Step |
Hyp |
Ref |
Expression |
1 |
|
fourierdlem35.a |
⊢ ( 𝜑 → 𝐴 ∈ ℝ ) |
2 |
|
fourierdlem35.b |
⊢ ( 𝜑 → 𝐵 ∈ ℝ ) |
3 |
|
fourierdlem35.altb |
⊢ ( 𝜑 → 𝐴 < 𝐵 ) |
4 |
|
fourierdlem35.t |
⊢ 𝑇 = ( 𝐵 − 𝐴 ) |
5 |
|
fourierdlem35.5 |
⊢ ( 𝜑 → 𝑋 ∈ ℝ ) |
6 |
|
fourierdlem35.i |
⊢ ( 𝜑 → 𝐼 ∈ ℤ ) |
7 |
|
fourierdlem35.j |
⊢ ( 𝜑 → 𝐽 ∈ ℤ ) |
8 |
|
fourierdlem35.iel |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐼 · 𝑇 ) ) ∈ ( 𝐴 (,] 𝐵 ) ) |
9 |
|
fourierdlem35.jel |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐽 · 𝑇 ) ) ∈ ( 𝐴 (,] 𝐵 ) ) |
10 |
|
neqne |
⊢ ( ¬ 𝐼 = 𝐽 → 𝐼 ≠ 𝐽 ) |
11 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → 𝐴 ∈ ℝ ) |
12 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → 𝐵 ∈ ℝ ) |
13 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → 𝐴 < 𝐵 ) |
14 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → 𝑋 ∈ ℝ ) |
15 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → 𝐼 ∈ ℤ ) |
16 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → 𝐽 ∈ ℤ ) |
17 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → 𝐼 < 𝐽 ) |
18 |
|
iocssicc |
⊢ ( 𝐴 (,] 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) |
19 |
18 8
|
sseldi |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐼 · 𝑇 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
20 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → ( 𝑋 + ( 𝐼 · 𝑇 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
21 |
18 9
|
sseldi |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐽 · 𝑇 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
22 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → ( 𝑋 + ( 𝐽 · 𝑇 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
23 |
11 12 13 4 14 15 16 17 20 22
|
fourierdlem6 |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → 𝐽 = ( 𝐼 + 1 ) ) |
24 |
23
|
orcd |
⊢ ( ( 𝜑 ∧ 𝐼 < 𝐽 ) → ( 𝐽 = ( 𝐼 + 1 ) ∨ 𝐼 = ( 𝐽 + 1 ) ) ) |
25 |
24
|
adantlr |
⊢ ( ( ( 𝜑 ∧ 𝐼 ≠ 𝐽 ) ∧ 𝐼 < 𝐽 ) → ( 𝐽 = ( 𝐼 + 1 ) ∨ 𝐼 = ( 𝐽 + 1 ) ) ) |
26 |
|
simpll |
⊢ ( ( ( 𝜑 ∧ 𝐼 ≠ 𝐽 ) ∧ ¬ 𝐼 < 𝐽 ) → 𝜑 ) |
27 |
7
|
zred |
⊢ ( 𝜑 → 𝐽 ∈ ℝ ) |
28 |
26 27
