Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem20.m |
⊢ ( 𝜑 → 𝑀 ∈ ℕ ) |
2 |
|
fourierdlem20.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
3 |
|
fourierdlem20.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
4 |
|
fourierdlem20.aleb |
⊢ ( 𝜑 → 𝐴 ≤ 𝐵 ) |
5 |
|
fourierdlem20.q |
⊢ ( 𝜑 → 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ ) |
6 |
|
fourierdlem20.q0 |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ≤ 𝐴 ) |
7 |
|
fourierdlem20.qm |
⊢ ( 𝜑 → 𝐵 ≤ ( 𝑄 ‘ 𝑀 ) ) |
8 |
|
fourierdlem20.j |
⊢ ( 𝜑 → 𝐽 ∈ ( 0 ..^ 𝑁 ) ) |
9 |
|
fourierdlem20.t |
⊢ 𝑇 = ( { 𝐴 , 𝐵 } ∪ ( ran 𝑄 ∩ ( 𝐴 (,) 𝐵 ) ) ) |
10 |
|
fourierdlem20.s |
⊢ ( 𝜑 → 𝑆 Isom < , < ( ( 0 ... 𝑁 ) , 𝑇 ) ) |
11 |
|
fourierdlem20.i |
⊢ 𝐼 = sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } , ℝ , < ) |
12 |
|
ssrab2 |
⊢ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ⊆ ( 0 ..^ 𝑀 ) |
13 |
|
fzossfz |
⊢ ( 0 ..^ 𝑀 ) ⊆ ( 0 ... 𝑀 ) |
14 |
|
fzssz |
⊢ ( 0 ... 𝑀 ) ⊆ ℤ |
15 |
13 14
|
sstri |
⊢ ( 0 ..^ 𝑀 ) ⊆ ℤ |
16 |
12 15
|
sstri |
⊢ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ⊆ ℤ |
17 |
16
|
a1i |
⊢ ( 𝜑 → { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ⊆ ℤ ) |
18 |
|
0z |
⊢ 0 ∈ ℤ |
19 |
|
0le0 |
⊢ 0 ≤ 0 |
20 |
|
eluz2 |
⊢ ( 0 ∈ ( ℤ≥ ‘ 0 ) ↔ ( 0 ∈ ℤ ∧ 0 ∈ ℤ ∧ 0 ≤ 0 ) ) |
21 |
18 18 19 20
|
mpbir3an |
⊢ 0 ∈ ( ℤ≥ ‘ 0 ) |
22 |
21
|
a1i |
⊢ ( 𝜑 → 0 ∈ ( ℤ≥ ‘ 0 ) ) |
23 |
1
|
nnzd |
⊢ ( 𝜑 → 𝑀 ∈ ℤ ) |
24 |
1
|
nngt0d |
⊢ ( 𝜑 → 0 < 𝑀 ) |
25 |
|
elfzo2 |
⊢ ( 0 ∈ ( 0 ..^ 𝑀 ) ↔ ( 0 ∈ ( ℤ≥ ‘ 0 ) ∧ 𝑀 ∈ ℤ ∧ 0 < 𝑀 ) ) |
26 |
22 23 24 25
|
syl3anbrc |
⊢ ( 𝜑 → 0 ∈ ( 0 ..^ 𝑀 ) ) |
27 |
13 26
|
sseldi |
⊢ ( 𝜑 → 0 ∈ ( 0 ... 𝑀 ) ) |
28 |
5 27
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ∈ ℝ ) |
29 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
30 |
3
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
31 |
|
lbicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
32 |
29 30 4 31
|
syl3anc |
⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) |
33 |
|
ubicc2 |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
34 |
29 30 4 33
|
syl3anc |
⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) |
35 |
32 34
|
jca |
⊢ ( 𝜑 → ( 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) ) |
36 |
|
prssg |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ) → ( ( 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ { 𝐴 , 𝐵 } ⊆ ( 𝐴 [,] 𝐵 ) ) ) |
37 |
29 30 36
|
syl2anc |
⊢ ( 𝜑 → ( ( 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ∧ 𝐵 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ { 𝐴 , 𝐵 } ⊆ ( 𝐴 [,] 𝐵 ) ) ) |
38 |
35 37
|
mpbid |
⊢ ( 𝜑 → { 𝐴 , 𝐵 } ⊆ ( 𝐴 [,] 𝐵 ) ) |
39 |
|
inss2 |
⊢ ( ran 𝑄 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 (,) 𝐵 ) |
40 |
|
ioossicc |
⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
41 |
39 40
|
sstri |
⊢ ( ran 𝑄 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) |
42 |
41
|
a1i |
⊢ ( 𝜑 → ( ran 𝑄 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
43 |
38 42
|
unssd |
⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ∪ ( ran 𝑄 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) ) |
44 |
9 43
|
eqsstrid |
⊢ ( 𝜑 → 𝑇 ⊆ ( 𝐴 [,] 𝐵 ) ) |
45 |
2 3
|
iccssred |
⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) |
46 |
44 45
|
sstrd |
⊢ ( 𝜑 → 𝑇 ⊆ ℝ ) |
47 |
|
