Step |
Hyp |
Ref |
Expression |
1 |
|
oveq1 |
⊢ ( 𝑁 = ( ( 2 · 𝑀 ) + 1 ) → ( 𝑁 / 4 ) = ( ( ( 2 · 𝑀 ) + 1 ) / 4 ) ) |
2 |
|
2cnd |
⊢ ( 𝑀 ∈ ℤ → 2 ∈ ℂ ) |
3 |
|
zcn |
⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℂ ) |
4 |
2 3
|
mulcld |
⊢ ( 𝑀 ∈ ℤ → ( 2 · 𝑀 ) ∈ ℂ ) |
5 |
|
1cnd |
⊢ ( 𝑀 ∈ ℤ → 1 ∈ ℂ ) |
6 |
|
4cn |
⊢ 4 ∈ ℂ |
7 |
|
4ne0 |
⊢ 4 ≠ 0 |
8 |
6 7
|
pm3.2i |
⊢ ( 4 ∈ ℂ ∧ 4 ≠ 0 ) |
9 |
8
|
a1i |
⊢ ( 𝑀 ∈ ℤ → ( 4 ∈ ℂ ∧ 4 ≠ 0 ) ) |
10 |
|
divdir |
⊢ ( ( ( 2 · 𝑀 ) ∈ ℂ ∧ 1 ∈ ℂ ∧ ( 4 ∈ ℂ ∧ 4 ≠ 0 ) ) → ( ( ( 2 · 𝑀 ) + 1 ) / 4 ) = ( ( ( 2 · 𝑀 ) / 4 ) + ( 1 / 4 ) ) ) |
11 |
4 5 9 10
|
syl3anc |
⊢ ( 𝑀 ∈ ℤ → ( ( ( 2 · 𝑀 ) + 1 ) / 4 ) = ( ( ( 2 · 𝑀 ) / 4 ) + ( 1 / 4 ) ) ) |
12 |
|
2t2e4 |
⊢ ( 2 · 2 ) = 4 |
13 |
12
|
eqcomi |
⊢ 4 = ( 2 · 2 ) |
14 |
13
|
a1i |
⊢ ( 𝑀 ∈ ℤ → 4 = ( 2 · 2 ) ) |
15 |
14
|
oveq2d |
⊢ ( 𝑀 ∈ ℤ → ( ( 2 · 𝑀 ) / 4 ) = ( ( 2 · 𝑀 ) / ( 2 · 2 ) ) ) |
16 |
|
2ne0 |
⊢ 2 ≠ 0 |
17 |
16
|
a1i |
⊢ ( 𝑀 ∈ ℤ → 2 ≠ 0 ) |
18 |
3 2 2 17 17
|
divcan5d |
⊢ ( 𝑀 ∈ ℤ → ( ( 2 · 𝑀 ) / ( 2 · 2 ) ) = ( 𝑀 / 2 ) ) |
19 |
15 18
|
eqtrd |
⊢ ( 𝑀 ∈ ℤ → ( ( 2 · 𝑀 ) / 4 ) = ( 𝑀 / 2 ) ) |
20 |
19
|
oveq1d |
⊢ ( 𝑀 ∈ ℤ → ( ( ( 2 · 𝑀 ) / 4 ) + ( 1 / 4 ) ) = ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) |
21 |
11 20
|
eqtrd |
⊢ ( 𝑀 ∈ ℤ → ( ( ( 2 · 𝑀 ) + 1 ) / 4 ) = ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) |
22 |
1 21
|
sylan9eqr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 = ( ( 2 · 𝑀 ) + 1 ) ) → ( 𝑁 / 4 ) = ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) |
23 |
22
|
fveq2d |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 = ( ( 2 · 𝑀 ) + 1 ) ) → ( ⌊ ‘ ( 𝑁 / 4 ) ) = ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) ) |
24 |
|
iftrue |
⊢ ( 2 ∥ 𝑀 → if ( 2 ∥ 𝑀 , ( 𝑀 / 2 ) , ( ( 𝑀 − 1 ) / 2 ) ) = ( 𝑀 / 2 ) ) |
25 |
24
|
adantr |
⊢ ( ( 2 ∥ 𝑀 ∧ 𝑀 ∈ ℤ ) → if ( 2 ∥ 𝑀 , ( 𝑀 / 2 ) , ( ( 𝑀 − 1 ) / 2 ) ) = ( 𝑀 / 2 ) ) |
26 |
|
1re |
⊢ 1 ∈ ℝ |
27 |
|
0le1 |
⊢ 0 ≤ 1 |
28 |
|
4re |
⊢ 4 ∈ ℝ |
29 |
|
4pos |
⊢ 0 < 4 |
30 |
|
divge0 |
⊢ ( ( ( 1 ∈ ℝ ∧ 0 ≤ 1 ) ∧ ( 4 ∈ ℝ ∧ 0 < 4 ) ) → 0 ≤ ( 1 / 4 ) ) |
