑔 𝑥 ≤ 𝑦) | 
| 208 | 207 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ ran ran 𝑔 𝑥 ≤ 𝑦) | 
| 209 | 143 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → Rel ran 𝑔) | 
| 210 |  | ffn 6736 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (𝑔:𝑠⟶(𝑋 × ℝ+) → 𝑔 Fn 𝑠) | 
| 211 |  | fnfvelrn 7100 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((𝑔 Fn 𝑠 ∧ 𝑏 ∈ 𝑠) → (𝑔‘𝑏) ∈ ran 𝑔) | 
| 212 | 210, 211 | sylan 580 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → (𝑔‘𝑏) ∈ ran 𝑔) | 
| 213 |  | 2ndrn 8066 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((Rel ran
𝑔 ∧ (𝑔‘𝑏) ∈ ran 𝑔) → (2nd ‘(𝑔‘𝑏)) ∈ ran ran 𝑔) | 
| 214 | 209, 212,
213 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → (2nd ‘(𝑔‘𝑏)) ∈ ran ran 𝑔) | 
| 215 |  | infrelb 12253 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((ran ran
𝑔 ⊆ ℝ ∧
∃𝑥 ∈ ℝ
∀𝑦 ∈ ran ran
𝑔 𝑥 ≤ 𝑦 ∧ (2nd ‘(𝑔‘𝑏)) ∈ ran ran 𝑔) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd
‘(𝑔‘𝑏))) | 
| 216 | 198, 208,
214, 215 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd
‘(𝑔‘𝑏))) | 
| 217 | 216 | adantll 714 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏 ∈ 𝑠) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd
‘(𝑔‘𝑏))) | 
| 218 | 217 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → inf(ran ran 𝑔, ℝ, < ) ≤ (2nd
‘(𝑔‘𝑏))) | 
| 219 | 18 | ad3antrrr 730 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝐶 ∈ (∞Met‘𝑋)) | 
| 220 |  | xmetcl 24341 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → (𝑥𝐶𝑤) ∈
ℝ*) | 
| 221 | 220 | 3expb 1121 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑥𝐶𝑤) ∈
ℝ*) | 
| 222 | 219, 221 | sylan 580 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑥𝐶𝑤) ∈
ℝ*) | 
| 223 | 222 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (𝑥𝐶𝑤) ∈
ℝ*) | 
| 224 |  | simplr 769 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 𝑔:𝑠⟶(𝑋 ×
ℝ+)) | 
| 225 |  | simpl 482 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏) → 𝑏 ∈ 𝑠) | 
| 226 | 214 | ne0d 4342 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → ran ran 𝑔 ≠ ∅) | 
| 227 |  | infrecl 12250 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((ran ran
𝑔 ⊆ ℝ ∧ ran
ran 𝑔 ≠ ∅ ∧
∃𝑥 ∈ ℝ
∀𝑦 ∈ ran ran
𝑔 𝑥 ≤ 𝑦) → inf(ran ran 𝑔, ℝ, < ) ∈
ℝ) | 
| 228 | 198, 226,
208, 227 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → inf(ran ran 𝑔, ℝ, < ) ∈
ℝ) | 
| 229 | 228 | rexrd 11311 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → inf(ran ran 𝑔, ℝ, < ) ∈
ℝ*) | 
| 230 | 224, 225,
229 | syl2an 596 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → inf(ran ran 𝑔, ℝ, < ) ∈
ℝ*) | 
| 231 |  | simpr 484 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑔:𝑠⟶(𝑋 ×
ℝ+)) | 
| 232 | 231 | ffvelcdmda 7104 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏 ∈ 𝑠) → (𝑔‘𝑏) ∈ (𝑋 ×
ℝ+)) | 
| 233 |  | xp2nd 8047 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑔‘𝑏) ∈ (𝑋 × ℝ+) →
(2nd ‘(𝑔‘𝑏)) ∈
ℝ+) | 
| 234 | 232, 233 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏 ∈ 𝑠) → (2nd ‘(𝑔‘𝑏)) ∈
ℝ+) | 
| 235 | 234 | rpxrd 13078 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏 ∈ 𝑠) → (2nd ‘(𝑔‘𝑏)) ∈
ℝ*) | 
| 236 | 235 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (2nd ‘(𝑔‘𝑏)) ∈
ℝ*) | 
| 237 |  | xrltletr 13199 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝑥𝐶𝑤) ∈ ℝ* ∧ inf(ran
ran 𝑔, ℝ, < )
∈ ℝ* ∧ (2nd ‘(𝑔‘𝑏)) ∈ ℝ*) →
(((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) ∧ inf(ran ran 𝑔, ℝ, < ) ≤
(2nd ‘(𝑔‘𝑏))) → (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏)))) | 
| 238 | 223, 230,
236, 237 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) ∧ inf(ran ran 𝑔, ℝ, < ) ≤
(2nd ‘(𝑔‘𝑏))) → (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏)))) | 
| 239 | 218, 238 | mpan2d 694 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏)))) | 
| 240 | 239 | adantlr 715 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏)))) | 
| 241 | 18 | ad6antr 736 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → 𝐶 ∈ (∞Met‘𝑋)) | 
| 242 |  | simpllr 776 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) → 𝑔:𝑠⟶(𝑋 ×
ℝ+)) | 
| 243 |  | ffvelcdm 7101 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → (𝑔‘𝑏) ∈ (𝑋 ×
ℝ+)) | 
| 244 |  | xp1st 8046 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((𝑔‘𝑏) ∈ (𝑋 × ℝ+) →
(1st ‘(𝑔‘𝑏)) ∈ 𝑋) | 
| 245 | 243, 244 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → (1st ‘(𝑔‘𝑏)) ∈ 𝑋) | 
| 246 | 242, 225,
245 | syl2an 596 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (1st ‘(𝑔‘𝑏)) ∈ 𝑋) | 
| 247 |  | simpr 484 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → 𝑤 ∈ 𝑋) | 
| 248 | 247 | ad3antlr 731 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → 𝑤 ∈ 𝑋) | 
