| Step | Hyp | Ref
| Expression |
| 1 | | resqcl 14164 |
. . . . . . . 8
⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈
ℝ) |
| 2 | 1 | adantr 480 |
. . . . . . 7
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (𝑅↑2) ∈
ℝ) |
| 3 | 2 | adantr 480 |
. . . . . 6
⊢ (((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑅↑2) ∈ ℝ) |
| 4 | | renegcl 11572 |
. . . . . . . . . 10
⊢ (𝑅 ∈ ℝ → -𝑅 ∈
ℝ) |
| 5 | | iccssre 13469 |
. . . . . . . . . 10
⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (-𝑅[,]𝑅) ⊆ ℝ) |
| 6 | 4, 5 | mpancom 688 |
. . . . . . . . 9
⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℝ) |
| 7 | 6 | sselda 3983 |
. . . . . . . 8
⊢ ((𝑅 ∈ ℝ ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → 𝑡 ∈ ℝ) |
| 8 | 7 | resqcld 14165 |
. . . . . . 7
⊢ ((𝑅 ∈ ℝ ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑡↑2) ∈ ℝ) |
| 9 | 8 | adantlr 715 |
. . . . . 6
⊢ (((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → (𝑡↑2) ∈ ℝ) |
| 10 | 3, 9 | resubcld 11691 |
. . . . 5
⊢ (((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((𝑅↑2) − (𝑡↑2)) ∈ ℝ) |
| 11 | | elicc2 13452 |
. . . . . . . . 9
⊢ ((-𝑅 ∈ ℝ ∧ 𝑅 ∈ ℝ) → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅))) |
| 12 | 4, 11 | mpancom 688 |
. . . . . . . 8
⊢ (𝑅 ∈ ℝ → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅))) |
| 13 | 12 | adantr 480 |
. . . . . . 7
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↔ (𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅))) |
| 14 | 1 | 3ad2ant1 1134 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → (𝑅↑2) ∈ ℝ) |
| 15 | | resqcl 14164 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ℝ → (𝑡↑2) ∈
ℝ) |
| 16 | 15 | 3ad2ant3 1136 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → (𝑡↑2) ∈ ℝ) |
| 17 | 14, 16 | subge0d 11853 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (𝑡↑2) ≤ (𝑅↑2))) |
| 18 | | absresq 15341 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ℝ →
((abs‘𝑡)↑2) =
(𝑡↑2)) |
| 19 | 18 | 3ad2ant3 1136 |
. . . . . . . . . . . . . 14
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡)↑2) = (𝑡↑2)) |
| 20 | 19 | breq1d 5153 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → (((abs‘𝑡)↑2) ≤ (𝑅↑2) ↔ (𝑡↑2) ≤ (𝑅↑2))) |
| 21 | 17, 20 | bitr4d 282 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔
((abs‘𝑡)↑2) ≤
(𝑅↑2))) |
| 22 | | recn 11245 |
. . . . . . . . . . . . . . 15
⊢ (𝑡 ∈ ℝ → 𝑡 ∈
ℂ) |
| 23 | 22 | abscld 15475 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ ℝ →
(abs‘𝑡) ∈
ℝ) |
| 24 | 23 | 3ad2ant3 1136 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → (abs‘𝑡) ∈
ℝ) |
| 25 | | simp1 1137 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → 𝑅 ∈ ℝ) |
| 26 | 22 | absge0d 15483 |
. . . . . . . . . . . . . 14
⊢ (𝑡 ∈ ℝ → 0 ≤
(abs‘𝑡)) |
| 27 | 26 | 3ad2ant3 1136 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤
(abs‘𝑡)) |
| 28 | | simp2 1138 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → 0 ≤ 𝑅) |
| 29 | 24, 25, 27, 28 | le2sqd 14296 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ ((abs‘𝑡)↑2) ≤ (𝑅↑2))) |
| 30 | | simp3 1139 |
. . . . . . . . . . . . 13
