Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem25.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
2 |
|
fourierdlem25.qf |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
3 |
|
fourierdlem25.cel |
⊢ ( 𝜑 → 𝐶 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) |
4 |
|
fourierdlem25.cnel |
⊢ ( 𝜑 → ¬ 𝐶 ∈ ran 𝑄 ) |
5 |
|
fourierdlem25.i |
⊢ 𝐼 = sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } , ℝ , < ) |
6 |
|
ssrab2 |
⊢ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ⊆ ( 0 ..^ 𝑀 ) |
7 |
|
ltso |
⊢ < Or ℝ |
8 |
7
|
a1i |
⊢ ( 𝜑 → < Or ℝ ) |
9 |
|
fzofi |
⊢ ( 0 ..^ 𝑀 ) ∈ Fin |
10 |
|
ssfi |
⊢ ( ( ( 0 ..^ 𝑀 ) ∈ Fin ∧ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ⊆ ( 0 ..^ 𝑀 ) ) → { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ∈ Fin ) |
11 |
9 6 10
|
mp2an |
⊢ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ∈ Fin |
12 |
11
|
a1i |
⊢ ( 𝜑 → { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ∈ Fin ) |
13 |
|
0zd |
⊢ ( 𝜑 → 0 ∈ ℤ ) |
14 |
1
|
nnzd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
15 |
1
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑀 ) |
16 |
|
fzolb |
⊢ ( 0 ∈ ( 0 ..^ 𝑀 ) ↔ ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) ) |
17 |
13 14 15 16
|
syl3anbrc |
⊢ ( 𝜑 → 0 ∈ ( 0 ..^ 𝑀 ) ) |
18 |
|
elfzofz |
⊢ ( 0 ∈ ( 0 ..^ 𝑀 ) → 0 ∈ ( 0 ... 𝑀 ) ) |
19 |
17 18
|
syl |
⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑀 ) ) |
20 |
2 19
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ∈ ℝ ) |
21 |
1
|
nnnn0d |
⊢ ( 𝜑 → 𝑀 ∈ ℕ0 ) |
22 |
|
nn0uz |
⊢ ℕ0 = ( ℤ≥ ‘ 0 ) |
23 |
21 22
|
eleqtrdi |
⊢ ( 𝜑 → 𝑀 ∈ ( ℤ≥ ‘ 0 ) ) |
24 |
|
eluzfz2 |
⊢ ( 𝑀 ∈ ( ℤ≥ ‘ 0 ) → 𝑀 ∈ ( 0 ... 𝑀 ) ) |
25 |
23 24
|
syl |
⊢ ( 𝜑 → 𝑀 ∈ ( 0 ... 𝑀 ) ) |
26 |
2 25
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) ∈ ℝ ) |
27 |
20 26
|
iccssred |
⊢ ( 𝜑 → ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ⊆ ℝ ) |
28 |
27 3
|
sseldd |
⊢ ( 𝜑 → 𝐶 ∈ ℝ ) |
29 |
20
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ∈ ℝ* ) |
30 |
26
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝑀 ) ∈ ℝ* ) |
31 |
|
iccgelb |
⊢ ( ( ( 𝑄 ‘ 0 ) ∈ ℝ* ∧ ( 𝑄 ‘ 𝑀 ) ∈ ℝ* ∧ 𝐶 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → ( 𝑄 ‘ 0 ) ≤ 𝐶 ) |
32 |
29 30 3 31
|
syl3anc |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ≤ 𝐶 ) |
33 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐶 = ( 𝑄 ‘ 0 ) ) → 𝐶 = ( 𝑄 ‘ 0 ) ) |
34 |
2
|
ffnd |
⊢ ( 𝜑 → 𝑄 Fn ( 0 ... 𝑀 ) ) |
35 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 = ( 𝑄 ‘ 0 ) ) → 𝑄 Fn ( 0 ... 𝑀 ) ) |
36 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐶 = ( 𝑄 ‘ 0 ) ) → 0 ∈ ( 0 ... 𝑀 ) ) |
37 |
|
fnfvelrn |
⊢ ( ( 𝑄 Fn ( 0 ... 𝑀 ) ∧ 0 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 0 ) ∈ ran 𝑄 ) |
38 |
35 36 37
|
syl2anc |
⊢ ( ( 𝜑 ∧ 𝐶 = ( 𝑄 ‘ 0 ) ) → ( 𝑄 ‘ 0 ) ∈ ran 𝑄 ) |
39 |
33 38
|
eqeltrd |
⊢ ( ( 𝜑 ∧ 𝐶 = ( 𝑄 ‘ 0 ) ) → 𝐶 ∈ ran 𝑄 ) |
40 |
4 39
|
mtand |
⊢ ( 𝜑 → ¬ 𝐶 = ( 𝑄 ‘ 0 ) ) |
41 |
40
|
neqned |
⊢ ( 𝜑 → 𝐶 ≠ ( 𝑄 ‘ 0 ) ) |
42 |
20 28 32 41
|
leneltd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) < 𝐶 ) |
43 |
|
fveq2 |
⊢ ( 𝑘 = 0 → ( 𝑄 ‘ 𝑘 ) = ( 𝑄 ‘ 0 ) ) |
44 |
43
|
breq1d |
⊢ ( 𝑘 = 0 → ( ( 𝑄 ‘ 𝑘 ) < 𝐶 ↔ ( 𝑄 ‘ 0 ) < 𝐶 ) ) |
45 |
44
|
elrab |
⊢ ( 0 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ↔ ( 0 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝑄 ‘ 0 ) < 𝐶 ) ) |
46 |
17 42 45
|
sylanbrc |
⊢ ( 𝜑 → 0 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ) |
47 |
46
|
ne0d |
⊢ ( 𝜑 → { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ≠ ∅ ) |
48 |
|
fzossfz |
⊢ ( 0 ..^ 𝑀 ) ⊆ ( 0 ... 𝑀 ) |
