Step |
Hyp |
Ref |
Expression |
1 |
|
evengpop3 |
|- ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> E. o e. GoldbachOddW N = ( o + 3 ) ) ) |
2 |
1
|
imp |
|- ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) -> E. o e. GoldbachOddW N = ( o + 3 ) ) |
3 |
|
simplll |
|- ( ( ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) /\ o e. GoldbachOddW ) /\ N = ( o + 3 ) ) -> A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) ) |
4 |
|
6nn |
|- 6 e. NN |
5 |
4
|
nnzi |
|- 6 e. ZZ |
6 |
5
|
a1i |
|- ( N e. ( ZZ>= ` 9 ) -> 6 e. ZZ ) |
7 |
|
3z |
|- 3 e. ZZ |
8 |
7
|
a1i |
|- ( N e. ( ZZ>= ` 9 ) -> 3 e. ZZ ) |
9 |
|
6p3e9 |
|- ( 6 + 3 ) = 9 |
10 |
9
|
eqcomi |
|- 9 = ( 6 + 3 ) |
11 |
10
|
fveq2i |
|- ( ZZ>= ` 9 ) = ( ZZ>= ` ( 6 + 3 ) ) |
12 |
11
|
eleq2i |
|- ( N e. ( ZZ>= ` 9 ) <-> N e. ( ZZ>= ` ( 6 + 3 ) ) ) |
13 |
12
|
biimpi |
|- ( N e. ( ZZ>= ` 9 ) -> N e. ( ZZ>= ` ( 6 + 3 ) ) ) |
14 |
|
eluzsub |
|- ( ( 6 e. ZZ /\ 3 e. ZZ /\ N e. ( ZZ>= ` ( 6 + 3 ) ) ) -> ( N - 3 ) e. ( ZZ>= ` 6 ) ) |
15 |
6 8 13 14
|
syl3anc |
|- ( N e. ( ZZ>= ` 9 ) -> ( N - 3 ) e. ( ZZ>= ` 6 ) ) |
16 |
15
|
adantr |
|- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> ( N - 3 ) e. ( ZZ>= ` 6 ) ) |
17 |
16
|
ad3antlr |
|- ( ( ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) /\ o e. GoldbachOddW ) /\ N = ( o + 3 ) ) -> ( N - 3 ) e. ( ZZ>= ` 6 ) ) |
18 |
|
3odd |
|- 3 e. Odd |
19 |
18
|
a1i |
|- ( N e. ( ZZ>= ` 9 ) -> 3 e. Odd ) |
20 |
19
|
anim1i |
|- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> ( 3 e. Odd /\ N e. Even ) ) |
21 |
20
|
adantl |
|- ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) -> ( 3 e. Odd /\ N e. Even ) ) |
22 |
21
|
ancomd |
|- ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) -> ( N e. Even /\ 3 e. Odd ) ) |
23 |
22
|
adantr |
|- ( ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) /\ o e. GoldbachOddW ) -> ( N e. Even /\ 3 e. Odd ) ) |
24 |
23
|
adantr |
|- ( ( ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) /\ o e. GoldbachOddW ) /\ N = ( o + 3 ) ) -> ( N e. Even /\ 3 e. Odd ) ) |
25 |
|
emoo |
|- ( ( N e. Even /\ 3 e. Odd ) -> ( N - 3 ) e. Odd ) |
26 |
24 25
|
syl |
|- ( ( ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) /\ o e. GoldbachOddW ) /\ N = ( o + 3 ) ) -> ( N - 3 ) e. Odd ) |
27 |
|
nnsum4primesodd |
|- ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( ( ( N - 3 ) e. ( ZZ>= ` 6 ) /\ ( N - 3 ) e. Odd ) -> E. g e. ( Prime ^m ( 1 ... 3 ) ) ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) ) |
28 |
27
|
imp |
|- ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( ( N - 3 ) e. ( ZZ>= ` 6 ) /\ ( N - 3 ) e. Odd ) ) -> E. g e. ( Prime ^m ( 1 ... 3 ) ) ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) |
29 |
3 17 26 28
|
syl12anc |
|- ( ( ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) /\ o e. GoldbachOddW ) /\ N = ( o + 3 ) ) -> E. g e. ( Prime ^m ( 1 ... 3 ) ) ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) |
30 |
|
simpr |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> g : ( 1 ... 3 ) --> Prime ) |
31 |
|
4z |
|- 4 e. ZZ |
32 |
|
fzonel |
|- -. 4 e. ( 1 ..^ 4 ) |
33 |
|
fzoval |
|- ( 4 e. ZZ -> ( 1 ..^ 4 ) = ( 1 ... ( 4 - 1 ) ) ) |
34 |
31 33
|
ax-mp |
|- ( 1 ..^ 4 ) = ( 1 ... ( 4 - 1 ) ) |
35 |
|
4cn |
|- 4 e. CC |
36 |
|
ax-1cn |
|- 1 e. CC |
37 |
|
3cn |
|- 3 e. CC |
38 |
35 36 37
|
3pm3.2i |
|- ( 4 e. CC /\ 1 e. CC /\ 3 e. CC ) |
39 |
|
3p1e4 |
|- ( 3 + 1 ) = 4 |
40 |
|
subadd2 |
|- ( ( 4 e. CC /\ 1 e. CC /\ 3 e. CC ) -> ( ( 4 - 1 ) = 3 <-> ( 3 + 1 ) = 4 ) ) |
41 |
39 40
|
mpbiri |
|- ( ( 4 e. CC /\ 1 e. CC /\ 3 e. CC ) -> ( 4 - 1 ) = 3 ) |
42 |
38 41
|
ax-mp |
|- ( 4 - 1 ) = 3 |
43 |
42
|
oveq2i |
|- ( 1 ... ( 4 - 1 ) ) = ( 1 ... 3 ) |
44 |
34 43
|
eqtri |
|- ( 1 ..^ 4 ) = ( 1 ... 3 ) |
45 |
44
|
eqcomi |
|- ( 1 ... 3 ) = ( 1 ..^ 4 ) |
46 |
45
|
eleq2i |
|- ( 4 e. ( 1 ... 3 ) <-> 4 e. ( 1 ..^ 4 ) ) |
47 |
32 46
|
mtbir |
|- -. 4 e. ( 1 ... 3 ) |
48 |
31 47
|
pm3.2i |
|- ( 4 e. ZZ /\ -. 4 e. ( 1 ... 3 ) ) |
49 |
48
|
a1i |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( 4 e. ZZ /\ -. 4 e. ( 1 ... 3 ) ) ) |
50 |
|
3prm |
|- 3 e. Prime |
51 |
50
|
a1i |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> 3 e. Prime ) |
52 |
|
fsnunf |
|- ( ( g : ( 1 ... 3 ) --> Prime /\ ( 4 e. ZZ /\ -. 4 e. ( 1 ... 3 ) ) /\ 3 e. Prime ) -> ( g u. { <. 4 , 3 >. } ) : ( ( 1 ... 3 ) u. { 4 } ) --> Prime ) |
53 |
30 49 51 52
|
syl3anc |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( g u. { <. 4 , 3 >. } ) : ( ( 1 ... 3 ) u. { 4 } ) --> Prime ) |
54 |
|
fzval3 |
|- ( 4 e. ZZ -> ( 1 ... 4 ) = ( 1 ..^ ( 4 + 1 ) ) ) |
55 |
31 54
|
ax-mp |
|- ( 1 ... 4 ) = ( 1 ..^ ( 4 + 1 ) ) |
56 |
|
1z |
|- 1 e. ZZ |
57 |
|
1re |
|- 1 e. RR |
58 |
|
4re |
|- 4 e. RR |
59 |
|
1lt4 |
|- 1 < 4 |
60 |
57 58 59
|
ltleii |
|- 1 <_ 4 |
61 |
|
eluz2 |
|- ( 4 e. ( ZZ>= ` 1 ) <-> ( 1 e. ZZ /\ 4 e. ZZ /\ 1 <_ 4 ) ) |
62 |
56 31 60 61
|
mpbir3an |
|- 4 e. ( ZZ>= ` 1 ) |
63 |
|
fzosplitsn |
|- ( 4 e. ( ZZ>= ` 1 ) -> ( 1 ..^ ( 4 + 1 ) ) = ( ( 1 ..^ 4 ) u. { 4 } ) ) |
64 |
62 63
|
ax-mp |
|- ( 1 ..^ ( 4 + 1 ) ) = ( ( 1 ..^ 4 ) u. { 4 } ) |
65 |
44
|
uneq1i |
|- ( ( 1 ..^ 4 ) u. { 4 } ) = ( ( 1 ... 3 ) u. { 4 } ) |
66 |
55 64 65
|
3eqtri |
|- ( 1 ... 4 ) = ( ( 1 ... 3 ) u. { 4 } ) |
67 |
66
|
feq2i |
|- ( ( g u. { <. 4 , 3 >. } ) : ( 1 ... 4 ) --> Prime <-> ( g u. { <. 4 , 3 >. } ) : ( ( 1 ... 3 ) u. { 4 } ) --> Prime ) |
68 |
53 67
|
sylibr |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( g u. { <. 4 , 3 >. } ) : ( 1 ... 4 ) --> Prime ) |
69 |
|
prmex |
|- Prime e. _V |
