Proof of Theorem ntrivcvgtail
Step | Hyp | Ref
| Expression |
1 | | fclim 15262 |
. . . . . . . 8
⊢ ⇝
:dom ⇝ ⟶ℂ |
2 | | ffun 6603 |
. . . . . . . 8
⊢ ( ⇝
:dom ⇝ ⟶ℂ → Fun ⇝ ) |
3 | 1, 2 | ax-mp 5 |
. . . . . . 7
⊢ Fun
⇝ |
4 | | ntrivcvgtail.3 |
. . . . . . 7
⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ 𝑋) |
5 | | funbrfv 6820 |
. . . . . . 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 3011 |
. . . . 5
⊢ (𝜑 → ( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0) |
9 | 4, 6 | breqtrrd 5102 |
. . . . 5
⊢ (𝜑 → seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹))) |
10 | 8, 9 | jca 512 |
. . . 4
⊢ (𝜑 → (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹)))) |
11 | 10 | adantr 481 |
. . 3
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹)))) |
12 | | seqeq1 13724 |
. . . . . . 7
⊢ (𝑁 = 𝑀 → seq𝑁( · , 𝐹) = seq𝑀( · , 𝐹)) |
13 | 12 | fveq2d 6778 |
. . . . . 6
⊢ (𝑁 = 𝑀 → ( ⇝ ‘seq𝑁( · , 𝐹)) = ( ⇝ ‘seq𝑀( · , 𝐹))) |
14 | 13 | neeq1d 3003 |
. . . . 5
⊢ (𝑁 = 𝑀 → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ↔ ( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0)) |
15 | 12, 13 | breq12d 5087 |
. . . . 5
⊢ (𝑁 = 𝑀 → (seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)) ↔ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹)))) |
16 | 14, 15 | anbi12d 631 |
. . . 4
⊢ (𝑁 = 𝑀 → ((( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))) ↔ (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹))))) |
17 | 16 | adantl 482 |
. . 3
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → ((( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹))) ↔ (( ⇝ ‘seq𝑀( · , 𝐹)) ≠ 0 ∧ seq𝑀( · , 𝐹) ⇝ ( ⇝ ‘seq𝑀( · , 𝐹))))) |
18 | 11, 17 | mpbird 256 |
. 2
⊢ ((𝜑 ∧ 𝑁 = 𝑀) → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)))) |
19 | | ntrivcvgtail.1 |
. . . . . 6
⊢ 𝑍 =
(ℤ≥‘𝑀) |
20 | | simpr 485 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (𝑁 − 1) ∈
(ℤ≥‘𝑀)) |
21 | 20, 19 | eleqtrrdi 2850 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (𝑁 − 1) ∈ 𝑍) |
22 | | ntrivcvgtail.5 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
23 | 22 | adantlr 712 |
. . . . . 6
⊢ (((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) ∧ 𝑘 ∈ 𝑍) → (𝐹‘𝑘) ∈ ℂ) |
24 | 4 | adantr 481 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → seq𝑀( · , 𝐹) ⇝ 𝑋) |
25 | 7 | adantr 481 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → 𝑋 ≠ 0) |
26 | 19, 21, 24, 25, 23 | ntrivcvgfvn0 15611 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (seq𝑀( · , 𝐹)‘(𝑁 − 1)) ≠ 0) |
27 | 19, 21, 23, 24, 26 | clim2div 15601 |
. . . . 5
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1)))) |
28 | | funbrfv 6820 |
. . . . 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 15208 |
. . . . . . 7
