Proof of Theorem ntrivcvgtail
| Step | Hyp | Ref
| Expression |
| 1 | | fclim 15589 |
. . . . . . . 8
⊢ ⇝
:dom ⇝ ⟶ℂ |
| 2 | | ffun 6739 |
. . . . . . . 8
⊢ ( ⇝
:dom ⇝ ⟶ℂ → Fun ⇝ ) |
| 3 | 1, 2 | ax-mp 5 |
. . . . . . 7
⊢ Fun
⇝ |
| 4 | | ntrivcvgtail.3 |
. . . . . . 7
⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) |
| 5 | | funbrfv 6957 |
. . . . . . 7
⊢ (Fun
⇝ → (seq𝑀(
· , 𝐹) ⇝ 𝑋 → ( ⇝
‘seq𝑀( · ,
𝐹)) = 𝑋)) |
| 6 | 3, 4, 5 | mpsyl 68 |
. . . . . 6
⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) = 𝑋) |
| 7 | | ntrivcvgtail.4 |
. . . . . 6
⊢ (𝜑 → 𝑋 ≠ 0) |
| 8 | 6, 7 | eqnetrd 3008 |
. . . . 5
⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0) |
| 9 | 4, 6 | breqtrrd 5171 |
. . . . 5
⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹))) |
| 10 | 8, 9 | jca 511 |
. . . 4
⊢ (𝜑 → (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹)))) |
| 11 | 10 | adantr 480 |
. . 3
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹)))) |
| 12 | | seqeq1 14045 |
. . . . . . 7
⊢ (𝑁 = 𝑀 → seq𝑁( · , 𝐹) = seq𝑀( · , 𝐹)) |
| 13 | 12 | fveq2d 6910 |
. . . . . 6
⊢ (𝑁 = 𝑀 → ( ⇝ ‘seq𝑁( · , 𝐹)) = ( ⇝ ‘seq𝑀( · , 𝐹))) |
| 14 | 13 | neeq1d 3000 |
. . . . 5
⊢ (𝑁 = 𝑀 → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ↔ ( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0)) |
| 15 | 12, 13 | breq12d 5156 |
. . . . 5
⊢ (𝑁 = 𝑀 → (seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)) ↔ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹)))) |
| 16 | 14, 15 | anbi12d 632 |
. . . 4
⊢ (𝑁 = 𝑀 → ((( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))) ↔ (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹))))) |
| 17 | 16 | adantl 481 |
. . 3
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → ((( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))) ↔ (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹))))) |
| 18 | 11, 17 | mpbird 257 |
. 2
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)))) |
| 19 | | ntrivcvgtail.1 |
. . . . . 6
⊢ 𝑍 =
(ℤ≥‘𝑀) |
| 20 | | simpr 484 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (𝑁 − 1) ∈
(ℤ≥‘𝑀)) |
| 21 | 20, 19 | eleqtrrdi 2852 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (𝑁 − 1) ∈ 𝑍) |
| 22 | | ntrivcvgtail.5 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
| 23 | 22 | adantlr 715 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
| 24 | 4 | adantr 480 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → seq𝑀( · , 𝐹) ⇝ 𝑋) |
| 25 | 7 | adantr 480 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → 𝑋 ≠ 0) |
| 26 | 19, 21, 24, 25, 23 | ntrivcvgfvn0 15935 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (seq𝑀( · , 𝐹)‘(𝑁 − 1)) ≠ 0) |
| 27 | 19, 21, 23, 24, 26 | clim2div 15925 |
. . . . 5
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1)))) |
| 28 | | funbrfv 6957 |
. . . . 5
⊢ (Fun
⇝ → (seq((𝑁
− 1) + 1)( · , 𝐹) ⇝ (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1))) → ( ⇝
‘seq((𝑁 − 1) +
1)( · , 𝐹)) = (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1))))) |
| 29 | 3, 27, 28 | mpsyl 68 |
. . . 4
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → ( ⇝ ‘seq((𝑁 − 1) + 1)( · ,
𝐹)) = (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1)))) |
| 30 | | climcl 15535 |
