Proof of Theorem muinv
| Step | Hyp | Ref
| Expression |
| 1 | | muinv.1 |
. . 3
⊢ (𝜑 → 𝐹:ℕ⟶ℂ) |
| 2 | 1 | feqmptd 6977 |
. 2
⊢ (𝜑 → 𝐹 = (𝑚 ∈ ℕ ↦ (𝐹‘𝑚))) |
| 3 | | muinv.2 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐺 = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (𝐹‘𝑘))) |
| 4 | 3 | ad2antrr 726 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝐺 = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (𝐹‘𝑘))) |
| 5 | 4 | fveq1d 6908 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝐺‘(𝑚 / 𝑗)) = ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (𝐹‘𝑘))‘(𝑚 / 𝑗))) |
| 6 | | breq1 5146 |
. . . . . . . . . . . . . 14
⊢ (𝑥 = 𝑗 → (𝑥 ∥ 𝑚 ↔ 𝑗 ∥ 𝑚)) |
| 7 | 6 | elrab 3692 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ↔ (𝑗 ∈ ℕ ∧ 𝑗 ∥ 𝑚)) |
| 8 | 7 | simprbi 496 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} → 𝑗 ∥ 𝑚) |
| 9 | 8 | adantl 481 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝑗 ∥ 𝑚) |
| 10 | | elrabi 3687 |
. . . . . . . . . . . . . 14
⊢ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} → 𝑗 ∈ ℕ) |
| 11 | 10 | adantl 481 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝑗 ∈ ℕ) |
| 12 | 11 | nnzd 12640 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝑗 ∈ ℤ) |
| 13 | 11 | nnne0d 12316 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝑗 ≠ 0) |
| 14 | | nnz 12634 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℤ) |
| 15 | 14 | ad2antlr 727 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝑚 ∈ ℤ) |
| 16 | | dvdsval2 16293 |
. . . . . . . . . . . 12
⊢ ((𝑗 ∈ ℤ ∧ 𝑗 ≠ 0 ∧ 𝑚 ∈ ℤ) → (𝑗 ∥ 𝑚 ↔ (𝑚 / 𝑗) ∈ ℤ)) |
| 17 | 12, 13, 15, 16 | syl3anc 1373 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝑗 ∥ 𝑚 ↔ (𝑚 / 𝑗) ∈ ℤ)) |
| 18 | 9, 17 | mpbid 232 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝑚 / 𝑗) ∈ ℤ) |
| 19 | | nnre 12273 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℝ) |
| 20 | | nngt0 12297 |
. . . . . . . . . . . . 13
⊢ (𝑚 ∈ ℕ → 0 <
𝑚) |
| 21 | 19, 20 | jca 511 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ ℕ → (𝑚 ∈ ℝ ∧ 0 <
𝑚)) |
| 22 | 21 | ad2antlr 727 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝑚 ∈ ℝ ∧ 0 < 𝑚)) |
| 23 | | nnre 12273 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → 𝑗 ∈
ℝ) |
| 24 | | nngt0 12297 |
. . . . . . . . . . . . 13
⊢ (𝑗 ∈ ℕ → 0 <
𝑗) |
| 25 | 23, 24 | jca 511 |
. . . . . . . . . . . 12
⊢ (𝑗 ∈ ℕ → (𝑗 ∈ ℝ ∧ 0 <
𝑗)) |
| 26 | 11, 25 | syl 17 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝑗 ∈ ℝ ∧ 0 < 𝑗)) |
| 27 | | divgt0 12136 |
. . . . . . . . . . 11
⊢ (((𝑚 ∈ ℝ ∧ 0 <
𝑚) ∧ (𝑗 ∈ ℝ ∧ 0 <
𝑗)) → 0 < (𝑚 / 𝑗)) |
| 28 | 22, 26, 27 | syl2anc 584 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 0 < (𝑚 / 𝑗)) |
| 29 | | elnnz 12623 |
. . . . . . . . . 10
⊢ ((𝑚 / 𝑗) ∈ ℕ ↔ ((𝑚 / 𝑗) ∈ ℤ ∧ 0 < (𝑚 / 𝑗))) |
| 30 | 18, 28, 29 | sylanbrc 583 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝑚 / 𝑗) ∈ ℕ) |