31 |
26 27 28 29 30
|
mp4an |
⊢ 0 ≤ ( 1 / 4 ) |
32 |
|
1lt4 |
⊢ 1 < 4 |
33 |
|
recgt1 |
⊢ ( ( 4 ∈ ℝ ∧ 0 < 4 ) → ( 1 < 4 ↔ ( 1 / 4 ) < 1 ) ) |
34 |
28 29 33
|
mp2an |
⊢ ( 1 < 4 ↔ ( 1 / 4 ) < 1 ) |
35 |
32 34
|
mpbi |
⊢ ( 1 / 4 ) < 1 |
36 |
31 35
|
pm3.2i |
⊢ ( 0 ≤ ( 1 / 4 ) ∧ ( 1 / 4 ) < 1 ) |
37 |
|
evend2 |
⊢ ( 𝑀 ∈ ℤ → ( 2 ∥ 𝑀 ↔ ( 𝑀 / 2 ) ∈ ℤ ) ) |
38 |
37
|
biimpac |
⊢ ( ( 2 ∥ 𝑀 ∧ 𝑀 ∈ ℤ ) → ( 𝑀 / 2 ) ∈ ℤ ) |
39 |
|
4nn |
⊢ 4 ∈ ℕ |
40 |
|
nnrecre |
⊢ ( 4 ∈ ℕ → ( 1 / 4 ) ∈ ℝ ) |
41 |
39 40
|
ax-mp |
⊢ ( 1 / 4 ) ∈ ℝ |
42 |
|
flbi2 |
⊢ ( ( ( 𝑀 / 2 ) ∈ ℤ ∧ ( 1 / 4 ) ∈ ℝ ) → ( ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) = ( 𝑀 / 2 ) ↔ ( 0 ≤ ( 1 / 4 ) ∧ ( 1 / 4 ) < 1 ) ) ) |
43 |
38 41 42
|
sylancl |
⊢ ( ( 2 ∥ 𝑀 ∧ 𝑀 ∈ ℤ ) → ( ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) = ( 𝑀 / 2 ) ↔ ( 0 ≤ ( 1 / 4 ) ∧ ( 1 / 4 ) < 1 ) ) ) |
44 |
36 43
|
mpbiri |
⊢ ( ( 2 ∥ 𝑀 ∧ 𝑀 ∈ ℤ ) → ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) = ( 𝑀 / 2 ) ) |
45 |
25 44
|
eqtr4d |
⊢ ( ( 2 ∥ 𝑀 ∧ 𝑀 ∈ ℤ ) → if ( 2 ∥ 𝑀 , ( 𝑀 / 2 ) , ( ( 𝑀 − 1 ) / 2 ) ) = ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) ) |
46 |
|
iffalse |
⊢ ( ¬ 2 ∥ 𝑀 → if ( 2 ∥ 𝑀 , ( 𝑀 / 2 ) , ( ( 𝑀 − 1 ) / 2 ) ) = ( ( 𝑀 − 1 ) / 2 ) ) |
47 |
46
|
adantr |
⊢ ( ( ¬ 2 ∥ 𝑀 ∧ 𝑀 ∈ ℤ ) → if ( 2 ∥ 𝑀 , ( 𝑀 / 2 ) , ( ( 𝑀 − 1 ) / 2 ) ) = ( ( 𝑀 − 1 ) / 2 ) ) |
48 |
|
odd2np1 |
⊢ ( 𝑀 ∈ ℤ → ( ¬ 2 ∥ 𝑀 ↔ ∃ 𝑥 ∈ ℤ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) ) |
49 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
50 |
|
2cnne0 |
⊢ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) |
51 |
|
divcan5 |
⊢ ( ( 1 ∈ ℂ ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) → ( ( 2 · 1 ) / ( 2 · 2 ) ) = ( 1 / 2 ) ) |
52 |
49 50 50 51
|
mp3an |
⊢ ( ( 2 · 1 ) / ( 2 · 2 ) ) = ( 1 / 2 ) |
53 |
|
2t1e2 |
⊢ ( 2 · 1 ) = 2 |
54 |
53 12
|
oveq12i |
⊢ ( ( 2 · 1 ) / ( 2 · 2 ) ) = ( 2 / 4 ) |
55 |
52 54
|
eqtr3i |
⊢ ( 1 / 2 ) = ( 2 / 4 ) |
56 |
55
|
oveq1i |
⊢ ( ( 1 / 2 ) + ( 1 / 4 ) ) = ( ( 2 / 4 ) + ( 1 / 4 ) ) |
57 |
|
2cn |
⊢ 2 ∈ ℂ |
58 |
