Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem12.1 |
⊢ 𝑃 = ( 𝑚 ∈ ℕ ↦ { 𝑝 ∈ ( ℝ ↑m ( 0 ... 𝑚 ) ) ∣ ( ( ( 𝑝 ‘ 0 ) = 𝐴 ∧ ( 𝑝 ‘ 𝑚 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑚 ) ( 𝑝 ‘ 𝑖 ) < ( 𝑝 ‘ ( 𝑖 + 1 ) ) ) } ) |
2 |
|
fourierdlem12.2 |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
3 |
|
fourierdlem12.3 |
⊢ ( 𝜑 → 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ) |
4 |
|
fourierdlem12.4 |
⊢ ( 𝜑 → 𝑋 ∈ ran 𝑄 ) |
5 |
1
|
fourierdlem2 |
⊢ ( 𝑀 ∈ ℕ → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
6 |
2 5
|
syl |
⊢ ( 𝜑 → ( 𝑄 ∈ ( 𝑃 ‘ 𝑀 ) ↔ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) ) |
7 |
3 6
|
mpbid |
⊢ ( 𝜑 → ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ∧ ( ( ( 𝑄 ‘ 0 ) = 𝐴 ∧ ( 𝑄 ‘ 𝑀 ) = 𝐵 ) ∧ ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
8 |
7
|
simpld |
⊢ ( 𝜑 → 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) ) |
9 |
|
elmapi |
⊢ ( 𝑄 ∈ ( ℝ ↑m ( 0 ... 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
10 |
|
ffn |
⊢ ( 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ → 𝑄 Fn ( 0 ... 𝑀 ) ) |
11 |
8 9 10
|
3syl |
⊢ ( 𝜑 → 𝑄 Fn ( 0 ... 𝑀 ) ) |
12 |
|
fvelrnb |
⊢ ( 𝑄 Fn ( 0 ... 𝑀 ) → ( 𝑋 ∈ ran 𝑄 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ) |
13 |
11 12
|
syl |
⊢ ( 𝜑 → ( 𝑋 ∈ ran 𝑄 ↔ ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ) |
14 |
4 13
|
mpbid |
⊢ ( 𝜑 → ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = 𝑋 ) |
15 |
14
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = 𝑋 ) |
16 |
8 9
|
syl |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
17 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
18 |
|
fzofzp1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
19 |
18
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑖 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
20 |
17 19
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
21 |
20
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
22 |
21
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑖 < 𝑗 ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ ) |
23 |
|
frn |
⊢ ( 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ → ran 𝑄 ⊆ ℝ ) |
24 |
16 23
|
syl |
⊢ ( 𝜑 → ran 𝑄 ⊆ ℝ ) |
25 |
24 4
|
sseldd |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
26 |
25
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → 𝑋 ∈ ℝ ) |
27 |
26
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑖 < 𝑗 ) → 𝑋 ∈ ℝ ) |
28 |
17
|
ffvelrnda |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ ) |
29 |
28
|
3adant3 |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ ) |
30 |
29
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑖 < 𝑗 ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ ) |
31 |
|
simpr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → 𝑖 < 𝑗 ) |
32 |
|
elfzoelz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ℤ ) |
33 |
32
|
ad2antrr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → 𝑖 ∈ ℤ ) |
34 |
|
elfzelz |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℤ ) |
35 |
34
|
ad2antlr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → 𝑗 ∈ ℤ ) |
36 |
|
zltp1le |
⊢ ( ( 𝑖 ∈ ℤ ∧ 𝑗 ∈ ℤ ) → ( 𝑖 < 𝑗 ↔ ( 𝑖 + 1 ) ≤ 𝑗 ) ) |
37 |
33 35 36
|
syl2anc |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → ( 𝑖 < 𝑗 ↔ ( 𝑖 + 1 ) ≤ 𝑗 ) ) |
38 |
31 37
|
mpbid |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → ( 𝑖 + 1 ) ≤ 𝑗 ) |
39 |
33
|