49 |
|
fzssz |
⊢ ( 0 ... 𝑀 ) ⊆ ℤ |
50 |
|
zssre |
⊢ ℤ ⊆ ℝ |
51 |
49 50
|
sstri |
⊢ ( 0 ... 𝑀 ) ⊆ ℝ |
52 |
48 51
|
sstri |
⊢ ( 0 ..^ 𝑀 ) ⊆ ℝ |
53 |
6 52
|
sstri |
⊢ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ⊆ ℝ |
54 |
53
|
a1i |
⊢ ( 𝜑 → { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ⊆ ℝ ) |
55 |
|
fisupcl |
⊢ ( ( < Or ℝ ∧ ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ∈ Fin ∧ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ≠ ∅ ∧ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ⊆ ℝ ) ) → sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } , ℝ , < ) ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ) |
56 |
8 12 47 54 55
|
syl13anc |
⊢ ( 𝜑 → sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } , ℝ , < ) ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ) |
57 |
6 56
|
sselid |
⊢ ( 𝜑 → sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } , ℝ , < ) ∈ ( 0 ..^ 𝑀 ) ) |
58 |
5 57
|
eqeltrid |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ..^ 𝑀 ) ) |
59 |
48 58
|
sselid |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ... 𝑀 ) ) |
60 |
2 59
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝐼 ) ∈ ℝ ) |
61 |
60
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝐼 ) ∈ ℝ* ) |
62 |
|
fzofzp1 |
⊢ ( 𝐼 ∈ ( 0 ..^ 𝑀 ) → ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
63 |
58 62
|
syl |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
64 |
2 63
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) |
65 |
64
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ* ) |
66 |
5 56
|
eqeltrid |
⊢ ( 𝜑 → 𝐼 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ) |
67 |
|
fveq2 |
⊢ ( 𝑘 = 𝐼 → ( 𝑄 ‘ 𝑘 ) = ( 𝑄 ‘ 𝐼 ) ) |
68 |
67
|
breq1d |
⊢ ( 𝑘 = 𝐼 → ( ( 𝑄 ‘ 𝑘 ) < 𝐶 ↔ ( 𝑄 ‘ 𝐼 ) < 𝐶 ) ) |
69 |
68
|
elrab |
⊢ ( 𝐼 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ↔ ( 𝐼 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝑄 ‘ 𝐼 ) < 𝐶 ) ) |
70 |
66 69
|
sylib |
⊢ ( 𝜑 → ( 𝐼 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝑄 ‘ 𝐼 ) < 𝐶 ) ) |
71 |
70
|
simprd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝐼 ) < 𝐶 ) |
72 |
52 58
|
sselid |
⊢ ( 𝜑 → 𝐼 ∈ ℝ ) |
73 |
|
ltp1 |
⊢ ( 𝐼 ∈ ℝ → 𝐼 < ( 𝐼 + 1 ) ) |
74 |
|
id |
⊢ ( 𝐼 ∈ ℝ → 𝐼 ∈ ℝ ) |
75 |
|
peano2re |
⊢ ( 𝐼 ∈ ℝ → ( 𝐼 + 1 ) ∈ ℝ ) |
76 |
74 75
|
ltnled |
⊢ ( 𝐼 ∈ ℝ → ( 𝐼 < ( 𝐼 + 1 ) ↔ ¬ ( 𝐼 + 1 ) ≤ 𝐼 ) ) |
77 |
73 76
|
mpbid |
⊢ ( 𝐼 ∈ ℝ → ¬ ( 𝐼 + 1 ) ≤ 𝐼 ) |
78 |
72 77
|
syl |
⊢ ( 𝜑 → ¬ ( 𝐼 + 1 ) ≤ 𝐼 ) |
79 |
48 49
|
sstri |
⊢ ( 0 ..^ 𝑀 ) ⊆ ℤ |
80 |
6 79
|
sstri |
⊢ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ⊆ ℤ |
81 |
80
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ⊆ ℤ ) |
82 |
|
elrabi |
⊢ ( ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } → ℎ ∈ ( 0 ..^ 𝑀 ) ) |
83 |
|
elfzo0le |
⊢ ( ℎ ∈ ( 0 ..^ 𝑀 ) → ℎ ≤ 𝑀 ) |
84 |
82 83
|
syl |
⊢ ( ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } → ℎ ≤ 𝑀 ) |
85 |
84
|
adantl |
⊢ ( ( 𝜑 ∧ ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ) → ℎ ≤ 𝑀 ) |
86 |
85
|
ralrimiva |
⊢ ( 𝜑 → ∀ ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ℎ ≤ 𝑀 ) |
87 |
|
breq2 |
⊢ ( 𝑚 = 𝑀 → ( ℎ ≤ 𝑚 ↔ ℎ ≤ 𝑀 ) ) |
88 |
87
|
ralbidv |
⊢ ( 𝑚 = 𝑀 → ( ∀ ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ℎ ≤ 𝑚 ↔ ∀ ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ℎ ≤ 𝑀 ) ) |
89 |
88
|
rspcev |
⊢ ( ( 𝑀 ∈ ℤ ∧ ∀ ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ℎ ≤ 𝑀 ) → ∃ 𝑚 ∈ ℤ ∀ ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ℎ ≤ 𝑚 ) |
90 |
14 86 89
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑚 ∈ ℤ ∀ ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ℎ ≤ 𝑚 ) |