70 |
|
ovex |
|- ( 1 ... 4 ) e. _V |
71 |
69 70
|
pm3.2i |
|- ( Prime e. _V /\ ( 1 ... 4 ) e. _V ) |
72 |
|
elmapg |
|- ( ( Prime e. _V /\ ( 1 ... 4 ) e. _V ) -> ( ( g u. { <. 4 , 3 >. } ) e. ( Prime ^m ( 1 ... 4 ) ) <-> ( g u. { <. 4 , 3 >. } ) : ( 1 ... 4 ) --> Prime ) ) |
73 |
71 72
|
mp1i |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( g u. { <. 4 , 3 >. } ) e. ( Prime ^m ( 1 ... 4 ) ) <-> ( g u. { <. 4 , 3 >. } ) : ( 1 ... 4 ) --> Prime ) ) |
74 |
68 73
|
mpbird |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( g u. { <. 4 , 3 >. } ) e. ( Prime ^m ( 1 ... 4 ) ) ) |
75 |
74
|
adantr |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) -> ( g u. { <. 4 , 3 >. } ) e. ( Prime ^m ( 1 ... 4 ) ) ) |
76 |
|
fveq1 |
|- ( f = ( g u. { <. 4 , 3 >. } ) -> ( f ` k ) = ( ( g u. { <. 4 , 3 >. } ) ` k ) ) |
77 |
76
|
adantr |
|- ( ( f = ( g u. { <. 4 , 3 >. } ) /\ k e. ( 1 ... 4 ) ) -> ( f ` k ) = ( ( g u. { <. 4 , 3 >. } ) ` k ) ) |
78 |
77
|
sumeq2dv |
|- ( f = ( g u. { <. 4 , 3 >. } ) -> sum_ k e. ( 1 ... 4 ) ( f ` k ) = sum_ k e. ( 1 ... 4 ) ( ( g u. { <. 4 , 3 >. } ) ` k ) ) |
79 |
78
|
eqeq2d |
|- ( f = ( g u. { <. 4 , 3 >. } ) -> ( N = sum_ k e. ( 1 ... 4 ) ( f ` k ) <-> N = sum_ k e. ( 1 ... 4 ) ( ( g u. { <. 4 , 3 >. } ) ` k ) ) ) |
80 |
79
|
adantl |
|- ( ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) /\ f = ( g u. { <. 4 , 3 >. } ) ) -> ( N = sum_ k e. ( 1 ... 4 ) ( f ` k ) <-> N = sum_ k e. ( 1 ... 4 ) ( ( g u. { <. 4 , 3 >. } ) ` k ) ) ) |
81 |
62
|
a1i |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> 4 e. ( ZZ>= ` 1 ) ) |
82 |
66
|
eleq2i |
|- ( k e. ( 1 ... 4 ) <-> k e. ( ( 1 ... 3 ) u. { 4 } ) ) |
83 |
|
elun |
|- ( k e. ( ( 1 ... 3 ) u. { 4 } ) <-> ( k e. ( 1 ... 3 ) \/ k e. { 4 } ) ) |
84 |
|
velsn |
|- ( k e. { 4 } <-> k = 4 ) |
85 |
84
|
orbi2i |
|- ( ( k e. ( 1 ... 3 ) \/ k e. { 4 } ) <-> ( k e. ( 1 ... 3 ) \/ k = 4 ) ) |
86 |
82 83 85
|
3bitri |
|- ( k e. ( 1 ... 4 ) <-> ( k e. ( 1 ... 3 ) \/ k = 4 ) ) |
87 |
|
elfz2 |
|- ( k e. ( 1 ... 3 ) <-> ( ( 1 e. ZZ /\ 3 e. ZZ /\ k e. ZZ ) /\ ( 1 <_ k /\ k <_ 3 ) ) ) |
88 |
|
3re |
|- 3 e. RR |
89 |
88 58
|
pm3.2i |
|- ( 3 e. RR /\ 4 e. RR ) |
90 |
|
3lt4 |
|- 3 < 4 |
91 |
|
ltnle |
|- ( ( 3 e. RR /\ 4 e. RR ) -> ( 3 < 4 <-> -. 4 <_ 3 ) ) |
92 |
90 91
|
mpbii |
|- ( ( 3 e. RR /\ 4 e. RR ) -> -. 4 <_ 3 ) |
93 |
89 92
|
ax-mp |
|- -. 4 <_ 3 |
94 |
|
breq1 |
|- ( k = 4 -> ( k <_ 3 <-> 4 <_ 3 ) ) |
95 |
94
|
eqcoms |
|- ( 4 = k -> ( k <_ 3 <-> 4 <_ 3 ) ) |
96 |
93 95
|
mtbiri |
|- ( 4 = k -> -. k <_ 3 ) |
97 |
96
|
a1i |
|- ( k e. ZZ -> ( 4 = k -> -. k <_ 3 ) ) |
98 |
97
|
necon2ad |
|- ( k e. ZZ -> ( k <_ 3 -> 4 =/= k ) ) |
99 |
98
|
adantld |
|- ( k e. ZZ -> ( ( 1 <_ k /\ k <_ 3 ) -> 4 =/= k ) ) |
100 |
99
|
3ad2ant3 |
