Step |
Hyp |
Ref |
Expression |
1 |
|
fsumcvg3.1 |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
2 |
|
fsumcvg3.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
3 |
|
fsumcvg3.3 |
⊢ ( 𝜑 → 𝐴 ∈ Fin ) |
4 |
|
fsumcvg3.4 |
⊢ ( 𝜑 → 𝐴 ⊆ 𝑍 ) |
5 |
|
fsumcvg3.5 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
6 |
|
fsumcvg3.6 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
7 |
|
sseq1 |
⊢ ( 𝐴 = ∅ → ( 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ↔ ∅ ⊆ ( 𝑀 ... 𝑛 ) ) ) |
8 |
7
|
rexbidv |
⊢ ( 𝐴 = ∅ → ( ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ↔ ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∅ ⊆ ( 𝑀 ... 𝑛 ) ) ) |
9 |
4
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → 𝐴 ⊆ 𝑍 ) |
10 |
9 1
|
sseqtrdi |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → 𝐴 ⊆ ( ℤ≥ ‘ 𝑀 ) ) |
11 |
|
ltso |
⊢ < Or ℝ |
12 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → 𝐴 ∈ Fin ) |
13 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → 𝐴 ≠ ∅ ) |
14 |
|
uzssz |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℤ |
15 |
|
zssre |
⊢ ℤ ⊆ ℝ |
16 |
14 15
|
sstri |
⊢ ( ℤ≥ ‘ 𝑀 ) ⊆ ℝ |
17 |
1 16
|
eqsstri |
⊢ 𝑍 ⊆ ℝ |
18 |
9 17
|
sstrdi |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → 𝐴 ⊆ ℝ ) |
19 |
12 13 18
|
3jca |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ( 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ 𝐴 ⊆ ℝ ) ) |
20 |
|
fisupcl |
⊢ ( ( < Or ℝ ∧ ( 𝐴 ∈ Fin ∧ 𝐴 ≠ ∅ ∧ 𝐴 ⊆ ℝ ) ) → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) |
21 |
11 19 20
|
sylancr |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → sup ( 𝐴 , ℝ , < ) ∈ 𝐴 ) |
22 |
10 21
|
sseldd |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → sup ( 𝐴 , ℝ , < ) ∈ ( ℤ≥ ‘ 𝑀 ) ) |
23 |
|
fimaxre2 |
⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ∈ Fin ) → ∃ 𝑘 ∈ ℝ ∀ 𝑛 ∈ 𝐴 𝑛 ≤ 𝑘 ) |
24 |
18 12 23
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑘 ∈ ℝ ∀ 𝑛 ∈ 𝐴 𝑛 ≤ 𝑘 ) |
25 |
18 13 24
|
3jca |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑘 ∈ ℝ ∀ 𝑛 ∈ 𝐴 𝑛 ≤ 𝑘 ) ) |
26 |
|
suprub |
⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑘 ∈ ℝ ∀ 𝑛 ∈ 𝐴 𝑛 ≤ 𝑘 ) ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ≤ sup ( 𝐴 , ℝ , < ) ) |
27 |
25 26
|
sylan |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ≤ sup ( 𝐴 , ℝ , < ) ) |
28 |
10
|
sselda |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
29 |
14 22
|
sselid |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → sup ( 𝐴 , ℝ , < ) ∈ ℤ ) |
30 |
29
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) ∧ 𝑘 ∈ 𝐴 ) → sup ( 𝐴 , ℝ , < ) ∈ ℤ ) |
31 |
|
elfz5 |
⊢ ( ( 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ sup ( 𝐴 , ℝ , < ) ∈ ℤ ) → ( 𝑘 ∈ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ↔ 𝑘 ≤ sup ( 𝐴 , ℝ , < ) ) ) |
32 |
28 30 31
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) ∧ 𝑘 ∈ 𝐴 ) → ( 𝑘 ∈ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ↔ 𝑘 ≤ sup ( 𝐴 , ℝ , < ) ) ) |
33 |
27 32
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) ∧ 𝑘 ∈ 𝐴 ) → 𝑘 ∈ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) |
34 |
33
|
ex |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ( 𝑘 ∈ 𝐴 → 𝑘 ∈ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) ) |
35 |
34
|
ssrdv |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → 𝐴 ⊆ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) |
36 |
|
oveq2 |
⊢ ( 𝑛 = sup ( 𝐴 , ℝ , < ) → ( 𝑀 ... 𝑛 ) = ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) |
37 |
36
|
sseq2d |
⊢ ( 𝑛 = sup ( 𝐴 , ℝ , < ) → ( 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ↔ 𝐴 ⊆ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) ) |
38 |
37
|
rspcev |
⊢ ( ( sup ( 𝐴 , ℝ , < ) ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐴 ⊆ ( 𝑀 ... sup ( 𝐴 , ℝ , < ) ) ) → ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ) |
39 |
22 35 38
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐴 ≠ ∅ ) → ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ) |
40 |
|
uzid |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
41 |
2 40
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
42 |
|
0ss |
⊢ ∅ ⊆ ( 𝑀 ... 𝑀 ) |
43 |
|
oveq2 |
⊢ ( 𝑛 = 𝑀 → ( 𝑀 ... 𝑛 ) = ( 𝑀 ... 𝑀 ) ) |
44 |
43
|
sseq2d |
⊢ ( 𝑛 = 𝑀 → ( ∅ ⊆ ( 𝑀 ... 𝑛 ) ↔ ∅ ⊆ ( 𝑀 ... 𝑀 ) ) ) |
45 |
44
|
rspcev |
⊢ ( ( 𝑀 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ ∅ ⊆ ( 𝑀 ... 𝑀 ) ) → ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∅ ⊆ ( 𝑀 ... 𝑛 ) ) |
46 |
41 42 45
|
sylancl |
⊢ ( 𝜑 → ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∅ ⊆ ( 𝑀 ... 𝑛 ) ) |
47 |
8 39 46
|
pm2.61ne |
⊢ ( 𝜑 → ∃ 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ) |
48 |
1
|
eleq2i |
⊢ ( 𝑘 ∈ 𝑍 ↔ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
49 |
48 5
|
sylan2br |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
50 |
49
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ) ) ∧ 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) → ( 𝐹 ‘ 𝑘 ) = if ( 𝑘 ∈ 𝐴 , 𝐵 , 0 ) ) |
51 |
|
simprl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ) ) → 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
52 |
6
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ) ) ∧ 𝑘 ∈ 𝐴 ) → 𝐵 ∈ ℂ ) |
53 |
|
simprr |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ) ) → 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ) |
54 |
50 51 52 53
|
fsumcvg2 |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ) ) → seq 𝑀 ( + , 𝐹 ) ⇝ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) ) |
55 |
|
climrel |
⊢ Rel ⇝ |
56 |
55
|
releldmi |
⊢ ( seq 𝑀 ( + , 𝐹 ) ⇝ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑛 ) → seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
57 |
54 56
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑛 ∈ ( ℤ≥ ‘ 𝑀 ) ∧ 𝐴 ⊆ ( 𝑀 ... 𝑛 ) ) ) → seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |
58 |
47 57
|
rexlimddv |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) ∈ dom ⇝ ) |