peano2zd |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → ( 𝑖 + 1 ) ∈ ℤ ) |
40 |
|
eluz |
⊢ ( ( ( 𝑖 + 1 ) ∈ ℤ ∧ 𝑗 ∈ ℤ ) → ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑖 + 1 ) ≤ 𝑗 ) ) |
41 |
39 35 40
|
syl2anc |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → ( 𝑗 ∈ ( ℤ≥ ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑖 + 1 ) ≤ 𝑗 ) ) |
42 |
38 41
|
mpbird |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → 𝑗 ∈ ( ℤ≥ ‘ ( 𝑖 + 1 ) ) ) |
43 |
42
|
adantlll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → 𝑗 ∈ ( ℤ≥ ‘ ( 𝑖 + 1 ) ) ) |
44 |
17
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
45 |
|
0red |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 0 ∈ ℝ ) |
46 |
|
elfzelz |
⊢ ( 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) → 𝑤 ∈ ℤ ) |
47 |
46
|
zred |
⊢ ( 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) → 𝑤 ∈ ℝ ) |
48 |
47
|
adantl |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑤 ∈ ℝ ) |
49 |
32
|
peano2zd |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ℤ ) |
50 |
49
|
zred |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → ( 𝑖 + 1 ) ∈ ℝ ) |
51 |
50
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → ( 𝑖 + 1 ) ∈ ℝ ) |
52 |
32
|
zred |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ℝ ) |
53 |
52
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑖 ∈ ℝ ) |
54 |
|
elfzole1 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 0 ≤ 𝑖 ) |
55 |
54
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 0 ≤ 𝑖 ) |
56 |
53
|
ltp1d |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑖 < ( 𝑖 + 1 ) ) |
57 |
45 53 51 55 56
|
lelttrd |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 0 < ( 𝑖 + 1 ) ) |
58 |
|
elfzle1 |
⊢ ( 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) → ( 𝑖 + 1 ) ≤ 𝑤 ) |
59 |
58
|
adantl |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → ( 𝑖 + 1 ) ≤ 𝑤 ) |
60 |
45 51 48 57 59
|
ltletrd |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 0 < 𝑤 ) |
61 |
45 48 60
|
ltled |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 0 ≤ 𝑤 ) |
62 |
61
|
adantlr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 0 ≤ 𝑤 ) |
63 |
47
|
adantl |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑤 ∈ ℝ ) |
64 |
34
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ∈ ℝ ) |
65 |
64
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑗 ∈ ℝ ) |
66 |
|
elfzel2 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ℤ ) |
67 |
66
|
zred |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑀 ∈ ℝ ) |
68 |
67
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑀 ∈ ℝ ) |
69 |
|
elfzle2 |
⊢ ( 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) → 𝑤 ≤ 𝑗 ) |
70 |
69
|
adantl |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑤 ≤ 𝑗 ) |
71 |
|
elfzle2 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 𝑗 ≤ 𝑀 ) |
72 |
71
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑗 ≤ 𝑀 ) |
73 |
63 65 68 70 72
|
letrd |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑤 ≤ 𝑀 ) |
74 |
73
|
adantll |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑤 ≤ 𝑀 ) |
75 |
46
|
adantl |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑤 ∈ ℤ ) |
76 |
|
0zd |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 0 ∈ ℤ ) |
77 |
66
|
ad2antlr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑀 ∈ ℤ ) |
78 |
|
elfz |