isof1o |
⊢ ( 𝑆 Isom < , < ( ( 0 ... 𝑁 ) , 𝑇 ) → 𝑆 : ( 0 ... 𝑁 ) –1-1-onto→ 𝑇 ) |
48 |
|
f1of |
⊢ ( 𝑆 : ( 0 ... 𝑁 ) –1-1-onto→ 𝑇 → 𝑆 : ( 0 ... 𝑁 ) ⟶ 𝑇 ) |
49 |
10 47 48
|
3syl |
⊢ ( 𝜑 → 𝑆 : ( 0 ... 𝑁 ) ⟶ 𝑇 ) |
50 |
|
elfzofz |
⊢ ( 𝐽 ∈ ( 0 ..^ 𝑁 ) → 𝐽 ∈ ( 0 ... 𝑁 ) ) |
51 |
8 50
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ ( 0 ... 𝑁 ) ) |
52 |
49 51
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐽 ) ∈ 𝑇 ) |
53 |
46 52
|
sseldd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐽 ) ∈ ℝ ) |
54 |
44 52
|
sseldd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐽 ) ∈ ( 𝐴 [,] 𝐵 ) ) |
55 |
|
iccgelb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝑆 ‘ 𝐽 ) ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ ( 𝑆 ‘ 𝐽 ) ) |
56 |
29 30 54 55
|
syl3anc |
⊢ ( 𝜑 → 𝐴 ≤ ( 𝑆 ‘ 𝐽 ) ) |
57 |
28 2 53 6 56
|
letrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 0 ) ≤ ( 𝑆 ‘ 𝐽 ) ) |
58 |
|
fveq2 |
⊢ ( 𝑘 = 0 → ( 𝑄 ‘ 𝑘 ) = ( 𝑄 ‘ 0 ) ) |
59 |
58
|
breq1d |
⊢ ( 𝑘 = 0 → ( ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) ↔ ( 𝑄 ‘ 0 ) ≤ ( 𝑆 ‘ 𝐽 ) ) ) |
60 |
59
|
elrab |
⊢ ( 0 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ↔ ( 0 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝑄 ‘ 0 ) ≤ ( 𝑆 ‘ 𝐽 ) ) ) |
61 |
26 57 60
|
sylanbrc |
⊢ ( 𝜑 → 0 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ) |
62 |
61
|
ne0d |
⊢ ( 𝜑 → { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ≠ ∅ ) |
63 |
1
|
nnred |
⊢ ( 𝜑 → 𝑀 ∈ ℝ ) |
64 |
12
|
sseli |
⊢ ( 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } → 𝑗 ∈ ( 0 ..^ 𝑀 ) ) |
65 |
|
elfzo0le |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → 𝑗 ≤ 𝑀 ) |
66 |
64 65
|
syl |
⊢ ( 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } → 𝑗 ≤ 𝑀 ) |
67 |
66
|
adantl |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ) → 𝑗 ≤ 𝑀 ) |
68 |
67
|
ralrimiva |
⊢ ( 𝜑 → ∀ 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } 𝑗 ≤ 𝑀 ) |
69 |
|
breq2 |
⊢ ( 𝑥 = 𝑀 → ( 𝑗 ≤ 𝑥 ↔ 𝑗 ≤ 𝑀 ) ) |
70 |
69
|
ralbidv |
⊢ ( 𝑥 = 𝑀 → ( ∀ 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } 𝑗 ≤ 𝑥 ↔ ∀ 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } 𝑗 ≤ 𝑀 ) ) |
71 |
70
|
rspcev |
⊢ ( ( 𝑀 ∈ ℝ ∧ ∀ 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } 𝑗 ≤ 𝑀 ) → ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } 𝑗 ≤ 𝑥 ) |
72 |
63 68 71
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } 𝑗 ≤ 𝑥 ) |
73 |
|
suprzcl |
⊢ ( ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ⊆ ℤ ∧ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } 𝑗 ≤ 𝑥 ) → sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } , ℝ , < ) ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ) |
74 |
17 62 72 73
|
syl3anc |
⊢ ( 𝜑 → sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } , ℝ , < ) ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ) |
75 |
12 74
|
sseldi |
⊢ ( 𝜑 → sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } , ℝ , < ) ∈ ( 0 ..^ 𝑀 ) ) |
76 |
11 75
|
eqeltrid |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ..^ 𝑀 ) ) |
77 |
13 76
|
sseldi |
⊢ ( 𝜑 → 𝐼 ∈ ( 0 ... 𝑀 ) ) |
78 |
5 77
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝐼 ) ∈ ℝ ) |
79 |
78
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝐼 ) ∈ ℝ* ) |
80 |
|
fzofzp1 |
⊢ ( 𝐼 ∈ ( 0 ..^ 𝑀 ) → ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
