Step |
Hyp |
Ref |
Expression |
1 |
|
7re |
⊢ 7 ∈ ℝ |
2 |
1
|
rexri |
⊢ 7 ∈ ℝ* |
3 |
|
1nn0 |
⊢ 1 ∈ ℕ0 |
4 |
|
3nn |
⊢ 3 ∈ ℕ |
5 |
3 4
|
decnncl |
⊢ ; 1 3 ∈ ℕ |
6 |
5
|
nnrei |
⊢ ; 1 3 ∈ ℝ |
7 |
6
|
rexri |
⊢ ; 1 3 ∈ ℝ* |
8 |
|
elico1 |
⊢ ( ( 7 ∈ ℝ* ∧ ; 1 3 ∈ ℝ* ) → ( 𝑁 ∈ ( 7 [,) ; 1 3 ) ↔ ( 𝑁 ∈ ℝ* ∧ 7 ≤ 𝑁 ∧ 𝑁 < ; 1 3 ) ) ) |
9 |
2 7 8
|
mp2an |
⊢ ( 𝑁 ∈ ( 7 [,) ; 1 3 ) ↔ ( 𝑁 ∈ ℝ* ∧ 7 ≤ 𝑁 ∧ 𝑁 < ; 1 3 ) ) |
10 |
|
7nn |
⊢ 7 ∈ ℕ |
11 |
10
|
nnzi |
⊢ 7 ∈ ℤ |
12 |
|
oddz |
⊢ ( 𝑁 ∈ Odd → 𝑁 ∈ ℤ ) |
13 |
|
zltp1le |
⊢ ( ( 7 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 7 < 𝑁 ↔ ( 7 + 1 ) ≤ 𝑁 ) ) |
14 |
|
7p1e8 |
⊢ ( 7 + 1 ) = 8 |
15 |
14
|
breq1i |
⊢ ( ( 7 + 1 ) ≤ 𝑁 ↔ 8 ≤ 𝑁 ) |
16 |
15
|
a1i |
⊢ ( ( 7 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ( 7 + 1 ) ≤ 𝑁 ↔ 8 ≤ 𝑁 ) ) |
17 |
|
8re |
⊢ 8 ∈ ℝ |
18 |
17
|
a1i |
⊢ ( 7 ∈ ℤ → 8 ∈ ℝ ) |
19 |
|
zre |
⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ ) |
20 |
|
leloe |
⊢ ( ( 8 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( 8 ≤ 𝑁 ↔ ( 8 < 𝑁 ∨ 8 = 𝑁 ) ) ) |
21 |
18 19 20
|
syl2an |
⊢ ( ( 7 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 8 ≤ 𝑁 ↔ ( 8 < 𝑁 ∨ 8 = 𝑁 ) ) ) |
22 |
13 16 21
|
3bitrd |
⊢ ( ( 7 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 7 < 𝑁 ↔ ( 8 < 𝑁 ∨ 8 = 𝑁 ) ) ) |
23 |
11 12 22
|
sylancr |
⊢ ( 𝑁 ∈ Odd → ( 7 < 𝑁 ↔ ( 8 < 𝑁 ∨ 8 = 𝑁 ) ) ) |
24 |
|
8nn |
⊢ 8 ∈ ℕ |
25 |
24
|
nnzi |
⊢ 8 ∈ ℤ |
26 |
|
zltp1le |
⊢ ( ( 8 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 8 < 𝑁 ↔ ( 8 + 1 ) ≤ 𝑁 ) ) |
27 |
25 12 26
|
sylancr |
⊢ ( 𝑁 ∈ Odd → ( 8 < 𝑁 ↔ ( 8 + 1 ) ≤ 𝑁 ) ) |
28 |
|
8p1e9 |
⊢ ( 8 + 1 ) = 9 |
29 |
28
|
breq1i |
⊢ ( ( 8 + 1 ) ≤ 𝑁 ↔ 9 ≤ 𝑁 ) |
30 |
29
|
a1i |
⊢ ( 𝑁 ∈ Odd → ( ( 8 + 1 ) ≤ 𝑁 ↔ 9 ≤ 𝑁 ) ) |
31 |
|
9re |
⊢ 9 ∈ ℝ |
32 |
31
|
a1i |
⊢ ( 𝑁 ∈ Odd → 9 ∈ ℝ ) |
33 |
12
|
zred |
⊢ ( 𝑁 ∈ Odd → 𝑁 ∈ ℝ ) |
34 |
32 33
|
leloed |
⊢ ( 𝑁 ∈ Odd → ( 9 ≤ 𝑁 ↔ ( 9 < 𝑁 ∨ 9 = 𝑁 ) ) ) |
35 |
27 30 34
|
3bitrd |
⊢ ( 𝑁 ∈ Odd → ( 8 < 𝑁 ↔ ( 9 < 𝑁 ∨ 9 = 𝑁 ) ) ) |
36 |
|
9nn |
⊢ 9 ∈ ℕ |
37 |
36
|
nnzi |
⊢ 9 ∈ ℤ |
38 |
|
zltp1le |