⊢ ( ( 𝑤 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑤 ∈ ( 0 ... 𝑀 ) ↔ ( 0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀 ) ) ) |
79 |
75 76 77 78
|
syl3anc |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → ( 𝑤 ∈ ( 0 ... 𝑀 ) ↔ ( 0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀 ) ) ) |
80 |
62 74 79
|
mpbir2and |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑤 ∈ ( 0 ... 𝑀 ) ) |
81 |
80
|
adantlll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → 𝑤 ∈ ( 0 ... 𝑀 ) ) |
82 |
44 81
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → ( 𝑄 ‘ 𝑤 ) ∈ ℝ ) |
83 |
82
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... 𝑗 ) ) → ( 𝑄 ‘ 𝑤 ) ∈ ℝ ) |
84 |
|
simp-4l |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝜑 ) |
85 |
|
0red |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 0 ∈ ℝ ) |
86 |
|
elfzelz |
⊢ ( 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) → 𝑤 ∈ ℤ ) |
87 |
86
|
zred |
⊢ ( 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) → 𝑤 ∈ ℝ ) |
88 |
87
|
adantl |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑤 ∈ ℝ ) |
89 |
|
0red |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 0 ∈ ℝ ) |
90 |
50
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → ( 𝑖 + 1 ) ∈ ℝ ) |
91 |
87
|
adantl |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑤 ∈ ℝ ) |
92 |
|
0red |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 0 ∈ ℝ ) |
93 |
52
|
ltp1d |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 < ( 𝑖 + 1 ) ) |
94 |
92 52 50 54 93
|
lelttrd |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 0 < ( 𝑖 + 1 ) ) |
95 |
94
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 0 < ( 𝑖 + 1 ) ) |
96 |
|
elfzle1 |
⊢ ( 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) → ( 𝑖 + 1 ) ≤ 𝑤 ) |
97 |
96
|
adantl |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → ( 𝑖 + 1 ) ≤ 𝑤 ) |
98 |
89 90 91 95 97
|
ltletrd |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 0 < 𝑤 ) |
99 |
98
|
adantlr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 0 < 𝑤 ) |
100 |
85 88 99
|
ltled |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 0 ≤ 𝑤 ) |
101 |
100
|
adantlll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 0 ≤ 𝑤 ) |
102 |
101
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 0 ≤ 𝑤 ) |
103 |
87
|
adantl |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑤 ∈ ℝ ) |
104 |
|
peano2rem |
⊢ ( 𝑗 ∈ ℝ → ( 𝑗 − 1 ) ∈ ℝ ) |
105 |
64 104
|
syl |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) ∈ ℝ ) |
106 |
105
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → ( 𝑗 − 1 ) ∈ ℝ ) |
107 |
67
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑀 ∈ ℝ ) |
108 |
|
elfzle2 |
⊢ ( 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) → 𝑤 ≤ ( 𝑗 − 1 ) ) |
109 |
108
|
adantl |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑤 ≤ ( 𝑗 − 1 ) ) |
110 |
|
zlem1lt |
⊢ ( ( 𝑗 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑗 ≤ 𝑀 ↔ ( 𝑗 − 1 ) < 𝑀 ) ) |
111 |
34 66 110
|
syl2anc |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 ≤ 𝑀 ↔ ( 𝑗 − 1 ) < 𝑀 ) ) |
112 |
71 111
|
mpbid |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → ( 𝑗 − 1 ) < 𝑀 ) |
113 |
112
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → ( 𝑗 − 1 ) < 𝑀 ) |
114 |
103 106 107 109 113
|
lelttrd |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑤 < 𝑀 ) |
115 |
114
|
adantlr |
⊢ ( ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑤 < 𝑀 ) |
116 |
115
|