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝐼 ≠ 𝐽 ) ∧ ¬ 𝐼 < 𝐽 ) → 𝐽 ∈ ℝ ) |
29 |
6
|
zred |
⊢ ( 𝜑 → 𝐼 ∈ ℝ ) |
30 |
26 29
|
syl |
⊢ ( ( ( 𝜑 ∧ 𝐼 ≠ 𝐽 ) ∧ ¬ 𝐼 < 𝐽 ) → 𝐼 ∈ ℝ ) |
31 |
|
id |
⊢ ( 𝐼 ≠ 𝐽 → 𝐼 ≠ 𝐽 ) |
32 |
31
|
necomd |
⊢ ( 𝐼 ≠ 𝐽 → 𝐽 ≠ 𝐼 ) |
33 |
32
|
ad2antlr |
⊢ ( ( ( 𝜑 ∧ 𝐼 ≠ 𝐽 ) ∧ ¬ 𝐼 < 𝐽 ) → 𝐽 ≠ 𝐼 ) |
34 |
|
simpr |
⊢ ( ( ( 𝜑 ∧ 𝐼 ≠ 𝐽 ) ∧ ¬ 𝐼 < 𝐽 ) → ¬ 𝐼 < 𝐽 ) |
35 |
28 30 33 34
|
lttri5d |
⊢ ( ( ( 𝜑 ∧ 𝐼 ≠ 𝐽 ) ∧ ¬ 𝐼 < 𝐽 ) → 𝐽 < 𝐼 ) |
36 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → 𝐴 ∈ ℝ ) |
37 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → 𝐵 ∈ ℝ ) |
38 |
3
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → 𝐴 < 𝐵 ) |
39 |
5
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → 𝑋 ∈ ℝ ) |
40 |
7
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → 𝐽 ∈ ℤ ) |
41 |
6
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → 𝐼 ∈ ℤ ) |
42 |
|
simpr |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → 𝐽 < 𝐼 ) |
43 |
21
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → ( 𝑋 + ( 𝐽 · 𝑇 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
44 |
19
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → ( 𝑋 + ( 𝐼 · 𝑇 ) ) ∈ ( 𝐴 [,] 𝐵 ) ) |
45 |
36 37 38 4 39 40 41 42 43 44
|
fourierdlem6 |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → 𝐼 = ( 𝐽 + 1 ) ) |
46 |
45
|
olcd |
⊢ ( ( 𝜑 ∧ 𝐽 < 𝐼 ) → ( 𝐽 = ( 𝐼 + 1 ) ∨ 𝐼 = ( 𝐽 + 1 ) ) ) |
47 |
26 35 46
|
syl2anc |
⊢ ( ( ( 𝜑 ∧ 𝐼 ≠ 𝐽 ) ∧ ¬ 𝐼 < 𝐽 ) → ( 𝐽 = ( 𝐼 + 1 ) ∨ 𝐼 = ( 𝐽 + 1 ) ) ) |
48 |
25 47
|
pm2.61dan |
⊢ ( ( 𝜑 ∧ 𝐼 ≠ 𝐽 ) → ( 𝐽 = ( 𝐼 + 1 ) ∨ 𝐼 = ( 𝐽 + 1 ) ) ) |
49 |
10 48
|
sylan2 |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝐽 ) → ( 𝐽 = ( 𝐼 + 1 ) ∨ 𝐼 = ( 𝐽 + 1 ) ) ) |
50 |
1
|
rexrd |
⊢ ( 𝜑 → 𝐴 ∈ ℝ* ) |
51 |
2
|
rexrd |
⊢ ( 𝜑 → 𝐵 ∈ ℝ* ) |
52 |
|
iocleub |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝑋 + ( 𝐽 · 𝑇 ) ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝑋 + ( 𝐽 · 𝑇 ) ) ≤ 𝐵 ) |
53 |
50 51 9 52
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐽 · 𝑇 ) ) ≤ 𝐵 ) |
54 |
53
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → ( 𝑋 + ( 𝐽 · 𝑇 ) ) ≤ 𝐵 ) |