| 249 |  | xmetcl 24341 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st
‘(𝑔‘𝑏)) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → ((1st ‘(𝑔‘𝑏))𝐶𝑤) ∈
ℝ*) | 
| 250 | 241, 246,
248, 249 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((1st ‘(𝑔‘𝑏))𝐶𝑤) ∈
ℝ*) | 
| 251 | 250 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → ((1st ‘(𝑔‘𝑏))𝐶𝑤) ∈
ℝ*) | 
| 252 | 243, 233 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → (2nd ‘(𝑔‘𝑏)) ∈
ℝ+) | 
| 253 | 224, 252 | sylan 580 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (2nd ‘(𝑔‘𝑏)) ∈
ℝ+) | 
| 254 | 253 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (2nd ‘(𝑔‘𝑏)) ∈
ℝ+) | 
| 255 | 254 | rpred 13077 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (2nd ‘(𝑔‘𝑏)) ∈ ℝ) | 
| 256 | 162 | ad3antlr 731 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → 𝑥 ∈ 𝑋) | 
| 257 |  | xmetcl 24341 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st
‘(𝑔‘𝑏)) ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈
ℝ*) | 
| 258 | 241, 246,
256, 257 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈
ℝ*) | 
| 259 | 252 | rpxrd 13078 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → (2nd ‘(𝑔‘𝑏)) ∈
ℝ*) | 
| 260 | 242, 225,
259 | syl2an 596 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (2nd ‘(𝑔‘𝑏)) ∈
ℝ*) | 
| 261 |  | eleq2 2830 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 44
⊢ (𝑏 = ((1st
‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) → (𝑥 ∈ 𝑏 ↔ 𝑥 ∈ ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))))) | 
| 262 | 18 | ad5antr 734 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 45
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → 𝐶 ∈ (∞Met‘𝑋)) | 
| 263 | 224, 245 | sylan 580 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 45
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (1st ‘(𝑔‘𝑏)) ∈ 𝑋) | 
| 264 | 253 | rpxrd 13078 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 45
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (2nd ‘(𝑔‘𝑏)) ∈
ℝ*) | 
| 265 |  | elbl 24398 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . 45
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st
‘(𝑔‘𝑏)) ∈ 𝑋 ∧ (2nd ‘(𝑔‘𝑏)) ∈ ℝ*) → (𝑥 ∈ ((1st
‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ↔ (𝑥 ∈ 𝑋 ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑥) < (2nd ‘(𝑔‘𝑏))))) | 
| 266 | 262, 263,
264, 265 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . 44
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (𝑥 ∈ ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ↔ (𝑥 ∈ 𝑋 ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑥) < (2nd ‘(𝑔‘𝑏))))) | 
| 267 | 261, 266 | sylan9bbr 510 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) → (𝑥 ∈ 𝑏 ↔ (𝑥 ∈ 𝑋 ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑥) < (2nd ‘(𝑔‘𝑏))))) | 
| 268 | 267 | biimpd 229 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) → (𝑥 ∈ 𝑏 → (𝑥 ∈ 𝑋 ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑥) < (2nd ‘(𝑔‘𝑏))))) | 
| 269 | 268 | an32s 652 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ 𝑏 ∈ 𝑠) → (𝑥 ∈ 𝑏 → (𝑥 ∈ 𝑋 ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑥) < (2nd ‘(𝑔‘𝑏))))) | 
| 270 | 269 | impr 454 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (𝑥 ∈ 𝑋 ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑥) < (2nd ‘(𝑔‘𝑏)))) | 
| 271 | 270 | simprd 495 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) < (2nd ‘(𝑔‘𝑏))) | 
| 272 | 258, 260,
271 | xrltled 13192 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔‘𝑏))) | 
| 273 | 224 | ffvelcdmda 7104 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (𝑔‘𝑏) ∈ (𝑋 ×
ℝ+)) | 
| 274 | 273, 244 | syl 17 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (1st ‘(𝑔‘𝑏)) ∈ 𝑋) | 
| 275 |  | simplrl 777 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → 𝑥 ∈ 𝑋) | 
| 276 | 262, 274,
275, 257 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈
ℝ*) | 
| 277 |  | xmetge0 24354 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (1st
‘(𝑔‘𝑏)) ∈ 𝑋 ∧ 𝑥 ∈ 𝑋) → 0 ≤ ((1st
‘(𝑔‘𝑏))𝐶𝑥)) | 
| 278 | 262, 274,
275, 277 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → 0 ≤ ((1st
‘(𝑔‘𝑏))𝐶𝑥)) | 
| 279 |  | xrrege0 13216 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢
(((((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ* ∧
(2nd ‘(𝑔‘𝑏)) ∈ ℝ) ∧ (0 ≤
((1st ‘(𝑔‘𝑏))𝐶𝑥) ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔‘𝑏)))) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ) | 
| 280 | 279 | an4s 660 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
(((((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ* ∧ 0 ≤