|- ( ( 1 e. ZZ /\ 3 e. ZZ /\ k e. ZZ ) -> ( ( 1 <_ k /\ k <_ 3 ) -> 4 =/= k ) ) |
101 |
100
|
imp |
|- ( ( ( 1 e. ZZ /\ 3 e. ZZ /\ k e. ZZ ) /\ ( 1 <_ k /\ k <_ 3 ) ) -> 4 =/= k ) |
102 |
87 101
|
sylbi |
|- ( k e. ( 1 ... 3 ) -> 4 =/= k ) |
103 |
102
|
adantr |
|- ( ( k e. ( 1 ... 3 ) /\ g : ( 1 ... 3 ) --> Prime ) -> 4 =/= k ) |
104 |
|
fvunsn |
|- ( 4 =/= k -> ( ( g u. { <. 4 , 3 >. } ) ` k ) = ( g ` k ) ) |
105 |
103 104
|
syl |
|- ( ( k e. ( 1 ... 3 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) = ( g ` k ) ) |
106 |
|
ffvelrn |
|- ( ( g : ( 1 ... 3 ) --> Prime /\ k e. ( 1 ... 3 ) ) -> ( g ` k ) e. Prime ) |
107 |
106
|
ancoms |
|- ( ( k e. ( 1 ... 3 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( g ` k ) e. Prime ) |
108 |
|
prmz |
|- ( ( g ` k ) e. Prime -> ( g ` k ) e. ZZ ) |
109 |
107 108
|
syl |
|- ( ( k e. ( 1 ... 3 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( g ` k ) e. ZZ ) |
110 |
109
|
zcnd |
|- ( ( k e. ( 1 ... 3 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( g ` k ) e. CC ) |
111 |
105 110
|
eqeltrd |
|- ( ( k e. ( 1 ... 3 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) e. CC ) |
112 |
111
|
ex |
|- ( k e. ( 1 ... 3 ) -> ( g : ( 1 ... 3 ) --> Prime -> ( ( g u. { <. 4 , 3 >. } ) ` k ) e. CC ) ) |
113 |
112
|
adantld |
|- ( k e. ( 1 ... 3 ) -> ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) e. CC ) ) |
114 |
|
fveq2 |
|- ( k = 4 -> ( ( g u. { <. 4 , 3 >. } ) ` k ) = ( ( g u. { <. 4 , 3 >. } ) ` 4 ) ) |
115 |
31
|
a1i |
|- ( g : ( 1 ... 3 ) --> Prime -> 4 e. ZZ ) |
116 |
7
|
a1i |
|- ( g : ( 1 ... 3 ) --> Prime -> 3 e. ZZ ) |
117 |
|
fdm |
|- ( g : ( 1 ... 3 ) --> Prime -> dom g = ( 1 ... 3 ) ) |
118 |
|
eleq2 |
|- ( dom g = ( 1 ... 3 ) -> ( 4 e. dom g <-> 4 e. ( 1 ... 3 ) ) ) |
119 |
47 118
|
mtbiri |
|- ( dom g = ( 1 ... 3 ) -> -. 4 e. dom g ) |
120 |
117 119
|
syl |
|- ( g : ( 1 ... 3 ) --> Prime -> -. 4 e. dom g ) |
121 |
|
fsnunfv |
|- ( ( 4 e. ZZ /\ 3 e. ZZ /\ -. 4 e. dom g ) -> ( ( g u. { <. 4 , 3 >. } ) ` 4 ) = 3 ) |
122 |
115 116 120 121
|
syl3anc |
|- ( g : ( 1 ... 3 ) --> Prime -> ( ( g u. { <. 4 , 3 >. } ) ` 4 ) = 3 ) |
123 |
122
|
adantl |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( g u. { <. 4 , 3 >. } ) ` 4 ) = 3 ) |
124 |
114 123
|
sylan9eq |
|- ( ( k = 4 /\ ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) = 3 ) |
125 |
124 37
|
eqeltrdi |
|- ( ( k = 4 /\ ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) e. CC ) |
126 |
125
|
ex |
|- ( k = 4 -> ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) e. CC ) ) |
127 |
113 126
|
jaoi |
|- ( ( k e. ( 1 ... 3 ) \/ k = 4 ) -> ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) e. CC ) ) |
128 |
127
|
com12 |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( k e. ( 1 ... 3 ) \/ k = 4 ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) e. CC ) ) |