55 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → 𝐴 ∈ ℝ ) |
56 |
2 1
|
resubcld |
⊢ ( 𝜑 → ( 𝐵 − 𝐴 ) ∈ ℝ ) |
57 |
4 56
|
eqeltrid |
⊢ ( 𝜑 → 𝑇 ∈ ℝ ) |
58 |
29 57
|
remulcld |
⊢ ( 𝜑 → ( 𝐼 · 𝑇 ) ∈ ℝ ) |
59 |
5 58
|
readdcld |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐼 · 𝑇 ) ) ∈ ℝ ) |
60 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → ( 𝑋 + ( 𝐼 · 𝑇 ) ) ∈ ℝ ) |
61 |
57
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → 𝑇 ∈ ℝ ) |
62 |
|
iocgtlb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝑋 + ( 𝐼 · 𝑇 ) ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 < ( 𝑋 + ( 𝐼 · 𝑇 ) ) ) |
63 |
50 51 8 62
|
syl3anc |
⊢ ( 𝜑 → 𝐴 < ( 𝑋 + ( 𝐼 · 𝑇 ) ) ) |
64 |
63
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → 𝐴 < ( 𝑋 + ( 𝐼 · 𝑇 ) ) ) |
65 |
55 60 61 64
|
ltadd1dd |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → ( 𝐴 + 𝑇 ) < ( ( 𝑋 + ( 𝐼 · 𝑇 ) ) + 𝑇 ) ) |
66 |
4
|
eqcomi |
⊢ ( 𝐵 − 𝐴 ) = 𝑇 |
67 |
2
|
recnd |
⊢ ( 𝜑 → 𝐵 ∈ ℂ ) |
68 |
1
|
recnd |
⊢ ( 𝜑 → 𝐴 ∈ ℂ ) |
69 |
57
|
recnd |
⊢ ( 𝜑 → 𝑇 ∈ ℂ ) |
70 |
67 68 69
|
subaddd |
⊢ ( 𝜑 → ( ( 𝐵 − 𝐴 ) = 𝑇 ↔ ( 𝐴 + 𝑇 ) = 𝐵 ) ) |
71 |
66 70
|
mpbii |
⊢ ( 𝜑 → ( 𝐴 + 𝑇 ) = 𝐵 ) |
72 |
71
|
eqcomd |
⊢ ( 𝜑 → 𝐵 = ( 𝐴 + 𝑇 ) ) |
73 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → 𝐵 = ( 𝐴 + 𝑇 ) ) |
74 |
5
|
recnd |
⊢ ( 𝜑 → 𝑋 ∈ ℂ ) |
75 |
58
|
recnd |
⊢ ( 𝜑 → ( 𝐼 · 𝑇 ) ∈ ℂ ) |
76 |
74 75 69
|
addassd |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝐼 · 𝑇 ) ) + 𝑇 ) = ( 𝑋 + ( ( 𝐼 · 𝑇 ) + 𝑇 ) ) ) |
77 |
76
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → ( ( 𝑋 + ( 𝐼 · 𝑇 ) ) + 𝑇 ) = ( 𝑋 + ( ( 𝐼 · 𝑇 ) + 𝑇 ) ) ) |
78 |
29
|
recnd |
⊢ ( 𝜑 → 𝐼 ∈ ℂ ) |
79 |
78 69
|
adddirp1d |
⊢ ( 𝜑 → ( ( 𝐼 + 1 ) · 𝑇 ) = ( ( 𝐼 · 𝑇 ) + 𝑇 ) ) |
80 |
79
|
eqcomd |
⊢ ( 𝜑 → ( ( 𝐼 · 𝑇 ) + 𝑇 ) = ( ( 𝐼 + 1 ) · 𝑇 ) ) |
81 |
80
|
oveq2d |
⊢ ( 𝜑 → ( 𝑋 + ( ( 𝐼 · 𝑇 ) + 𝑇 ) ) = ( 𝑋 + ( ( 𝐼 + 1 ) · 𝑇 ) ) ) |
82 |
81
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → ( 𝑋 + ( ( 𝐼 · 𝑇 ) + 𝑇 ) ) = ( 𝑋 + ( ( 𝐼 + 1 ) · 𝑇 ) ) ) |
83 |
|
oveq1 |
⊢ ( 𝐽 = ( 𝐼 + 1 ) → ( 𝐽 · 𝑇 ) = ( ( 𝐼 + 1 ) · 𝑇 ) ) |
84 |
83
|
eqcomd |
⊢ ( 𝐽 = ( 𝐼 + 1 ) → ( ( 𝐼 + 1 ) · 𝑇 ) = ( 𝐽 · 𝑇 ) ) |