((1st ‘(𝑔‘𝑏))𝐶𝑥)) ∧ ((2nd ‘(𝑔‘𝑏)) ∈ ℝ ∧ ((1st
‘(𝑔‘𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔‘𝑏)))) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ) | 
| 281 | 280 | ex 412 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ* ∧ 0 ≤
((1st ‘(𝑔‘𝑏))𝐶𝑥)) → (((2nd ‘(𝑔‘𝑏)) ∈ ℝ ∧ ((1st
‘(𝑔‘𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔‘𝑏))) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ)) | 
| 282 | 276, 278,
281 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (((2nd ‘(𝑔‘𝑏)) ∈ ℝ ∧ ((1st
‘(𝑔‘𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔‘𝑏))) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ)) | 
| 283 | 282 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (((2nd ‘(𝑔‘𝑏)) ∈ ℝ ∧ ((1st
‘(𝑔‘𝑏))𝐶𝑥) ≤ (2nd ‘(𝑔‘𝑏))) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ)) | 
| 284 | 255, 272,
283 | mp2and 699 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ) | 
| 285 | 284 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ) | 
| 286 |  | xrltle 13191 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (((𝑥𝐶𝑤) ∈ ℝ* ∧
(2nd ‘(𝑔‘𝑏)) ∈ ℝ*) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏)) → (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔‘𝑏)))) | 
| 287 | 223, 236,
286 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏)) → (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔‘𝑏)))) | 
| 288 |  | xmetge0 24354 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) → 0 ≤ (𝑥𝐶𝑤)) | 
| 289 | 288 | 3expb 1121 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 0 ≤ (𝑥𝐶𝑤)) | 
| 290 | 219, 289 | sylan 580 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → 0 ≤ (𝑥𝐶𝑤)) | 
| 291 | 290 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → 0 ≤ (𝑥𝐶𝑤)) | 
| 292 | 234 | rpred 13077 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑏 ∈ 𝑠) → (2nd ‘(𝑔‘𝑏)) ∈ ℝ) | 
| 293 | 292 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (2nd ‘(𝑔‘𝑏)) ∈ ℝ) | 
| 294 |  | xrrege0 13216 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢ ((((𝑥𝐶𝑤) ∈ ℝ* ∧
(2nd ‘(𝑔‘𝑏)) ∈ ℝ) ∧ (0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔‘𝑏)))) → (𝑥𝐶𝑤) ∈ ℝ) | 
| 295 | 294 | ex 412 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ (((𝑥𝐶𝑤) ∈ ℝ* ∧
(2nd ‘(𝑔‘𝑏)) ∈ ℝ) → ((0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔‘𝑏))) → (𝑥𝐶𝑤) ∈ ℝ)) | 
| 296 | 223, 293,
295 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((0 ≤ (𝑥𝐶𝑤) ∧ (𝑥𝐶𝑤) ≤ (2nd ‘(𝑔‘𝑏))) → (𝑥𝐶𝑤) ∈ ℝ)) | 
| 297 | 291, 296 | mpand 695 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑥𝐶𝑤) ≤ (2nd ‘(𝑔‘𝑏)) → (𝑥𝐶𝑤) ∈ ℝ)) | 
| 298 | 287, 297 | syld 47 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏)) → (𝑥𝐶𝑤) ∈ ℝ)) | 
| 299 | 298 | adantlr 715 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏)) → (𝑥𝐶𝑤) ∈ ℝ)) | 
| 300 | 299 | imp 406 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → (𝑥𝐶𝑤) ∈ ℝ) | 
| 301 | 285, 300 | readdcld 11290 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → (((1st ‘(𝑔‘𝑏))𝐶𝑥) + (𝑥𝐶𝑤)) ∈ ℝ) | 
| 302 | 301 | rexrd 11311 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → (((1st ‘(𝑔‘𝑏))𝐶𝑥) + (𝑥𝐶𝑤)) ∈
ℝ*) | 
| 303 | 254, 254 | rpaddcld 13092 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∈
ℝ+) | 
| 304 | 303 | rpxrd 13078 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∈
ℝ*) | 
| 305 | 304 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∈
ℝ*) | 
| 306 |  | xmettri 24361 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ ((1st
‘(𝑔‘𝑏)) ∈ 𝑋 ∧ 𝑤 ∈ 𝑋 ∧ 𝑥 ∈ 𝑋)) → ((1st ‘(𝑔‘𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔‘𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤))) | 
| 307 | 241, 246,
248, 256, 306 | syl13anc 1374 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((1st ‘(𝑔‘𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔‘𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤))) | 
| 308 | 307 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → ((1st ‘(𝑔‘𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔‘𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤))) | 
| 309 |  | rexadd 13274 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
((((1st ‘(𝑔‘𝑏))𝐶𝑥) ∈ ℝ ∧ (𝑥𝐶𝑤) ∈ ℝ) → (((1st
‘(𝑔‘𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)) = (((1st ‘(𝑔‘𝑏))𝐶𝑥) + (𝑥𝐶𝑤))) | 
| 310 | 285, 300,
309 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → (((1st ‘(𝑔‘𝑏))𝐶𝑥) +𝑒 (𝑥𝐶𝑤)) = (((1st ‘(𝑔‘𝑏))𝐶𝑥) + (𝑥𝐶𝑤))) | 
| 311 | 308, 310 | breqtrd 5169 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → ((1st ‘(𝑔‘𝑏))𝐶𝑤) ≤ (((1st ‘(𝑔‘𝑏))𝐶𝑥) + (𝑥𝐶𝑤))) | 
| 312 | 255 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → (2nd ‘(𝑔‘𝑏)) ∈ ℝ) | 