| 31 | | breq2 5147 |
. . . . . . . . . . . 12
⊢ (𝑛 = (𝑚 / 𝑗) → (𝑥 ∥ 𝑛 ↔ 𝑥 ∥ (𝑚 / 𝑗))) |
| 32 | 31 | rabbidv 3444 |
. . . . . . . . . . 11
⊢ (𝑛 = (𝑚 / 𝑗) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} = {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)}) |
| 33 | 32 | sumeq1d 15736 |
. . . . . . . . . 10
⊢ (𝑛 = (𝑚 / 𝑗) → Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (𝐹‘𝑘) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} (𝐹‘𝑘)) |
| 34 | | eqid 2737 |
. . . . . . . . . 10
⊢ (𝑛 ∈ ℕ ↦
Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (𝐹‘𝑘)) = (𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (𝐹‘𝑘)) |
| 35 | | sumex 15724 |
. . . . . . . . . 10
⊢
Σ𝑘 ∈
{𝑥 ∈ ℕ ∣
𝑥 ∥ (𝑚 / 𝑗)} (𝐹‘𝑘) ∈ V |
| 36 | 33, 34, 35 | fvmpt 7016 |
. . . . . . . . 9
⊢ ((𝑚 / 𝑗) ∈ ℕ → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (𝐹‘𝑘))‘(𝑚 / 𝑗)) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} (𝐹‘𝑘)) |
| 37 | 30, 36 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → ((𝑛 ∈ ℕ ↦ Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑛} (𝐹‘𝑘))‘(𝑚 / 𝑗)) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} (𝐹‘𝑘)) |
| 38 | 5, 37 | eqtrd 2777 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝐺‘(𝑚 / 𝑗)) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} (𝐹‘𝑘)) |
| 39 | 38 | oveq2d 7447 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → ((μ‘𝑗) · (𝐺‘(𝑚 / 𝑗))) = ((μ‘𝑗) · Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} (𝐹‘𝑘))) |
| 40 | | fzfid 14014 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (1...(𝑚 / 𝑗)) ∈ Fin) |
| 41 | | dvdsssfz1 16355 |
. . . . . . . . 9
⊢ ((𝑚 / 𝑗) ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} ⊆ (1...(𝑚 / 𝑗))) |
| 42 | 30, 41 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} ⊆ (1...(𝑚 / 𝑗))) |
| 43 | 40, 42 | ssfid 9301 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} ∈ Fin) |
| 44 | | mucl 27184 |
. . . . . . . . 9
⊢ (𝑗 ∈ ℕ →
(μ‘𝑗) ∈
ℤ) |
| 45 | 11, 44 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (μ‘𝑗) ∈ ℤ) |
| 46 | 45 | zcnd 12723 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (μ‘𝑗) ∈ ℂ) |
| 47 | 1 | ad2antrr 726 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝐹:ℕ⟶ℂ) |
| 48 | | elrabi 3687 |
. . . . . . . 8
⊢ (𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} → 𝑘 ∈ ℕ) |
| 49 | | ffvelcdm 7101 |
. . . . . . . 8
⊢ ((𝐹:ℕ⟶ℂ ∧
𝑘 ∈ ℕ) →
(𝐹‘𝑘) ∈ ℂ) |
| 50 | 47, 48, 49 | syl2an 596 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)}) → (𝐹‘𝑘) ∈ ℂ) |
| 51 | 43, 46, 50 | fsummulc2 15820 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → ((μ‘𝑗) · Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} (𝐹‘𝑘)) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} ((μ‘𝑗) · (𝐹‘𝑘))) |
| 52 | 39, 51 | eqtrd 2777 |
. . . . 5
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → ((μ‘𝑗) · (𝐺‘(𝑚 / 𝑗))) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} ((μ‘𝑗) · (𝐹‘𝑘))) |
