| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | stoweidlem62.4 | . . . . 5
⊢ 𝐻 = (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < ))) | 
| 2 |  | nfmpt1 5250 | . . . . 5
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < ))) | 
| 3 | 1, 2 | nfcxfr 2903 | . . . 4
⊢
Ⅎ𝑡𝐻 | 
| 4 |  | stoweidlem62.3 | . . . 4
⊢
Ⅎ𝑡𝜑 | 
| 5 |  | stoweidlem62.5 | . . . 4
⊢ 𝐾 = (topGen‘ran
(,)) | 
| 6 |  | stoweidlem62.7 | . . . 4
⊢ (𝜑 → 𝐽 ∈ Comp) | 
| 7 |  | stoweidlem62.6 | . . . 4
⊢ 𝑇 = ∪
𝐽 | 
| 8 |  | stoweidlem62.16 | . . . 4
⊢ (𝜑 → 𝑇 ≠ ∅) | 
| 9 |  | stoweidlem62.8 | . . . 4
⊢ 𝐶 = (𝐽 Cn 𝐾) | 
| 10 |  | stoweidlem62.9 | . . . 4
⊢ (𝜑 → 𝐴 ⊆ 𝐶) | 
| 11 |  | eleq1w 2824 | . . . . . . 7
⊢ (𝑔 = ℎ → (𝑔 ∈ 𝐴 ↔ ℎ ∈ 𝐴)) | 
| 12 | 11 | 3anbi3d 1444 | . . . . . 6
⊢ (𝑔 = ℎ → ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) ↔ (𝜑 ∧ 𝑓 ∈ 𝐴 ∧ ℎ ∈ 𝐴))) | 
| 13 |  | fveq1 6905 | . . . . . . . . 9
⊢ (𝑔 = ℎ → (𝑔‘𝑡) = (ℎ‘𝑡)) | 
| 14 | 13 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑔 = ℎ → ((𝑓‘𝑡) + (𝑔‘𝑡)) = ((𝑓‘𝑡) + (ℎ‘𝑡))) | 
| 15 | 14 | mpteq2dv 5244 | . . . . . . 7
⊢ (𝑔 = ℎ → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (ℎ‘𝑡)))) | 
| 16 | 15 | eleq1d 2826 | . . . . . 6
⊢ (𝑔 = ℎ → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (ℎ‘𝑡))) ∈ 𝐴)) | 
| 17 | 12, 16 | imbi12d 344 | . . . . 5
⊢ (𝑔 = ℎ → (((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (ℎ‘𝑡))) ∈ 𝐴))) | 
| 18 |  | stoweidlem62.10 | . . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) | 
| 19 | 17, 18 | chvarvv 1998 | . . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (ℎ‘𝑡))) ∈ 𝐴) | 
| 20 | 13 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑔 = ℎ → ((𝑓‘𝑡) · (𝑔‘𝑡)) = ((𝑓‘𝑡) · (ℎ‘𝑡))) | 
| 21 | 20 | mpteq2dv 5244 | . . . . . . 7
⊢ (𝑔 = ℎ → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (ℎ‘𝑡)))) | 
| 22 | 21 | eleq1d 2826 | . . . . . 6
⊢ (𝑔 = ℎ → ((𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴 ↔ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (ℎ‘𝑡))) ∈ 𝐴)) | 
| 23 | 12, 22 | imbi12d 344 | . . . . 5
⊢ (𝑔 = ℎ → (((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) ↔ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (ℎ‘𝑡))) ∈ 𝐴))) | 
| 24 |  | stoweidlem62.11 | . . . . 5
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴) | 
| 25 | 23, 24 | chvarvv 1998 | . . . 4
⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ ℎ ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (ℎ‘𝑡))) ∈ 𝐴) | 
| 26 |  | stoweidlem62.12 | . . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) | 
| 27 |  | stoweidlem62.13 | . . . 4
⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡)) | 
| 28 |  | stoweidlem62.1 | . . . . . 6
⊢
Ⅎ𝑡𝐹 | 
| 29 | 28 | nfrn 5963 | . . . . . . 7
⊢
Ⅎ𝑡ran
𝐹 | 
| 30 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑡ℝ | 
| 31 |  | nfcv 2905 | . . . . . . 7
⊢
Ⅎ𝑡
< | 
| 32 | 29, 30, 31 | nfinf 9522 | . . . . . 6
⊢
Ⅎ𝑡inf(ran 𝐹, ℝ, < ) | 
| 33 |  | eqid 2737 | . . . . . 6
⊢ (𝑇 × {-inf(ran 𝐹, ℝ, < )}) = (𝑇 × {-inf(ran 𝐹, ℝ, <
)}) | 
| 34 |  | cmptop 23403 | . . . . . . 7
⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top) | 
| 35 | 6, 34 | syl 17 | . . . . . 6
⊢ (𝜑 → 𝐽 ∈ Top) | 
| 36 |  | stoweidlem62.14 | . . . . . 6
⊢ (𝜑 → 𝐹 ∈ 𝐶) | 
| 37 | 36, 9 | eleqtrdi 2851 | . . . . . . . 8
⊢ (𝜑 → 𝐹 ∈ (𝐽 Cn 𝐾)) | 
| 38 | 28, 4, 7, 5, 6, 37,
8 | stoweidlem29 46044 | . . . . . . 7
⊢ (𝜑 → (inf(ran 𝐹, ℝ, < ) ∈ ran 𝐹 ∧ inf(ran 𝐹, ℝ, < ) ∈ ℝ ∧
∀𝑡 ∈ 𝑇 inf(ran 𝐹, ℝ, < ) ≤ (𝐹‘𝑡))) | 
| 39 | 38 | simp2d 1144 | . . . . . 6
⊢ (𝜑 → inf(ran 𝐹, ℝ, < ) ∈
ℝ) | 
| 40 | 28, 32, 4, 7, 33, 5,
35, 9, 36, 39 | stoweidlem47 46062 | . . . . 5
⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < ))) ∈ 𝐶) | 
| 41 | 1, 40 | eqeltrid 2845 | . . . 4
⊢ (𝜑 → 𝐻 ∈ 𝐶) | 
| 42 | 38 | simp3d 1145 | . . . . . . . . 9
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 inf(ran 𝐹, ℝ, < ) ≤ (𝐹‘𝑡)) | 
| 43 | 42 | r19.21bi 3251 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → inf(ran 𝐹, ℝ, < ) ≤ (𝐹‘𝑡)) | 
| 44 | 5, 7, 9, 36 | fcnre 45030 | . . . . . . . . . 10
⊢ (𝜑 → 𝐹:𝑇⟶ℝ) | 
| 45 | 44 | ffvelcdmda 7104 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) ∈ ℝ) | 
| 46 | 39 | adantr 480 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → inf(ran 𝐹, ℝ, < ) ∈
ℝ) | 
| 47 | 45, 46 | subge0d 11853 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < )) ↔ inf(ran 𝐹, ℝ, < ) ≤ (𝐹‘𝑡))) | 
| 48 | 43, 47 | mpbird 257 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < ))) | 
| 49 |  | simpr 484 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇) | 
| 50 | 45, 46 | resubcld 11691 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < )) ∈
ℝ) | 
| 51 | 1 | fvmpt2 7027 | . . . . . . . 8
⊢ ((𝑡 ∈ 𝑇 ∧ ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < )) ∈ ℝ) →
(𝐻‘𝑡) = ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < ))) | 
| 52 | 49, 50, 51 | syl2anc 584 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐻‘𝑡) = ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < ))) | 
| 53 | 48, 52 | breqtrrd 5171 | . . . . . 6
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝐻‘𝑡)) | 
| 54 | 53 | ex 412 | . . . . 5
⊢ (𝜑 → (𝑡 ∈ 𝑇 → 0 ≤ (𝐻‘𝑡))) | 
| 55 | 4, 54 | ralrimi 3257 | . . . 4
⊢ (𝜑 → ∀𝑡 ∈ 𝑇 0 ≤ (𝐻‘𝑡)) | 
| 56 |  | stoweidlem62.15 | . . . . 5
⊢ (𝜑 → 𝐸 ∈
ℝ+) | 
| 57 | 56 | rphalfcld 13089 | . . . 4
⊢ (𝜑 → (𝐸 / 2) ∈
ℝ+) | 
| 58 | 56 | rpred 13077 | . . . . . 6
⊢ (𝜑 → 𝐸 ∈ ℝ) | 
| 59 | 58 | rehalfcld 12513 | . . . . 5
⊢ (𝜑 → (𝐸 / 2) ∈ ℝ) | 
| 60 |  | 3re 12346 | . . . . . . 7
⊢ 3 ∈
ℝ | 
| 61 |  | 3ne0 12372 | . . . . . . 7
⊢ 3 ≠
0 | 
| 62 | 60, 61 | rereccli 12032 | . . . . . 6
⊢ (1 / 3)
∈ ℝ | 
| 63 | 62 | a1i 11 | . . . . 5
⊢ (𝜑 → (1 / 3) ∈
ℝ) | 
| 64 |  | rphalflt 13064 | . . . . . 6
⊢ (𝐸 ∈ ℝ+
→ (𝐸 / 2) < 𝐸) | 
| 65 | 56, 64 | syl 17 | . . . . 5
⊢ (𝜑 → (𝐸 / 2) < 𝐸) | 
| 66 |  | stoweidlem62.17 | . . . . 5
⊢ (𝜑 → 𝐸 < (1 / 3)) | 
| 67 | 59, 58, 63, 65, 66 | lttrd 11422 | . . . 4
⊢ (𝜑 → (𝐸 / 2) < (1 / 3)) | 
| 68 | 3, 4, 5, 6, 7, 8, 9, 10, 19, 25, 26, 27, 41, 55, 57, 67 | stoweidlem61 46076 | . . 3
⊢ (𝜑 → ∃ℎ ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2))) | 
| 69 |  | nfra1 3284 | . . . . . . 7
