Step |
Hyp |
Ref |
Expression |
1 |
|
sticksstones6.1 |
⊢ ( 𝜑 → 𝑁 ∈ ℕ0 ) |
2 |
|
sticksstones6.2 |
⊢ ( 𝜑 → 𝐾 ∈ ℕ0 ) |
3 |
|
sticksstones6.3 |
⊢ ( 𝜑 → 𝐺 : ( 1 ... ( 𝐾 + 1 ) ) ⟶ ℕ0 ) |
4 |
|
sticksstones6.4 |
⊢ ( 𝜑 → 𝑋 ∈ ( 1 ... 𝐾 ) ) |
5 |
|
sticksstones6.5 |
⊢ ( 𝜑 → 𝑌 ∈ ( 1 ... 𝐾 ) ) |
6 |
|
sticksstones6.6 |
⊢ ( 𝜑 → 𝑋 < 𝑌 ) |
7 |
|
sticksstones6.7 |
⊢ 𝐹 = ( 𝑥 ∈ ( 1 ... 𝐾 ) ↦ ( 𝑥 + Σ 𝑖 ∈ ( 1 ... 𝑥 ) ( 𝐺 ‘ 𝑖 ) ) ) |
8 |
|
elfznn |
⊢ ( 𝑋 ∈ ( 1 ... 𝐾 ) → 𝑋 ∈ ℕ ) |
9 |
4 8
|
syl |
⊢ ( 𝜑 → 𝑋 ∈ ℕ ) |
10 |
9
|
nnred |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
11 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... 𝑋 ) ∈ Fin ) |
12 |
|
1zzd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 1 ∈ ℤ ) |
13 |
2
|
nn0zd |
⊢ ( 𝜑 → 𝐾 ∈ ℤ ) |
14 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝐾 ∈ ℤ ) |
15 |
14
|
peano2zd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → ( 𝐾 + 1 ) ∈ ℤ ) |
16 |
|
elfznn |
⊢ ( 𝑖 ∈ ( 1 ... 𝑋 ) → 𝑖 ∈ ℕ ) |
17 |
16
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝑖 ∈ ℕ ) |
18 |
17
|
nnzd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝑖 ∈ ℤ ) |
19 |
17
|
nnge1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 1 ≤ 𝑖 ) |
20 |
17
|
nnred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝑖 ∈ ℝ ) |
21 |
14
|
zred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝐾 ∈ ℝ ) |
22 |
15
|
zred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → ( 𝐾 + 1 ) ∈ ℝ ) |
23 |
9
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝑋 ∈ ℕ ) |
24 |
23
|
nnred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝑋 ∈ ℝ ) |
25 |
|
elfzle2 |
⊢ ( 𝑖 ∈ ( 1 ... 𝑋 ) → 𝑖 ≤ 𝑋 ) |
26 |
25
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝑖 ≤ 𝑋 ) |
27 |
|
elfzle2 |
⊢ ( 𝑋 ∈ ( 1 ... 𝐾 ) → 𝑋 ≤ 𝐾 ) |
28 |
4 27
|
syl |
⊢ ( 𝜑 → 𝑋 ≤ 𝐾 ) |
29 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝑋 ≤ 𝐾 ) |
30 |
20 24 21 26 29
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝑖 ≤ 𝐾 ) |
31 |
21
|
lep1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝐾 ≤ ( 𝐾 + 1 ) ) |
32 |
20 21 22 30 31
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝑖 ≤ ( 𝐾 + 1 ) ) |
33 |
12 15 18 19 32
|
elfzd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → 𝑖 ∈ ( 1 ... ( 𝐾 + 1 ) ) ) |
34 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... ( 𝐾 + 1 ) ) ) → 𝐺 : ( 1 ... ( 𝐾 + 1 ) ) ⟶ ℕ0 ) |
35 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... ( 𝐾 + 1 ) ) ) → 𝑖 ∈ ( 1 ... ( 𝐾 + 1 ) ) ) |
36 |
34 35
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... ( 𝐾 + 1 ) ) ) → ( 𝐺 ‘ 𝑖 ) ∈ ℕ0 ) |
37 |
36
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) ∧ 𝑖 ∈ ( 1 ... ( 𝐾 + 1 ) ) ) → ( 𝐺 ‘ 𝑖 ) ∈ ℕ0 ) |
38 |
33 37
|
mpdan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑋 ) ) → ( 𝐺 ‘ 𝑖 ) ∈ ℕ0 ) |
39 |
11 38
|
fsumnn0cl |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) ∈ ℕ0 ) |
40 |
39
|
nn0red |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) ∈ ℝ ) |
41 |
|
elfznn |
⊢ ( 𝑌 ∈ ( 1 ... 𝐾 ) → 𝑌 ∈ ℕ ) |
42 |
5 41