57 49 6 7
|
divdiri |
⊢ ( ( 2 + 1 ) / 4 ) = ( ( 2 / 4 ) + ( 1 / 4 ) ) |
59 |
|
2p1e3 |
⊢ ( 2 + 1 ) = 3 |
60 |
59
|
oveq1i |
⊢ ( ( 2 + 1 ) / 4 ) = ( 3 / 4 ) |
61 |
56 58 60
|
3eqtr2i |
⊢ ( ( 1 / 2 ) + ( 1 / 4 ) ) = ( 3 / 4 ) |
62 |
61
|
a1i |
⊢ ( 𝑥 ∈ ℤ → ( ( 1 / 2 ) + ( 1 / 4 ) ) = ( 3 / 4 ) ) |
63 |
62
|
oveq2d |
⊢ ( 𝑥 ∈ ℤ → ( 𝑥 + ( ( 1 / 2 ) + ( 1 / 4 ) ) ) = ( 𝑥 + ( 3 / 4 ) ) ) |
64 |
63
|
fveq2d |
⊢ ( 𝑥 ∈ ℤ → ( ⌊ ‘ ( 𝑥 + ( ( 1 / 2 ) + ( 1 / 4 ) ) ) ) = ( ⌊ ‘ ( 𝑥 + ( 3 / 4 ) ) ) ) |
65 |
|
3re |
⊢ 3 ∈ ℝ |
66 |
|
0re |
⊢ 0 ∈ ℝ |
67 |
|
3pos |
⊢ 0 < 3 |
68 |
66 65 67
|
ltleii |
⊢ 0 ≤ 3 |
69 |
|
divge0 |
⊢ ( ( ( 3 ∈ ℝ ∧ 0 ≤ 3 ) ∧ ( 4 ∈ ℝ ∧ 0 < 4 ) ) → 0 ≤ ( 3 / 4 ) ) |
70 |
65 68 28 29 69
|
mp4an |
⊢ 0 ≤ ( 3 / 4 ) |
71 |
|
3lt4 |
⊢ 3 < 4 |
72 |
|
nnrp |
⊢ ( 4 ∈ ℕ → 4 ∈ ℝ+ ) |
73 |
39 72
|
ax-mp |
⊢ 4 ∈ ℝ+ |
74 |
|
divlt1lt |
⊢ ( ( 3 ∈ ℝ ∧ 4 ∈ ℝ+ ) → ( ( 3 / 4 ) < 1 ↔ 3 < 4 ) ) |
75 |
65 73 74
|
mp2an |
⊢ ( ( 3 / 4 ) < 1 ↔ 3 < 4 ) |
76 |
71 75
|
mpbir |
⊢ ( 3 / 4 ) < 1 |
77 |
70 76
|
pm3.2i |
⊢ ( 0 ≤ ( 3 / 4 ) ∧ ( 3 / 4 ) < 1 ) |
78 |
65 28 7
|
redivcli |
⊢ ( 3 / 4 ) ∈ ℝ |
79 |
|
flbi2 |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( 3 / 4 ) ∈ ℝ ) → ( ( ⌊ ‘ ( 𝑥 + ( 3 / 4 ) ) ) = 𝑥 ↔ ( 0 ≤ ( 3 / 4 ) ∧ ( 3 / 4 ) < 1 ) ) ) |
80 |
78 79
|
mpan2 |
⊢ ( 𝑥 ∈ ℤ → ( ( ⌊ ‘ ( 𝑥 + ( 3 / 4 ) ) ) = 𝑥 ↔ ( 0 ≤ ( 3 / 4 ) ∧ ( 3 / 4 ) < 1 ) ) ) |
81 |
77 80
|
mpbiri |
⊢ ( 𝑥 ∈ ℤ → ( ⌊ ‘ ( 𝑥 + ( 3 / 4 ) ) ) = 𝑥 ) |
82 |
64 81
|
eqtrd |
⊢ ( 𝑥 ∈ ℤ → ( ⌊ ‘ ( 𝑥 + ( ( 1 / 2 ) + ( 1 / 4 ) ) ) ) = 𝑥 ) |
83 |
82
|
adantr |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( ⌊ ‘ ( 𝑥 + ( ( 1 / 2 ) + ( 1 / 4 ) ) ) ) = 𝑥 ) |
84 |
|
oveq1 |
⊢ ( 𝑀 = ( ( 2 · 𝑥 ) + 1 ) → ( 𝑀 / 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) / 2 ) ) |
85 |
84
|
eqcoms |
⊢ ( ( ( 2 · 𝑥 ) + 1 ) = 𝑀 → ( 𝑀 / 2 ) = ( ( ( 2 · 𝑥 ) + 1 ) / 2 ) ) |
86 |
|
2z |
⊢ 2 ∈ ℤ |
87 |
86
|
a1i |
⊢ ( 𝑥 ∈ ℤ → 2 ∈ ℤ ) |
88 |
|
id |
⊢ ( 𝑥 ∈ ℤ → 𝑥 ∈ ℤ ) |
89 |
87 88
|
zmulcld |
⊢ ( 𝑥 ∈ ℤ → ( 2 · 𝑥 ) ∈ ℤ ) |
90 |
89
|
zcnd |
⊢ ( 𝑥 ∈ ℤ → ( 2 · 𝑥 ) ∈ ℂ ) |
91 |
|
