| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | nncn 12274 | . . . . . . . 8
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℂ) | 
| 2 | 1 | adantl 481 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℂ) | 
| 3 |  | divcnvlin.3 | . . . . . . . . 9
⊢ (𝜑 → 𝐴 ∈ ℂ) | 
| 4 |  | divcnvlin.4 | . . . . . . . . . 10
⊢ (𝜑 → 𝐵 ∈ ℤ) | 
| 5 | 4 | zcnd 12723 | . . . . . . . . 9
⊢ (𝜑 → 𝐵 ∈ ℂ) | 
| 6 | 3, 5 | subcld 11620 | . . . . . . . 8
⊢ (𝜑 → (𝐴 − 𝐵) ∈ ℂ) | 
| 7 | 6 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐴 − 𝐵) ∈ ℂ) | 
| 8 |  | nnne0 12300 | . . . . . . . 8
⊢ (𝑚 ∈ ℕ → 𝑚 ≠ 0) | 
| 9 | 8 | adantl 481 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ≠ 0) | 
| 10 | 2, 7, 2, 9 | divdird 12081 | . . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + (𝐴 − 𝐵)) / 𝑚) = ((𝑚 / 𝑚) + ((𝐴 − 𝐵) / 𝑚))) | 
| 11 | 2, 9 | dividd 12041 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝑚 / 𝑚) = 1) | 
| 12 | 11 | oveq1d 7446 | . . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 / 𝑚) + ((𝐴 − 𝐵) / 𝑚)) = (1 + ((𝐴 − 𝐵) / 𝑚))) | 
| 13 | 10, 12 | eqtrd 2777 | . . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝑚 + (𝐴 − 𝐵)) / 𝑚) = (1 + ((𝐴 − 𝐵) / 𝑚))) | 
| 14 | 13 | mpteq2dva 5242 | . . . 4
⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) = (𝑚 ∈ ℕ ↦ (1 + ((𝐴 − 𝐵) / 𝑚)))) | 
| 15 |  | nnuz 12921 | . . . . 5
⊢ ℕ =
(ℤ≥‘1) | 
| 16 |  | 1zzd 12648 | . . . . 5
⊢ (𝜑 → 1 ∈
ℤ) | 
| 17 |  | divcnv 15889 | . . . . . 6
⊢ ((𝐴 − 𝐵) ∈ ℂ → (𝑚 ∈ ℕ ↦ ((𝐴 − 𝐵) / 𝑚)) ⇝ 0) | 
| 18 | 6, 17 | syl 17 | . . . . 5
⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((𝐴 − 𝐵) / 𝑚)) ⇝ 0) | 
| 19 |  | 1cnd 11256 | . . . . 5
⊢ (𝜑 → 1 ∈
ℂ) | 
| 20 |  | nnex 12272 | . . . . . . 7
⊢ ℕ
∈ V | 
| 21 | 20 | mptex 7243 | . . . . . 6
⊢ (𝑚 ∈ ℕ ↦ (1 +
((𝐴 − 𝐵) / 𝑚))) ∈ V | 
| 22 | 21 | a1i 11 | . . . . 5
⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (1 + ((𝐴 − 𝐵) / 𝑚))) ∈ V) | 
| 23 | 7, 2, 9 | divcld 12043 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → ((𝐴 − 𝐵) / 𝑚) ∈ ℂ) | 
| 24 | 23 | fmpttd 7135 | . . . . . 6
⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((𝐴 − 𝐵) / 𝑚)):ℕ⟶ℂ) | 
| 25 | 24 | ffvelcdmda 7104 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ ((𝐴 − 𝐵) / 𝑚))‘𝑘) ∈ ℂ) | 
| 26 |  | oveq2 7439 | . . . . . . . . 9
⊢ (𝑚 = 𝑘 → ((𝐴 − 𝐵) / 𝑚) = ((𝐴 − 𝐵) / 𝑘)) | 
| 27 | 26 | oveq2d 7447 | . . . . . . . 8
⊢ (𝑚 = 𝑘 → (1 + ((𝐴 − 𝐵) / 𝑚)) = (1 + ((𝐴 − 𝐵) / 𝑘))) | 
| 28 |  | eqid 2737 | . . . . . . . 8
⊢ (𝑚 ∈ ℕ ↦ (1 +
((𝐴 − 𝐵) / 𝑚))) = (𝑚 ∈ ℕ ↦ (1 + ((𝐴 − 𝐵) / 𝑚))) | 
| 29 |  | ovex 7464 | . . . . . . . 8
⊢ (1 +
((𝐴 − 𝐵) / 𝑘)) ∈ V | 
| 30 | 27, 28, 29 | fvmpt 7016 | . . . . . . 7