81 |
76 80
|
syl |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
82 |
5 81
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) |
83 |
82
|
rexrd |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ* ) |
84 |
11 74
|
eqeltrid |
⊢ ( 𝜑 → 𝐼 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ) |
85 |
|
nfrab1 |
⊢ Ⅎ 𝑘 { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } |
86 |
|
nfcv |
⊢ Ⅎ 𝑘 ℝ |
87 |
|
nfcv |
⊢ Ⅎ 𝑘 < |
88 |
85 86 87
|
nfsup |
⊢ Ⅎ 𝑘 sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } , ℝ , < ) |
89 |
11 88
|
nfcxfr |
⊢ Ⅎ 𝑘 𝐼 |
90 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 0 ..^ 𝑀 ) |
91 |
|
nfcv |
⊢ Ⅎ 𝑘 𝑄 |
92 |
91 89
|
nffv |
⊢ Ⅎ 𝑘 ( 𝑄 ‘ 𝐼 ) |
93 |
|
nfcv |
⊢ Ⅎ 𝑘 ≤ |
94 |
|
nfcv |
⊢ Ⅎ 𝑘 ( 𝑆 ‘ 𝐽 ) |
95 |
92 93 94
|
nfbr |
⊢ Ⅎ 𝑘 ( 𝑄 ‘ 𝐼 ) ≤ ( 𝑆 ‘ 𝐽 ) |
96 |
|
fveq2 |
⊢ ( 𝑘 = 𝐼 → ( 𝑄 ‘ 𝑘 ) = ( 𝑄 ‘ 𝐼 ) ) |
97 |
96
|
breq1d |
⊢ ( 𝑘 = 𝐼 → ( ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) ↔ ( 𝑄 ‘ 𝐼 ) ≤ ( 𝑆 ‘ 𝐽 ) ) ) |
98 |
89 90 95 97
|
elrabf |
⊢ ( 𝐼 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ↔ ( 𝐼 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝑄 ‘ 𝐼 ) ≤ ( 𝑆 ‘ 𝐽 ) ) ) |
99 |
84 98
|
sylib |
⊢ ( 𝜑 → ( 𝐼 ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝑄 ‘ 𝐼 ) ≤ ( 𝑆 ‘ 𝐽 ) ) ) |
100 |
99
|
simprd |
⊢ ( 𝜑 → ( 𝑄 ‘ 𝐼 ) ≤ ( 𝑆 ‘ 𝐽 ) ) |
101 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ¬ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
102 |
83
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ* ) |
103 |
|
iccssxr |
⊢ ( 𝐴 [,] 𝐵 ) ⊆ ℝ* |
104 |
44 103
|
sstrdi |
⊢ ( 𝜑 → 𝑇 ⊆ ℝ* ) |
105 |
|
fzofzp1 |
⊢ ( 𝐽 ∈ ( 0 ..^ 𝑁 ) → ( 𝐽 + 1 ) ∈ ( 0 ... 𝑁 ) ) |
106 |
8 105
|
syl |
⊢ ( 𝜑 → ( 𝐽 + 1 ) ∈ ( 0 ... 𝑁 ) ) |
107 |
49 106
|
ffvelrnd |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ 𝑇 ) |
108 |
104 107
|
sseldd |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ ℝ* ) |
109 |
108
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ ℝ* ) |
110 |
|
xrltnle |
⊢ ( ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ* ∧ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ ℝ* ) → ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ↔ ¬ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) |
111 |
102 109 110
|
syl2anc |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ↔ ¬ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) |
112 |
101 111
|
mpbird |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
113 |
|
fzssz |
⊢ ( 0 ... 𝑁 ) ⊆ ℤ |
114 |
|
f1ofo |
⊢ ( 𝑆 : ( 0 ... 𝑁 ) –1-1-onto→ 𝑇 → 𝑆 : ( 0 ... 𝑁 ) –onto→ 𝑇 ) |
115 |
10 47 114
|
3syl |
⊢ ( 𝜑 → 𝑆 : ( 0 ... 𝑁 ) –onto→ 𝑇 ) |
116 |
115
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → 𝑆 : ( 0 ... 𝑁 ) –onto→ 𝑇 ) |
117 |
|
ffun |
⊢ ( 𝑄 : ( 0 ... 𝑀 ) ⟶ ℝ → Fun 𝑄 ) |
118 |
5 117
|
syl |
⊢ ( 𝜑 → Fun 𝑄 ) |
119 |
5
|
fdmd |
⊢ ( 𝜑 → dom 𝑄 = ( 0 ... 𝑀 ) ) |
120 |
119
|
eqcomd |
⊢ ( 𝜑 → ( 0 ... 𝑀 ) = dom 𝑄 ) |
121 |
81 120
|
eleqtrd |
⊢ ( 𝜑 → ( 𝐼 + 1 ) ∈ dom 𝑄 ) |
122 |
|
fvelrn |
⊢ ( ( Fun 𝑄 ∧ ( 𝐼 + 1 ) ∈ dom 𝑄 ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ran 𝑄 ) |
123 |
118 121 122
|
syl2anc |
⊢ ( 𝜑 → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ran 𝑄 ) |
124 |
123
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ran 𝑄 ) |
125 |
29