91 |
90
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ∃ 𝑚 ∈ ℤ ∀ ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ℎ ≤ 𝑚 ) |
92 |
|
elfzuz |
⊢ ( ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) → ( 𝐼 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
93 |
63 92
|
syl |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
94 |
93
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( 𝐼 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ) |
95 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → 𝑀 ∈ ℤ ) |
96 |
51 63
|
sselid |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ∈ ℝ ) |
97 |
96
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( 𝐼 + 1 ) ∈ ℝ ) |
98 |
95
|
zred |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → 𝑀 ∈ ℝ ) |
99 |
|
elfzle2 |
⊢ ( ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) → ( 𝐼 + 1 ) ≤ 𝑀 ) |
100 |
63 99
|
syl |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ≤ 𝑀 ) |
101 |
100
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( 𝐼 + 1 ) ≤ 𝑀 ) |
102 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) |
103 |
64
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) |
104 |
28
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → 𝐶 ∈ ℝ ) |
105 |
103 104
|
ltnled |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ↔ ¬ 𝐶 ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) |
106 |
102 105
|
mpbid |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ¬ 𝐶 ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
107 |
|
iccleub |
⊢ ( ( ( 𝑄 ‘ 0 ) ∈ ℝ* ∧ ( 𝑄 ‘ 𝑀 ) ∈ ℝ* ∧ 𝐶 ∈ ( ( 𝑄 ‘ 0 ) [,] ( 𝑄 ‘ 𝑀 ) ) ) → 𝐶 ≤ ( 𝑄 ‘ 𝑀 ) ) |
108 |
29 30 3 107
|
syl3anc |
⊢ ( 𝜑 → 𝐶 ≤ ( 𝑄 ‘ 𝑀 ) ) |
109 |
108
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑀 = ( 𝐼 + 1 ) ) → 𝐶 ≤ ( 𝑄 ‘ 𝑀 ) ) |
110 |
|
fveq2 |
⊢ ( 𝑀 = ( 𝐼 + 1 ) → ( 𝑄 ‘ 𝑀 ) = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
111 |
110
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑀 = ( 𝐼 + 1 ) ) → ( 𝑄 ‘ 𝑀 ) = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
112 |
109 111
|
breqtrd |
⊢ ( ( 𝜑 ∧ 𝑀 = ( 𝐼 + 1 ) ) → 𝐶 ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
113 |
112
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) ∧ 𝑀 = ( 𝐼 + 1 ) ) → 𝐶 ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
114 |
106 113
|
mtand |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ¬ 𝑀 = ( 𝐼 + 1 ) ) |
115 |
114
|
neqned |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → 𝑀 ≠ ( 𝐼 + 1 ) ) |
116 |
97 98 101 115
|
leneltd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( 𝐼 + 1 ) < 𝑀 ) |
117 |
|
elfzo2 |
⊢ ( ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ↔ ( ( 𝐼 + 1 ) ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑀 ∈ ℤ ∧ ( 𝐼 + 1 ) < 𝑀 ) ) |
118 |
94 95 116 117
|
syl3anbrc |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) |
119 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝐼 + 1 ) → ( 𝑄 ‘ 𝑘 ) = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
120 |
119
|
breq1d |
⊢ ( 𝑘 = ( 𝐼 + 1 ) → ( ( 𝑄 ‘ 𝑘 ) < 𝐶 ↔ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) ) |
121 |
120
|
elrab |
⊢ ( ( 𝐼 + 1 ) ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ↔ ( ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) ) |
122 |
118 102 121
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( 𝐼 + 1 ) ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ) |
123 |
|
suprzub |