| 313 | 271 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) < (2nd ‘(𝑔‘𝑏))) | 
| 314 |  | simpr 484 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) | 
| 315 | 285, 300,
312, 312, 313, 314 | lt2addd 11886 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → (((1st ‘(𝑔‘𝑏))𝐶𝑥) + (𝑥𝐶𝑤)) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) | 
| 316 | 251, 302,
305, 311, 315 | xrlelttrd 13202 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏))) → ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) | 
| 317 | 316 | ex 412 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏)) → ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))))) | 
| 318 | 252 | rpred 13077 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → (2nd ‘(𝑔‘𝑏)) ∈ ℝ) | 
| 319 | 318, 252 | ltaddrpd 13110 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧ 𝑏 ∈ 𝑠) → (2nd ‘(𝑔‘𝑏)) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) | 
| 320 | 242, 225,
319 | syl2an 596 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (2nd ‘(𝑔‘𝑏)) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) | 
| 321 | 258, 260,
304, 271, 320 | xrlttrd 13201 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((1st ‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) | 
| 322 | 317, 321 | jctild 525 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑥𝐶𝑤) < (2nd ‘(𝑔‘𝑏)) → (((1st ‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))))) | 
| 323 | 240, 322 | syld 47 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → (((1st
‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))))) | 
| 324 |  | simpll 767 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → (𝜑 ∧ 𝑓:𝑋⟶𝑌)) | 
| 325 |  | heicant.d | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ (𝜑 → 𝐷 ∈ (∞Met‘𝑌)) | 
| 326 |  | ffvelcdm 7101 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑓:𝑋⟶𝑌 ∧ 𝑥 ∈ 𝑋) → (𝑓‘𝑥) ∈ 𝑌) | 
| 327 |  | ffvelcdm 7101 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝑓:𝑋⟶𝑌 ∧ 𝑤 ∈ 𝑋) → (𝑓‘𝑤) ∈ 𝑌) | 
| 328 | 326, 327 | anim12dan 619 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝑓:𝑋⟶𝑌 ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑓‘𝑥) ∈ 𝑌 ∧ (𝑓‘𝑤) ∈ 𝑌)) | 
| 329 |  | xmetcl 24341 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘𝑥) ∈ 𝑌 ∧ (𝑓‘𝑤) ∈ 𝑌) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ∈
ℝ*) | 
| 330 | 329 | 3expb 1121 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ ((𝑓‘𝑥) ∈ 𝑌 ∧ (𝑓‘𝑤) ∈ 𝑌)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ∈
ℝ*) | 
| 331 | 325, 328,
330 | syl2an 596 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((𝜑 ∧ (𝑓:𝑋⟶𝑌 ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋))) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ∈
ℝ*) | 
| 332 | 331 | anassrs 467 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ∈
ℝ*) | 
| 333 | 324, 332 | sylan 580 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ∈
ℝ*) | 
| 334 | 333 | ad3antrrr 730 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ∈
ℝ*) | 
| 335 | 325 | ad5antr 734 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → 𝐷 ∈ (∞Met‘𝑌)) | 
| 336 |  | simp-5r 786 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → 𝑓:𝑋⟶𝑌) | 
| 337 | 336, 274 | ffvelcdmd 7105 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (𝑓‘(1st ‘(𝑔‘𝑏))) ∈ 𝑌) | 
| 338 |  | simpllr 776 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . 43
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) → 𝑓:𝑋⟶𝑌) | 
| 339 | 338 | ffvelcdmda 7104 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑥 ∈ 𝑋) → (𝑓‘𝑥) ∈ 𝑌) | 
| 340 | 339 | adantrr 717 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑓‘𝑥) ∈ 𝑌) | 
| 341 | 340 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (𝑓‘𝑥) ∈ 𝑌) | 
| 342 |  | xmetcl 24341 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔‘𝑏))) ∈ 𝑌 ∧ (𝑓‘𝑥) ∈ 𝑌) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈
ℝ*) | 
| 343 | 335, 337,
341, 342 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈
ℝ*) | 
| 344 | 9 | rpxrd 13078 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ (𝑑 ∈ ℝ+
→ (𝑑 / 2) ∈
ℝ*) | 
| 345 | 344 | ad4antlr 733 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (𝑑 / 2) ∈
ℝ*) | 
| 346 |  | xrltle 13191 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ ((((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ* ∧ (𝑑 / 2) ∈
ℝ*) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ≤ (𝑑 / 2))) | 
| 347 | 343, 345,
346 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ≤ (𝑑 / 2))) | 
| 348 |  | xmetge0 24354 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔‘𝑏))) ∈ 𝑌 ∧ (𝑓‘𝑥) ∈ 𝑌) → 0 ≤ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥))) | 