⊢ (𝑘 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (1 +
((𝐴 − 𝐵) / 𝑚)))‘𝑘) = (1 + ((𝐴 − 𝐵) / 𝑘))) | 
| 31 |  | eqid 2737 | . . . . . . . . 9
⊢ (𝑚 ∈ ℕ ↦ ((𝐴 − 𝐵) / 𝑚)) = (𝑚 ∈ ℕ ↦ ((𝐴 − 𝐵) / 𝑚)) | 
| 32 |  | ovex 7464 | . . . . . . . . 9
⊢ ((𝐴 − 𝐵) / 𝑘) ∈ V | 
| 33 | 26, 31, 32 | fvmpt 7016 | . . . . . . . 8
⊢ (𝑘 ∈ ℕ → ((𝑚 ∈ ℕ ↦ ((𝐴 − 𝐵) / 𝑚))‘𝑘) = ((𝐴 − 𝐵) / 𝑘)) | 
| 34 | 33 | oveq2d 7447 | . . . . . . 7
⊢ (𝑘 ∈ ℕ → (1 +
((𝑚 ∈ ℕ ↦
((𝐴 − 𝐵) / 𝑚))‘𝑘)) = (1 + ((𝐴 − 𝐵) / 𝑘))) | 
| 35 | 30, 34 | eqtr4d 2780 | . . . . . 6
⊢ (𝑘 ∈ ℕ → ((𝑚 ∈ ℕ ↦ (1 +
((𝐴 − 𝐵) / 𝑚)))‘𝑘) = (1 + ((𝑚 ∈ ℕ ↦ ((𝐴 − 𝐵) / 𝑚))‘𝑘))) | 
| 36 | 35 | adantl 481 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ ℕ) → ((𝑚 ∈ ℕ ↦ (1 + ((𝐴 − 𝐵) / 𝑚)))‘𝑘) = (1 + ((𝑚 ∈ ℕ ↦ ((𝐴 − 𝐵) / 𝑚))‘𝑘))) | 
| 37 | 15, 16, 18, 19, 22, 25, 36 | climaddc2 15672 | . . . 4
⊢ (𝜑 → (𝑚 ∈ ℕ ↦ (1 + ((𝐴 − 𝐵) / 𝑚))) ⇝ (1 + 0)) | 
| 38 | 14, 37 | eqbrtrd 5165 | . . 3
⊢ (𝜑 → (𝑚 ∈ ℕ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ⇝ (1 + 0)) | 
| 39 |  | nnssz 12635 | . . . . . . 7
⊢ ℕ
⊆ ℤ | 
| 40 |  | resmpt 6055 | . . . . . . 7
⊢ (ℕ
⊆ ℤ → ((𝑚
∈ ℤ ↦ ((𝑚
+ (𝐴 − 𝐵)) / 𝑚)) ↾ ℕ) = (𝑚 ∈ ℕ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚))) | 
| 41 | 39, 40 | ax-mp 5 | . . . . . 6
⊢ ((𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ↾ ℕ) = (𝑚 ∈ ℕ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) | 
| 42 | 15 | reseq2i 5994 | . . . . . 6
⊢ ((𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ↾ ℕ) = ((𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ↾
(ℤ≥‘1)) | 
| 43 | 41, 42 | eqtr3i 2767 | . . . . 5
⊢ (𝑚 ∈ ℕ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) = ((𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ↾
(ℤ≥‘1)) | 
| 44 |  | 1p0e1 12390 | . . . . 5
⊢ (1 + 0) =
1 | 
| 45 | 43, 44 | breq12i 5152 | . . . 4
⊢ ((𝑚 ∈ ℕ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ⇝ (1 + 0) ↔ ((𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ↾ (ℤ≥‘1))
⇝ 1) | 
| 46 |  | 1z 12647 | . . . . 5
⊢ 1 ∈
ℤ | 
| 47 |  | zex 12622 | . . . . . 6
⊢ ℤ
∈ V | 
| 48 | 47 | mptex 7243 | . . . . 5
⊢ (𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ∈ V | 
| 49 |  | climres 15611 | . . . . 5
⊢ ((1
∈ ℤ ∧ (𝑚
∈ ℤ ↦ ((𝑚
+ (𝐴 − 𝐵)) / 𝑚)) ∈ V) → (((𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ↾ (ℤ≥‘1))
⇝ 1 ↔ (𝑚 ∈
ℤ ↦ ((𝑚 +
(𝐴 − 𝐵)) / 𝑚)) ⇝ 1)) | 
| 50 | 46, 48, 49 | mp2an 692 | . . . 4
⊢ (((𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ↾ (ℤ≥‘1))
⇝ 1 ↔ (𝑚 ∈
ℤ ↦ ((𝑚 +
(𝐴 − 𝐵)) / 𝑚)) ⇝ 1) | 
| 51 | 45, 50 | bitri 275 | . . 3
⊢ ((𝑚 ∈ ℕ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ⇝ (1 + 0) ↔ (𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ⇝ 1) | 