adantlll |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑤 < 𝑀 ) |
117 |
86
|
adantl |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑤 ∈ ℤ ) |
118 |
|
0zd |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 0 ∈ ℤ ) |
119 |
66
|
ad3antlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑀 ∈ ℤ ) |
120 |
|
elfzo |
⊢ ( ( 𝑤 ∈ ℤ ∧ 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → ( 𝑤 ∈ ( 0 ..^ 𝑀 ) ↔ ( 0 ≤ 𝑤 ∧ 𝑤 < 𝑀 ) ) ) |
121 |
117 118 119 120
|
syl3anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → ( 𝑤 ∈ ( 0 ..^ 𝑀 ) ↔ ( 0 ≤ 𝑤 ∧ 𝑤 < 𝑀 ) ) ) |
122 |
102 116 121
|
mpbir2and |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → 𝑤 ∈ ( 0 ..^ 𝑀 ) ) |
123 |
16
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 0 ..^ 𝑀 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
124 |
|
elfzofz |
⊢ ( 𝑤 ∈ ( 0 ..^ 𝑀 ) → 𝑤 ∈ ( 0 ... 𝑀 ) ) |
125 |
124
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 0 ..^ 𝑀 ) ) → 𝑤 ∈ ( 0 ... 𝑀 ) ) |
126 |
123 125
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑤 ) ∈ ℝ ) |
127 |
|
fzofzp1 |
⊢ ( 𝑤 ∈ ( 0 ..^ 𝑀 ) → ( 𝑤 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
128 |
127
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑤 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
129 |
123 128
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝑤 + 1 ) ) ∈ ℝ ) |
130 |
|
eleq1w |
⊢ ( 𝑖 = 𝑤 → ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ↔ 𝑤 ∈ ( 0 ..^ 𝑀 ) ) ) |
131 |
130
|
anbi2d |
⊢ ( 𝑖 = 𝑤 → ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ↔ ( 𝜑 ∧ 𝑤 ∈ ( 0 ..^ 𝑀 ) ) ) ) |
132 |
|
fveq2 |
⊢ ( 𝑖 = 𝑤 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝑤 ) ) |
133 |
|
oveq1 |
⊢ ( 𝑖 = 𝑤 → ( 𝑖 + 1 ) = ( 𝑤 + 1 ) ) |
134 |
133
|
fveq2d |
⊢ ( 𝑖 = 𝑤 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( 𝑤 + 1 ) ) ) |
135 |
132 134
|
breq12d |
⊢ ( 𝑖 = 𝑤 → ( ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ↔ ( 𝑄 ‘ 𝑤 ) < ( 𝑄 ‘ ( 𝑤 + 1 ) ) ) ) |
136 |
131 135
|
imbi12d |
⊢ ( 𝑖 = 𝑤 → ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( 𝜑 ∧ 𝑤 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑤 ) < ( 𝑄 ‘ ( 𝑤 + 1 ) ) ) ) ) |
137 |
7
|
simprrd |
⊢ ( 𝜑 → ∀ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
138 |
137
|
r19.21bi |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
139 |
136 138
|
chvarvv |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑤 ) < ( 𝑄 ‘ ( 𝑤 + 1 ) ) ) |
140 |
126 129 139
|
ltled |
⊢ ( ( 𝜑 ∧ 𝑤 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑤 ) ≤ ( 𝑄 ‘ ( 𝑤 + 1 ) ) ) |
141 |
84 122 140
|
syl2anc |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) ∧ 𝑤 ∈ ( ( 𝑖 + 1 ) ... ( 𝑗 − 1 ) ) ) → ( 𝑄 ‘ 𝑤 ) ≤ ( 𝑄 ‘ ( 𝑤 + 1 ) ) ) |
142 |
43 83 141
|
monoord |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑖 < 𝑗 ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≤ ( 𝑄 ‘ 𝑗 ) ) |
143 |
142
|
3adantl3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑖 < 𝑗 ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≤ ( 𝑄 ‘ 𝑗 ) ) |
144 |
16
|
ffvelrnda |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ ) |
145 |
144
|
3adant3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) → ( 𝑄 ‘ 𝑗 ) ∈ ℝ ) |
146 |
|
simp3 |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) → ( 𝑄 ‘ 𝑗 ) = 𝑋 ) |
147 |
145 146
|
eqled |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) → ( 𝑄 ‘ 𝑗 ) ≤ 𝑋 ) |
148 |
147
|
3adant1r |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) → ( 𝑄 ‘ 𝑗 ) ≤ 𝑋 ) |