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → 𝐴 ∈ ℝ* ) |
126 |
30
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → 𝐵 ∈ ℝ* ) |
127 |
82
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) |
128 |
45 54
|
sseldd |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐽 ) ∈ ℝ ) |
129 |
14
|
sseli |
⊢ ( 𝐼 ∈ ( 0 ... 𝑀 ) → 𝐼 ∈ ℤ ) |
130 |
|
zre |
⊢ ( 𝐼 ∈ ℤ → 𝐼 ∈ ℝ ) |
131 |
77 129 130
|
3syl |
⊢ ( 𝜑 → 𝐼 ∈ ℝ ) |
132 |
131
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → 𝐼 ∈ ℝ ) |
133 |
132
|
ltp1d |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → 𝐼 < ( 𝐼 + 1 ) ) |
134 |
133
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) ∧ ¬ ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → 𝐼 < ( 𝐼 + 1 ) ) |
135 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) ∧ ¬ ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) |
136 |
128
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) ∧ ¬ ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ( 𝑆 ‘ 𝐽 ) ∈ ℝ ) |
137 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) ∧ ¬ ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ¬ ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
138 |
135 136 137
|
nltled |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) ∧ ¬ ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) |
139 |
131
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → 𝐼 ∈ ℝ ) |
140 |
|
1red |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → 1 ∈ ℝ ) |
141 |
139 140
|
readdcld |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝐼 + 1 ) ∈ ℝ ) |
142 |
|
elfzoelz |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → 𝑗 ∈ ℤ ) |
143 |
142
|
zred |
⊢ ( 𝑗 ∈ ( 0 ..^ 𝑀 ) → 𝑗 ∈ ℝ ) |
144 |
143
|
ssriv |
⊢ ( 0 ..^ 𝑀 ) ⊆ ℝ |
145 |
12 144
|
sstri |
⊢ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ⊆ ℝ |
146 |
145
|
a1i |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ⊆ ℝ ) |
147 |
62
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ≠ ∅ ) |
148 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } 𝑗 ≤ 𝑥 ) |
149 |
82
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) |
150 |
128
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑆 ‘ 𝐽 ) ∈ ℝ ) |
151 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → 𝐵 ∈ ℝ ) |
152 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) |
153 |
46 107
|
sseldd |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ ℝ ) |
154 |
153
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ ℝ ) |
155 |
|
elfzoelz |
⊢ ( 𝐽 ∈ ( 0 ..^ 𝑁 ) → 𝐽 ∈ ℤ ) |
156 |
|
zre |
⊢ ( 𝐽 ∈ ℤ → 𝐽 ∈ ℝ ) |
157 |
8 155 156
|
3syl |
⊢ ( 𝜑 → 𝐽 ∈ ℝ ) |
158 |
157
|
ltp1d |
⊢ ( 𝜑 → 𝐽 < ( 𝐽 + 1 ) ) |
159 |
|
isorel |
⊢ ( ( 𝑆 Isom < , < ( ( 0 ... 𝑁 ) , 𝑇 ) ∧ ( 𝐽 ∈ ( 0 ... 𝑁 ) ∧ ( 𝐽 + 1 ) ∈ ( 0 ... 𝑁 ) ) ) → ( 𝐽 < ( 𝐽 + 1 ) ↔ ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
160 |
10 51 106 159
|
syl12anc |
⊢ ( 𝜑 → ( 𝐽 < ( 𝐽 + 1 ) ↔ ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
161 |
158 160
|
mpbid |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
162 |
161
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
163 |
44 107
|
sseldd |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
164 |
|
iccleub |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ 𝐵 ) |
165 |
29 30 163 164
|