⊢ (seq𝑀( · , 𝐹) ⇝ 𝑋 → 𝑋 ∈ ℂ) |
31 | 4, 30 | syl 17 |
. . . . . 6
⊢ (𝜑 → 𝑋 ∈ ℂ) |
32 | 31 | adantr 481 |
. . . . 5
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → 𝑋 ∈ ℂ) |
33 | | ntrivcvgtail.2 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ 𝑍) |
34 | | eluzel2 12587 |
. . . . . . . . . 10
⊢ (𝑁 ∈
(ℤ≥‘𝑀) → 𝑀 ∈ ℤ) |
35 | 34, 19 | eleq2s 2857 |
. . . . . . . . 9
⊢ (𝑁 ∈ 𝑍 → 𝑀 ∈ ℤ) |
36 | 33, 35 | syl 17 |
. . . . . . . 8
⊢ (𝜑 → 𝑀 ∈ ℤ) |
37 | 19, 36, 22 | prodf 15599 |
. . . . . . 7
⊢ (𝜑 → seq𝑀( · , 𝐹):𝑍⟶ℂ) |
38 | 19 | feq2i 6592 |
. . . . . . 7
⊢ (seq𝑀( · , 𝐹):𝑍⟶ℂ ↔ seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ) |
39 | 37, 38 | sylib 217 |
. . . . . 6
⊢ (𝜑 → seq𝑀( · , 𝐹):(ℤ≥‘𝑀)⟶ℂ) |
40 | 39 | ffvelrnda 6961 |
. . . . 5
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (seq𝑀( · , 𝐹)‘(𝑁 − 1)) ∈
ℂ) |
41 | 32, 40, 25, 26 | divne0d 11767 |
. . . 4
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (𝑋 / (seq𝑀( · , 𝐹)‘(𝑁 − 1))) ≠ 0) |
42 | 29, 41 | eqnetrd 3011 |
. . 3
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → ( ⇝ ‘seq((𝑁 − 1) + 1)( · ,
𝐹)) ≠
0) |
43 | 27, 29 | breqtrrd 5102 |
. . 3
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ ( ⇝
‘seq((𝑁 − 1) +
1)( · , 𝐹))) |
44 | | uzssz 12603 |
. . . . . . . . . . . 12
⊢
(ℤ≥‘𝑀) ⊆ ℤ |
45 | 19, 44 | eqsstri 3955 |
. . . . . . . . . . 11
⊢ 𝑍 ⊆
ℤ |
46 | 45, 33 | sselid 3919 |
. . . . . . . . . 10
⊢ (𝜑 → 𝑁 ∈ ℤ) |
47 | 46 | zcnd 12427 |
. . . . . . . . 9
⊢ (𝜑 → 𝑁 ∈ ℂ) |
48 | 47 | adantr 481 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → 𝑁 ∈ ℂ) |
49 | | 1cnd 10970 |
. . . . . . . 8
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → 1 ∈ ℂ) |
50 | 48, 49 | npcand 11336 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → ((𝑁 − 1) + 1) = 𝑁) |
51 | 50 | seqeq1d 13727 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → seq((𝑁 − 1) + 1)( · , 𝐹) = seq𝑁( · , 𝐹)) |
52 | 51 | fveq2d 6778 |
. . . . 5
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → ( ⇝ ‘seq((𝑁 − 1) + 1)( · ,
𝐹)) = ( ⇝
‘seq𝑁( · ,
𝐹))) |
53 | 52 | neeq1d 3003 |
. . . 4
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (( ⇝ ‘seq((𝑁 − 1) + 1)( · ,
𝐹)) ≠ 0 ↔ ( ⇝
‘seq𝑁( · ,
𝐹)) ≠
0)) |
54 | 51, 52 | breq12d 5087 |
. . . 4
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (seq((𝑁 − 1) + 1)( · , 𝐹) ⇝ ( ⇝
‘seq((𝑁 − 1) +
1)( · , 𝐹)) ↔
seq𝑁( · , 𝐹) ⇝ ( ⇝
‘seq𝑁( · ,
𝐹)))) |
55 | 53, 54 | anbi12d 631 |
. . 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 709 |
. 2
⊢ ((𝜑 ∧ (𝑁 − 1) ∈
(ℤ≥‘𝑀)) → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)))) |
57 | 33, 19 | eleqtrdi 2849 |
. . 3
⊢ (𝜑 → 𝑁 ∈ (ℤ≥‘𝑀)) |
58 | | uzm1 12616 |
. . 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 956 |
1
⊢ (𝜑 → (( ⇝ ‘seq𝑁( · , 𝐹)) ≠ 0 ∧ seq𝑁( · , 𝐹) ⇝ ( ⇝ ‘seq𝑁( · , 𝐹)))) |