85 |
84
|
oveq2d |
⊢ ( 𝐽 = ( 𝐼 + 1 ) → ( 𝑋 + ( ( 𝐼 + 1 ) · 𝑇 ) ) = ( 𝑋 + ( 𝐽 · 𝑇 ) ) ) |
86 |
85
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → ( 𝑋 + ( ( 𝐼 + 1 ) · 𝑇 ) ) = ( 𝑋 + ( 𝐽 · 𝑇 ) ) ) |
87 |
77 82 86
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → ( 𝑋 + ( 𝐽 · 𝑇 ) ) = ( ( 𝑋 + ( 𝐼 · 𝑇 ) ) + 𝑇 ) ) |
88 |
65 73 87
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → 𝐵 < ( 𝑋 + ( 𝐽 · 𝑇 ) ) ) |
89 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → 𝐵 ∈ ℝ ) |
90 |
27 57
|
remulcld |
⊢ ( 𝜑 → ( 𝐽 · 𝑇 ) ∈ ℝ ) |
91 |
5 90
|
readdcld |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐽 · 𝑇 ) ) ∈ ℝ ) |
92 |
91
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → ( 𝑋 + ( 𝐽 · 𝑇 ) ) ∈ ℝ ) |
93 |
89 92
|
ltnled |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → ( 𝐵 < ( 𝑋 + ( 𝐽 · 𝑇 ) ) ↔ ¬ ( 𝑋 + ( 𝐽 · 𝑇 ) ) ≤ 𝐵 ) ) |
94 |
88 93
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐽 = ( 𝐼 + 1 ) ) → ¬ ( 𝑋 + ( 𝐽 · 𝑇 ) ) ≤ 𝐵 ) |
95 |
54 94
|
pm2.65da |
⊢ ( 𝜑 → ¬ 𝐽 = ( 𝐼 + 1 ) ) |
96 |
|
iocleub |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝑋 + ( 𝐼 · 𝑇 ) ) ∈ ( 𝐴 (,] 𝐵 ) ) → ( 𝑋 + ( 𝐼 · 𝑇 ) ) ≤ 𝐵 ) |
97 |
50 51 8 96
|
syl3anc |
⊢ ( 𝜑 → ( 𝑋 + ( 𝐼 · 𝑇 ) ) ≤ 𝐵 ) |
98 |
97
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → ( 𝑋 + ( 𝐼 · 𝑇 ) ) ≤ 𝐵 ) |
99 |
1
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → 𝐴 ∈ ℝ ) |
100 |
91
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → ( 𝑋 + ( 𝐽 · 𝑇 ) ) ∈ ℝ ) |
101 |
57
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → 𝑇 ∈ ℝ ) |
102 |
|
iocgtlb |
⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ ( 𝑋 + ( 𝐽 · 𝑇 ) ) ∈ ( 𝐴 (,] 𝐵 ) ) → 𝐴 < ( 𝑋 + ( 𝐽 · 𝑇 ) ) ) |
103 |
50 51 9 102
|
syl3anc |
⊢ ( 𝜑 → 𝐴 < ( 𝑋 + ( 𝐽 · 𝑇 ) ) ) |
104 |
103
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → 𝐴 < ( 𝑋 + ( 𝐽 · 𝑇 ) ) ) |
105 |
99 100 101 104
|
ltadd1dd |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → ( 𝐴 + 𝑇 ) < ( ( 𝑋 + ( 𝐽 · 𝑇 ) ) + 𝑇 ) ) |
106 |
72
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → 𝐵 = ( 𝐴 + 𝑇 ) ) |
107 |
90
|
recnd |
⊢ ( 𝜑 → ( 𝐽 · 𝑇 ) ∈ ℂ ) |