|
syl |
⊢ ( 𝜑 → 𝑌 ∈ ℕ ) |
43 |
42
|
nnred |
⊢ ( 𝜑 → 𝑌 ∈ ℝ ) |
44 |
|
fzfid |
⊢ ( 𝜑 → ( ( 𝑋 + 1 ) ... 𝑌 ) ∈ Fin ) |
45 |
|
1zzd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 1 ∈ ℤ ) |
46 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝐾 ∈ ℤ ) |
47 |
46
|
peano2zd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → ( 𝐾 + 1 ) ∈ ℤ ) |
48 |
|
elfzelz |
⊢ ( 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) → 𝑖 ∈ ℤ ) |
49 |
48
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝑖 ∈ ℤ ) |
50 |
|
1red |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 1 ∈ ℝ ) |
51 |
10
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝑋 ∈ ℝ ) |
52 |
51 50
|
readdcld |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → ( 𝑋 + 1 ) ∈ ℝ ) |
53 |
49
|
zred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝑖 ∈ ℝ ) |
54 |
|
1red |
⊢ ( 𝜑 → 1 ∈ ℝ ) |
55 |
10 54
|
readdcld |
⊢ ( 𝜑 → ( 𝑋 + 1 ) ∈ ℝ ) |
56 |
9
|
nnge1d |
⊢ ( 𝜑 → 1 ≤ 𝑋 ) |
57 |
10
|
ltp1d |
⊢ ( 𝜑 → 𝑋 < ( 𝑋 + 1 ) ) |
58 |
10 55 57
|
ltled |
⊢ ( 𝜑 → 𝑋 ≤ ( 𝑋 + 1 ) ) |
59 |
54 10 55 56 58
|
letrd |
⊢ ( 𝜑 → 1 ≤ ( 𝑋 + 1 ) ) |
60 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 1 ≤ ( 𝑋 + 1 ) ) |
61 |
|
elfzle1 |
⊢ ( 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) → ( 𝑋 + 1 ) ≤ 𝑖 ) |
62 |
61
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → ( 𝑋 + 1 ) ≤ 𝑖 ) |
63 |
50 52 53 60 62
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 1 ≤ 𝑖 ) |
64 |
43
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝑌 ∈ ℝ ) |
65 |
47
|
zred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → ( 𝐾 + 1 ) ∈ ℝ ) |
66 |
|
elfzle2 |
⊢ ( 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) → 𝑖 ≤ 𝑌 ) |
67 |
66
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝑖 ≤ 𝑌 ) |
68 |
46
|
zred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝐾 ∈ ℝ ) |
69 |
|
elfzle2 |
⊢ ( 𝑌 ∈ ( 1 ... 𝐾 ) → 𝑌 ≤ 𝐾 ) |
70 |
5 69
|
syl |
⊢ ( 𝜑 → 𝑌 ≤ 𝐾 ) |
71 |
70
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝑌 ≤ 𝐾 ) |
72 |
68
|
lep1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝐾 ≤ ( 𝐾 + 1 ) ) |
73 |
64 68 65 71 72
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝑌 ≤ ( 𝐾 + 1 ) ) |
74 |
53 64 65 67 73
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝑖 ≤ ( 𝐾 + 1 ) ) |
75 |
45 47 49 63 74
|
elfzd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 𝑖 ∈ ( 1 ... ( 𝐾 + 1 ) ) ) |
76 |
36
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) ∧ 𝑖 ∈ ( 1 ... ( 𝐾 + 1 ) ) ) → ( 𝐺 ‘ 𝑖 ) ∈ ℕ0 ) |
77 |
75 76
|
mpdan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → ( 𝐺 ‘ 𝑖 ) ∈ ℕ0 ) |
78 |
77
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → ( 𝐺 ‘ 𝑖 ) ∈ ℝ ) |
79 |
44 78
|
fsumrecl |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ∈ ℝ ) |
80 |
40 79
|
readdcld |
⊢ ( 𝜑 → ( Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) + Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ∈ ℝ ) |
81 |
77
|
nn0ge0d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ) → 0 ≤ ( 𝐺 ‘ 𝑖 ) ) |
82 |
