Step |
Hyp |
Ref |
Expression |
1 |
|
fmul01.1 |
⊢ Ⅎ 𝑖 𝐵 |
2 |
|
fmul01.2 |
⊢ Ⅎ 𝑖 𝜑 |
3 |
|
fmul01.3 |
⊢ 𝐴 = seq 𝐿 ( · , 𝐵 ) |
4 |
|
fmul01.4 |
⊢ ( 𝜑 → 𝐿 ∈ ℤ ) |
5 |
|
fmul01.5 |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 𝐿 ) ) |
6 |
|
fmul01.6 |
⊢ ( 𝜑 → 𝐾 ∈ ( 𝐿 ... 𝑀 ) ) |
7 |
|
fmul01.7 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝑖 ) ∈ ℝ ) |
8 |
|
fmul01.8 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) → 0 ≤ ( 𝐵 ‘ 𝑖 ) ) |
9 |
|
fmul01.9 |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝑖 ) ≤ 1 ) |
10 |
|
fveq2 |
⊢ ( 𝑘 = 𝐿 → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝐿 ) ) |
11 |
10
|
breq2d |
⊢ ( 𝑘 = 𝐿 → ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ↔ 0 ≤ ( 𝐴 ‘ 𝐿 ) ) ) |
12 |
10
|
breq1d |
⊢ ( 𝑘 = 𝐿 → ( ( 𝐴 ‘ 𝑘 ) ≤ 1 ↔ ( 𝐴 ‘ 𝐿 ) ≤ 1 ) ) |
13 |
11 12
|
anbi12d |
⊢ ( 𝑘 = 𝐿 → ( ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ∧ ( 𝐴 ‘ 𝑘 ) ≤ 1 ) ↔ ( 0 ≤ ( 𝐴 ‘ 𝐿 ) ∧ ( 𝐴 ‘ 𝐿 ) ≤ 1 ) ) ) |
14 |
13
|
imbi2d |
⊢ ( 𝑘 = 𝐿 → ( ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ∧ ( 𝐴 ‘ 𝑘 ) ≤ 1 ) ) ↔ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝐿 ) ∧ ( 𝐴 ‘ 𝐿 ) ≤ 1 ) ) ) ) |
15 |
|
fveq2 |
⊢ ( 𝑘 = 𝑗 → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝑗 ) ) |
16 |
15
|
breq2d |
⊢ ( 𝑘 = 𝑗 → ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ↔ 0 ≤ ( 𝐴 ‘ 𝑗 ) ) ) |
17 |
15
|
breq1d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝐴 ‘ 𝑘 ) ≤ 1 ↔ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) |
18 |
16 17
|
anbi12d |
⊢ ( 𝑘 = 𝑗 → ( ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ∧ ( 𝐴 ‘ 𝑘 ) ≤ 1 ) ↔ ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ) |
19 |
18
|
imbi2d |
⊢ ( 𝑘 = 𝑗 → ( ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ∧ ( 𝐴 ‘ 𝑘 ) ≤ 1 ) ) ↔ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ) ) |
20 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) |
21 |
20
|
breq2d |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ↔ 0 ≤ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) ) |
22 |
20
|
breq1d |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( ( 𝐴 ‘ 𝑘 ) ≤ 1 ↔ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) ) |
23 |
21 22
|
anbi12d |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ∧ ( 𝐴 ‘ 𝑘 ) ≤ 1 ) ↔ ( 0 ≤ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ∧ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) ) ) |
24 |
23
|
imbi2d |
⊢ ( 𝑘 = ( 𝑗 + 1 ) → ( ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ∧ ( 𝐴 ‘ 𝑘 ) ≤ 1 ) ) ↔ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ∧ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) ) ) ) |
25 |
|
fveq2 |
⊢ ( 𝑘 = 𝐾 → ( 𝐴 ‘ 𝑘 ) = ( 𝐴 ‘ 𝐾 ) ) |
26 |
25
|
breq2d |
⊢ ( 𝑘 = 𝐾 → ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ↔ 0 ≤ ( 𝐴 ‘ 𝐾 ) ) ) |
27 |
25
|
breq1d |
⊢ ( 𝑘 = 𝐾 → ( ( 𝐴 ‘ 𝑘 ) ≤ 1 ↔ ( 𝐴 ‘ 𝐾 ) ≤ 1 ) ) |
28 |
26 27
|
anbi12d |
⊢ ( 𝑘 = 𝐾 → ( ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ∧ ( 𝐴 ‘ 𝑘 ) ≤ 1 ) ↔ ( 0 ≤ ( 𝐴 ‘ 𝐾 ) ∧ ( 𝐴 ‘ 𝐾 ) ≤ 1 ) ) ) |
29 |
28
|
imbi2d |
⊢ ( 𝑘 = 𝐾 → ( ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑘 ) ∧ ( 𝐴 ‘ 𝑘 ) ≤ 1 ) ) ↔ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝐾 ) ∧ ( 𝐴 ‘ 𝐾 ) ≤ 1 ) ) ) ) |
30 |
|
eluzelz |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝐿 ) → 𝑀 ∈ ℤ ) |