149 |
148
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑖 < 𝑗 ) → ( 𝑄 ‘ 𝑗 ) ≤ 𝑋 ) |
150 |
22 30 27 143 149
|
letrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑖 < 𝑗 ) → ( 𝑄 ‘ ( 𝑖 + 1 ) ) ≤ 𝑋 ) |
151 |
22 27 150
|
lensymd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑖 < 𝑗 ) → ¬ 𝑋 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) |
152 |
151
|
intnand |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑖 < 𝑗 ) → ¬ ( ( 𝑄 ‘ 𝑖 ) < 𝑋 ∧ 𝑋 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
153 |
64
|
ad2antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑖 < 𝑗 ) → 𝑗 ∈ ℝ ) |
154 |
52
|
ad3antlr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑖 < 𝑗 ) → 𝑖 ∈ ℝ ) |
155 |
|
simpr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑖 < 𝑗 ) → ¬ 𝑖 < 𝑗 ) |
156 |
153 154 155
|
nltled |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ ¬ 𝑖 < 𝑗 ) → 𝑗 ≤ 𝑖 ) |
157 |
156
|
3adantl3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ ¬ 𝑖 < 𝑗 ) → 𝑗 ≤ 𝑖 ) |
158 |
|
eqcom |
⊢ ( ( 𝑄 ‘ 𝑗 ) = 𝑋 ↔ 𝑋 = ( 𝑄 ‘ 𝑗 ) ) |
159 |
158
|
biimpi |
⊢ ( ( 𝑄 ‘ 𝑗 ) = 𝑋 → 𝑋 = ( 𝑄 ‘ 𝑗 ) ) |
160 |
159
|
adantr |
⊢ ( ( ( 𝑄 ‘ 𝑗 ) = 𝑋 ∧ 𝑗 ≤ 𝑖 ) → 𝑋 = ( 𝑄 ‘ 𝑗 ) ) |
161 |
160
|
3ad2antl3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑗 ≤ 𝑖 ) → 𝑋 = ( 𝑄 ‘ 𝑗 ) ) |
162 |
34
|
ad2antlr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑗 ≤ 𝑖 ) → 𝑗 ∈ ℤ ) |
163 |
32
|
ad2antrr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑗 ≤ 𝑖 ) → 𝑖 ∈ ℤ ) |
164 |
|
simpr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑗 ≤ 𝑖 ) → 𝑗 ≤ 𝑖 ) |
165 |
|
eluz2 |
⊢ ( 𝑖 ∈ ( ℤ≥ ‘ 𝑗 ) ↔ ( 𝑗 ∈ ℤ ∧ 𝑖 ∈ ℤ ∧ 𝑗 ≤ 𝑖 ) ) |
166 |
162 163 164 165
|
syl3anbrc |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑗 ≤ 𝑖 ) → 𝑖 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
167 |
166
|
adantlll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑗 ≤ 𝑖 ) → 𝑖 ∈ ( ℤ≥ ‘ 𝑗 ) ) |
168 |
17
|
ad2antrr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
169 |
|
0zd |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 0 ∈ ℤ ) |
170 |
66
|
ad2antlr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑀 ∈ ℤ ) |
171 |
|
elfzelz |
⊢ ( 𝑤 ∈ ( 𝑗 ... 𝑖 ) → 𝑤 ∈ ℤ ) |
172 |
171
|
adantl |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑤 ∈ ℤ ) |
173 |
169 170 172
|
3jca |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ ) ) |
174 |
|
0red |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 0 ∈ ℝ ) |
175 |
64
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑗 ∈ ℝ ) |
176 |
171
|
zred |
⊢ ( 𝑤 ∈ ( 𝑗 ... 𝑖 ) → 𝑤 ∈ ℝ ) |
177 |
176
|
adantl |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑤 ∈ ℝ ) |
178 |
|
elfzle1 |
⊢ ( 𝑗 ∈ ( 0 ... 𝑀 ) → 0 ≤ 𝑗 ) |
179 |
178
|
adantr |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 0 ≤ 𝑗 ) |
180 |
|
elfzle1 |
⊢ ( 𝑤 ∈ ( 𝑗 ... 𝑖 ) → 𝑗 ≤ 𝑤 ) |
181 |
180
|
adantl |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑗 ≤ 𝑤 ) |
182 |
174 175 177 179 181
|
letrd |
⊢ ( ( 𝑗 ∈ ( 0 ... 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 0 ≤ 𝑤 ) |
183 |
182
|
adantll |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 0 ≤ 𝑤 ) |
184 |
176
|
adantl |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑤 ∈ ℝ ) |
185 |
|
elfzoel2 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑀 ∈ ℤ ) |
186 |
185
|
zred |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑀 ∈ ℝ ) |
187 |
186