| 349 | 335, 337,
341, 348 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → 0 ≤ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥))) | 
| 350 | 9 | rpred 13077 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢ (𝑑 ∈ ℝ+
→ (𝑑 / 2) ∈
ℝ) | 
| 351 | 350 | ad4antlr 733 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (𝑑 / 2) ∈ ℝ) | 
| 352 |  | xrrege0 13216 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
(((((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ) ∧ (0
≤ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ≤ (𝑑 / 2))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ) | 
| 353 | 352 | ex 412 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ) → ((0
≤ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ≤ (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ)) | 
| 354 | 343, 351,
353 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → ((0 ≤ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ≤ (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ)) | 
| 355 | 349, 354 | mpand 695 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ≤ (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ)) | 
| 356 | 347, 355 | syld 47 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ)) | 
| 357 | 356 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ)) | 
| 358 | 357 | imp 406 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ) | 
| 359 | 338 | ffvelcdmda 7104 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 42
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ 𝑤 ∈ 𝑋) → (𝑓‘𝑤) ∈ 𝑌) | 
| 360 | 359 | adantrl 716 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑓‘𝑤) ∈ 𝑌) | 
| 361 | 360 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (𝑓‘𝑤) ∈ 𝑌) | 
| 362 |  | xmetcl 24341 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔‘𝑏))) ∈ 𝑌 ∧ (𝑓‘𝑤) ∈ 𝑌) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈
ℝ*) | 
| 363 | 335, 337,
361, 362 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈
ℝ*) | 
| 364 |  | xrltle 13191 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢ ((((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ* ∧ (𝑑 / 2) ∈
ℝ*) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ≤ (𝑑 / 2))) | 
| 365 | 363, 345,
364 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ≤ (𝑑 / 2))) | 
| 366 |  | xmetge0 24354 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘(1st ‘(𝑔‘𝑏))) ∈ 𝑌 ∧ (𝑓‘𝑤) ∈ 𝑌) → 0 ≤ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) | 
| 367 | 335, 337,
361, 366 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → 0 ≤ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) | 
| 368 |  | xrrege0 13216 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . 41
⊢
(((((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ) ∧ (0
≤ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ≤ (𝑑 / 2))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ) | 
| 369 | 368 | ex 412 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 40
⊢ ((((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ* ∧ (𝑑 / 2) ∈ ℝ) → ((0
≤ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ≤ (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ)) | 
| 370 | 363, 351,
369 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → ((0 ≤ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ≤ (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ)) | 
| 371 | 367, 370 | mpand 695 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ≤ (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ)) | 
| 372 | 365, 371 | syld 47 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ)) | 
| 373 | 372 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ)) | 
| 374 | 373 | imp 406 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ) | 
| 375 |  | readdcl 11238 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ ((((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ) → (((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) ∈ ℝ) | 
| 376 | 358, 374,
375 | syl2an 596 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
(((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2)) ∧ (((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) ∈ ℝ) | 
| 377 | 376 | anandis 678 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) ∈ ℝ) | 
| 378 | 377 | rexrd 11311 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) ∈
ℝ*) | 
| 379 |  | rpxr 13044 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢ (𝑑 ∈ ℝ+
→ 𝑑 ∈
ℝ*) | 
| 380 | 379 | ad6antlr 737 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → 𝑑 ∈ ℝ*) | 