⊢ ( ( 9 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 9 < 𝑁 ↔ ( 9 + 1 ) ≤ 𝑁 ) ) |
39 |
37 12 38
|
sylancr |
⊢ ( 𝑁 ∈ Odd → ( 9 < 𝑁 ↔ ( 9 + 1 ) ≤ 𝑁 ) ) |
40 |
|
9p1e10 |
⊢ ( 9 + 1 ) = ; 1 0 |
41 |
40
|
breq1i |
⊢ ( ( 9 + 1 ) ≤ 𝑁 ↔ ; 1 0 ≤ 𝑁 ) |
42 |
41
|
a1i |
⊢ ( 𝑁 ∈ Odd → ( ( 9 + 1 ) ≤ 𝑁 ↔ ; 1 0 ≤ 𝑁 ) ) |
43 |
|
10re |
⊢ ; 1 0 ∈ ℝ |
44 |
43
|
a1i |
⊢ ( 𝑁 ∈ Odd → ; 1 0 ∈ ℝ ) |
45 |
44 33
|
leloed |
⊢ ( 𝑁 ∈ Odd → ( ; 1 0 ≤ 𝑁 ↔ ( ; 1 0 < 𝑁 ∨ ; 1 0 = 𝑁 ) ) ) |
46 |
39 42 45
|
3bitrd |
⊢ ( 𝑁 ∈ Odd → ( 9 < 𝑁 ↔ ( ; 1 0 < 𝑁 ∨ ; 1 0 = 𝑁 ) ) ) |
47 |
|
10nn |
⊢ ; 1 0 ∈ ℕ |
48 |
47
|
nnzi |
⊢ ; 1 0 ∈ ℤ |
49 |
|
zltp1le |
⊢ ( ( ; 1 0 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ; 1 0 < 𝑁 ↔ ( ; 1 0 + 1 ) ≤ 𝑁 ) ) |
50 |
48 12 49
|
sylancr |
⊢ ( 𝑁 ∈ Odd → ( ; 1 0 < 𝑁 ↔ ( ; 1 0 + 1 ) ≤ 𝑁 ) ) |
51 |
|
dec10p |
⊢ ( ; 1 0 + 1 ) = ; 1 1 |
52 |
51
|
breq1i |
⊢ ( ( ; 1 0 + 1 ) ≤ 𝑁 ↔ ; 1 1 ≤ 𝑁 ) |
53 |
52
|
a1i |
⊢ ( 𝑁 ∈ Odd → ( ( ; 1 0 + 1 ) ≤ 𝑁 ↔ ; 1 1 ≤ 𝑁 ) ) |
54 |
|
1nn |
⊢ 1 ∈ ℕ |
55 |
3 54
|
decnncl |
⊢ ; 1 1 ∈ ℕ |
56 |
55
|
nnrei |
⊢ ; 1 1 ∈ ℝ |
57 |
56
|
a1i |
⊢ ( 𝑁 ∈ Odd → ; 1 1 ∈ ℝ ) |
58 |
57 33
|
leloed |
⊢ ( 𝑁 ∈ Odd → ( ; 1 1 ≤ 𝑁 ↔ ( ; 1 1 < 𝑁 ∨ ; 1 1 = 𝑁 ) ) ) |
59 |
50 53 58
|
3bitrd |
⊢ ( 𝑁 ∈ Odd → ( ; 1 0 < 𝑁 ↔ ( ; 1 1 < 𝑁 ∨ ; 1 1 = 𝑁 ) ) ) |
60 |
55
|
nnzi |
⊢ ; 1 1 ∈ ℤ |
61 |
|
zltp1le |
⊢ ( ( ; 1 1 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ; 1 1 < 𝑁 ↔ ( ; 1 1 + 1 ) ≤ 𝑁 ) ) |
62 |
60 12 61
|
sylancr |
⊢ ( 𝑁 ∈ Odd → ( ; 1 1 < 𝑁 ↔ ( ; 1 1 + 1 ) ≤ 𝑁 ) ) |
63 |
51
|
eqcomi |
⊢ ; 1 1 = ( ; 1 0 + 1 ) |
64 |
63
|
oveq1i |
⊢ ( ; 1 1 + 1 ) = ( ( ; 1 0 + 1 ) + 1 ) |
65 |
47
|
nncni |
⊢ ; 1 0 ∈ ℂ |
66 |
|
ax-1cn |
⊢ 1 ∈ ℂ |
67 |
65 66 66
|
addassi |
⊢ ( ( ; 1 0 + 1 ) + 1 ) = ( ; 1 0 + ( 1 + 1 ) ) |
68 |
|
1p1e2 |
⊢ ( 1 + 1 ) = 2 |
69 |
68
|
oveq2i |
⊢ ( ; 1 0 + ( 1 + 1 ) ) = ( ; 1 0 + 2 ) |
70 |
|
dec10p |
⊢ ( ; 1 0 + 2 ) = ; 1 2 |
71 |
69 70
|
eqtri |
⊢ ( ; 1 0 + ( 1 + 1 ) ) = ; 1 2 |
72 |
64 67 71
|
3eqtri |
⊢ ( ; 1 1 + 1 ) = ; 1 2 |
73 |
72
|
breq1i |
⊢ ( ( ; 1 1 + 1 ) ≤ 𝑁 ↔ ; 1 2 ≤ 𝑁 ) |
74 |
73
|
a1i |