129 |
86 128
|
syl5bi |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( k e. ( 1 ... 4 ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) e. CC ) ) |
130 |
129
|
imp |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ k e. ( 1 ... 4 ) ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) e. CC ) |
131 |
81 130 114
|
fsumm1 |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> sum_ k e. ( 1 ... 4 ) ( ( g u. { <. 4 , 3 >. } ) ` k ) = ( sum_ k e. ( 1 ... ( 4 - 1 ) ) ( ( g u. { <. 4 , 3 >. } ) ` k ) + ( ( g u. { <. 4 , 3 >. } ) ` 4 ) ) ) |
132 |
131
|
adantr |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) -> sum_ k e. ( 1 ... 4 ) ( ( g u. { <. 4 , 3 >. } ) ` k ) = ( sum_ k e. ( 1 ... ( 4 - 1 ) ) ( ( g u. { <. 4 , 3 >. } ) ` k ) + ( ( g u. { <. 4 , 3 >. } ) ` 4 ) ) ) |
133 |
42
|
eqcomi |
|- 3 = ( 4 - 1 ) |
134 |
133
|
oveq2i |
|- ( 1 ... 3 ) = ( 1 ... ( 4 - 1 ) ) |
135 |
134
|
a1i |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( 1 ... 3 ) = ( 1 ... ( 4 - 1 ) ) ) |
136 |
102
|
adantl |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ k e. ( 1 ... 3 ) ) -> 4 =/= k ) |
137 |
136 104
|
syl |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ k e. ( 1 ... 3 ) ) -> ( ( g u. { <. 4 , 3 >. } ) ` k ) = ( g ` k ) ) |
138 |
137
|
eqcomd |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ k e. ( 1 ... 3 ) ) -> ( g ` k ) = ( ( g u. { <. 4 , 3 >. } ) ` k ) ) |
139 |
135 138
|
sumeq12dv |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> sum_ k e. ( 1 ... 3 ) ( g ` k ) = sum_ k e. ( 1 ... ( 4 - 1 ) ) ( ( g u. { <. 4 , 3 >. } ) ` k ) ) |
140 |
139
|
eqeq2d |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) <-> ( N - 3 ) = sum_ k e. ( 1 ... ( 4 - 1 ) ) ( ( g u. { <. 4 , 3 >. } ) ` k ) ) ) |
141 |
140
|
biimpa |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) -> ( N - 3 ) = sum_ k e. ( 1 ... ( 4 - 1 ) ) ( ( g u. { <. 4 , 3 >. } ) ` k ) ) |
142 |
141
|
eqcomd |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) -> sum_ k e. ( 1 ... ( 4 - 1 ) ) ( ( g u. { <. 4 , 3 >. } ) ` k ) = ( N - 3 ) ) |
143 |
142
|
oveq1d |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) -> ( sum_ k e. ( 1 ... ( 4 - 1 ) ) ( ( g u. { <. 4 , 3 >. } ) ` k ) + ( ( g u. { <. 4 , 3 >. } ) ` 4 ) ) = ( ( N - 3 ) + ( ( g u. { <. 4 , 3 >. } ) ` 4 ) ) ) |
144 |
31
|
a1i |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> 4 e. ZZ ) |
145 |
7
|
a1i |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> 3 e. ZZ ) |
146 |
120
|
adantl |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> -. 4 e. dom g ) |
147 |
144 145 146 121
|
syl3anc |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( g u. { <. 4 , 3 >. } ) ` 4 ) = 3 ) |
148 |
147