| 381 |  | xmettri 24361 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ ((𝑓‘𝑥) ∈ 𝑌 ∧ (𝑓‘𝑤) ∈ 𝑌 ∧ (𝑓‘(1st ‘(𝑔‘𝑏))) ∈ 𝑌)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ≤ (((𝑓‘𝑥)𝐷(𝑓‘(1st ‘(𝑔‘𝑏)))) +𝑒 ((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)))) | 
| 382 | 335, 341,
361, 337, 381 | syl13anc 1374 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ≤ (((𝑓‘𝑥)𝐷(𝑓‘(1st ‘(𝑔‘𝑏)))) +𝑒 ((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)))) | 
| 383 | 382 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ≤ (((𝑓‘𝑥)𝐷(𝑓‘(1st ‘(𝑔‘𝑏)))) +𝑒 ((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)))) | 
| 384 | 383 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ≤ (((𝑓‘𝑥)𝐷(𝑓‘(1st ‘(𝑔‘𝑏)))) +𝑒 ((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)))) | 
| 385 |  | xmetsym 24357 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ ((𝐷 ∈ (∞Met‘𝑌) ∧ (𝑓‘𝑥) ∈ 𝑌 ∧ (𝑓‘(1st ‘(𝑔‘𝑏))) ∈ 𝑌) → ((𝑓‘𝑥)𝐷(𝑓‘(1st ‘(𝑔‘𝑏)))) = ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥))) | 
| 386 | 335, 341,
337, 385 | syl3anc 1373 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → ((𝑓‘𝑥)𝐷(𝑓‘(1st ‘(𝑔‘𝑏)))) = ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥))) | 
| 387 | 386 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑓‘𝑥)𝐷(𝑓‘(1st ‘(𝑔‘𝑏)))) = ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥))) | 
| 388 | 387 | adantr 480 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ((𝑓‘𝑥)𝐷(𝑓‘(1st ‘(𝑔‘𝑏)))) = ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥))) | 
| 389 | 388 | oveq1d 7446 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (((𝑓‘𝑥)𝐷(𝑓‘(1st ‘(𝑔‘𝑏)))) +𝑒 ((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) = (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) +𝑒 ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)))) | 
| 390 |  | rexadd 13274 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢ ((((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ) → (((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) +𝑒 ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) = (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)))) | 
| 391 | 358, 374,
390 | syl2an 596 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
(((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2)) ∧ (((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) +𝑒 ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) = (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)))) | 
| 392 | 391 | anandis 678 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) +𝑒 ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) = (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)))) | 
| 393 | 389, 392 | eqtrd 2777 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (((𝑓‘𝑥)𝐷(𝑓‘(1st ‘(𝑔‘𝑏)))) +𝑒 ((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) = (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)))) | 
| 394 | 384, 393 | breqtrd 5169 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) ≤ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)))) | 
| 395 |  | lt2add 11748 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 39
⊢
(((((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ) ∧ ((𝑑 / 2) ∈ ℝ ∧ (𝑑 / 2) ∈ ℝ)) →
((((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) < ((𝑑 / 2) + (𝑑 / 2)))) | 
| 396 | 395 | expcom 413 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . 38
⊢ (((𝑑 / 2) ∈ ℝ ∧
(𝑑 / 2) ∈ ℝ)
→ ((((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ) → ((((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) < ((𝑑 / 2) + (𝑑 / 2))))) | 
| 397 | 351, 351,
396 | syl2anc 584 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . 37
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → ((((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) ∈ ℝ ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) ∈ ℝ) → ((((𝑓‘(1st
‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) < ((𝑑 / 2) + (𝑑 / 2))))) | 
| 398 | 356, 372,
397 | syl2and 608 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 36
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → ((((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → ((((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) < ((𝑑 / 2) + (𝑑 / 2))))) | 
| 399 | 398 | pm2.43d 53 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢
((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 ∈ 𝑠) → ((((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) < ((𝑑 / 2) + (𝑑 / 2)))) | 
| 400 | 399 | ad2ant2r 747 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) < ((𝑑 / 2) + (𝑑 / 2)))) | 
| 401 | 400 | imp 406 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) < ((𝑑 / 2) + (𝑑 / 2))) | 
| 402 |  | rpcn 13045 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. 35