| 53 | 52 | sumeq2dv 15738 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ((μ‘𝑗) · (𝐺‘(𝑚 / 𝑗))) = Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} ((μ‘𝑗) · (𝐹‘𝑘))) |
| 54 | | simpr 484 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℕ) |
| 55 | 46 | adantrr 717 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)})) → (μ‘𝑗) ∈ ℂ) |
| 56 | 50 | anasss 466 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)})) → (𝐹‘𝑘) ∈ ℂ) |
| 57 | 55, 56 | mulcld 11281 |
. . . . 5
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ (𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)})) → ((μ‘𝑗) · (𝐹‘𝑘)) ∈ ℂ) |
| 58 | 54, 57 | fsumdvdsdiag 27227 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑗)} ((μ‘𝑗) · (𝐹‘𝑘)) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} ((μ‘𝑗) · (𝐹‘𝑘))) |
| 59 | | ssrab2 4080 |
. . . . . . . . . 10
⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ⊆ ℕ |
| 60 | | dvdsdivcl 16353 |
. . . . . . . . . . 11
⊢ ((𝑚 ∈ ℕ ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝑚 / 𝑘) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) |
| 61 | 60 | adantll 714 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝑚 / 𝑘) ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) |
| 62 | 59, 61 | sselid 3981 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝑚 / 𝑘) ∈ ℕ) |
| 63 | | musum 27234 |
. . . . . . . . 9
⊢ ((𝑚 / 𝑘) ∈ ℕ → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} (μ‘𝑗) = if((𝑚 / 𝑘) = 1, 1, 0)) |
| 64 | 62, 63 | syl 17 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} (μ‘𝑗) = if((𝑚 / 𝑘) = 1, 1, 0)) |
| 65 | 64 | oveq1d 7446 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} (μ‘𝑗) · (𝐹‘𝑘)) = (if((𝑚 / 𝑘) = 1, 1, 0) · (𝐹‘𝑘))) |
| 66 | | fzfid 14014 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (1...(𝑚 / 𝑘)) ∈ Fin) |
| 67 | | dvdsssfz1 16355 |
. . . . . . . . . 10
⊢ ((𝑚 / 𝑘) ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} ⊆ (1...(𝑚 / 𝑘))) |
| 68 | 62, 67 | syl 17 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} ⊆ (1...(𝑚 / 𝑘))) |
| 69 | 66, 68 | ssfid 9301 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} ∈ Fin) |
| 70 | 1 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝐹:ℕ⟶ℂ) |
| 71 | | elrabi 3687 |
. . . . . . . . 9
⊢ (𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} → 𝑘 ∈ ℕ) |
| 72 | 70, 71, 49 | syl2an 596 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝐹‘𝑘) ∈ ℂ) |
| 73 | | ssrab2 4080 |
. . . . . . . . . . 11
⊢ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} ⊆ ℕ |
| 74 | | simpr 484 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)}) → 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)}) |
| 75 | 73, 74 | sselid 3981 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)}) → 𝑗 ∈ ℕ) |
| 76 | 75, 44 | syl 17 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)}) → (μ‘𝑗) ∈ ℤ) |