. . . . . . 7
⊢ (seq𝑀( · , 𝐹) ⇝ 𝑋 → 𝑋 ∈ ℂ) |
| 31 | 4, 30 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ ℂ) |
| 32 | 31 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → 𝑋 ∈ ℂ) |
| 33 | | ntrivcvgtail.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ 𝑍) |
| 34 | | eluzel2 12883 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
| 35 | 34, 19 | eleq2s 2859 |
. . . . . . . . 9
⊢ (𝑁 ∈ 𝑍 → 𝑀 ∈ ℤ) |
| 36 | 33, 35 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
| 37 | 19, 36, 22 | prodf 15923 |
. . . . . . 7
⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ) |
| 38 | 19 | feq2i 6728 |
. . . . . . 7
⊢ (seq𝑀( · , 𝐹):𝑍⟶ℂ ↔ seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ) |
| 39 | 37, 38 | sylib 218 |
. . . . . 6
⊢ (𝜑 → seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ) |
| 40 | 39 | ffvelcdmda 7104 |
. . . . 5
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (seq𝑀( · , 𝐹)‘(𝑁 − 1)) ∈
ℂ) |
| 41 | 32, 40, 25, 26 | divne0d 12059 |
. . . 4
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1))) ≠ 0) |
| 42 | 29, 41 | eqnetrd 3008 |
. . 3
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → ( ⇝ ‘seq((𝑁 − 1) + 1)( · ,
𝐹)) ≠
0) |
| 43 | 27, 29 | breqtrrd 5171 |
. . 3
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ ( ⇝
‘seq((𝑁 − 1) +
1)( · , 𝐹))) |
| 44 | | uzssz 12899 |
. . . . . . . . . . . 12
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
| 45 | 19, 44 | eqsstri 4030 |
. . . . . . . . . . 11
⊢ 𝑍 ⊆
ℤ |
| 46 | 45, 33 | sselid 3981 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℤ) |
| 47 | 46 | zcnd 12723 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℂ) |
| 48 | 47 | adantr 480 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → 𝑁 ∈ ℂ) |
| 49 | | 1cnd 11256 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → 1 ∈ ℂ) |
| 50 | 48, 49 | npcand 11624 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → ((𝑁 − 1) + 1) = 𝑁) |
| 51 | 50 | seqeq1d 14048 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → seq((𝑁 − 1) + 1)( · , 𝐹) = seq𝑁( · , 𝐹)) |
| 52 | 51 | fveq2d 6910 |
. . . . 5
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → ( ⇝ ‘seq((𝑁 − 1) + 1)( · ,
𝐹)) = ( ⇝
‘seq𝑁( · ,
𝐹))) |
| 53 | 52 | neeq1d 3000 |
. . . 4
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (( ⇝ ‘seq((𝑁 − 1) + 1)( · ,
𝐹)) ≠ 0 ↔ ( ⇝
‘seq𝑁( · ,
𝐹)) ≠
0)) |
| 54 | 51, 52 | breq12d 5156 |
. . . 4
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ ( ⇝
‘seq((𝑁 − 1) +
1)( · , 𝐹)) ↔
seq𝑁( · , 𝐹) ⇝ ( ⇝
‘seq𝑁( · ,
𝐹)))) |
| 55 | 53, 54 | anbi12d 632 |
. . 3
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → ((( ⇝ ‘seq((𝑁 − 1) + 1)( · ,
𝐹)) ≠ 0 ∧ seq((𝑁 − 1) + 1)( · ,
𝐹) ⇝ ( ⇝
‘seq((𝑁 − 1) +
1)( · , 𝐹))) ↔
(( ⇝ ‘seq𝑁(
· , 𝐹)) ≠ 0 ∧
seq𝑁( · , 𝐹) ⇝ ( ⇝
‘seq𝑁( · ,
𝐹))))) |
| 56 | 42, 43, 55 | mpbi2and 712 |
. 2
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)))) |
| 57 | 33, 19 | eleqtrdi 2851 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
| 58 | | uzm1 12916 |
. . 3
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈
(ℤ≥‘𝑀))) |
| 59 | 57, 58 | syl 17 |
. 2
⊢ (𝜑 → (𝑁 = 𝑀 ∨ (𝑁 − 1) ∈
(ℤ≥‘𝑀))) |
| 60 | 18, 56, 59 | mpjaodan 961 |
1
⊢ (𝜑 → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)))) |