Step |
Hyp |
Ref |
Expression |
1 |
|
sge0seq.m |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
2 |
|
sge0seq.z |
⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 ) |
3 |
|
sge0seq.f |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( 0 [,) +∞ ) ) |
4 |
|
sge0seq.g |
⊢ 𝐺 = seq 𝑀 ( + , 𝐹 ) |
5 |
|
rge0ssre |
⊢ ( 0 [,) +∞ ) ⊆ ℝ |
6 |
3
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 0 [,) +∞ ) ) |
7 |
5 6
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
8 |
|
readdcl |
⊢ ( ( 𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ) → ( 𝑘 + 𝑖 ) ∈ ℝ ) |
9 |
8
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑘 ∈ ℝ ∧ 𝑖 ∈ ℝ ) ) → ( 𝑘 + 𝑖 ) ∈ ℝ ) |
10 |
2 1 7 9
|
seqf |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) : 𝑍 ⟶ ℝ ) |
11 |
4
|
a1i |
⊢ ( 𝜑 → 𝐺 = seq 𝑀 ( + , 𝐹 ) ) |
12 |
11
|
feq1d |
⊢ ( 𝜑 → ( 𝐺 : 𝑍 ⟶ ℝ ↔ seq 𝑀 ( + , 𝐹 ) : 𝑍 ⟶ ℝ ) ) |
13 |
10 12
|
mpbird |
⊢ ( 𝜑 → 𝐺 : 𝑍 ⟶ ℝ ) |
14 |
13
|
frnd |
⊢ ( 𝜑 → ran 𝐺 ⊆ ℝ ) |
15 |
|
ressxr |
⊢ ℝ ⊆ ℝ* |
16 |
15
|
a1i |
⊢ ( 𝜑 → ℝ ⊆ ℝ* ) |
17 |
14 16
|
sstrd |
⊢ ( 𝜑 → ran 𝐺 ⊆ ℝ* ) |
18 |
2
|
fvexi |
⊢ 𝑍 ∈ V |
19 |
18
|
a1i |
⊢ ( 𝜑 → 𝑍 ∈ V ) |
20 |
|
icossicc |
⊢ ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) |
21 |
20
|
a1i |
⊢ ( 𝜑 → ( 0 [,) +∞ ) ⊆ ( 0 [,] +∞ ) ) |
22 |
3 21
|
fssd |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ( 0 [,] +∞ ) ) |
23 |
19 22
|
sge0xrcl |
⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) ∈ ℝ* ) |
24 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → 𝑧 ∈ ran 𝐺 ) |
25 |
13
|
ffnd |
⊢ ( 𝜑 → 𝐺 Fn 𝑍 ) |
26 |
|
fvelrnb |
⊢ ( 𝐺 Fn 𝑍 → ( 𝑧 ∈ ran 𝐺 ↔ ∃ 𝑗 ∈ 𝑍 ( 𝐺 ‘ 𝑗 ) = 𝑧 ) ) |
27 |
25 26
|
syl |
⊢ ( 𝜑 → ( 𝑧 ∈ ran 𝐺 ↔ ∃ 𝑗 ∈ 𝑍 ( 𝐺 ‘ 𝑗 ) = 𝑧 ) ) |
28 |
27
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → ( 𝑧 ∈ ran 𝐺 ↔ ∃ 𝑗 ∈ 𝑍 ( 𝐺 ‘ 𝑗 ) = 𝑧 ) ) |
29 |
24 28
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → ∃ 𝑗 ∈ 𝑍 ( 𝐺 ‘ 𝑗 ) = 𝑧 ) |
30 |
20 6
|
sseldi |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 0 [,] +∞ ) ) |
31 |
|
elfzuz |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → 𝑘 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
32 |
31 2
|
eleqtrrdi |
⊢ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) → 𝑘 ∈ 𝑍 ) |
33 |
32
|
ssriv |
⊢ ( 𝑀 ... 𝑗 ) ⊆ 𝑍 |
34 |
33
|
a1i |
⊢ ( 𝜑 → ( 𝑀 ... 𝑗 ) ⊆ 𝑍 ) |
35 |
19 30 34
|
sge0lessmpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) ≤ ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) |
36 |
35
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) ≤ ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) |
37 |
|
fzfid |
⊢ ( 𝜑 → ( 𝑀 ... 𝑗 ) ∈ Fin ) |
38 |
32 6
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 0 [,) +∞ ) ) |
39 |
37 38
|
sge0fsummpt |
⊢ ( 𝜑 → ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) = Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) |
40 |
39
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) = Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) |
41 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → 𝜑 ) |
42 |
32
|
adantl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → 𝑘 ∈ 𝑍 ) |
43 |
|
eqidd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
44 |
41 42 43
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) = ( 𝐹 ‘ 𝑘 ) ) |
45 |
2
|
eleq2i |
⊢ ( 𝑗 ∈ 𝑍 ↔ 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
46 |
45
|
biimpi |
⊢ ( 𝑗 ∈ 𝑍 → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
47 |
46
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ ( ℤ≥ ‘ 𝑀 ) ) |
48 |
7
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
49 |
41 42 48
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℂ ) |
50 |
44 47 49
|
fsumser |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ) |
51 |
50
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ) |
52 |
40 51
|
eqtrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) = ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ) |
53 |
4
|
eqcomi |
⊢ seq 𝑀 ( + , 𝐹 ) = 𝐺 |
54 |
53
|
fveq1i |
⊢ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) |
55 |
54
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) = ( 𝐺 ‘ 𝑗 ) ) |
56 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( 𝐺 ‘ 𝑗 ) = 𝑧 ) |
57 |
52 55 56
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → 𝑧 = ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) |
58 |
3
|
feqmptd |
⊢ ( 𝜑 → 𝐹 = ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) |
59 |
58
|
fveq2d |
⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) = ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) |
60 |
59
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( Σ^ ‘ 𝐹 ) = ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) |