| 77 | 76 | zcnd 12723 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) ∧ 𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)}) → (μ‘𝑗) ∈ ℂ) |
| 78 | 69, 72, 77 | fsummulc1 15821 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} (μ‘𝑗) · (𝐹‘𝑘)) = Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} ((μ‘𝑗) · (𝐹‘𝑘))) |
| 79 | | ovif 7531 |
. . . . . . . 8
⊢
(if((𝑚 / 𝑘) = 1, 1, 0) · (𝐹‘𝑘)) = if((𝑚 / 𝑘) = 1, (1 · (𝐹‘𝑘)), (0 · (𝐹‘𝑘))) |
| 80 | | nncn 12274 |
. . . . . . . . . . . 12
⊢ (𝑚 ∈ ℕ → 𝑚 ∈
ℂ) |
| 81 | 80 | ad2antlr 727 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝑚 ∈ ℂ) |
| 82 | 71 | adantl 481 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝑘 ∈ ℕ) |
| 83 | 82 | nncnd 12282 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝑘 ∈ ℂ) |
| 84 | | 1cnd 11256 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 1 ∈ ℂ) |
| 85 | 82 | nnne0d 12316 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → 𝑘 ≠ 0) |
| 86 | 81, 83, 84, 85 | divmuld 12065 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → ((𝑚 / 𝑘) = 1 ↔ (𝑘 · 1) = 𝑚)) |
| 87 | 83 | mulridd 11278 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (𝑘 · 1) = 𝑘) |
| 88 | 87 | eqeq1d 2739 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → ((𝑘 · 1) = 𝑚 ↔ 𝑘 = 𝑚)) |
| 89 | 86, 88 | bitrd 279 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → ((𝑚 / 𝑘) = 1 ↔ 𝑘 = 𝑚)) |
| 90 | 72 | mullidd 11279 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (1 · (𝐹‘𝑘)) = (𝐹‘𝑘)) |
| 91 | 72 | mul02d 11459 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (0 · (𝐹‘𝑘)) = 0) |
| 92 | 89, 90, 91 | ifbieq12d 4554 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → if((𝑚 / 𝑘) = 1, (1 · (𝐹‘𝑘)), (0 · (𝐹‘𝑘))) = if(𝑘 = 𝑚, (𝐹‘𝑘), 0)) |
| 93 | 79, 92 | eqtrid 2789 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → (if((𝑚 / 𝑘) = 1, 1, 0) · (𝐹‘𝑘)) = if(𝑘 = 𝑚, (𝐹‘𝑘), 0)) |
| 94 | 65, 78, 93 | 3eqtr3d 2785 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} ((μ‘𝑗) · (𝐹‘𝑘)) = if(𝑘 = 𝑚, (𝐹‘𝑘), 0)) |
| 95 | 94 | sumeq2dv 15738 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} ((μ‘𝑗) · (𝐹‘𝑘)) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}if(𝑘 = 𝑚, (𝐹‘𝑘), 0)) |
| 96 | | breq1 5146 |
. . . . . . . 8
⊢ (𝑥 = 𝑚 → (𝑥 ∥ 𝑚 ↔ 𝑚 ∥ 𝑚)) |
| 97 | 54 | nnzd 12640 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ ℤ) |
| 98 | | iddvds 16307 |
. . . . . . . . 9
⊢ (𝑚 ∈ ℤ → 𝑚 ∥ 𝑚) |
| 99 | 97, 98 | syl 17 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∥ 𝑚) |
| 100 | 96, 54, 99 | elrabd 3694 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → 𝑚 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) |
| 101 | 100 | snssd 4809 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑚} ⊆ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) |
| 102 | 101 | sselda 3983 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑚}) → 𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}) |