⊢ (𝑑 ∈ ℝ+
→ 𝑑 ∈
ℂ) | 
| 403 | 402 | 2halvesd 12512 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
⊢ (𝑑 ∈ ℝ+
→ ((𝑑 / 2) + (𝑑 / 2)) = 𝑑) | 
| 404 | 403 | ad6antlr 737 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ((𝑑 / 2) + (𝑑 / 2)) = 𝑑) | 
| 405 | 401, 404 | breqtrd 5169 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
32
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) + ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤))) < 𝑑) | 
| 406 | 334, 378,
380, 394, 405 | xrlelttrd 13202 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
31
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) ∧ (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) | 
| 407 | 406 | ex 412 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)) | 
| 408 | 323, 407 | imim12d 81 | . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (((((1st ‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 409 | 196, 408 | sylanl1 680 | . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
⊢
(((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ 𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (((((1st ‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 410 | 409 | adantlrr 721 | . . . . . . . . . . . . . . . . . . . . . . . . . . 27
⊢
(((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → (((((1st ‘(𝑔‘𝑏))𝐶𝑥) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) ∧ ((1st ‘(𝑔‘𝑏))𝐶𝑤) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏)))) → (((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑥)) < (𝑑 / 2) ∧ ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑤)) < (𝑑 / 2))) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 411 | 193, 410 | mpd 15 | . . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢
(((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ∧ (𝑏 ∈ 𝑠 ∧ 𝑥 ∈ 𝑏)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)) | 
| 412 | 411 | exp32 420 | . . . . . . . . . . . . . . . . . . . . . . . . 25
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → (𝑏 ∈ 𝑠 → (𝑥 ∈ 𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)))) | 
| 413 | 174, 412 | sylan2 593 | . . . . . . . . . . . . . . . . . . . . . . . 24
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ (∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))) ∧ 𝑏 ∈ 𝑠)) → (𝑏 ∈ 𝑠 → (𝑥 ∈ 𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)))) | 
| 414 | 413 | expr 456 | . . . . . . . . . . . . . . . . . . . . . . 23
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → (𝑏 ∈ 𝑠 → (𝑏 ∈ 𝑠 → (𝑥 ∈ 𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))))) | 
| 415 | 414 | pm2.43d 53 | . . . . . . . . . . . . . . . . . . . . . 22
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) ∧ ∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → (𝑏 ∈ 𝑠 → (𝑥 ∈ 𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)))) | 
| 416 | 415 | an32s 652 | . . . . . . . . . . . . . . . . . . . . 21
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑏 ∈ 𝑠 → (𝑥 ∈ 𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)))) | 
| 417 | 172, 173,
416 | rexlimd 3266 | . . . . . . . . . . . . . . . . . . . 20
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (∃𝑏 ∈ 𝑠 𝑥 ∈ 𝑏 → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 418 | 167, 417 | mpd 15 | . . . . . . . . . . . . . . . . . . 19
⊢
((((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) ∧ (𝑥 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)) | 
| 419 | 418 | ralrimivva 3202 | . . . . . . . . . . . . . . . . . 18
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)) | 
| 420 |  | breq2 5147 | . . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑧 = inf(ran ran 𝑔, ℝ, < ) → ((𝑥𝐶𝑤) < 𝑧 ↔ (𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ))) | 
| 421 | 420 | imbi1d 341 | . . . . . . . . . . . . . . . . . . . 20
⊢ (𝑧 = inf(ran ran 𝑔, ℝ, < ) →
(((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) ↔ ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 422 | 421 | 2ralbidv 3221 | . . . . . . . . . . . . . . . . . . 19
⊢ (𝑧 = inf(ran ran 𝑔, ℝ, < ) →
(∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) ↔ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 423 | 422 | rspcev 3622 | . . . . . . . . . . . . . . . . . 18
⊢ ((inf(ran
ran 𝑔, ℝ, < )
∈ ℝ+ ∧ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < inf(ran ran 𝑔, ℝ, < ) → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)) | 
| 424 | 158, 419,
423 | syl2anc 584 | . . . . . . . . . . . . . . . . 17
⊢
(((((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) ∧ 𝑔:𝑠⟶(𝑋 × ℝ+)) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)) | 