⊢ ( ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ⊆ ℤ ∧ ∃ 𝑚 ∈ ℤ ∀ ℎ ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ℎ ≤ 𝑚 ∧ ( 𝐼 + 1 ) ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } ) → ( 𝐼 + 1 ) ≤ sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } , ℝ , < ) ) |
124 |
81 91 122 123
|
syl3anc |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( 𝐼 + 1 ) ≤ sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) < 𝐶 } , ℝ , < ) ) |
125 |
124 5
|
breqtrrdi |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) → ( 𝐼 + 1 ) ≤ 𝐼 ) |
126 |
78 125
|
mtand |
⊢ ( 𝜑 → ¬ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ) |
127 |
|
eqcom |
⊢ ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ↔ 𝐶 = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
128 |
127
|
biimpi |
⊢ ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 → 𝐶 = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
129 |
128
|
adantl |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) → 𝐶 = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
130 |
34
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) → 𝑄 Fn ( 0 ... 𝑀 ) ) |
131 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) → ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
132 |
|
fnfvelrn |
⊢ ( ( 𝑄 Fn ( 0 ... 𝑀 ) ∧ ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ran 𝑄 ) |
133 |
130 131 132
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ran 𝑄 ) |
134 |
129 133
|
eqeltrd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) → 𝐶 ∈ ran 𝑄 ) |
135 |
4 134
|
mtand |
⊢ ( 𝜑 → ¬ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) |
136 |
126 135
|
jca |
⊢ ( 𝜑 → ( ¬ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ∧ ¬ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) ) |
137 |
|
pm4.56 |
⊢ ( ( ¬ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ∧ ¬ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) ↔ ¬ ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ∨ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) ) |
138 |
136 137
|
sylib |
⊢ ( 𝜑 → ¬ ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ∨ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) ) |
139 |
64 28
|
leloed |
⊢ ( 𝜑 → ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ 𝐶 ↔ ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐶 ∨ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = 𝐶 ) ) ) |
140 |
138 139
|
mtbird |
⊢ ( 𝜑 → ¬ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ 𝐶 ) |
141 |
28 64
|
ltnled |
⊢ ( 𝜑 → ( 𝐶 < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ↔ ¬ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ 𝐶 ) ) |
142 |
140 141
|
mpbird |
⊢ ( 𝜑 → 𝐶 < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
143 |
61 65 28 71 142
|
eliood |
⊢ ( 𝜑 → 𝐶 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) |
144 |
|
fveq2 |
⊢ ( 𝑗 = 𝐼 → ( 𝑄 ‘ 𝑗 ) = ( 𝑄 ‘ 𝐼 ) ) |
145 |
|
oveq1 |
⊢ ( 𝑗 = 𝐼 → ( 𝑗 + 1 ) = ( 𝐼 + 1 ) ) |
146 |
145
|
fveq2d |
⊢ ( 𝑗 = 𝐼 → ( 𝑄 ‘ ( 𝑗 + 1 ) ) = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
147 |
144 146
|
oveq12d |
⊢ ( 𝑗 = 𝐼 → ( ( 𝑄 ‘ 𝑗 ) (,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) = ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) |
148 |
147
|
eleq2d |
⊢ ( 𝑗 = 𝐼 → ( 𝐶 ∈ ( ( 𝑄 ‘ 𝑗 ) (,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ↔ 𝐶 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) ) |
149 |
148
|
rspcev |
⊢ ( ( 𝐼 ∈ ( 0 ..^ 𝑀 ) ∧ 𝐶 ∈ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → ∃ 𝑗 ∈ ( 0 ..^ 𝑀 ) 𝐶 ∈ ( ( 𝑄 ‘ 𝑗 ) (,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |
150 |
58 143 149
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑗 ∈ ( 0 ..^ 𝑀 ) 𝐶 ∈ ( ( 𝑄 ‘ 𝑗 ) (,) ( 𝑄 ‘ ( 𝑗 + 1 ) ) ) ) |