|
oveq2d |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( N - 3 ) + ( ( g u. { <. 4 , 3 >. } ) ` 4 ) ) = ( ( N - 3 ) + 3 ) ) |
149 |
|
eluzelcn |
|- ( N e. ( ZZ>= ` 9 ) -> N e. CC ) |
150 |
37
|
a1i |
|- ( N e. ( ZZ>= ` 9 ) -> 3 e. CC ) |
151 |
149 150
|
npcand |
|- ( N e. ( ZZ>= ` 9 ) -> ( ( N - 3 ) + 3 ) = N ) |
152 |
151
|
adantr |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( N - 3 ) + 3 ) = N ) |
153 |
148 152
|
eqtrd |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( N - 3 ) + ( ( g u. { <. 4 , 3 >. } ) ` 4 ) ) = N ) |
154 |
153
|
adantr |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) -> ( ( N - 3 ) + ( ( g u. { <. 4 , 3 >. } ) ` 4 ) ) = N ) |
155 |
132 143 154
|
3eqtrrd |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) -> N = sum_ k e. ( 1 ... 4 ) ( ( g u. { <. 4 , 3 >. } ) ` k ) ) |
156 |
75 80 155
|
rspcedvd |
|- ( ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) /\ ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) |
157 |
156
|
ex |
|- ( ( N e. ( ZZ>= ` 9 ) /\ g : ( 1 ... 3 ) --> Prime ) -> ( ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) ) |
158 |
157
|
expcom |
|- ( g : ( 1 ... 3 ) --> Prime -> ( N e. ( ZZ>= ` 9 ) -> ( ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) ) ) |
159 |
|
elmapi |
|- ( g e. ( Prime ^m ( 1 ... 3 ) ) -> g : ( 1 ... 3 ) --> Prime ) |
160 |
158 159
|
syl11 |
|- ( N e. ( ZZ>= ` 9 ) -> ( g e. ( Prime ^m ( 1 ... 3 ) ) -> ( ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) ) ) |
161 |
160
|
rexlimdv |
|- ( N e. ( ZZ>= ` 9 ) -> ( E. g e. ( Prime ^m ( 1 ... 3 ) ) ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) ) |
162 |
161
|
adantr |
|- ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> ( E. g e. ( Prime ^m ( 1 ... 3 ) ) ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) ) |
163 |
162
|
ad3antlr |
|- ( ( ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) /\ o e. GoldbachOddW ) /\ N = ( o + 3 ) ) -> ( E. g e. ( Prime ^m ( 1 ... 3 ) ) ( N - 3 ) = sum_ k e. ( 1 ... 3 ) ( g ` k ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) ) |
164 |
29 163
|
mpd |
|- ( ( ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) /\ o e. GoldbachOddW ) /\ N = ( o + 3 ) ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) |
165 |
164
|
rexlimdva2 |
|- ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) -> ( E. o e. GoldbachOddW N = ( o + 3 ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) ) |
166 |
2 165
|
mpd |
|- ( ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) /\ ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) |
167 |
166
|
ex |
|- ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( ( N e. ( ZZ>= ` 9 ) /\ N e. Even ) -> E. f e. ( Prime ^m ( 1 ... 4 ) ) N = sum_ k e. ( 1 ... 4 ) ( f ` k ) ) ) |