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑀 ∈ ℝ ) |
188 |
52
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑖 ∈ ℝ ) |
189 |
|
elfzle2 |
⊢ ( 𝑤 ∈ ( 𝑗 ... 𝑖 ) → 𝑤 ≤ 𝑖 ) |
190 |
189
|
adantl |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑤 ≤ 𝑖 ) |
191 |
|
elfzolt2 |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 < 𝑀 ) |
192 |
191
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑖 < 𝑀 ) |
193 |
184 188 187 190 192
|
lelttrd |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑤 < 𝑀 ) |
194 |
184 187 193
|
ltled |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑤 ≤ 𝑀 ) |
195 |
194
|
adantlr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑤 ≤ 𝑀 ) |
196 |
173 183 195
|
jca32 |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → ( ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ ) ∧ ( 0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀 ) ) ) |
197 |
196
|
adantlll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → ( ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ ) ∧ ( 0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀 ) ) ) |
198 |
|
elfz2 |
⊢ ( 𝑤 ∈ ( 0 ... 𝑀 ) ↔ ( ( 0 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑤 ∈ ℤ ) ∧ ( 0 ≤ 𝑤 ∧ 𝑤 ≤ 𝑀 ) ) ) |
199 |
197 198
|
sylibr |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → 𝑤 ∈ ( 0 ... 𝑀 ) ) |
200 |
168 199
|
ffvelrnd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → ( 𝑄 ‘ 𝑤 ) ∈ ℝ ) |
201 |
200
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑗 ≤ 𝑖 ) ∧ 𝑤 ∈ ( 𝑗 ... 𝑖 ) ) → ( 𝑄 ‘ 𝑤 ) ∈ ℝ ) |
202 |
|
simplll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝜑 ) |
203 |
|
0red |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 0 ∈ ℝ ) |
204 |
64
|
ad2antlr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑗 ∈ ℝ ) |
205 |
|
elfzelz |
⊢ ( 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) → 𝑤 ∈ ℤ ) |
206 |
205
|
zred |
⊢ ( 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) → 𝑤 ∈ ℝ ) |
207 |
206
|
adantl |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑤 ∈ ℝ ) |
208 |
178
|
ad2antlr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 0 ≤ 𝑗 ) |
209 |
|
elfzle1 |
⊢ ( 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) → 𝑗 ≤ 𝑤 ) |
210 |
209
|
adantl |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑗 ≤ 𝑤 ) |
211 |
203 204 207 208 210
|
letrd |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 0 ≤ 𝑤 ) |
212 |
206
|
adantl |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑤 ∈ ℝ ) |
213 |
52
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑖 ∈ ℝ ) |
214 |
186
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑀 ∈ ℝ ) |
215 |
|
peano2rem |
⊢ ( 𝑖 ∈ ℝ → ( 𝑖 − 1 ) ∈ ℝ ) |
216 |
213 215
|
syl |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → ( 𝑖 − 1 ) ∈ ℝ ) |
217 |
|
elfzle2 |
⊢ ( 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) → 𝑤 ≤ ( 𝑖 − 1 ) ) |
218 |
217
|
adantl |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑤 ≤ ( 𝑖 − 1 ) ) |
219 |
213
|
ltm1d |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → ( 𝑖 − 1 ) < 𝑖 ) |
220 |
212 216 213 218 219
|
lelttrd |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑤 < 𝑖 ) |
221 |
191
|
adantr |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑖 < 𝑀 ) |
222 |
212 213 214 220 221
|
lttrd |
⊢ ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑤 < 𝑀 ) |
223 |
222
|
adantlr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑤 < 𝑀 ) |