| 103 | 102, 72 | syldan 591 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑚}) → (𝐹‘𝑘) ∈ ℂ) |
| 104 | | 0cn 11253 |
. . . . . . 7
⊢ 0 ∈
ℂ |
| 105 | | ifcl 4571 |
. . . . . . 7
⊢ (((𝐹‘𝑘) ∈ ℂ ∧ 0 ∈ ℂ)
→ if(𝑘 = 𝑚, (𝐹‘𝑘), 0) ∈ ℂ) |
| 106 | 103, 104,
105 | sylancl 586 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ {𝑚}) → if(𝑘 = 𝑚, (𝐹‘𝑘), 0) ∈ ℂ) |
| 107 | | eldifsni 4790 |
. . . . . . . . 9
⊢ (𝑘 ∈ ({𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ∖ {𝑚}) → 𝑘 ≠ 𝑚) |
| 108 | 107 | adantl 481 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ ({𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ∖ {𝑚})) → 𝑘 ≠ 𝑚) |
| 109 | 108 | neneqd 2945 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ ({𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ∖ {𝑚})) → ¬ 𝑘 = 𝑚) |
| 110 | 109 | iffalsed 4536 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑚 ∈ ℕ) ∧ 𝑘 ∈ ({𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ∖ {𝑚})) → if(𝑘 = 𝑚, (𝐹‘𝑘), 0) = 0) |
| 111 | | fzfid 14014 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (1...𝑚) ∈ Fin) |
| 112 | | dvdsssfz1 16355 |
. . . . . . . 8
⊢ (𝑚 ∈ ℕ → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ⊆ (1...𝑚)) |
| 113 | 112 | adantl 481 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ⊆ (1...𝑚)) |
| 114 | 111, 113 | ssfid 9301 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ∈ Fin) |
| 115 | 101, 106,
110, 114 | fsumss 15761 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ {𝑚}if(𝑘 = 𝑚, (𝐹‘𝑘), 0) = Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}if(𝑘 = 𝑚, (𝐹‘𝑘), 0)) |
| 116 | 1 | ffvelcdmda 7104 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → (𝐹‘𝑚) ∈ ℂ) |
| 117 | | iftrue 4531 |
. . . . . . . 8
⊢ (𝑘 = 𝑚 → if(𝑘 = 𝑚, (𝐹‘𝑘), 0) = (𝐹‘𝑘)) |
| 118 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑘 = 𝑚 → (𝐹‘𝑘) = (𝐹‘𝑚)) |
| 119 | 117, 118 | eqtrd 2777 |
. . . . . . 7
⊢ (𝑘 = 𝑚 → if(𝑘 = 𝑚, (𝐹‘𝑘), 0) = (𝐹‘𝑚)) |
| 120 | 119 | sumsn 15782 |
. . . . . 6
⊢ ((𝑚 ∈ ℕ ∧ (𝐹‘𝑚) ∈ ℂ) → Σ𝑘 ∈ {𝑚}if(𝑘 = 𝑚, (𝐹‘𝑘), 0) = (𝐹‘𝑚)) |
| 121 | 54, 116, 120 | syl2anc 584 |
. . . . 5
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ {𝑚}if(𝑘 = 𝑚, (𝐹‘𝑘), 0) = (𝐹‘𝑚)) |
| 122 | 95, 115, 121 | 3eqtr2d 2783 |
. . . 4
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑘 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚}Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ (𝑚 / 𝑘)} ((μ‘𝑗) · (𝐹‘𝑘)) = (𝐹‘𝑚)) |
| 123 | 53, 58, 122 | 3eqtrd 2781 |
. . 3
⊢ ((𝜑 ∧ 𝑚 ∈ ℕ) → Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ((μ‘𝑗) · (𝐺‘(𝑚 / 𝑗))) = (𝐹‘𝑚)) |
| 124 | 123 | mpteq2dva 5242 |
. 2
⊢ (𝜑 → (𝑚 ∈ ℕ ↦ Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ((μ‘𝑗) · (𝐺‘(𝑚 / 𝑗)))) = (𝑚 ∈ ℕ ↦ (𝐹‘𝑚))) |
| 125 | 2, 124 | eqtr4d 2780 |
1
⊢ (𝜑 → 𝐹 = (𝑚 ∈ ℕ ↦ Σ𝑗 ∈ {𝑥 ∈ ℕ ∣ 𝑥 ∥ 𝑚} ((μ‘𝑗) · (𝐺‘(𝑚 / 𝑗))))) |