1cnd |
⊢ ( 𝑥 ∈ ℤ → 1 ∈ ℂ ) |
92 |
50
|
a1i |
⊢ ( 𝑥 ∈ ℤ → ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) |
93 |
|
divdir |
⊢ ( ( ( 2 · 𝑥 ) ∈ ℂ ∧ 1 ∈ ℂ ∧ ( 2 ∈ ℂ ∧ 2 ≠ 0 ) ) → ( ( ( 2 · 𝑥 ) + 1 ) / 2 ) = ( ( ( 2 · 𝑥 ) / 2 ) + ( 1 / 2 ) ) ) |
94 |
90 91 92 93
|
syl3anc |
⊢ ( 𝑥 ∈ ℤ → ( ( ( 2 · 𝑥 ) + 1 ) / 2 ) = ( ( ( 2 · 𝑥 ) / 2 ) + ( 1 / 2 ) ) ) |
95 |
|
zcn |
⊢ ( 𝑥 ∈ ℤ → 𝑥 ∈ ℂ ) |
96 |
|
2cnd |
⊢ ( 𝑥 ∈ ℤ → 2 ∈ ℂ ) |
97 |
16
|
a1i |
⊢ ( 𝑥 ∈ ℤ → 2 ≠ 0 ) |
98 |
95 96 97
|
divcan3d |
⊢ ( 𝑥 ∈ ℤ → ( ( 2 · 𝑥 ) / 2 ) = 𝑥 ) |
99 |
98
|
oveq1d |
⊢ ( 𝑥 ∈ ℤ → ( ( ( 2 · 𝑥 ) / 2 ) + ( 1 / 2 ) ) = ( 𝑥 + ( 1 / 2 ) ) ) |
100 |
94 99
|
eqtrd |
⊢ ( 𝑥 ∈ ℤ → ( ( ( 2 · 𝑥 ) + 1 ) / 2 ) = ( 𝑥 + ( 1 / 2 ) ) ) |
101 |
85 100
|
sylan9eqr |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( 𝑀 / 2 ) = ( 𝑥 + ( 1 / 2 ) ) ) |
102 |
101
|
oveq1d |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) = ( ( 𝑥 + ( 1 / 2 ) ) + ( 1 / 4 ) ) ) |
103 |
|
halfcn |
⊢ ( 1 / 2 ) ∈ ℂ |
104 |
103
|
a1i |
⊢ ( 𝑥 ∈ ℤ → ( 1 / 2 ) ∈ ℂ ) |
105 |
6 7
|
reccli |
⊢ ( 1 / 4 ) ∈ ℂ |
106 |
105
|
a1i |
⊢ ( 𝑥 ∈ ℤ → ( 1 / 4 ) ∈ ℂ ) |
107 |
95 104 106
|
addassd |
⊢ ( 𝑥 ∈ ℤ → ( ( 𝑥 + ( 1 / 2 ) ) + ( 1 / 4 ) ) = ( 𝑥 + ( ( 1 / 2 ) + ( 1 / 4 ) ) ) ) |
108 |
107
|
adantr |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( ( 𝑥 + ( 1 / 2 ) ) + ( 1 / 4 ) ) = ( 𝑥 + ( ( 1 / 2 ) + ( 1 / 4 ) ) ) ) |
109 |
102 108
|
eqtrd |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) = ( 𝑥 + ( ( 1 / 2 ) + ( 1 / 4 ) ) ) ) |
110 |
109
|
fveq2d |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) = ( ⌊ ‘ ( 𝑥 + ( ( 1 / 2 ) + ( 1 / 4 ) ) ) ) ) |
111 |
|
oveq1 |
⊢ ( 𝑀 = ( ( 2 · 𝑥 ) + 1 ) → ( 𝑀 − 1 ) = ( ( ( 2 · 𝑥 ) + 1 ) − 1 ) ) |
112 |
111
|
eqcoms |
⊢ ( ( ( 2 · 𝑥 ) + 1 ) = 𝑀 → ( 𝑀 − 1 ) = ( ( ( 2 · 𝑥 ) + 1 ) − 1 ) ) |
113 |
|
pncan1 |
⊢ ( ( 2 · 𝑥 ) ∈ ℂ → ( ( ( 2 · 𝑥 ) + 1 ) − 1 ) = ( 2 · 𝑥 ) ) |
114 |
90 113
|
syl |
⊢ ( 𝑥 ∈ ℤ → ( ( ( 2 · 𝑥 ) + 1 ) − 1 ) = ( 2 · 𝑥 ) ) |
115 |
112 114
|
sylan9eqr |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( 𝑀 − 1 ) = ( 2 · 𝑥 ) ) |