31 |
5 30
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
32 |
4
|
zred |
⊢ ( 𝜑 → 𝐿 ∈ ℝ ) |
33 |
32
|
leidd |
⊢ ( 𝜑 → 𝐿 ≤ 𝐿 ) |
34 |
|
eluz |
⊢ ( ( 𝐿 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑀 ∈ ( ℤ≥ ‘ 𝐿 ) ↔ 𝐿 ≤ 𝑀 ) ) |
35 |
4 31 34
|
syl2anc |
⊢ ( 𝜑 → ( 𝑀 ∈ ( ℤ≥ ‘ 𝐿 ) ↔ 𝐿 ≤ 𝑀 ) ) |
36 |
5 35
|
mpbid |
⊢ ( 𝜑 → 𝐿 ≤ 𝑀 ) |
37 |
4 31 4 33 36
|
elfzd |
⊢ ( 𝜑 → 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) |
38 |
37
|
ancli |
⊢ ( 𝜑 → ( 𝜑 ∧ 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) ) |
39 |
|
nfv |
⊢ Ⅎ 𝑖 𝐿 ∈ ( 𝐿 ... 𝑀 ) |
40 |
2 39
|
nfan |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) |
41 |
|
nfcv |
⊢ Ⅎ 𝑖 0 |
42 |
|
nfcv |
⊢ Ⅎ 𝑖 ≤ |
43 |
|
nfcv |
⊢ Ⅎ 𝑖 𝐿 |
44 |
1 43
|
nffv |
⊢ Ⅎ 𝑖 ( 𝐵 ‘ 𝐿 ) |
45 |
41 42 44
|
nfbr |
⊢ Ⅎ 𝑖 0 ≤ ( 𝐵 ‘ 𝐿 ) |
46 |
40 45
|
nfim |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) → 0 ≤ ( 𝐵 ‘ 𝐿 ) ) |
47 |
|
eleq1 |
⊢ ( 𝑖 = 𝐿 → ( 𝑖 ∈ ( 𝐿 ... 𝑀 ) ↔ 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) ) |
48 |
47
|
anbi2d |
⊢ ( 𝑖 = 𝐿 → ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) ↔ ( 𝜑 ∧ 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) ) ) |
49 |
|
fveq2 |
⊢ ( 𝑖 = 𝐿 → ( 𝐵 ‘ 𝑖 ) = ( 𝐵 ‘ 𝐿 ) ) |
50 |
49
|
breq2d |
⊢ ( 𝑖 = 𝐿 → ( 0 ≤ ( 𝐵 ‘ 𝑖 ) ↔ 0 ≤ ( 𝐵 ‘ 𝐿 ) ) ) |
51 |
48 50
|
imbi12d |
⊢ ( 𝑖 = 𝐿 → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) → 0 ≤ ( 𝐵 ‘ 𝑖 ) ) ↔ ( ( 𝜑 ∧ 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) → 0 ≤ ( 𝐵 ‘ 𝐿 ) ) ) ) |
52 |
46 51 8
|
vtoclg1f |
⊢ ( 𝐿 ∈ ( 𝐿 ... 𝑀 ) → ( ( 𝜑 ∧ 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) → 0 ≤ ( 𝐵 ‘ 𝐿 ) ) ) |
53 |
37 38 52
|
sylc |
⊢ ( 𝜑 → 0 ≤ ( 𝐵 ‘ 𝐿 ) ) |
54 |
3
|
fveq1i |
⊢ ( 𝐴 ‘ 𝐿 ) = ( seq 𝐿 ( · , 𝐵 ) ‘ 𝐿 ) |
55 |
|
seq1 |
⊢ ( 𝐿 ∈ ℤ → ( seq 𝐿 ( · , 𝐵 ) ‘ 𝐿 ) = ( 𝐵 ‘ 𝐿 ) ) |
56 |
4 55
|
syl |
⊢ ( 𝜑 → ( seq 𝐿 ( · , 𝐵 ) ‘ 𝐿 ) = ( 𝐵 ‘ 𝐿 ) ) |
57 |
54 56
|
syl5eq |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝐿 ) = ( 𝐵 ‘ 𝐿 ) ) |
58 |
53 57
|
breqtrrd |
⊢ ( 𝜑 → 0 ≤ ( 𝐴 ‘ 𝐿 ) ) |
59 |
|
nfcv |
⊢ Ⅎ 𝑖 1 |
60 |
44 42 59
|
nfbr |
⊢ Ⅎ 𝑖 ( 𝐵 ‘ 𝐿 ) ≤ 1 |
61 |
40 60
|
nfim |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝐿 ) ≤ 1 ) |
62 |
49
|
breq1d |
⊢ ( 𝑖 = 𝐿 → ( ( 𝐵 ‘ 𝑖 ) ≤ 1 ↔ ( 𝐵 ‘ 𝐿 ) ≤ 1 ) ) |
63 |
48 62
|
imbi12d |
⊢ ( 𝑖 = 𝐿 → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝑖 ) ≤ 1 ) ↔ ( ( 𝜑 ∧ 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝐿 ) ≤ 1 ) ) ) |
64 |
61 63 9
|
vtoclg1f |
⊢ ( 𝐿 ∈ ( 𝐿 ... 𝑀 ) → ( ( 𝜑 ∧ 𝐿 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝐿 ) ≤ 1 ) ) |
65 |
37 38 64
|
sylc |
⊢ ( 𝜑 → ( 𝐵 ‘ 𝐿 ) ≤ 1 ) |
66 |
57 65
|
eqbrtrd |
⊢ ( 𝜑 → ( 𝐴 ‘ 𝐿 ) ≤ 1 ) |
67 |
58 66
|
jca |
⊢ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝐿 ) ∧ ( 𝐴 ‘ 𝐿 ) ≤ 1 ) ) |
68 |
67
|
a1i |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝐿 ) → ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝐿 ) ∧ ( 𝐴 ‘ 𝐿 ) ≤ 1 ) ) ) |
69 |
|
elfzouz |
⊢ ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝐿 ) ) |
70 |
69
|
3ad2ant1 |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝐿 ) ) |
71 |
|
simpl3 |
⊢ ( ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) ∧ 𝑘 ∈ ( 𝐿 ... 𝑗 ) ) → 𝜑 ) |
72 |
|
elfzouz2 |
⊢ ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) → 𝑀 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