⊢ ( 𝑁 ∈ Odd → ( ( ; 1 1 + 1 ) ≤ 𝑁 ↔ ; 1 2 ≤ 𝑁 ) ) |
75 |
|
2nn |
⊢ 2 ∈ ℕ |
76 |
3 75
|
decnncl |
⊢ ; 1 2 ∈ ℕ |
77 |
76
|
nnrei |
⊢ ; 1 2 ∈ ℝ |
78 |
77
|
a1i |
⊢ ( 𝑁 ∈ Odd → ; 1 2 ∈ ℝ ) |
79 |
78 33
|
leloed |
⊢ ( 𝑁 ∈ Odd → ( ; 1 2 ≤ 𝑁 ↔ ( ; 1 2 < 𝑁 ∨ ; 1 2 = 𝑁 ) ) ) |
80 |
62 74 79
|
3bitrd |
⊢ ( 𝑁 ∈ Odd → ( ; 1 1 < 𝑁 ↔ ( ; 1 2 < 𝑁 ∨ ; 1 2 = 𝑁 ) ) ) |
81 |
76
|
nnzi |
⊢ ; 1 2 ∈ ℤ |
82 |
|
zltp1le |
⊢ ( ( ; 1 2 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( ; 1 2 < 𝑁 ↔ ( ; 1 2 + 1 ) ≤ 𝑁 ) ) |
83 |
81 12 82
|
sylancr |
⊢ ( 𝑁 ∈ Odd → ( ; 1 2 < 𝑁 ↔ ( ; 1 2 + 1 ) ≤ 𝑁 ) ) |
84 |
70
|
eqcomi |
⊢ ; 1 2 = ( ; 1 0 + 2 ) |
85 |
84
|
oveq1i |
⊢ ( ; 1 2 + 1 ) = ( ( ; 1 0 + 2 ) + 1 ) |
86 |
|
2cn |
⊢ 2 ∈ ℂ |
87 |
65 86 66
|
addassi |
⊢ ( ( ; 1 0 + 2 ) + 1 ) = ( ; 1 0 + ( 2 + 1 ) ) |
88 |
|
2p1e3 |
⊢ ( 2 + 1 ) = 3 |
89 |
88
|
oveq2i |
⊢ ( ; 1 0 + ( 2 + 1 ) ) = ( ; 1 0 + 3 ) |
90 |
|
dec10p |
⊢ ( ; 1 0 + 3 ) = ; 1 3 |
91 |
89 90
|
eqtri |
⊢ ( ; 1 0 + ( 2 + 1 ) ) = ; 1 3 |
92 |
85 87 91
|
3eqtri |
⊢ ( ; 1 2 + 1 ) = ; 1 3 |
93 |
92
|
breq1i |
⊢ ( ( ; 1 2 + 1 ) ≤ 𝑁 ↔ ; 1 3 ≤ 𝑁 ) |
94 |
93
|
a1i |
⊢ ( 𝑁 ∈ Odd → ( ( ; 1 2 + 1 ) ≤ 𝑁 ↔ ; 1 3 ≤ 𝑁 ) ) |
95 |
6
|
a1i |
⊢ ( 𝑁 ∈ Odd → ; 1 3 ∈ ℝ ) |
96 |
95 33
|
lenltd |
⊢ ( 𝑁 ∈ Odd → ( ; 1 3 ≤ 𝑁 ↔ ¬ 𝑁 < ; 1 3 ) ) |
97 |
83 94 96
|
3bitrd |
⊢ ( 𝑁 ∈ Odd → ( ; 1 2 < 𝑁 ↔ ¬ 𝑁 < ; 1 3 ) ) |
98 |
|
pm2.21 |
⊢ ( ¬ 𝑁 < ; 1 3 → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) |
99 |
97 98
|
syl6bi |
⊢ ( 𝑁 ∈ Odd → ( ; 1 2 < 𝑁 → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
100 |
99
|
com12 |
⊢ ( ; 1 2 < 𝑁 → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
101 |
|
eleq1 |
⊢ ( ; 1 2 = 𝑁 → ( ; 1 2 ∈ Odd ↔ 𝑁 ∈ Odd ) ) |
102 |
|
6p6e12 |
⊢ ( 6 + 6 ) = ; 1 2 |
103 |
|
6even |
⊢ 6 ∈ Even |
104 |
|
epee |
⊢ ( ( 6 ∈ Even ∧ 6 ∈ Even ) → ( 6 + 6 ) ∈ Even ) |
105 |
103 103 104
|
mp2an |
⊢ ( 6 + 6 ) ∈ Even |
106 |
102 105
|
eqeltrri |
⊢ ; 1 2 ∈ Even |
107 |
|
evennodd |
⊢ ( ; 1 2 ∈ Even → ¬ ; 1 2 ∈ Odd ) |
108 |
106 107
|
ax-mp |
⊢ ¬ ; 1 2 ∈ Odd |
109 |
108
|
pm2.21i |