116 |
115
|
oveq1d |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( ( 𝑀 − 1 ) / 2 ) = ( ( 2 · 𝑥 ) / 2 ) ) |
117 |
98
|
adantr |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( ( 2 · 𝑥 ) / 2 ) = 𝑥 ) |
118 |
116 117
|
eqtrd |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( ( 𝑀 − 1 ) / 2 ) = 𝑥 ) |
119 |
83 110 118
|
3eqtr4rd |
⊢ ( ( 𝑥 ∈ ℤ ∧ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 ) → ( ( 𝑀 − 1 ) / 2 ) = ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) ) |
120 |
119
|
ex |
⊢ ( 𝑥 ∈ ℤ → ( ( ( 2 · 𝑥 ) + 1 ) = 𝑀 → ( ( 𝑀 − 1 ) / 2 ) = ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) ) ) |
121 |
120
|
adantl |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑥 ∈ ℤ ) → ( ( ( 2 · 𝑥 ) + 1 ) = 𝑀 → ( ( 𝑀 − 1 ) / 2 ) = ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) ) ) |
122 |
121
|
rexlimdva |
⊢ ( 𝑀 ∈ ℤ → ( ∃ 𝑥 ∈ ℤ ( ( 2 · 𝑥 ) + 1 ) = 𝑀 → ( ( 𝑀 − 1 ) / 2 ) = ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) ) ) |
123 |
48 122
|
sylbid |
⊢ ( 𝑀 ∈ ℤ → ( ¬ 2 ∥ 𝑀 → ( ( 𝑀 − 1 ) / 2 ) = ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) ) ) |
124 |
123
|
impcom |
⊢ ( ( ¬ 2 ∥ 𝑀 ∧ 𝑀 ∈ ℤ ) → ( ( 𝑀 − 1 ) / 2 ) = ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) ) |
125 |
47 124
|
eqtrd |
⊢ ( ( ¬ 2 ∥ 𝑀 ∧ 𝑀 ∈ ℤ ) → if ( 2 ∥ 𝑀 , ( 𝑀 / 2 ) , ( ( 𝑀 − 1 ) / 2 ) ) = ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) ) |
126 |
45 125
|
pm2.61ian |
⊢ ( 𝑀 ∈ ℤ → if ( 2 ∥ 𝑀 , ( 𝑀 / 2 ) , ( ( 𝑀 − 1 ) / 2 ) ) = ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) ) |
127 |
126
|
eqcomd |
⊢ ( 𝑀 ∈ ℤ → ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) = if ( 2 ∥ 𝑀 , ( 𝑀 / 2 ) , ( ( 𝑀 − 1 ) / 2 ) ) ) |
128 |
127
|
adantr |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 = ( ( 2 · 𝑀 ) + 1 ) ) → ( ⌊ ‘ ( ( 𝑀 / 2 ) + ( 1 / 4 ) ) ) = if ( 2 ∥ 𝑀 , ( 𝑀 / 2 ) , ( ( 𝑀 − 1 ) / 2 ) ) ) |
129 |
23 128
|
eqtrd |
⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 = ( ( 2 · 𝑀 ) + 1 ) ) → ( ⌊ ‘ ( 𝑁 / 4 ) ) = if ( 2 ∥ 𝑀 , ( 𝑀 / 2 ) , ( ( 𝑀 − 1 ) / 2 ) ) ) |