73 |
|
fzss2 |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 𝑗 ) → ( 𝐿 ... 𝑗 ) ⊆ ( 𝐿 ... 𝑀 ) ) |
74 |
72 73
|
syl |
⊢ ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) → ( 𝐿 ... 𝑗 ) ⊆ ( 𝐿 ... 𝑀 ) ) |
75 |
74
|
3ad2ant1 |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( 𝐿 ... 𝑗 ) ⊆ ( 𝐿 ... 𝑀 ) ) |
76 |
75
|
sselda |
⊢ ( ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) ∧ 𝑘 ∈ ( 𝐿 ... 𝑗 ) ) → 𝑘 ∈ ( 𝐿 ... 𝑀 ) ) |
77 |
|
nfv |
⊢ Ⅎ 𝑖 𝑘 ∈ ( 𝐿 ... 𝑀 ) |
78 |
2 77
|
nfan |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ 𝑘 ∈ ( 𝐿 ... 𝑀 ) ) |
79 |
|
nfcv |
⊢ Ⅎ 𝑖 𝑘 |
80 |
1 79
|
nffv |
⊢ Ⅎ 𝑖 ( 𝐵 ‘ 𝑘 ) |
81 |
80
|
nfel1 |
⊢ Ⅎ 𝑖 ( 𝐵 ‘ 𝑘 ) ∈ ℝ |
82 |
78 81
|
nfim |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) |
83 |
|
eleq1 |
⊢ ( 𝑖 = 𝑘 → ( 𝑖 ∈ ( 𝐿 ... 𝑀 ) ↔ 𝑘 ∈ ( 𝐿 ... 𝑀 ) ) ) |
84 |
83
|
anbi2d |
⊢ ( 𝑖 = 𝑘 → ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) ↔ ( 𝜑 ∧ 𝑘 ∈ ( 𝐿 ... 𝑀 ) ) ) ) |
85 |
|
fveq2 |
⊢ ( 𝑖 = 𝑘 → ( 𝐵 ‘ 𝑖 ) = ( 𝐵 ‘ 𝑘 ) ) |
86 |
85
|
eleq1d |
⊢ ( 𝑖 = 𝑘 → ( ( 𝐵 ‘ 𝑖 ) ∈ ℝ ↔ ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) ) |
87 |
84 86
|
imbi12d |
⊢ ( 𝑖 = 𝑘 → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝑖 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) ) ) |
88 |
82 87 7
|
chvarfv |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) |
89 |
71 76 88
|
syl2anc |
⊢ ( ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) ∧ 𝑘 ∈ ( 𝐿 ... 𝑗 ) ) → ( 𝐵 ‘ 𝑘 ) ∈ ℝ ) |
90 |
|
remulcl |
⊢ ( ( 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ ) → ( 𝑘 · 𝑙 ) ∈ ℝ ) |
91 |
90
|
adantl |
⊢ ( ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) ∧ ( 𝑘 ∈ ℝ ∧ 𝑙 ∈ ℝ ) ) → ( 𝑘 · 𝑙 ) ∈ ℝ ) |
92 |
70 89 91
|
seqcl |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) ∈ ℝ ) |
93 |
|
simp3 |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 𝜑 ) |
94 |
|
fzofzp1 |
⊢ ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) → ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) |
95 |
94
|
3ad2ant1 |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) |
96 |
|
nfv |
⊢ Ⅎ 𝑖 ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) |
97 |
2 96
|
nfan |
⊢ Ⅎ 𝑖 ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) |
98 |
|
nfcv |
⊢ Ⅎ 𝑖 ( 𝑗 + 1 ) |
99 |
1 98
|
nffv |
⊢ Ⅎ 𝑖 ( 𝐵 ‘ ( 𝑗 + 1 ) ) |
100 |
99
|
nfel1 |
⊢ Ⅎ 𝑖 ( 𝐵 ‘ ( 𝑗 + 1 ) ) ∈ ℝ |
101 |
97 100
|
nfim |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) |
102 |
|
eleq1 |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( 𝑖 ∈ ( 𝐿 ... 𝑀 ) ↔ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) ) |
103 |
102
|
anbi2d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) ↔ ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) ) ) |
104 |
|
fveq2 |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( 𝐵 ‘ 𝑖 ) = ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) |
105 |
104
|
eleq1d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( 𝐵 ‘ 𝑖 ) ∈ ℝ ↔ ( 𝐵 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) ) |
106 |
103 105
|
imbi12d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝑖 ) ∈ ℝ ) ↔ ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) ) ) |
107 |
101 106 7
|
vtoclg1f |
⊢ ( ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) → ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) ) |