224 |
205
|
adantl |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑤 ∈ ℤ ) |
225 |
|
0zd |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 0 ∈ ℤ ) |
226 |
185
|
ad2antrr |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑀 ∈ ℤ ) |
227 |
224 225 226 120
|
syl3anc |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → ( 𝑤 ∈ ( 0 ..^ 𝑀 ) ↔ ( 0 ≤ 𝑤 ∧ 𝑤 < 𝑀 ) ) ) |
228 |
211 223 227
|
mpbir2and |
⊢ ( ( ( 𝑖 ∈ ( 0 ..^ 𝑀 ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑤 ∈ ( 0 ..^ 𝑀 ) ) |
229 |
228
|
adantlll |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → 𝑤 ∈ ( 0 ..^ 𝑀 ) ) |
230 |
202 229 140
|
syl2anc |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → ( 𝑄 ‘ 𝑤 ) ≤ ( 𝑄 ‘ ( 𝑤 + 1 ) ) ) |
231 |
230
|
adantlr |
⊢ ( ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑗 ≤ 𝑖 ) ∧ 𝑤 ∈ ( 𝑗 ... ( 𝑖 − 1 ) ) ) → ( 𝑄 ‘ 𝑤 ) ≤ ( 𝑄 ‘ ( 𝑤 + 1 ) ) ) |
232 |
167 201 231
|
monoord |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ) ∧ 𝑗 ≤ 𝑖 ) → ( 𝑄 ‘ 𝑗 ) ≤ ( 𝑄 ‘ 𝑖 ) ) |
233 |
232
|
3adantl3 |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑗 ≤ 𝑖 ) → ( 𝑄 ‘ 𝑗 ) ≤ ( 𝑄 ‘ 𝑖 ) ) |
234 |
161 233
|
eqbrtrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑗 ≤ 𝑖 ) → 𝑋 ≤ ( 𝑄 ‘ 𝑖 ) ) |
235 |
25
|
adantr |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑋 ∈ ℝ ) |
236 |
|
elfzofz |
⊢ ( 𝑖 ∈ ( 0 ..^ 𝑀 ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
237 |
236
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → 𝑖 ∈ ( 0 ... 𝑀 ) ) |
238 |
17 237
|
ffvelrnd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑖 ) ∈ ℝ ) |
239 |
235 238
|
lenltd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑋 ≤ ( 𝑄 ‘ 𝑖 ) ↔ ¬ ( 𝑄 ‘ 𝑖 ) < 𝑋 ) ) |
240 |
239
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ≤ 𝑖 ) → ( 𝑋 ≤ ( 𝑄 ‘ 𝑖 ) ↔ ¬ ( 𝑄 ‘ 𝑖 ) < 𝑋 ) ) |
241 |
240
|
3ad2antl1 |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑗 ≤ 𝑖 ) → ( 𝑋 ≤ ( 𝑄 ‘ 𝑖 ) ↔ ¬ ( 𝑄 ‘ 𝑖 ) < 𝑋 ) ) |
242 |
234 241
|
mpbid |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ 𝑗 ≤ 𝑖 ) → ¬ ( 𝑄 ‘ 𝑖 ) < 𝑋 ) |
243 |
157 242
|
syldan |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ ¬ 𝑖 < 𝑗 ) → ¬ ( 𝑄 ‘ 𝑖 ) < 𝑋 ) |
244 |
243
|
intnanrd |
⊢ ( ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) ∧ ¬ 𝑖 < 𝑗 ) → ¬ ( ( 𝑄 ‘ 𝑖 ) < 𝑋 ∧ 𝑋 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
245 |
152 244
|
pm2.61dan |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) → ¬ ( ( 𝑄 ‘ 𝑖 ) < 𝑋 ∧ 𝑋 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
246 |
245
|
intnand |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) → ¬ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑋 ∈ ℝ* ) ∧ ( ( 𝑄 ‘ 𝑖 ) < 𝑋 ∧ 𝑋 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
247 |
|
elioo3g |
⊢ ( 𝑋 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( ( 𝑄 ‘ 𝑖 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝑖 + 1 ) ) ∈ ℝ* ∧ 𝑋 ∈ ℝ* ) ∧ ( ( 𝑄 ‘ 𝑖 ) < 𝑋 ∧ 𝑋 < ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
248 |
246 247
|
sylnibr |
⊢ ( ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) ∧ 𝑗 ∈ ( 0 ... 𝑀 ) ∧ ( 𝑄 ‘ 𝑗 ) = 𝑋 ) → ¬ 𝑋 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
249 |
248
|
rexlimdv3a |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ( ∃ 𝑗 ∈ ( 0 ... 𝑀 ) ( 𝑄 ‘ 𝑗 ) = 𝑋 → ¬ 𝑋 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) ) |
250 |
15 249
|
mpd |
⊢ ( ( 𝜑 ∧ 𝑖 ∈ ( 0 ..^ 𝑀 ) ) → ¬ 𝑋 ∈ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |