| Step | Hyp | Ref
| Expression |
| 1 | | mbflim.1 |
. . 3
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 2 | | mbflim.2 |
. . 3
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 3 | | mbflim.4 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶) |
| 4 | 1 | fvexi 6920 |
. . . . . 6
⊢ 𝑍 ∈ V |
| 5 | 4 | mptex 7243 |
. . . . 5
⊢ (𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵)) ∈ V |
| 6 | 5 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵)) ∈ V) |
| 7 | 2 | adantr 480 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑀 ∈ ℤ) |
| 8 | | mbflim.5 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn) |
| 9 | | mbflim.6 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ 𝑉) |
| 10 | 9 | anassrs 467 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
| 11 | 8, 10 | mbfmptcl 25671 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℂ) |
| 12 | 11 | an32s 652 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝐵 ∈ ℂ) |
| 13 | 12 | fmpttd 7135 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ 𝐵):𝑍⟶ℂ) |
| 14 | 13 | ffvelcdmda 7104 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) ∈ ℂ) |
| 15 | | simpr 484 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ 𝑍) |
| 16 | 12 | recld 15233 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (ℜ‘𝐵) ∈ ℝ) |
| 17 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵)) = (𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵)) |
| 18 | 17 | fvmpt2 7027 |
. . . . . . . . 9
⊢ ((𝑛 ∈ 𝑍 ∧ (ℜ‘𝐵) ∈ ℝ) → ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑛) = (ℜ‘𝐵)) |
| 19 | 15, 16, 18 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑛) = (ℜ‘𝐵)) |
| 20 | | eqid 2737 |
. . . . . . . . . . 11
⊢ (𝑛 ∈ 𝑍 ↦ 𝐵) = (𝑛 ∈ 𝑍 ↦ 𝐵) |
| 21 | 20 | fvmpt2 7027 |
. . . . . . . . . 10
⊢ ((𝑛 ∈ 𝑍 ∧ 𝐵 ∈ ℂ) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵) |
| 22 | 15, 12, 21 | syl2anc 584 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛) = 𝐵) |
| 23 | 22 | fveq2d 6910 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)) = (ℜ‘𝐵)) |
| 24 | 19, 23 | eqtr4d 2780 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑛) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))) |
| 25 | 24 | ralrimiva 3146 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑛) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))) |
| 26 | | nffvmpt1 6917 |
. . . . . . . 8
⊢
Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑘) |
| 27 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑛ℜ |
| 28 | | nffvmpt1 6917 |
. . . . . . . . 9
⊢
Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘) |
| 29 | 27, 28 | nffv 6916 |
. . . . . . . 8
⊢
Ⅎ𝑛(ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) |
| 30 | 26, 29 | nfeq 2919 |
. . . . . . 7
⊢
Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑘) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) |
| 31 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑘((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑛) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)) |
| 32 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑘) = ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑛)) |
| 33 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))) |
| 34 | 32, 33 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑘 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑘) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) ↔ ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑛) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)))) |
| 35 | 30, 31, 34 | cbvralw 3306 |
. . . . . 6
⊢
(∀𝑘 ∈
𝑍 ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑘) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) ↔ ∀𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑛) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))) |
| 36 | 25, 35 | sylibr 234 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑘) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))) |
| 37 | 36 | r19.21bi 3251 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵))‘𝑘) = (ℜ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))) |
| 38 | 1, 3, 6, 7, 14, 37 | climre 15642 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ (ℜ‘𝐵)) ⇝ (ℜ‘𝐶)) |
| 39 | 11 | ismbfcn2 25673 |