| 52 | 38, 51 | sylib 218 | . 2
⊢ (𝜑 → (𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ⇝ 1) | 
| 53 |  | divcnvlin.1 | . . 3
⊢ 𝑍 =
(ℤ≥‘𝑀) | 
| 54 |  | divcnvlin.2 | . . 3
⊢ (𝜑 → 𝑀 ∈ ℤ) | 
| 55 |  | divcnvlin.5 | . . 3
⊢ (𝜑 → 𝐹 ∈ 𝑉) | 
| 56 | 48 | a1i 11 | . . 3
⊢ (𝜑 → (𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ∈ V) | 
| 57 |  | eluzelz 12888 | . . . . . . . . 9
⊢ (𝑘 ∈
(ℤ≥‘𝑀) → 𝑘 ∈ ℤ) | 
| 58 | 57, 53 | eleq2s 2859 | . . . . . . . 8
⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℤ) | 
| 59 | 58 | zcnd 12723 | . . . . . . 7
⊢ (𝑘 ∈ 𝑍 → 𝑘 ∈ ℂ) | 
| 60 | 59 | adantl 481 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℂ) | 
| 61 | 4 | adantr 480 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℤ) | 
| 62 | 61 | zcnd 12723 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐵 ∈ ℂ) | 
| 63 | 3 | adantr 480 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝐴 ∈ ℂ) | 
| 64 | 60, 62, 63 | ppncand 11660 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑘 + 𝐵) + (𝐴 − 𝐵)) = (𝑘 + 𝐴)) | 
| 65 | 64 | oveq1d 7446 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (((𝑘 + 𝐵) + (𝐴 − 𝐵)) / (𝑘 + 𝐵)) = ((𝑘 + 𝐴) / (𝑘 + 𝐵))) | 
| 66 | 58 | adantl 481 | . . . . . 6
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → 𝑘 ∈ ℤ) | 
| 67 | 66, 61 | zaddcld 12726 | . . . . 5
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝑘 + 𝐵) ∈ ℤ) | 
| 68 |  | oveq1 7438 | . . . . . . 7
⊢ (𝑚 = (𝑘 + 𝐵) → (𝑚 + (𝐴 − 𝐵)) = ((𝑘 + 𝐵) + (𝐴 − 𝐵))) | 
| 69 |  | id 22 | . . . . . . 7
⊢ (𝑚 = (𝑘 + 𝐵) → 𝑚 = (𝑘 + 𝐵)) | 
| 70 | 68, 69 | oveq12d 7449 | . . . . . 6
⊢ (𝑚 = (𝑘 + 𝐵) → ((𝑚 + (𝐴 − 𝐵)) / 𝑚) = (((𝑘 + 𝐵) + (𝐴 − 𝐵)) / (𝑘 + 𝐵))) | 
| 71 |  | eqid 2737 | . . . . . 6
⊢ (𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) = (𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) | 
| 72 |  | ovex 7464 | . . . . . 6
⊢ (((𝑘 + 𝐵) + (𝐴 − 𝐵)) / (𝑘 + 𝐵)) ∈ V | 
| 73 | 70, 71, 72 | fvmpt 7016 | . . . . 5
⊢ ((𝑘 + 𝐵) ∈ ℤ → ((𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚))‘(𝑘 + 𝐵)) = (((𝑘 + 𝐵) + (𝐴 − 𝐵)) / (𝑘 + 𝐵))) | 
| 74 | 67, 73 | syl 17 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚))‘(𝑘 + 𝐵)) = (((𝑘 + 𝐵) + (𝐴 − 𝐵)) / (𝑘 + 𝐵))) | 
| 75 |  | divcnvlin.6 | . . . 4
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) = ((𝑘 + 𝐴) / (𝑘 + 𝐵))) | 
| 76 | 65, 74, 75 | 3eqtr4d 2787 | . . 3
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → ((𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚))‘(𝑘 + 𝐵)) = (𝐹‘𝑘)) | 
| 77 | 53, 54, 4, 55, 56, 76 | climshft2 15618 | . 2
⊢ (𝜑 → (𝐹 ⇝ 1 ↔ (𝑚 ∈ ℤ ↦ ((𝑚 + (𝐴 − 𝐵)) / 𝑚)) ⇝ 1)) | 
| 78 | 52, 77 | mpbird 257 | 1
⊢ (𝜑 → 𝐹 ⇝ 1) |