61 |
57 60
|
breq12d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → ( 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ↔ ( Σ^ ‘ ( 𝑘 ∈ ( 𝑀 ... 𝑗 ) ↦ ( 𝐹 ‘ 𝑘 ) ) ) ≤ ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) ) |
62 |
36 61
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ∧ ( 𝐺 ‘ 𝑗 ) = 𝑧 ) → 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ) |
63 |
62
|
3exp |
⊢ ( 𝜑 → ( 𝑗 ∈ 𝑍 → ( ( 𝐺 ‘ 𝑗 ) = 𝑧 → 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ) ) ) |
64 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → ( 𝑗 ∈ 𝑍 → ( ( 𝐺 ‘ 𝑗 ) = 𝑧 → 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ) ) ) |
65 |
64
|
rexlimdv |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → ( ∃ 𝑗 ∈ 𝑍 ( 𝐺 ‘ 𝑗 ) = 𝑧 → 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ) ) |
66 |
29 65
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ran 𝐺 ) → 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ) |
67 |
66
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑧 ∈ ran 𝐺 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ) |
68 |
|
nfv |
⊢ Ⅎ 𝑘 ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) |
69 |
18
|
a1i |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → 𝑍 ∈ V ) |
70 |
6
|
ad4ant14 |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) ∧ 𝑘 ∈ 𝑍 ) → ( 𝐹 ‘ 𝑘 ) ∈ ( 0 [,) +∞ ) ) |
71 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → 𝑧 ∈ ℝ ) |
72 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → 𝑧 < ( Σ^ ‘ 𝐹 ) ) |
73 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ( Σ^ ‘ 𝐹 ) = ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) |
74 |
72 73
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → 𝑧 < ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) |
75 |
74
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → 𝑧 < ( Σ^ ‘ ( 𝑘 ∈ 𝑍 ↦ ( 𝐹 ‘ 𝑘 ) ) ) ) |
76 |
68 69 70 71 75
|
sge0gtfsumgt |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ∃ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) |
77 |
1
|
3ad2ant1 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → 𝑀 ∈ ℤ ) |
78 |
|
elpwinss |
⊢ ( 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) → 𝑤 ⊆ 𝑍 ) |
79 |
78
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → 𝑤 ⊆ 𝑍 ) |
80 |
|
elinel2 |
⊢ ( 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) → 𝑤 ∈ Fin ) |
81 |
80
|
3ad2ant2 |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → 𝑤 ∈ Fin ) |
82 |
77 2 79 81
|
uzfissfz |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → ∃ 𝑗 ∈ 𝑍 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) |
83 |
82
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → ∃ 𝑗 ∈ 𝑍 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) |
84 |
|
simpl1r |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → 𝑧 ∈ ℝ ) |
85 |
80
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) → 𝑤 ∈ Fin ) |
86 |
58 7
|
fmpt3d |
⊢ ( 𝜑 → 𝐹 : 𝑍 ⟶ ℝ ) |
87 |
86
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑤 ) → 𝐹 : 𝑍 ⟶ ℝ ) |
88 |
78
|
sselda |
⊢ ( ( 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑘 ∈ 𝑤 ) → 𝑘 ∈ 𝑍 ) |
89 |
88
|
adantll |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑤 ) → 𝑘 ∈ 𝑍 ) |
90 |
87 89
|
ffvelrnd |
⊢ ( ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) ∧ 𝑘 ∈ 𝑤 ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
91 |
85 90
|
fsumrecl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
92 |
91
|
ad4ant13 |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
93 |
92
|
3adantl3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
94 |
32 7
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
95 |
37 94
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
96 |
95
|
ad3antrrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
97 |
96
|
3adantl3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
98 |
|
simpl3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) |
99 |
37
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → ( 𝑀 ... 𝑗 ) ∈ Fin ) |
100 |
94
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → ( 𝐹 ‘ 𝑘 ) ∈ ℝ ) |
101 |
|
0xr |
⊢ 0 ∈ ℝ* |
102 |
101
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 0 ∈ ℝ* ) |
103 |
|
pnfxr |
⊢ +∞ ∈ ℝ* |
104 |
103
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → +∞ ∈ ℝ* ) |
105 |
|
icogelb |
⊢ ( ( 0 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ ( 𝐹 ‘ 𝑘 ) ∈ ( 0 [,) +∞ ) ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) ) |
106 |
102 104 6 105
|
syl3anc |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ 𝑍 ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) ) |
107 |
32 106
|
sylan2 |
⊢ ( ( 𝜑 ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) ) |