⊢ ( ; 1 2 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) |
110 |
101 109
|
syl6bir |
⊢ ( ; 1 2 = 𝑁 → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
111 |
100 110
|
jaoi |
⊢ ( ( ; 1 2 < 𝑁 ∨ ; 1 2 = 𝑁 ) → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
112 |
111
|
com12 |
⊢ ( 𝑁 ∈ Odd → ( ( ; 1 2 < 𝑁 ∨ ; 1 2 = 𝑁 ) → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
113 |
80 112
|
sylbid |
⊢ ( 𝑁 ∈ Odd → ( ; 1 1 < 𝑁 → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
114 |
113
|
com12 |
⊢ ( ; 1 1 < 𝑁 → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
115 |
|
11gbo |
⊢ ; 1 1 ∈ GoldbachOdd |
116 |
|
eleq1 |
⊢ ( ; 1 1 = 𝑁 → ( ; 1 1 ∈ GoldbachOdd ↔ 𝑁 ∈ GoldbachOdd ) ) |
117 |
115 116
|
mpbii |
⊢ ( ; 1 1 = 𝑁 → 𝑁 ∈ GoldbachOdd ) |
118 |
117
|
2a1d |
⊢ ( ; 1 1 = 𝑁 → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
119 |
114 118
|
jaoi |
⊢ ( ( ; 1 1 < 𝑁 ∨ ; 1 1 = 𝑁 ) → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
120 |
119
|
com12 |
⊢ ( 𝑁 ∈ Odd → ( ( ; 1 1 < 𝑁 ∨ ; 1 1 = 𝑁 ) → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
121 |
59 120
|
sylbid |
⊢ ( 𝑁 ∈ Odd → ( ; 1 0 < 𝑁 → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
122 |
121
|
com12 |
⊢ ( ; 1 0 < 𝑁 → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
123 |
|
eleq1 |
⊢ ( ; 1 0 = 𝑁 → ( ; 1 0 ∈ Odd ↔ 𝑁 ∈ Odd ) ) |
124 |
|
5p5e10 |
⊢ ( 5 + 5 ) = ; 1 0 |
125 |
|
5odd |
⊢ 5 ∈ Odd |
126 |
|
opoeALTV |
⊢ ( ( 5 ∈ Odd ∧ 5 ∈ Odd ) → ( 5 + 5 ) ∈ Even ) |
127 |
125 125 126
|
mp2an |
⊢ ( 5 + 5 ) ∈ Even |
128 |
124 127
|
eqeltrri |
⊢ ; 1 0 ∈ Even |
129 |
|
evennodd |
⊢ ( ; 1 0 ∈ Even → ¬ ; 1 0 ∈ Odd ) |
130 |
128 129
|
ax-mp |
⊢ ¬ ; 1 0 ∈ Odd |
131 |
130
|
pm2.21i |
⊢ ( ; 1 0 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) |
132 |
123 131
|
syl6bir |
⊢ ( ; 1 0 = 𝑁 → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
133 |
122 132
|
jaoi |
⊢ ( ( ; 1 0 < 𝑁 ∨ ; 1 0 = 𝑁 ) → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
134 |
133
|
com12 |
⊢ ( 𝑁 ∈ Odd → ( ( ; 1 0 < 𝑁 ∨ ; 1 0 = 𝑁 ) → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
135 |