44 78 81
|
fsumge0 |
⊢ ( 𝜑 → 0 ≤ Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) |
83 |
40 79
|
addge01d |
⊢ ( 𝜑 → ( 0 ≤ Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ↔ Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) ≤ ( Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) + Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ) ) |
84 |
82 83
|
mpbid |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) ≤ ( Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) + Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ) |
85 |
10 40 43 80 6 84
|
ltleaddd |
⊢ ( 𝜑 → ( 𝑋 + Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) ) < ( 𝑌 + ( Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) + Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ) ) |
86 |
7
|
a1i |
⊢ ( 𝜑 → 𝐹 = ( 𝑥 ∈ ( 1 ... 𝐾 ) ↦ ( 𝑥 + Σ 𝑖 ∈ ( 1 ... 𝑥 ) ( 𝐺 ‘ 𝑖 ) ) ) ) |
87 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → 𝑥 = 𝑋 ) |
88 |
87
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 1 ... 𝑥 ) = ( 1 ... 𝑋 ) ) |
89 |
88
|
sumeq1d |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → Σ 𝑖 ∈ ( 1 ... 𝑥 ) ( 𝐺 ‘ 𝑖 ) = Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) ) |
90 |
87 89
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑋 ) → ( 𝑥 + Σ 𝑖 ∈ ( 1 ... 𝑥 ) ( 𝐺 ‘ 𝑖 ) ) = ( 𝑋 + Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) ) ) |
91 |
9
|
nnnn0d |
⊢ ( 𝜑 → 𝑋 ∈ ℕ0 ) |
92 |
91 39
|
nn0addcld |
⊢ ( 𝜑 → ( 𝑋 + Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) ) ∈ ℕ0 ) |
93 |
86 90 4 92
|
fvmptd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) = ( 𝑋 + Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) ) ) |
94 |
93
|
eqcomd |
⊢ ( 𝜑 → ( 𝑋 + Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) ) = ( 𝐹 ‘ 𝑋 ) ) |
95 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑌 ) → 𝑥 = 𝑌 ) |
96 |
95
|
oveq2d |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑌 ) → ( 1 ... 𝑥 ) = ( 1 ... 𝑌 ) ) |
97 |
96
|
sumeq1d |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑌 ) → Σ 𝑖 ∈ ( 1 ... 𝑥 ) ( 𝐺 ‘ 𝑖 ) = Σ 𝑖 ∈ ( 1 ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) |
98 |
95 97
|
oveq12d |
⊢ ( ( 𝜑 ∧ 𝑥 = 𝑌 ) → ( 𝑥 + Σ 𝑖 ∈ ( 1 ... 𝑥 ) ( 𝐺 ‘ 𝑖 ) ) = ( 𝑌 + Σ 𝑖 ∈ ( 1 ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ) |
99 |
42
|
nnnn0d |
⊢ ( 𝜑 → 𝑌 ∈ ℕ0 ) |
100 |
|
fzfid |
⊢ ( 𝜑 → ( 1 ... 𝑌 ) ∈ Fin ) |
101 |
|
1zzd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 1 ∈ ℤ ) |
102 |
13
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝐾 ∈ ℤ ) |
103 |
102
|
peano2zd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → ( 𝐾 + 1 ) ∈ ℤ ) |
104 |
|
elfzelz |
⊢ ( 𝑖 ∈ ( 1 ... 𝑌 ) → 𝑖 ∈ ℤ ) |
105 |
104
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝑖 ∈ ℤ ) |
106 |
|
elfzle1 |
⊢ ( 𝑖 ∈ ( 1 ... 𝑌 ) → 1 ≤ 𝑖 ) |
107 |
106
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 1 ≤ 𝑖 ) |
108 |
105
|
zred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝑖 ∈ ℝ ) |
109 |
43
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝑌 ∈ ℝ ) |