108 |
107
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) ∧ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ) → 0 ≤ ( 𝐹 ‘ 𝑘 ) ) |
109 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) |
110 |
99 100 108 109
|
fsumless |
⊢ ( ( 𝜑 ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ≤ Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) |
111 |
110
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ≤ Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) |
112 |
111
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ≤ Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) |
113 |
84 93 97 98 112
|
ltletrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) ∧ 𝑤 ⊆ ( 𝑀 ... 𝑗 ) ) → 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) |
114 |
113
|
ex |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → ( 𝑤 ⊆ ( 𝑀 ... 𝑗 ) → 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) ) |
115 |
114
|
reximdv |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → ( ∃ 𝑗 ∈ 𝑍 𝑤 ⊆ ( 𝑀 ... 𝑗 ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) ) |
116 |
83 115
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) ∧ 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) |
117 |
116
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) → ( 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) ) ) |
118 |
117
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ( 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) → ( 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) ) ) |
119 |
118
|
rexlimdv |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ( ∃ 𝑤 ∈ ( 𝒫 𝑍 ∩ Fin ) 𝑧 < Σ 𝑘 ∈ 𝑤 ( 𝐹 ‘ 𝑘 ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) ) |
120 |
76 119
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) |
121 |
10
|
ffnd |
⊢ ( 𝜑 → seq 𝑀 ( + , 𝐹 ) Fn 𝑍 ) |
122 |
121
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → seq 𝑀 ( + , 𝐹 ) Fn 𝑍 ) |
123 |
47 45
|
sylibr |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝑗 ∈ 𝑍 ) |
124 |
|
fnfvelrn |
⊢ ( ( seq 𝑀 ( + , 𝐹 ) Fn 𝑍 ∧ 𝑗 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ∈ ran seq 𝑀 ( + , 𝐹 ) ) |
125 |
122 123 124
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ∈ ran seq 𝑀 ( + , 𝐹 ) ) |
126 |
4
|
a1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → 𝐺 = seq 𝑀 ( + , 𝐹 ) ) |
127 |
126
|
rneqd |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ran 𝐺 = ran seq 𝑀 ( + , 𝐹 ) ) |
128 |
50 127
|
eleq12d |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → ( Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ran 𝐺 ↔ ( seq 𝑀 ( + , 𝐹 ) ‘ 𝑗 ) ∈ ran seq 𝑀 ( + , 𝐹 ) ) ) |
129 |
125 128
|
mpbird |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ 𝑍 ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ran 𝐺 ) |
130 |
129
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ran 𝐺 ) |
131 |
130
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) → Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ran 𝐺 ) |
132 |
|
simp3 |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) → 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) |
133 |
|
breq2 |
⊢ ( 𝑦 = Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) → ( 𝑧 < 𝑦 ↔ 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) ) |
134 |
133
|
rspcev |
⊢ ( ( Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ∈ ran 𝐺 ∧ 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) |
135 |
131 132 134
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑗 ∈ 𝑍 ∧ 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) |
136 |
135
|
3exp |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( 𝑗 ∈ 𝑍 → ( 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) ) ) |
137 |
136
|
rexlimdv |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) ) |
138 |
137
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ( ∃ 𝑗 ∈ 𝑍 𝑧 < Σ 𝑘 ∈ ( 𝑀 ... 𝑗 ) ( 𝐹 ‘ 𝑘 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) ) |
139 |
120 138
|
mpd |
⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) ∧ 𝑧 < ( Σ^ ‘ 𝐹 ) ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) |
140 |
139
|
ex |
⊢ ( ( 𝜑 ∧ 𝑧 ∈ ℝ ) → ( 𝑧 < ( Σ^ ‘ 𝐹 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) ) |
141 |
140
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑧 ∈ ℝ ( 𝑧 < ( Σ^ ‘ 𝐹 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) ) |
142 |
|
supxr2 |
⊢ ( ( ( ran 𝐺 ⊆ ℝ* ∧ ( Σ^ ‘ 𝐹 ) ∈ ℝ* ) ∧ ( ∀ 𝑧 ∈ ran 𝐺 𝑧 ≤ ( Σ^ ‘ 𝐹 ) ∧ ∀ 𝑧 ∈ ℝ ( 𝑧 < ( Σ^ ‘ 𝐹 ) → ∃ 𝑦 ∈ ran 𝐺 𝑧 < 𝑦 ) ) ) → sup ( ran 𝐺 , ℝ* , < ) = ( Σ^ ‘ 𝐹 ) ) |
143 |
17 23 67 141 142
|
syl22anc |
⊢ ( 𝜑 → sup ( ran 𝐺 , ℝ* , < ) = ( Σ^ ‘ 𝐹 ) ) |
144 |
143
|
eqcomd |
⊢ ( 𝜑 → ( Σ^ ‘ 𝐹 ) = sup ( ran 𝐺 , ℝ* , < ) ) |