syl3anc |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ 𝐵 ) |
166 |
165
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ 𝐵 ) |
167 |
150 154 151 162 166
|
ltletrd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑆 ‘ 𝐽 ) < 𝐵 ) |
168 |
149 150 151 152 167
|
lelttrd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐵 ) |
169 |
168
|
adantr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐵 ) |
170 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → 𝐵 ∈ ℝ ) |
171 |
82
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) |
172 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → 𝐵 ≤ ( 𝑄 ‘ 𝑀 ) ) |
173 |
23
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → 𝑀 ∈ ℤ ) |
174 |
81
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐼 + 1 ) ∈ ( 0 ... 𝑀 ) ) |
175 |
|
fzval3 |
⊢ ( 𝑀 ∈ ℤ → ( 0 ... 𝑀 ) = ( 0 ..^ ( 𝑀 + 1 ) ) ) |
176 |
23 175
|
syl |
⊢ ( 𝜑 → ( 0 ... 𝑀 ) = ( 0 ..^ ( 𝑀 + 1 ) ) ) |
177 |
176
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 0 ... 𝑀 ) = ( 0 ..^ ( 𝑀 + 1 ) ) ) |
178 |
174 177
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐼 + 1 ) ∈ ( 0 ..^ ( 𝑀 + 1 ) ) ) |
179 |
|
simpr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) |
180 |
178 179
|
jca |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( ( 𝐼 + 1 ) ∈ ( 0 ..^ ( 𝑀 + 1 ) ) ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) ) |
181 |
|
elfzonelfzo |
⊢ ( 𝑀 ∈ ℤ → ( ( ( 𝐼 + 1 ) ∈ ( 0 ..^ ( 𝑀 + 1 ) ) ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐼 + 1 ) ∈ ( 𝑀 ..^ ( 𝑀 + 1 ) ) ) ) |
182 |
173 180 181
|
sylc |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐼 + 1 ) ∈ ( 𝑀 ..^ ( 𝑀 + 1 ) ) ) |
183 |
|
fzval3 |
⊢ ( 𝑀 ∈ ℤ → ( 𝑀 ... 𝑀 ) = ( 𝑀 ..^ ( 𝑀 + 1 ) ) ) |
184 |
23 183
|
syl |
⊢ ( 𝜑 → ( 𝑀 ... 𝑀 ) = ( 𝑀 ..^ ( 𝑀 + 1 ) ) ) |
185 |
184
|
eqcomd |
⊢ ( 𝜑 → ( 𝑀 ..^ ( 𝑀 + 1 ) ) = ( 𝑀 ... 𝑀 ) ) |
186 |
185
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑀 ..^ ( 𝑀 + 1 ) ) = ( 𝑀 ... 𝑀 ) ) |
187 |
182 186
|
eleqtrd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐼 + 1 ) ∈ ( 𝑀 ... 𝑀 ) ) |
188 |
|
elfz1eq |
⊢ ( ( 𝐼 + 1 ) ∈ ( 𝑀 ... 𝑀 ) → ( 𝐼 + 1 ) = 𝑀 ) |
189 |
187 188
|
syl |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝐼 + 1 ) = 𝑀 ) |
190 |
189
|
eqcomd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → 𝑀 = ( 𝐼 + 1 ) ) |
191 |
190
|
fveq2d |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ( 𝑄 ‘ 𝑀 ) = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
192 |
172 191
|
breqtrd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → 𝐵 ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
193 |
170 171 192
|
lensymd |
⊢ ( ( 𝜑 ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ¬ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐵 ) |
194 |
193
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) ∧ ¬ ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) → ¬ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐵 ) |
195 |
169 194
|
condan |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ) |
196 |
|
nfcv |
⊢ Ⅎ 𝑘 + |
197 |
|
nfcv |
⊢ Ⅎ 𝑘 1 |
198 |
89 196 197
|
nfov |
⊢ Ⅎ 𝑘 ( 𝐼 + 1 ) |
199 |
91 198
|
nffv |
⊢ Ⅎ 𝑘 ( 𝑄 ‘ ( 𝐼 + 1 ) ) |
200 |
199 93 94
|
nfbr |
⊢ Ⅎ 𝑘 ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) |
201 |
|
fveq2 |
⊢ ( 𝑘 = ( 𝐼 + 1 ) → ( 𝑄 ‘ 𝑘 ) = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
202 |
201
|
breq1d |
⊢ ( 𝑘 = ( 𝐼 + 1 ) → ( ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) ↔ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) ) |
203 |
198 90 200 202
|
elrabf |
⊢ ( ( 𝐼 + 1 ) ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ↔ ( ( 𝐼 + 1 ) ∈ ( 0 ..^ 𝑀 ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) ) |
204 |
195 152 203
|
sylanbrc |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝐼 + 1 ) ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ) |
205 |
|
suprub |
⊢ ( ( ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ⊆ ℝ ∧ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑗 ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } 𝑗 ≤ 𝑥 ) ∧ ( 𝐼 + 1 ) ∈ { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } ) → ( 𝐼 + 1 ) ≤ sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } , ℝ , < ) ) |
206 |
146 147 148 204 205
|
syl31anc |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝐼 + 1 ) ≤ sup ( { 𝑘 ∈ ( 0 ..^ 𝑀 ) ∣ ( 𝑄 ‘ 𝑘 ) ≤ ( 𝑆 ‘ 𝐽 ) } , ℝ , < ) ) |
207 |
206 11
|
breqtrrdi |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ( 𝐼 + 1 ) ≤ 𝐼 ) |
208 |
141 139 207
|
lensymd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ¬ 𝐼 < ( 𝐼 + 1 ) ) |
209 |
208
|
adantlr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ≤ ( 𝑆 ‘ 𝐽 ) ) → ¬ 𝐼 < ( 𝐼 + 1 ) ) |
210 |
138 209
|
syldan |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) ∧ ¬ ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ¬ 𝐼 < ( 𝐼 + 1 ) ) |
211 |
134 210
|
condan |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ ) → ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
212 |
82 211
|
mpdan |
⊢ ( 𝜑 → ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
213 |
2 128 82 56 212
|
lelttrd |
⊢ ( 𝜑 → 𝐴 < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
214 |
213
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → 𝐴 < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
215 |
153
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∈ ℝ ) |
216 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → 𝐵 ∈ ℝ ) |
217 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
218 |
165
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ 𝐵 ) |
219 |
127 215 216 217 218
|
ltletrd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) < 𝐵 ) |
220 |
125 126 127 214 219
|
eliood |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ( 𝐴 (,) 𝐵 ) ) |
221 |
124 220
|
elind |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ( ran 𝑄 ∩ ( 𝐴 (,) 𝐵 ) ) ) |
222 |
|
elun2 |
⊢ ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ( ran 𝑄 ∩ ( 𝐴 (,) 𝐵 ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ( { 𝐴 , 𝐵 } ∪ ( ran 𝑄 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) |
223 |
221 222
|
syl |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ( { 𝐴 , 𝐵 } ∪ ( ran 𝑄 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) |
224 |
223 9
|
eleqtrrdi |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ 𝑇 ) |
225 |
|
foelrn |
⊢ ( ( 𝑆 : ( 0 ... 𝑁 ) –onto→ 𝑇 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ 𝑇 ) → ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) |
226 |
116 224 225
|
syl2anc |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) |
227 |
212
|