108 |
74 107 69
|
addassd |
⊢ ( 𝜑 → ( ( 𝑋 + ( 𝐽 · 𝑇 ) ) + 𝑇 ) = ( 𝑋 + ( ( 𝐽 · 𝑇 ) + 𝑇 ) ) ) |
109 |
108
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → ( ( 𝑋 + ( 𝐽 · 𝑇 ) ) + 𝑇 ) = ( 𝑋 + ( ( 𝐽 · 𝑇 ) + 𝑇 ) ) ) |
110 |
27
|
recnd |
⊢ ( 𝜑 → 𝐽 ∈ ℂ ) |
111 |
110 69
|
adddirp1d |
⊢ ( 𝜑 → ( ( 𝐽 + 1 ) · 𝑇 ) = ( ( 𝐽 · 𝑇 ) + 𝑇 ) ) |
112 |
111
|
eqcomd |
⊢ ( 𝜑 → ( ( 𝐽 · 𝑇 ) + 𝑇 ) = ( ( 𝐽 + 1 ) · 𝑇 ) ) |
113 |
112
|
oveq2d |
⊢ ( 𝜑 → ( 𝑋 + ( ( 𝐽 · 𝑇 ) + 𝑇 ) ) = ( 𝑋 + ( ( 𝐽 + 1 ) · 𝑇 ) ) ) |
114 |
113
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → ( 𝑋 + ( ( 𝐽 · 𝑇 ) + 𝑇 ) ) = ( 𝑋 + ( ( 𝐽 + 1 ) · 𝑇 ) ) ) |
115 |
|
oveq1 |
⊢ ( 𝐼 = ( 𝐽 + 1 ) → ( 𝐼 · 𝑇 ) = ( ( 𝐽 + 1 ) · 𝑇 ) ) |
116 |
115
|
eqcomd |
⊢ ( 𝐼 = ( 𝐽 + 1 ) → ( ( 𝐽 + 1 ) · 𝑇 ) = ( 𝐼 · 𝑇 ) ) |
117 |
116
|
oveq2d |
⊢ ( 𝐼 = ( 𝐽 + 1 ) → ( 𝑋 + ( ( 𝐽 + 1 ) · 𝑇 ) ) = ( 𝑋 + ( 𝐼 · 𝑇 ) ) ) |
118 |
117
|
adantl |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → ( 𝑋 + ( ( 𝐽 + 1 ) · 𝑇 ) ) = ( 𝑋 + ( 𝐼 · 𝑇 ) ) ) |
119 |
109 114 118
|
3eqtrrd |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → ( 𝑋 + ( 𝐼 · 𝑇 ) ) = ( ( 𝑋 + ( 𝐽 · 𝑇 ) ) + 𝑇 ) ) |
120 |
105 106 119
|
3brtr4d |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → 𝐵 < ( 𝑋 + ( 𝐼 · 𝑇 ) ) ) |
121 |
2
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → 𝐵 ∈ ℝ ) |
122 |
59
|
adantr |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → ( 𝑋 + ( 𝐼 · 𝑇 ) ) ∈ ℝ ) |
123 |
121 122
|
ltnled |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → ( 𝐵 < ( 𝑋 + ( 𝐼 · 𝑇 ) ) ↔ ¬ ( 𝑋 + ( 𝐼 · 𝑇 ) ) ≤ 𝐵 ) ) |
124 |
120 123
|
mpbid |
⊢ ( ( 𝜑 ∧ 𝐼 = ( 𝐽 + 1 ) ) → ¬ ( 𝑋 + ( 𝐼 · 𝑇 ) ) ≤ 𝐵 ) |
125 |
98 124
|
pm2.65da |
⊢ ( 𝜑 → ¬ 𝐼 = ( 𝐽 + 1 ) ) |
126 |
95 125
|
jca |
⊢ ( 𝜑 → ( ¬ 𝐽 = ( 𝐼 + 1 ) ∧ ¬ 𝐼 = ( 𝐽 + 1 ) ) ) |
127 |
126
|
adantr |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝐽 ) → ( ¬ 𝐽 = ( 𝐼 + 1 ) ∧ ¬ 𝐼 = ( 𝐽 + 1 ) ) ) |
128 |
|
pm4.56 |
⊢ ( ( ¬ 𝐽 = ( 𝐼 + 1 ) ∧ ¬ 𝐼 = ( 𝐽 + 1 ) ) ↔ ¬ ( 𝐽 = ( 𝐼 + 1 ) ∨ 𝐼 = ( 𝐽 + 1 ) ) ) |
129 |
127 128
|
sylib |
⊢ ( ( 𝜑 ∧ ¬ 𝐼 = 𝐽 ) → ¬ ( 𝐽 = ( 𝐼 + 1 ) ∨ 𝐼 = ( 𝐽 + 1 ) ) ) |
130 |
49 129
|
condan |
⊢ ( 𝜑 → 𝐼 = 𝐽 ) |