⊢
Ⅎ𝑡∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2)) | 
| 70 | 4, 69 | nfan 1899 | . . . . . 6
⊢
Ⅎ𝑡(𝜑 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2))) | 
| 71 |  | rsp 3247 | . . . . . . 7
⊢
(∀𝑡 ∈
𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2)) → (𝑡 ∈ 𝑇 → (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2)))) | 
| 72 | 56 | rpcnd 13079 | . . . . . . . . . 10
⊢ (𝜑 → 𝐸 ∈ ℂ) | 
| 73 |  | 2cnd 12344 | . . . . . . . . . 10
⊢ (𝜑 → 2 ∈
ℂ) | 
| 74 |  | 2ne0 12370 | . . . . . . . . . . 11
⊢ 2 ≠
0 | 
| 75 | 74 | a1i 11 | . . . . . . . . . 10
⊢ (𝜑 → 2 ≠ 0) | 
| 76 | 72, 73, 75 | divcan2d 12045 | . . . . . . . . 9
⊢ (𝜑 → (2 · (𝐸 / 2)) = 𝐸) | 
| 77 | 76 | breq2d 5155 | . . . . . . . 8
⊢ (𝜑 → ((abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2)) ↔ (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) | 
| 78 | 77 | biimpd 229 | . . . . . . 7
⊢ (𝜑 → ((abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2)) → (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) | 
| 79 | 71, 78 | sylan9r 508 | . . . . . 6
⊢ ((𝜑 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2))) → (𝑡 ∈ 𝑇 → (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) | 
| 80 | 70, 79 | ralrimi 3257 | . . . . 5
⊢ ((𝜑 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2))) → ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸) | 
| 81 | 80 | ex 412 | . . . 4
⊢ (𝜑 → (∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2)) → ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) | 
| 82 | 81 | reximdv 3170 | . . 3
⊢ (𝜑 → (∃ℎ ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < (2 · (𝐸 / 2)) → ∃ℎ ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) | 
| 83 | 68, 82 | mpd 15 | . 2
⊢ (𝜑 → ∃ℎ ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸) | 
| 84 |  | nfmpt1 5250 | . . 3
⊢
Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) + inf(ran 𝐹, ℝ, < ))) | 
| 85 |  | nfcv 2905 | . . 3
⊢
Ⅎ𝑡ℎ | 
| 86 |  | nfv 1914 | . . . . 5
⊢
Ⅎ𝑡 ℎ ∈ 𝐴 | 
| 87 |  | nfra1 3284 | . . . . 5
⊢
Ⅎ𝑡∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸 | 
| 88 | 86, 87 | nfan 1899 | . . . 4
⊢
Ⅎ𝑡(ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸) | 
| 89 | 4, 88 | nfan 1899 | . . 3
⊢
Ⅎ𝑡(𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) | 
| 90 |  | eqid 2737 | . . 3
⊢ (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) + inf(ran 𝐹, ℝ, < ))) = (𝑡 ∈ 𝑇 ↦ ((ℎ‘𝑡) + inf(ran 𝐹, ℝ, < ))) | 
| 91 | 44 | adantr 480 | . . 3
⊢ ((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) → 𝐹:𝑇⟶ℝ) | 
| 92 | 39 | adantr 480 | . . 3
⊢ ((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) → inf(ran 𝐹, ℝ, < ) ∈
ℝ) | 
| 93 | 18 | 3adant1r 1178 | . . 3
⊢ (((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴) | 
| 94 | 26 | adantlr 715 | . . 3
⊢ (((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴) | 
| 95 |  | stoweidlem62.2 | . . . . 5
⊢
Ⅎ𝑓𝜑 | 
| 96 | 10 | sseld 3982 | . . . . . . . 8
⊢ (𝜑 → (𝑓 ∈ 𝐴 → 𝑓 ∈ 𝐶)) | 
| 97 | 9 | eleq2i 2833 | . . . . . . . 8
⊢ (𝑓 ∈ 𝐶 ↔ 𝑓 ∈ (𝐽 Cn 𝐾)) | 