adantr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝑆 ‘ 𝐽 ) < ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
228 |
|
simpr |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) |
229 |
227 228
|
breqtrd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ 𝑗 ) ) |
230 |
229
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ 𝑗 ) ) |
231 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → 𝑆 Isom < , < ( ( 0 ... 𝑁 ) , 𝑇 ) ) |
232 |
51
|
anim1i |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐽 ∈ ( 0 ... 𝑁 ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ) |
233 |
232
|
adantr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝐽 ∈ ( 0 ... 𝑁 ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ) |
234 |
|
isorel |
⊢ ( ( 𝑆 Isom < , < ( ( 0 ... 𝑁 ) , 𝑇 ) ∧ ( 𝐽 ∈ ( 0 ... 𝑁 ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ) → ( 𝐽 < 𝑗 ↔ ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ 𝑗 ) ) ) |
235 |
231 233 234
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝐽 < 𝑗 ↔ ( 𝑆 ‘ 𝐽 ) < ( 𝑆 ‘ 𝑗 ) ) ) |
236 |
230 235
|
mpbird |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → 𝐽 < 𝑗 ) |
237 |
236
|
adantllr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → 𝐽 < 𝑗 ) |
238 |
|
eqcom |
⊢ ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ↔ ( 𝑆 ‘ 𝑗 ) = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
239 |
238
|
biimpi |
⊢ ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) → ( 𝑆 ‘ 𝑗 ) = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
240 |
239
|
adantl |
⊢ ( ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝑆 ‘ 𝑗 ) = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
241 |
|
simpl |
⊢ ( ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
242 |
240 241
|
eqbrtrd |
⊢ ( ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝑆 ‘ 𝑗 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
243 |
242
|
adantll |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝑆 ‘ 𝑗 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
244 |
243
|
adantlr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝑆 ‘ 𝑗 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) |
245 |
10
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝑆 Isom < , < ( ( 0 ... 𝑁 ) , 𝑇 ) ) |
246 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → 𝑗 ∈ ( 0 ... 𝑁 ) ) |
247 |
106
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝐽 + 1 ) ∈ ( 0 ... 𝑁 ) ) |
248 |
|
isorel |
⊢ ( ( 𝑆 Isom < , < ( ( 0 ... 𝑁 ) , 𝑇 ) ∧ ( 𝑗 ∈ ( 0 ... 𝑁 ) ∧ ( 𝐽 + 1 ) ∈ ( 0 ... 𝑁 ) ) ) → ( 𝑗 < ( 𝐽 + 1 ) ↔ ( 𝑆 ‘ 𝑗 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
249 |
245 246 247 248
|
syl12anc |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( 𝑗 < ( 𝐽 + 1 ) ↔ ( 𝑆 ‘ 𝑗 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
250 |
249
|
adantr |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝑗 < ( 𝐽 + 1 ) ↔ ( 𝑆 ‘ 𝑗 ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ) |
251 |
244 250
|
mpbird |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → 𝑗 < ( 𝐽 + 1 ) ) |
252 |
237 251
|
jca |
⊢ ( ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) ) → ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) |
253 |
252
|
ex |