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → 𝑡 ∈ ℝ) |
| 31 | 30, 25 | absled 15469 |
. . . . . . . . . . . 12
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → ((abs‘𝑡) ≤ 𝑅 ↔ (-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅))) |
| 32 | 21, 29, 31 | 3bitr2d 307 |
. . . . . . . . . . 11
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → (0 ≤ ((𝑅↑2) − (𝑡↑2)) ↔ (-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅))) |
| 33 | 32 | biimprd 248 |
. . . . . . . . . 10
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅 ∧ 𝑡 ∈ ℝ) → ((-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2)))) |
| 34 | 33 | 3expa 1119 |
. . . . . . . . 9
⊢ (((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) ∧ 𝑡 ∈ ℝ) → ((-𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2)))) |
| 35 | 34 | exp4b 430 |
. . . . . . . 8
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (𝑡 ∈ ℝ → (-𝑅 ≤ 𝑡 → (𝑡 ≤ 𝑅 → 0 ≤ ((𝑅↑2) − (𝑡↑2)))))) |
| 36 | 35 | 3impd 1349 |
. . . . . . 7
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → ((𝑡 ∈ ℝ ∧ -𝑅 ≤ 𝑡 ∧ 𝑡 ≤ 𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2)))) |
| 37 | 13, 36 | sylbid 240 |
. . . . . 6
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) → 0 ≤ ((𝑅↑2) − (𝑡↑2)))) |
| 38 | 37 | imp 406 |
. . . . 5
⊢ (((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → 0 ≤ ((𝑅↑2) − (𝑡↑2))) |
| 39 | | elrege0 13494 |
. . . . 5
⊢ (((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞)
↔ (((𝑅↑2) −
(𝑡↑2)) ∈ ℝ
∧ 0 ≤ ((𝑅↑2)
− (𝑡↑2)))) |
| 40 | 10, 38, 39 | sylanbrc 583 |
. . . 4
⊢ (((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((𝑅↑2) − (𝑡↑2)) ∈
(0[,)+∞)) |
| 41 | | fvres 6925 |
. . . 4
⊢ (((𝑅↑2) − (𝑡↑2)) ∈ (0[,)+∞)
→ ((√ ↾ (0[,)+∞))‘((𝑅↑2) − (𝑡↑2))) = (√‘((𝑅↑2) − (𝑡↑2)))) |
| 42 | 40, 41 | syl 17 |
. . 3
⊢ (((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅)) → ((√ ↾
(0[,)+∞))‘((𝑅↑2) − (𝑡↑2))) = (√‘((𝑅↑2) − (𝑡↑2)))) |
| 43 | 42 | mpteq2dva 5242 |
. 2
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾
(0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2))))) |
| 44 | | eqid 2737 |
. . . . . . 7
⊢
(TopOpen‘ℂfld) =
(TopOpen‘ℂfld) |
| 45 | 44 | cnfldtopon 24803 |
. . . . . 6
⊢
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ) |
| 46 | | ax-resscn 11212 |
. . . . . . 7
⊢ ℝ
⊆ ℂ |
| 47 | 6, 46 | sstrdi 3996 |
. . . . . 6
⊢ (𝑅 ∈ ℝ → (-𝑅[,]𝑅) ⊆ ℂ) |
| 48 | | resttopon 23169 |
. . . . . 6
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ (-𝑅[,]𝑅) ⊆ ℂ) →
((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅))) |
| 49 | 45, 47, 48 | sylancr 587 |
. . . . 5
⊢ (𝑅 ∈ ℝ →
((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅))) |
| 50 | 49 | adantr 480 |
. . . 4
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) →
((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) ∈ (TopOn‘(-𝑅[,]𝑅))) |
| 51 | 47 | resmptd 6058 |
. . . . . . 7
⊢ (𝑅 ∈ ℝ → ((𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ↾ (-𝑅[,]𝑅)) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2)))) |
| 52 | 45 | a1i 11 |
. . . . . . . . 9
⊢ (𝑅 ∈ ℝ →
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ)) |
| 53 | | recn 11245 |