108 |
107
|
anabsi7 |
⊢ ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) |
109 |
93 95 108
|
syl2anc |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( 𝐵 ‘ ( 𝑗 + 1 ) ) ∈ ℝ ) |
110 |
|
pm3.35 |
⊢ ( ( 𝜑 ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ) → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) |
111 |
110
|
ancoms |
⊢ ( ( ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) |
112 |
|
simpl |
⊢ ( ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) → 0 ≤ ( 𝐴 ‘ 𝑗 ) ) |
113 |
111 112
|
syl |
⊢ ( ( ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 0 ≤ ( 𝐴 ‘ 𝑗 ) ) |
114 |
113
|
3adant1 |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 0 ≤ ( 𝐴 ‘ 𝑗 ) ) |
115 |
3
|
fveq1i |
⊢ ( 𝐴 ‘ 𝑗 ) = ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) |
116 |
114 115
|
breqtrdi |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 0 ≤ ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) ) |
117 |
|
simp1 |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ) |
118 |
94
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ) → ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) |
119 |
|
simpl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ) → 𝜑 ) |
120 |
119 118
|
jca |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ) → ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) ) |
121 |
41 42 99
|
nfbr |
⊢ Ⅎ 𝑖 0 ≤ ( 𝐵 ‘ ( 𝑗 + 1 ) ) |
122 |
97 121
|
nfim |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) → 0 ≤ ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) |
123 |
104
|
breq2d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( 0 ≤ ( 𝐵 ‘ 𝑖 ) ↔ 0 ≤ ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) ) |
124 |
103 123
|
imbi12d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) → 0 ≤ ( 𝐵 ‘ 𝑖 ) ) ↔ ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) → 0 ≤ ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) ) ) |
125 |
122 124 8
|
vtoclg1f |
⊢ ( ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) → ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) → 0 ≤ ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) ) |
126 |
118 120 125
|
sylc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ) → 0 ≤ ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) |
127 |
93 117 126
|
syl2anc |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 0 ≤ ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) |
128 |
92 109 116 127
|
mulge0d |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 0 ≤ ( ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) · ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) ) |
129 |
|
seqp1 |
⊢ ( 𝑗 ∈ ( ℤ≥ ‘ 𝐿 ) → ( seq 𝐿 ( · , 𝐵 ) ‘ ( 𝑗 + 1 ) ) = ( ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) · ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) ) |
130 |
70 129
|
syl |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( seq 𝐿 ( · , 𝐵 ) ‘ ( 𝑗 + 1 ) ) = ( ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) · ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) ) |
131 |
128 130
|
breqtrrd |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 0 ≤ ( seq 𝐿 ( · , 𝐵 ) ‘ ( 𝑗 + 1 ) ) ) |
132 |
3
|
fveq1i |
⊢ ( 𝐴 ‘ ( 𝑗 + 1 ) ) = ( seq 𝐿 ( · , 𝐵 ) ‘ ( 𝑗 + 1 ) ) |
133 |
131 132
|