⊢ ( ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ∧ 𝑗 ∈ ( 0 ... 𝑁 ) ) → ( ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) → ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) ) |
254 |
253
|
reximdva |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝑄 ‘ ( 𝐼 + 1 ) ) = ( 𝑆 ‘ 𝑗 ) → ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) ) |
255 |
226 254
|
mpd |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) |
256 |
|
ssrexv |
⊢ ( ( 0 ... 𝑁 ) ⊆ ℤ → ( ∃ 𝑗 ∈ ( 0 ... 𝑁 ) ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) → ∃ 𝑗 ∈ ℤ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) ) |
257 |
113 255 256
|
mpsyl |
⊢ ( ( 𝜑 ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) < ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) → ∃ 𝑗 ∈ ℤ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) |
258 |
112 257
|
syldan |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ∃ 𝑗 ∈ ℤ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) |
259 |
|
simplr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℤ ) ∧ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) → 𝑗 ∈ ℤ ) |
260 |
8 155
|
syl |
⊢ ( 𝜑 → 𝐽 ∈ ℤ ) |
261 |
260
|
ad2antrr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℤ ) ∧ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) → 𝐽 ∈ ℤ ) |
262 |
|
simprl |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℤ ) ∧ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) → 𝐽 < 𝑗 ) |
263 |
|
simprr |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℤ ) ∧ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) → 𝑗 < ( 𝐽 + 1 ) ) |
264 |
|
btwnnz |
⊢ ( ( 𝐽 ∈ ℤ ∧ 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) → ¬ 𝑗 ∈ ℤ ) |
265 |
261 262 263 264
|
syl3anc |
⊢ ( ( ( 𝜑 ∧ 𝑗 ∈ ℤ ) ∧ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) → ¬ 𝑗 ∈ ℤ ) |
266 |
259 265
|
pm2.65da |
⊢ ( ( 𝜑 ∧ 𝑗 ∈ ℤ ) → ¬ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) |
267 |
266
|
nrexdv |
⊢ ( 𝜑 → ¬ ∃ 𝑗 ∈ ℤ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) |
268 |
267
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) → ¬ ∃ 𝑗 ∈ ℤ ( 𝐽 < 𝑗 ∧ 𝑗 < ( 𝐽 + 1 ) ) ) |
269 |
258 268
|
condan |
⊢ ( 𝜑 → ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
270 |
|
ioossioo |
⊢ ( ( ( ( 𝑄 ‘ 𝐼 ) ∈ ℝ* ∧ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ∈ ℝ* ) ∧ ( ( 𝑄 ‘ 𝐼 ) ≤ ( 𝑆 ‘ 𝐽 ) ∧ ( 𝑆 ‘ ( 𝐽 + 1 ) ) ≤ ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) |
271 |
79 83 100 269 270
|
syl22anc |
⊢ ( 𝜑 → ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) |
272 |
|
fveq2 |
⊢ ( 𝑖 = 𝐼 → ( 𝑄 ‘ 𝑖 ) = ( 𝑄 ‘ 𝐼 ) ) |
273 |
|
oveq1 |
⊢ ( 𝑖 = 𝐼 → ( 𝑖 + 1 ) = ( 𝐼 + 1 ) ) |
274 |
273
|
fveq2d |
⊢ ( 𝑖 = 𝐼 → ( 𝑄 ‘ ( 𝑖 + 1 ) ) = ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) |
275 |
272 274
|
oveq12d |
⊢ ( 𝑖 = 𝐼 → ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) = ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) |
276 |
275
|
sseq2d |
⊢ ( 𝑖 = 𝐼 → ( ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ↔ ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) ) |
277 |
276
|
rspcev |
⊢ ( ( 𝐼 ∈ ( 0 ..^ 𝑀 ) ∧ ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝐼 ) (,) ( 𝑄 ‘ ( 𝐼 + 1 ) ) ) ) → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |
278 |
76 271 277
|
syl2anc |
⊢ ( 𝜑 → ∃ 𝑖 ∈ ( 0 ..^ 𝑀 ) ( ( 𝑆 ‘ 𝐽 ) (,) ( 𝑆 ‘ ( 𝐽 + 1 ) ) ) ⊆ ( ( 𝑄 ‘ 𝑖 ) (,) ( 𝑄 ‘ ( 𝑖 + 1 ) ) ) ) |