| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | prodfn0.1 | . . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) | 
| 2 |  | eluzfz2 13573 | . . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑁 ∈ (𝑀...𝑁)) | 
| 3 | 1, 2 | syl 17 | . 2
⊢ (𝜑 → 𝑁 ∈ (𝑀...𝑁)) | 
| 4 |  | fveq2 6905 | . . . . 5
⊢ (𝑚 = 𝑀 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑀)) | 
| 5 | 4 | neeq1d 2999 | . . . 4
⊢ (𝑚 = 𝑀 → ((seq𝑀( · , 𝐹)‘𝑚) ≠ 0 ↔ (seq𝑀( · , 𝐹)‘𝑀) ≠ 0)) | 
| 6 | 5 | imbi2d 340 | . . 3
⊢ (𝑚 = 𝑀 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) ≠ 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) ≠ 0))) | 
| 7 |  | fveq2 6905 | . . . . 5
⊢ (𝑚 = 𝑛 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑛)) | 
| 8 | 7 | neeq1d 2999 | . . . 4
⊢ (𝑚 = 𝑛 → ((seq𝑀( · , 𝐹)‘𝑚) ≠ 0 ↔ (seq𝑀( · , 𝐹)‘𝑛) ≠ 0)) | 
| 9 | 8 | imbi2d 340 | . . 3
⊢ (𝑚 = 𝑛 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) ≠ 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑛) ≠ 0))) | 
| 10 |  | fveq2 6905 | . . . . 5
⊢ (𝑚 = (𝑛 + 1) → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘(𝑛 + 1))) | 
| 11 | 10 | neeq1d 2999 | . . . 4
⊢ (𝑚 = (𝑛 + 1) → ((seq𝑀( · , 𝐹)‘𝑚) ≠ 0 ↔ (seq𝑀( · , 𝐹)‘(𝑛 + 1)) ≠ 0)) | 
| 12 | 11 | imbi2d 340 | . . 3
⊢ (𝑚 = (𝑛 + 1) → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) ≠ 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) ≠ 0))) | 
| 13 |  | fveq2 6905 | . . . . 5
⊢ (𝑚 = 𝑁 → (seq𝑀( · , 𝐹)‘𝑚) = (seq𝑀( · , 𝐹)‘𝑁)) | 
| 14 | 13 | neeq1d 2999 | . . . 4
⊢ (𝑚 = 𝑁 → ((seq𝑀( · , 𝐹)‘𝑚) ≠ 0 ↔ (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)) | 
| 15 | 14 | imbi2d 340 | . . 3
⊢ (𝑚 = 𝑁 → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑚) ≠ 0) ↔ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0))) | 
| 16 |  | eluzfz1 13572 | . . . 4
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ (𝑀...𝑁)) | 
| 17 |  | elfzelz 13565 | . . . . . . . 8
⊢ (𝑀 ∈ (𝑀...𝑁) → 𝑀 ∈ ℤ) | 
| 18 | 17 | adantl 481 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → 𝑀 ∈ ℤ) | 
| 19 |  | seq1 14056 | . . . . . . 7
⊢ (𝑀 ∈ ℤ → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀)) | 
| 20 | 18, 19 | syl 17 | . . . . . 6
⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑀) = (𝐹‘𝑀)) | 
| 21 |  | fveq2 6905 | . . . . . . . . . 10
⊢ (𝑘 = 𝑀 → (𝐹‘𝑘) = (𝐹‘𝑀)) | 
| 22 | 21 | neeq1d 2999 | . . . . . . . . 9
⊢ (𝑘 = 𝑀 → ((𝐹‘𝑘) ≠ 0 ↔ (𝐹‘𝑀) ≠ 0)) | 
| 23 | 22 | imbi2d 340 | . . . . . . . 8
⊢ (𝑘 = 𝑀 → ((𝜑 → (𝐹‘𝑘) ≠ 0) ↔ (𝜑 → (𝐹‘𝑀) ≠ 0))) | 
| 24 |  | prodfn0.3 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ≠ 0) | 
| 25 | 24 | expcom 413 | . . . . . . . 8
⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) ≠ 0)) | 