| 425 | 424 | expl 457 | . . . . . . . . . . . . . . . 16
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) → ((𝑔:𝑠⟶(𝑋 × ℝ+) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 426 | 425 | exlimdv 1933 | . . . . . . . . . . . . . . 15
⊢
(((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) ∧ ∪ (MetOpen‘𝐶) = ∪ 𝑠) → (∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2)))) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 427 | 426 | expimpd 453 | . . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ Fin) → ((∪ (MetOpen‘𝐶) = ∪ 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 428 | 102, 427 | sylan2 593 | . . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) ∧ 𝑠 ∈ (𝒫
(MetOpen‘𝐶) ∩
Fin)) → ((∪ (MetOpen‘𝐶) = ∪ 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 429 | 428 | rexlimdva 3155 | . . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) →
(∃𝑠 ∈ (𝒫
(MetOpen‘𝐶) ∩
Fin)(∪ (MetOpen‘𝐶) = ∪ 𝑠 ∧ ∃𝑔(𝑔:𝑠⟶(𝑋 × ℝ+) ∧
∀𝑏 ∈ 𝑠 (𝑏 = ((1st ‘(𝑔‘𝑏))(ball‘𝐶)(2nd ‘(𝑔‘𝑏))) ∧ ∀𝑐 ∈ 𝑋 (((1st ‘(𝑔‘𝑏))𝐶𝑐) < ((2nd ‘(𝑔‘𝑏)) + (2nd ‘(𝑔‘𝑏))) → ((𝑓‘(1st ‘(𝑔‘𝑏)))𝐷(𝑓‘𝑐)) < (𝑑 / 2))))) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 430 | 101, 429 | syld 47 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) →
(∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < (𝑑 / 2)) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 431 | 15, 430 | syl5 34 | . . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑓:𝑋⟶𝑌) ∧ 𝑑 ∈ ℝ+) → (((𝑑 / 2) ∈ ℝ+
∧ ∀𝑦 ∈
ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 432 | 431 | exp4b 430 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑓:𝑋⟶𝑌) → (𝑑 ∈ ℝ+ → ((𝑑 / 2) ∈ ℝ+
→ (∀𝑦 ∈
ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))))) | 
| 433 | 9, 432 | mpdi 45 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑓:𝑋⟶𝑌) → (𝑑 ∈ ℝ+ →
(∀𝑦 ∈
ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)))) | 
| 434 | 433 | ralrimiv 3145 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑓:𝑋⟶𝑌) → ∀𝑑 ∈ ℝ+ (∀𝑦 ∈ ℝ+
∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 435 |  | r19.21v 3180 | . . . . . . 7
⊢
(∀𝑑 ∈
ℝ+ (∀𝑦 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) → ∃𝑧 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)) ↔ (∀𝑦 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) → ∀𝑑 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 436 | 434, 435 | sylib 218 | . . . . . 6
⊢ ((𝜑 ∧ 𝑓:𝑋⟶𝑌) → (∀𝑦 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) → ∀𝑑 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑))) | 
| 437 | 8, 436 | impbid2 226 | . . . . 5
⊢ ((𝜑 ∧ 𝑓:𝑋⟶𝑌) → (∀𝑑 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) ↔ ∀𝑦 ∈ ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦))) | 
| 438 |  | ralcom 3289 | . . . . 5
⊢
(∀𝑦 ∈
ℝ+ ∀𝑥 ∈ 𝑋 ∃𝑧 ∈ ℝ+ ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)) | 
| 439 | 437, 438 | bitrdi 287 | . . . 4
⊢ ((𝜑 ∧ 𝑓:𝑋⟶𝑌) → (∀𝑑 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑) ↔ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦))) | 
| 440 | 439 | pm5.32da 579 | . . 3
⊢ (𝜑 → ((𝑓:𝑋⟶𝑌 ∧ ∀𝑑 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)) ↔ (𝑓:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)))) | 
| 441 |  | eqid 2737 | . . . 4
⊢
(metUnif‘𝐶) =
(metUnif‘𝐶) | 
| 442 |  | eqid 2737 | . . . 4
⊢
(metUnif‘𝐷) =
(metUnif‘𝐷) | 
| 443 |  | heicant.y | . . . 4
⊢ (𝜑 → 𝑌 ≠ ∅) | 
| 444 |  | xmetpsmet 24358 | . . . . 5
⊢ (𝐶 ∈ (∞Met‘𝑋) → 𝐶 ∈ (PsMet‘𝑋)) | 
| 445 | 18, 444 | syl 17 | . . . 4
⊢ (𝜑 → 𝐶 ∈ (PsMet‘𝑋)) | 
| 446 |  | xmetpsmet 24358 | . . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑌) → 𝐷 ∈ (PsMet‘𝑌)) | 
| 447 | 325, 446 | syl 17 | . . . 4
⊢ (𝜑 → 𝐷 ∈ (PsMet‘𝑌)) | 
| 448 | 441, 442,
128, 443, 445, 447 | metucn 24584 | . . 3
⊢ (𝜑 → (𝑓 ∈ ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) ↔ (𝑓:𝑋⟶𝑌 ∧ ∀𝑑 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑥 ∈ 𝑋 ∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑑)))) | 
| 449 |  | eqid 2737 | . . . . 5
⊢
(MetOpen‘𝐷) =
(MetOpen‘𝐷) | 
| 450 | 23, 449 | metcn 24556 | . . . 4
⊢ ((𝐶 ∈ (∞Met‘𝑋) ∧ 𝐷 ∈ (∞Met‘𝑌)) → (𝑓 ∈ ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)) ↔ (𝑓:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)))) | 
| 451 | 18, 325, 450 | syl2anc 584 | . . 3
⊢ (𝜑 → (𝑓 ∈ ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)) ↔ (𝑓:𝑋⟶𝑌 ∧ ∀𝑥 ∈ 𝑋 ∀𝑦 ∈ ℝ+ ∃𝑧 ∈ ℝ+
∀𝑤 ∈ 𝑋 ((𝑥𝐶𝑤) < 𝑧 → ((𝑓‘𝑥)𝐷(𝑓‘𝑤)) < 𝑦)))) | 
| 452 | 440, 448,
451 | 3bitr4d 311 | . 2
⊢ (𝜑 → (𝑓 ∈ ((metUnif‘𝐶) Cnu(metUnif‘𝐷)) ↔ 𝑓 ∈ ((MetOpen‘𝐶) Cn (MetOpen‘𝐷)))) | 
| 453 | 452 | eqrdv 2735 | 1
⊢ (𝜑 → ((metUnif‘𝐶)
Cnu(metUnif‘𝐷)) = ((MetOpen‘𝐶) Cn (MetOpen‘𝐷))) |