| 98 | 96, 97 | imbitrdi 251 | . . . . . . 7
⊢ (𝜑 → (𝑓 ∈ 𝐴 → 𝑓 ∈ (𝐽 Cn 𝐾))) | 
| 99 |  | eqid 2737 | . . . . . . . 8
⊢ ∪ 𝐽 =
∪ 𝐽 | 
| 100 |  | uniretop 24783 | . . . . . . . . 9
⊢ ℝ =
∪ (topGen‘ran (,)) | 
| 101 | 5 | unieqi 4919 | . . . . . . . . 9
⊢ ∪ 𝐾 =
∪ (topGen‘ran (,)) | 
| 102 | 100, 101 | eqtr4i 2768 | . . . . . . . 8
⊢ ℝ =
∪ 𝐾 | 
| 103 | 99, 102 | cnf 23254 | . . . . . . 7
⊢ (𝑓 ∈ (𝐽 Cn 𝐾) → 𝑓:∪ 𝐽⟶ℝ) | 
| 104 | 98, 103 | syl6 35 | . . . . . 6
⊢ (𝜑 → (𝑓 ∈ 𝐴 → 𝑓:∪ 𝐽⟶ℝ)) | 
| 105 |  | feq2 6717 | . . . . . . 7
⊢ (𝑇 = ∪
𝐽 → (𝑓:𝑇⟶ℝ ↔ 𝑓:∪ 𝐽⟶ℝ)) | 
| 106 | 7, 105 | mp1i 13 | . . . . . 6
⊢ (𝜑 → (𝑓:𝑇⟶ℝ ↔ 𝑓:∪ 𝐽⟶ℝ)) | 
| 107 | 104, 106 | sylibrd 259 | . . . . 5
⊢ (𝜑 → (𝑓 ∈ 𝐴 → 𝑓:𝑇⟶ℝ)) | 
| 108 | 95, 107 | ralrimi 3257 | . . . 4
⊢ (𝜑 → ∀𝑓 ∈ 𝐴 𝑓:𝑇⟶ℝ) | 
| 109 | 108 | adantr 480 | . . 3
⊢ ((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) → ∀𝑓 ∈ 𝐴 𝑓:𝑇⟶ℝ) | 
| 110 |  | simprl 771 | . . 3
⊢ ((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) → ℎ ∈ 𝐴) | 
| 111 | 52 | eqcomd 2743 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < )) = (𝐻‘𝑡)) | 
| 112 | 111 | oveq2d 7447 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((ℎ‘𝑡) − ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < ))) = ((ℎ‘𝑡) − (𝐻‘𝑡))) | 
| 113 | 112 | fveq2d 6910 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (abs‘((ℎ‘𝑡) − ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < )))) = (abs‘((ℎ‘𝑡) − (𝐻‘𝑡)))) | 
| 114 | 113 | adantlr 715 | . . . . . 6
⊢ (((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) ∧ 𝑡 ∈ 𝑇) → (abs‘((ℎ‘𝑡) − ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < )))) = (abs‘((ℎ‘𝑡) − (𝐻‘𝑡)))) | 
| 115 |  | simplrr 778 | . . . . . . 7
⊢ (((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) ∧ 𝑡 ∈ 𝑇) → ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸) | 
| 116 |  | rspa 3248 | . . . . . . 7
⊢
((∀𝑡 ∈
𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸 ∧ 𝑡 ∈ 𝑇) → (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸) | 
| 117 | 115, 116 | sylancom 588 | . . . . . 6
⊢ (((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) ∧ 𝑡 ∈ 𝑇) → (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸) | 
| 118 | 114, 117 | eqbrtrd 5165 | . . . . 5
⊢ (((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) ∧ 𝑡 ∈ 𝑇) → (abs‘((ℎ‘𝑡) − ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < )))) < 𝐸) | 
| 119 | 118 | ex 412 | . . . 4
⊢ ((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) → (𝑡 ∈ 𝑇 → (abs‘((ℎ‘𝑡) − ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < )))) < 𝐸)) | 
| 120 | 89, 119 | ralrimi 3257 | . . 3
⊢ ((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) → ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − ((𝐹‘𝑡) − inf(ran 𝐹, ℝ, < )))) < 𝐸) | 
| 121 | 84, 85, 32, 89, 90, 91, 92, 93, 94, 109, 110, 120 | stoweidlem21 46036 | . 2
⊢ ((𝜑 ∧ (ℎ ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (abs‘((ℎ‘𝑡) − (𝐻‘𝑡))) < 𝐸)) → ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸) | 
| 122 | 83, 121 | rexlimddv 3161 | 1
⊢ (𝜑 → ∃𝑓 ∈ 𝐴 ∀𝑡 ∈ 𝑇 (abs‘((𝑓‘𝑡) − (𝐹‘𝑡))) < 𝐸) |