46 134
|
sylbid |
⊢ ( 𝑁 ∈ Odd → ( 9 < 𝑁 → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
136 |
135
|
com12 |
⊢ ( 9 < 𝑁 → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
137 |
|
9gbo |
⊢ 9 ∈ GoldbachOdd |
138 |
|
eleq1 |
⊢ ( 9 = 𝑁 → ( 9 ∈ GoldbachOdd ↔ 𝑁 ∈ GoldbachOdd ) ) |
139 |
137 138
|
mpbii |
⊢ ( 9 = 𝑁 → 𝑁 ∈ GoldbachOdd ) |
140 |
139
|
2a1d |
⊢ ( 9 = 𝑁 → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
141 |
136 140
|
jaoi |
⊢ ( ( 9 < 𝑁 ∨ 9 = 𝑁 ) → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
142 |
141
|
com12 |
⊢ ( 𝑁 ∈ Odd → ( ( 9 < 𝑁 ∨ 9 = 𝑁 ) → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
143 |
35 142
|
sylbid |
⊢ ( 𝑁 ∈ Odd → ( 8 < 𝑁 → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
144 |
143
|
com12 |
⊢ ( 8 < 𝑁 → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
145 |
|
eleq1 |
⊢ ( 8 = 𝑁 → ( 8 ∈ Odd ↔ 𝑁 ∈ Odd ) ) |
146 |
|
8even |
⊢ 8 ∈ Even |
147 |
|
evennodd |
⊢ ( 8 ∈ Even → ¬ 8 ∈ Odd ) |
148 |
146 147
|
ax-mp |
⊢ ¬ 8 ∈ Odd |
149 |
148
|
pm2.21i |
⊢ ( 8 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) |
150 |
145 149
|
syl6bir |
⊢ ( 8 = 𝑁 → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
151 |
144 150
|
jaoi |
⊢ ( ( 8 < 𝑁 ∨ 8 = 𝑁 ) → ( 𝑁 ∈ Odd → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
152 |
151
|
com12 |
⊢ ( 𝑁 ∈ Odd → ( ( 8 < 𝑁 ∨ 8 = 𝑁 ) → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
153 |
23 152
|
sylbid |
⊢ ( 𝑁 ∈ Odd → ( 7 < 𝑁 → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) ) |
154 |
153
|
imp |
⊢ ( ( 𝑁 ∈ Odd ∧ 7 < 𝑁 ) → ( 𝑁 < ; 1 3 → 𝑁 ∈ GoldbachOdd ) ) |
155 |
154
|
com12 |
⊢ ( 𝑁 < ; 1 3 → ( ( 𝑁 ∈ Odd ∧ 7 < 𝑁 ) → 𝑁 ∈ GoldbachOdd ) ) |
156 |
155
|
3ad2ant3 |
⊢ ( ( 𝑁 ∈ ℝ* ∧ 7 ≤ 𝑁 ∧ 𝑁 < ; 1 3 ) → ( ( 𝑁 ∈ Odd ∧ 7 < 𝑁 ) → 𝑁 ∈ GoldbachOdd ) ) |
157 |
156
|
com12 |
⊢ ( ( 𝑁 ∈ Odd ∧ 7 < 𝑁 ) → ( ( 𝑁 ∈ ℝ* ∧ 7 ≤ 𝑁 ∧ 𝑁 < ; 1 3 ) → 𝑁 ∈ GoldbachOdd ) ) |
158 |
9 157
|
syl5bi |
⊢ ( ( 𝑁 ∈ Odd ∧ 7 < 𝑁 ) → ( 𝑁 ∈ ( 7 [,) ; 1 3 ) → 𝑁 ∈ GoldbachOdd ) ) |
159 |
158
|
3impia |
⊢ ( ( 𝑁 ∈ Odd ∧ 7 < 𝑁 ∧ 𝑁 ∈ ( 7 [,) ; 1 3 ) ) → 𝑁 ∈ GoldbachOdd ) |