110 |
103
|
zred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → ( 𝐾 + 1 ) ∈ ℝ ) |
111 |
|
elfzle2 |
⊢ ( 𝑖 ∈ ( 1 ... 𝑌 ) → 𝑖 ≤ 𝑌 ) |
112 |
111
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝑖 ≤ 𝑌 ) |
113 |
102
|
zred |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝐾 ∈ ℝ ) |
114 |
70
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝑌 ≤ 𝐾 ) |
115 |
113
|
lep1d |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝐾 ≤ ( 𝐾 + 1 ) ) |
116 |
109 113 110 114 115
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝑌 ≤ ( 𝐾 + 1 ) ) |
117 |
108 109 110 112 116
|
letrd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝑖 ≤ ( 𝐾 + 1 ) ) |
118 |
101 103 105 107 117
|
elfzd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → 𝑖 ∈ ( 1 ... ( 𝐾 + 1 ) ) ) |
119 |
36
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) ∧ 𝑖 ∈ ( 1 ... ( 𝐾 + 1 ) ) ) → ( 𝐺 ‘ 𝑖 ) ∈ ℕ0 ) |
120 |
118 119
|
mpdan |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → ( 𝐺 ‘ 𝑖 ) ∈ ℕ0 ) |
121 |
100 120
|
fsumnn0cl |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 1 ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ∈ ℕ0 ) |
122 |
99 121
|
nn0addcld |
⊢ ( 𝜑 → ( 𝑌 + Σ 𝑖 ∈ ( 1 ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ∈ ℕ0 ) |
123 |
86 98 5 122
|
fvmptd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑌 ) = ( 𝑌 + Σ 𝑖 ∈ ( 1 ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ) |
124 |
|
fzdisj |
⊢ ( 𝑋 < ( 𝑋 + 1 ) → ( ( 1 ... 𝑋 ) ∩ ( ( 𝑋 + 1 ) ... 𝑌 ) ) = ∅ ) |
125 |
57 124
|
syl |
⊢ ( 𝜑 → ( ( 1 ... 𝑋 ) ∩ ( ( 𝑋 + 1 ) ... 𝑌 ) ) = ∅ ) |
126 |
|
1zzd |
⊢ ( 𝜑 → 1 ∈ ℤ ) |
127 |
99
|
nn0zd |
⊢ ( 𝜑 → 𝑌 ∈ ℤ ) |
128 |
91
|
nn0zd |
⊢ ( 𝜑 → 𝑋 ∈ ℤ ) |
129 |
10 43 6
|
ltled |
⊢ ( 𝜑 → 𝑋 ≤ 𝑌 ) |
130 |
126 127 128 56 129
|
elfzd |
⊢ ( 𝜑 → 𝑋 ∈ ( 1 ... 𝑌 ) ) |
131 |
|
fzsplit |
⊢ ( 𝑋 ∈ ( 1 ... 𝑌 ) → ( 1 ... 𝑌 ) = ( ( 1 ... 𝑋 ) ∪ ( ( 𝑋 + 1 ) ... 𝑌 ) ) ) |
132 |
130 131
|
syl |
⊢ ( 𝜑 → ( 1 ... 𝑌 ) = ( ( 1 ... 𝑋 ) ∪ ( ( 𝑋 + 1 ) ... 𝑌 ) ) ) |
133 |
120
|
nn0red |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → ( 𝐺 ‘ 𝑖 ) ∈ ℝ ) |
134 |
133
|
recnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 1 ... 𝑌 ) ) → ( 𝐺 ‘ 𝑖 ) ∈ ℂ ) |
135 |
125 132 100 134
|
fsumsplit |
⊢ ( 𝜑 → Σ 𝑖 ∈ ( 1 ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) = ( Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) + Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ) |
136 |
135
|
oveq2d |
⊢ ( 𝜑 → ( 𝑌 + Σ 𝑖 ∈ ( 1 ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) = ( 𝑌 + ( Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) + Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ) ) |
137 |
123 136
|
eqtrd |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑌 ) = ( 𝑌 + ( Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) + Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ) ) |
138 |
137
|
eqcomd |
⊢ ( 𝜑 → ( 𝑌 + ( Σ 𝑖 ∈ ( 1 ... 𝑋 ) ( 𝐺 ‘ 𝑖 ) + Σ 𝑖 ∈ ( ( 𝑋 + 1 ) ... 𝑌 ) ( 𝐺 ‘ 𝑖 ) ) ) = ( 𝐹 ‘ 𝑌 ) ) |
139 |
85 94 138
|
3brtr3d |
⊢ ( 𝜑 → ( 𝐹 ‘ 𝑋 ) < ( 𝐹 ‘ 𝑌 ) ) |