. . . . . . . . . . 11
⊢ (𝑅 ∈ ℝ → 𝑅 ∈
ℂ) |
| 54 | 53 | sqcld 14184 |
. . . . . . . . . 10
⊢ (𝑅 ∈ ℝ → (𝑅↑2) ∈
ℂ) |
| 55 | 52, 52, 54 | cnmptc 23670 |
. . . . . . . . 9
⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ (𝑅↑2)) ∈
((TopOpen‘ℂfld) Cn
(TopOpen‘ℂfld))) |
| 56 | 44 | sqcn 24900 |
. . . . . . . . . 10
⊢ (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈
((TopOpen‘ℂfld) Cn
(TopOpen‘ℂfld)) |
| 57 | 56 | a1i 11 |
. . . . . . . . 9
⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ (𝑡↑2)) ∈
((TopOpen‘ℂfld) Cn
(TopOpen‘ℂfld))) |
| 58 | 44 | subcn 24888 |
. . . . . . . . . 10
⊢ −
∈ (((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld)) |
| 59 | 58 | a1i 11 |
. . . . . . . . 9
⊢ (𝑅 ∈ ℝ → −
∈ (((TopOpen‘ℂfld) ×t
(TopOpen‘ℂfld)) Cn
(TopOpen‘ℂfld))) |
| 60 | 52, 55, 57, 59 | cnmpt12f 23674 |
. . . . . . . 8
⊢ (𝑅 ∈ ℝ → (𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ∈
((TopOpen‘ℂfld) Cn
(TopOpen‘ℂfld))) |
| 61 | 45 | toponunii 22922 |
. . . . . . . . 9
⊢ ℂ =
∪
(TopOpen‘ℂfld) |
| 62 | 61 | cnrest 23293 |
. . . . . . . 8
⊢ (((𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ∈
((TopOpen‘ℂfld) Cn
(TopOpen‘ℂfld)) ∧ (-𝑅[,]𝑅) ⊆ ℂ) → ((𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ↾ (-𝑅[,]𝑅)) ∈
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn
(TopOpen‘ℂfld))) |
| 63 | 60, 47, 62 | syl2anc 584 |
. . . . . . 7
⊢ (𝑅 ∈ ℝ → ((𝑡 ∈ ℂ ↦ ((𝑅↑2) − (𝑡↑2))) ↾ (-𝑅[,]𝑅)) ∈
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn
(TopOpen‘ℂfld))) |
| 64 | 51, 63 | eqeltrrd 2842 |
. . . . . 6
⊢ (𝑅 ∈ ℝ → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn
(TopOpen‘ℂfld))) |
| 65 | 64 | adantr 480 |
. . . . 5
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn
(TopOpen‘ℂfld))) |
| 66 | 45 | a1i 11 |
. . . . . 6
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) →
(TopOpen‘ℂfld) ∈
(TopOn‘ℂ)) |
| 67 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) = (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) |
| 68 | 67 | rnmpt 5968 |
. . . . . . 7
⊢ ran
(𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) = {𝑢 ∣ ∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2))} |
| 69 | | simp3 1139 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → 𝑢 = ((𝑅↑2) − (𝑡↑2))) |
| 70 | 40 | 3adant3 1133 |
. . . . . . . . . 10
⊢ (((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → ((𝑅↑2) − (𝑡↑2)) ∈
(0[,)+∞)) |
| 71 | 69, 70 | eqeltrd 2841 |
. . . . . . . . 9
⊢ (((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) ∧ 𝑡 ∈ (-𝑅[,]𝑅) ∧ 𝑢 = ((𝑅↑2) − (𝑡↑2))) → 𝑢 ∈ (0[,)+∞)) |
| 72 | 71 | rexlimdv3a 3159 |
. . . . . . . 8
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2)) → 𝑢 ∈ (0[,)+∞))) |
| 73 | 72 | abssdv 4068 |
. . . . . . 7
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → {𝑢 ∣ ∃𝑡 ∈ (-𝑅[,]𝑅)𝑢 = ((𝑅↑2) − (𝑡↑2))} ⊆
(0[,)+∞)) |
| 74 | 68, 73 | eqsstrid 4022 |
. . . . . 6
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → ran (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ⊆
(0[,)+∞)) |
| 75 | | rge0ssre 13496 |
. . . . . . . 8
⊢
(0[,)+∞) ⊆ ℝ |
| 76 | 75, 46 | sstri 3993 |
. . . . . . 7
⊢
(0[,)+∞) ⊆ ℂ |