breqtrrdi |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 0 ≤ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ) |
134 |
92 109
|
remulcld |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) · ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) ∈ ℝ ) |
135 |
|
1red |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → 1 ∈ ℝ ) |
136 |
93 95
|
jca |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) ) |
137 |
99 42 59
|
nfbr |
⊢ Ⅎ 𝑖 ( 𝐵 ‘ ( 𝑗 + 1 ) ) ≤ 1 |
138 |
97 137
|
nfim |
⊢ Ⅎ 𝑖 ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) |
139 |
104
|
breq1d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( 𝐵 ‘ 𝑖 ) ≤ 1 ↔ ( 𝐵 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) ) |
140 |
103 139
|
imbi12d |
⊢ ( 𝑖 = ( 𝑗 + 1 ) → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ 𝑖 ) ≤ 1 ) ↔ ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) ) ) |
141 |
138 140 9
|
vtoclg1f |
⊢ ( ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) → ( ( 𝜑 ∧ ( 𝑗 + 1 ) ∈ ( 𝐿 ... 𝑀 ) ) → ( 𝐵 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) ) |
142 |
95 136 141
|
sylc |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( 𝐵 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) |
143 |
109 135 92 116 142
|
lemul2ad |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) · ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) ≤ ( ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) · 1 ) ) |
144 |
92
|
recnd |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) ∈ ℂ ) |
145 |
144
|
mulid1d |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) · 1 ) = ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) ) |
146 |
143 145
|
breqtrd |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) · ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) ≤ ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) ) |
147 |
|
simp2 |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ) |
148 |
110
|
simprd |
⊢ ( ( 𝜑 ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ) → ( 𝐴 ‘ 𝑗 ) ≤ 1 ) |
149 |
93 147 148
|
syl2anc |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( 𝐴 ‘ 𝑗 ) ≤ 1 ) |
150 |
115 149
|
eqbrtrrid |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) ≤ 1 ) |
151 |
134 92 135 146 150
|
letrd |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( ( seq 𝐿 ( · , 𝐵 ) ‘ 𝑗 ) · ( 𝐵 ‘ ( 𝑗 + 1 ) ) ) ≤ 1 ) |
152 |
130 151
|
eqbrtrd |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( seq 𝐿 ( · , 𝐵 ) ‘ ( 𝑗 + 1 ) ) ≤ 1 ) |
153 |
132 152
|
eqbrtrid |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( 𝐴 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) |
154 |
133 153
|
jca |
⊢ ( ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) ∧ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) ∧ 𝜑 ) → ( 0 ≤ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ∧ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) ) |
155 |
154
|
3exp |
⊢ ( 𝑗 ∈ ( 𝐿 ..^ 𝑀 ) → ( ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝑗 ) ∧ ( 𝐴 ‘ 𝑗 ) ≤ 1 ) ) → ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ∧ ( 𝐴 ‘ ( 𝑗 + 1 ) ) ≤ 1 ) ) ) ) |
156 |
14 19 24 29 68 155
|
fzind2 |
⊢ ( 𝐾 ∈ ( 𝐿 ... 𝑀 ) → ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝐾 ) ∧ ( 𝐴 ‘ 𝐾 ) ≤ 1 ) ) ) |
157 |
6 156
|
mpcom |
⊢ ( 𝜑 → ( 0 ≤ ( 𝐴 ‘ 𝐾 ) ∧ ( 𝐴 ‘ 𝐾 ) ≤ 1 ) ) |