| 26 | 23, 25 | vtoclga 3576 | . . . . . . 7
⊢ (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑀) ≠ 0)) | 
| 27 | 26 | impcom 407 | . . . . . 6
⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → (𝐹‘𝑀) ≠ 0) | 
| 28 | 20, 27 | eqnetrd 3007 | . . . . 5
⊢ ((𝜑 ∧ 𝑀 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑀) ≠ 0) | 
| 29 | 28 | expcom 413 | . . . 4
⊢ (𝑀 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) ≠ 0)) | 
| 30 | 16, 29 | syl 17 | . . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑀) ≠ 0)) | 
| 31 |  | elfzouz 13704 | . . . . . . . . 9
⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (ℤ≥‘𝑀)) | 
| 32 | 31 | 3ad2ant2 1134 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) → 𝑛 ∈ (ℤ≥‘𝑀)) | 
| 33 |  | seqp1 14058 | . . . . . . . 8
⊢ (𝑛 ∈
(ℤ≥‘𝑀) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) | 
| 34 | 32, 33 | syl 17 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) = ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1)))) | 
| 35 |  | elfzofz 13716 | . . . . . . . . . 10
⊢ (𝑛 ∈ (𝑀..^𝑁) → 𝑛 ∈ (𝑀...𝑁)) | 
| 36 |  | elfzuz 13561 | . . . . . . . . . . . 12
⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑛 ∈ (ℤ≥‘𝑀)) | 
| 37 | 36 | adantl 481 | . . . . . . . . . . 11
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → 𝑛 ∈ (ℤ≥‘𝑀)) | 
| 38 |  | elfzuz3 13562 | . . . . . . . . . . . . . . 15
⊢ (𝑛 ∈ (𝑀...𝑁) → 𝑁 ∈ (ℤ≥‘𝑛)) | 
| 39 |  | fzss2 13605 | . . . . . . . . . . . . . . 15
⊢ (𝑁 ∈
(ℤ≥‘𝑛) → (𝑀...𝑛) ⊆ (𝑀...𝑁)) | 
| 40 | 38, 39 | syl 17 | . . . . . . . . . . . . . 14
⊢ (𝑛 ∈ (𝑀...𝑁) → (𝑀...𝑛) ⊆ (𝑀...𝑁)) | 
| 41 | 40 | sselda 3982 | . . . . . . . . . . . . 13
⊢ ((𝑛 ∈ (𝑀...𝑁) ∧ 𝑘 ∈ (𝑀...𝑛)) → 𝑘 ∈ (𝑀...𝑁)) | 
| 42 |  | prodfn0.2 | . . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑘 ∈ (𝑀...𝑁)) → (𝐹‘𝑘) ∈ ℂ) | 
| 43 | 41, 42 | sylan2 593 | . . . . . . . . . . . 12
⊢ ((𝜑 ∧ (𝑛 ∈ (𝑀...𝑁) ∧ 𝑘 ∈ (𝑀...𝑛))) → (𝐹‘𝑘) ∈ ℂ) | 
| 44 | 43 | anassrs 467 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) ∧ 𝑘 ∈ (𝑀...𝑛)) → (𝐹‘𝑘) ∈ ℂ) | 
| 45 |  | mulcl 11240 | . . . . . . . . . . . 12
⊢ ((𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ) → (𝑘 · 𝑥) ∈ ℂ) | 
| 46 | 45 | adantl 481 | . . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) ∧ (𝑘 ∈ ℂ ∧ 𝑥 ∈ ℂ)) → (𝑘 · 𝑥) ∈ ℂ) | 
| 47 | 37, 44, 46 | seqcl 14064 | . . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀...𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ) | 
| 48 | 35, 47 | sylan2 593 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ) | 
| 49 | 48 | 3adant3 1132 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) → (seq𝑀( · , 𝐹)‘𝑛) ∈ ℂ) | 
| 50 |  | fzofzp1 13804 | . . . . . . . . . . 11
⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝑛 + 1) ∈ (𝑀...𝑁)) | 
| 51 |  | fveq2 6905 | . . . . . . . . . . . . . 14
⊢ (𝑘 = (𝑛 + 1) → (𝐹‘𝑘) = (𝐹‘(𝑛 + 1))) | 
| 52 | 51 | eleq1d 2825 | . . . . . . . . . . . . 13
⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ∈ ℂ ↔ (𝐹‘(𝑛 + 1)) ∈ ℂ)) | 
| 53 | 52 | imbi2d 340 | . . . . . . . . . . . 12
⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) ∈ ℂ) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ))) | 
| 54 | 42 | expcom 413 | . . . . . . . . . . . 12
⊢ (𝑘 ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘𝑘) ∈ ℂ)) | 
| 55 | 53, 54 | vtoclga 3576 | . . . . . . . . . . 11
⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ)) | 
| 56 | 50, 55 | syl 17 | . . . . . . . . . 10
⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ∈ ℂ)) | 
| 57 | 56 | impcom 407 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ∈ ℂ) | 
| 58 | 57 | 3adant3 1132 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) → (𝐹‘(𝑛 + 1)) ∈ ℂ) | 
| 59 |  | simp3 1138 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) → (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) | 
| 60 | 51 | neeq1d 2999 | . . . . . . . . . . . . 13
⊢ (𝑘 = (𝑛 + 1) → ((𝐹‘𝑘) ≠ 0 ↔ (𝐹‘(𝑛 + 1)) ≠ 0)) | 
| 61 | 60 | imbi2d 340 | . . . . . . . . . . . 12
⊢ (𝑘 = (𝑛 + 1) → ((𝜑 → (𝐹‘𝑘) ≠ 0) ↔ (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0))) | 
| 62 | 61, 25 | vtoclga 3576 | . . . . . . . . . . 11
⊢ ((𝑛 + 1) ∈ (𝑀...𝑁) → (𝜑 → (𝐹‘(𝑛 + 1)) ≠ 0)) | 
| 63 | 62 | impcom 407 | . . . . . . . . . 10
⊢ ((𝜑 ∧ (𝑛 + 1) ∈ (𝑀...𝑁)) → (𝐹‘(𝑛 + 1)) ≠ 0) | 
| 64 | 50, 63 | sylan2 593 | . . . . . . . . 9
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁)) → (𝐹‘(𝑛 + 1)) ≠ 0) | 
| 65 | 64 | 3adant3 1132 | . . . . . . . 8
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) → (𝐹‘(𝑛 + 1)) ≠ 0) | 
| 66 | 49, 58, 59, 65 | mulne0d 11916 | . . . . . . 7
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) → ((seq𝑀( · , 𝐹)‘𝑛) · (𝐹‘(𝑛 + 1))) ≠ 0) | 
| 67 | 34, 66 | eqnetrd 3007 | . . . . . 6
⊢ ((𝜑 ∧ 𝑛 ∈ (𝑀..^𝑁) ∧ (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) ≠ 0) | 
| 68 | 67 | 3exp 1119 | . . . . 5
⊢ (𝜑 → (𝑛 ∈ (𝑀..^𝑁) → ((seq𝑀( · , 𝐹)‘𝑛) ≠ 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) ≠ 0))) | 
| 69 | 68 | com12 32 | . . . 4
⊢ (𝑛 ∈ (𝑀..^𝑁) → (𝜑 → ((seq𝑀( · , 𝐹)‘𝑛) ≠ 0 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) ≠ 0))) | 
| 70 | 69 | a2d 29 | . . 3
⊢ (𝑛 ∈ (𝑀..^𝑁) → ((𝜑 → (seq𝑀( · , 𝐹)‘𝑛) ≠ 0) → (𝜑 → (seq𝑀( · , 𝐹)‘(𝑛 + 1)) ≠ 0))) | 
| 71 | 6, 9, 12, 15, 30, 70 | fzind2 13825 | . 2
⊢ (𝑁 ∈ (𝑀...𝑁) → (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0)) | 
| 72 | 3, 71 | mpcom 38 | 1
⊢ (𝜑 → (seq𝑀( · , 𝐹)‘𝑁) ≠ 0) |