. . . . 5
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ 𝐵) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn))) |
| 40 | 8, 39 | mpbid 232 |
. . . 4
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn)) |
| 41 | 40 | simpld 494 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐵)) ∈ MblFn) |
| 42 | 11 | anasss 466 |
. . . 4
⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → 𝐵 ∈ ℂ) |
| 43 | 42 | recld 15233 |
. . 3
⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → (ℜ‘𝐵) ∈ ℝ) |
| 44 | 1, 2, 38, 41, 43 | mbflimlem 25702 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ MblFn) |
| 45 | 4 | mptex 7243 |
. . . . 5
⊢ (𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵)) ∈ V |
| 46 | 45 | a1i 11 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵)) ∈ V) |
| 47 | 12 | imcld 15234 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (ℑ‘𝐵) ∈ ℝ) |
| 48 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵)) = (𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵)) |
| 49 | 48 | fvmpt2 7027 |
. . . . . . . . 9
⊢ ((𝑛 ∈ 𝑍 ∧ (ℑ‘𝐵) ∈ ℝ) → ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑛) = (ℑ‘𝐵)) |
| 50 | 15, 47, 49 | syl2anc 584 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑛) = (ℑ‘𝐵)) |
| 51 | 22 | fveq2d 6910 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)) = (ℑ‘𝐵)) |
| 52 | 50, 51 | eqtr4d 2780 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑛 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑛) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))) |
| 53 | 52 | ralrimiva 3146 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑛) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))) |
| 54 | | nffvmpt1 6917 |
. . . . . . . 8
⊢
Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑘) |
| 55 | | nfcv 2905 |
. . . . . . . . 9
⊢
Ⅎ𝑛ℑ |
| 56 | 55, 28 | nffv 6916 |
. . . . . . . 8
⊢
Ⅎ𝑛(ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) |
| 57 | 54, 56 | nfeq 2919 |
. . . . . . 7
⊢
Ⅎ𝑛((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑘) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) |
| 58 | | nfv 1914 |
. . . . . . 7
⊢
Ⅎ𝑘((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑛) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)) |
| 59 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑘) = ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑛)) |
| 60 | | 2fveq3 6911 |
. . . . . . . 8
⊢ (𝑘 = 𝑛 → (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))) |
| 61 | 59, 60 | eqeq12d 2753 |
. . . . . . 7
⊢ (𝑘 = 𝑛 → (((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑘) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) ↔ ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑛) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛)))) |
| 62 | 57, 58, 61 | cbvralw 3306 |
. . . . . 6
⊢
(∀𝑘 ∈
𝑍 ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑘) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘)) ↔ ∀𝑛 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑛) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑛))) |
| 63 | 53, 62 | sylibr 234 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑘 ∈ 𝑍 ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑘) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))) |
| 64 | 63 | r19.21bi 3251 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑘 ∈ 𝑍) → ((𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵))‘𝑘) = (ℑ‘((𝑛 ∈ 𝑍 ↦ 𝐵)‘𝑘))) |
| 65 | 1, 3, 46, 7, 14, 64 | climim 15643 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑛 ∈ 𝑍 ↦ (ℑ‘𝐵)) ⇝ (ℑ‘𝐶)) |
| 66 | 40 | simprd 495 |
. . 3
⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐵)) ∈ MblFn) |
| 67 | 42 | imcld 15234 |
. . 3
⊢ ((𝜑 ∧ (𝑛 ∈ 𝑍 ∧ 𝑥 ∈ 𝐴)) → (ℑ‘𝐵) ∈ ℝ) |
| 68 | 1, 2, 65, 66, 67 | mbflimlem 25702 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈ MblFn) |
| 69 | | climcl 15535 |
. . . 4
⊢ ((𝑛 ∈ 𝑍 ↦ 𝐵) ⇝ 𝐶 → 𝐶 ∈ ℂ) |
| 70 | 3, 69 | syl 17 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐶 ∈ ℂ) |
| 71 | 70 | ismbfcn2 25673 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn ↔ ((𝑥 ∈ 𝐴 ↦ (ℜ‘𝐶)) ∈ MblFn ∧ (𝑥 ∈ 𝐴 ↦ (ℑ‘𝐶)) ∈ MblFn))) |
| 72 | 44, 68, 71 | mpbir2and 713 |
1
⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐶) ∈ MblFn) |