| 77 | 76 | a1i 11 |
. . . . . 6
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (0[,)+∞)
⊆ ℂ) |
| 78 | | cnrest2 23294 |
. . . . . 6
⊢
(((TopOpen‘ℂfld) ∈ (TopOn‘ℂ)
∧ ran (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ⊆ (0[,)+∞) ∧
(0[,)+∞) ⊆ ℂ) → ((𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn (TopOpen‘ℂfld))
↔ (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂfld)
↾t (0[,)+∞))))) |
| 79 | 66, 74, 77, 78 | syl3anc 1373 |
. . . . 5
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → ((𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn (TopOpen‘ℂfld))
↔ (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂfld)
↾t (0[,)+∞))))) |
| 80 | 65, 79 | mpbid 232 |
. . . 4
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((𝑅↑2) − (𝑡↑2))) ∈
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂfld)
↾t (0[,)+∞)))) |
| 81 | | ssid 4006 |
. . . . . . . 8
⊢ ℂ
⊆ ℂ |
| 82 | | cncfss 24925 |
. . . . . . . 8
⊢ ((ℝ
⊆ ℂ ∧ ℂ ⊆ ℂ) →
((0[,)+∞)–cn→ℝ)
⊆ ((0[,)+∞)–cn→ℂ)) |
| 83 | 46, 81, 82 | mp2an 692 |
. . . . . . 7
⊢
((0[,)+∞)–cn→ℝ) ⊆ ((0[,)+∞)–cn→ℂ) |
| 84 | | resqrtcn 26792 |
. . . . . . 7
⊢ (√
↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℝ) |
| 85 | 83, 84 | sselii 3980 |
. . . . . 6
⊢ (√
↾ (0[,)+∞)) ∈ ((0[,)+∞)–cn→ℂ) |
| 86 | | eqid 2737 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t
(0[,)+∞)) = ((TopOpen‘ℂfld) ↾t
(0[,)+∞)) |
| 87 | | eqid 2737 |
. . . . . . . 8
⊢
((TopOpen‘ℂfld) ↾t ℂ) =
((TopOpen‘ℂfld) ↾t
ℂ) |
| 88 | 44, 86, 87 | cncfcn 24936 |
. . . . . . 7
⊢
(((0[,)+∞) ⊆ ℂ ∧ ℂ ⊆ ℂ) →
((0[,)+∞)–cn→ℂ) =
(((TopOpen‘ℂfld) ↾t (0[,)+∞)) Cn
((TopOpen‘ℂfld) ↾t
ℂ))) |
| 89 | 76, 81, 88 | mp2an 692 |
. . . . . 6
⊢
((0[,)+∞)–cn→ℂ) =
(((TopOpen‘ℂfld) ↾t (0[,)+∞)) Cn
((TopOpen‘ℂfld) ↾t
ℂ)) |
| 90 | 85, 89 | eleqtri 2839 |
. . . . 5
⊢ (√
↾ (0[,)+∞)) ∈ (((TopOpen‘ℂfld)
↾t (0[,)+∞)) Cn ((TopOpen‘ℂfld)
↾t ℂ)) |
| 91 | 90 | a1i 11 |
. . . 4
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (√ ↾
(0[,)+∞)) ∈ (((TopOpen‘ℂfld)
↾t (0[,)+∞)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) |
| 92 | 50, 80, 91 | cnmpt11f 23672 |
. . 3
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾
(0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) ∈
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) |
| 93 | | eqid 2737 |
. . . . . 6
⊢
((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) = ((TopOpen‘ℂfld)
↾t (-𝑅[,]𝑅)) |
| 94 | 44, 93, 87 | cncfcn 24936 |
. . . . 5
⊢ (((-𝑅[,]𝑅) ⊆ ℂ ∧ ℂ ⊆
ℂ) → ((-𝑅[,]𝑅)–cn→ℂ) =
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) |
| 95 | 47, 81, 94 | sylancl 586 |
. . . 4
⊢ (𝑅 ∈ ℝ → ((-𝑅[,]𝑅)–cn→ℂ) =
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) |
| 96 | 95 | adantr 480 |
. . 3
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → ((-𝑅[,]𝑅)–cn→ℂ) =
(((TopOpen‘ℂfld) ↾t (-𝑅[,]𝑅)) Cn ((TopOpen‘ℂfld)
↾t ℂ))) |
| 97 | 92, 96 | eleqtrrd 2844 |
. 2
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ ((√ ↾
(0[,)+∞))‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ)) |
| 98 | 43, 97 | eqeltrrd 2842 |
1
⊢ ((𝑅 ∈ ℝ ∧ 0 ≤
𝑅) → (𝑡 ∈ (-𝑅[,]𝑅) ↦ (√‘((𝑅↑2) − (𝑡↑2)))) ∈ ((-𝑅[,]𝑅)–cn→ℂ)) |