Step |
Hyp |
Ref |
Expression |
1 |
|
breq2 |
|- ( m = N -> ( 5 < m <-> 5 < N ) ) |
2 |
|
eleq1 |
|- ( m = N -> ( m e. GoldbachOddW <-> N e. GoldbachOddW ) ) |
3 |
1 2
|
imbi12d |
|- ( m = N -> ( ( 5 < m -> m e. GoldbachOddW ) <-> ( 5 < N -> N e. GoldbachOddW ) ) ) |
4 |
3
|
rspcv |
|- ( N e. Odd -> ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( 5 < N -> N e. GoldbachOddW ) ) ) |
5 |
4
|
adantl |
|- ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( 5 < N -> N e. GoldbachOddW ) ) ) |
6 |
|
eluz2 |
|- ( N e. ( ZZ>= ` 6 ) <-> ( 6 e. ZZ /\ N e. ZZ /\ 6 <_ N ) ) |
7 |
|
5lt6 |
|- 5 < 6 |
8 |
|
5re |
|- 5 e. RR |
9 |
8
|
a1i |
|- ( N e. ZZ -> 5 e. RR ) |
10 |
|
6re |
|- 6 e. RR |
11 |
10
|
a1i |
|- ( N e. ZZ -> 6 e. RR ) |
12 |
|
zre |
|- ( N e. ZZ -> N e. RR ) |
13 |
|
ltletr |
|- ( ( 5 e. RR /\ 6 e. RR /\ N e. RR ) -> ( ( 5 < 6 /\ 6 <_ N ) -> 5 < N ) ) |
14 |
9 11 12 13
|
syl3anc |
|- ( N e. ZZ -> ( ( 5 < 6 /\ 6 <_ N ) -> 5 < N ) ) |
15 |
7 14
|
mpani |
|- ( N e. ZZ -> ( 6 <_ N -> 5 < N ) ) |
16 |
15
|
imp |
|- ( ( N e. ZZ /\ 6 <_ N ) -> 5 < N ) |
17 |
16
|
3adant1 |
|- ( ( 6 e. ZZ /\ N e. ZZ /\ 6 <_ N ) -> 5 < N ) |
18 |
6 17
|
sylbi |
|- ( N e. ( ZZ>= ` 6 ) -> 5 < N ) |
19 |
18
|
adantr |
|- ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> 5 < N ) |
20 |
|
pm2.27 |
|- ( 5 < N -> ( ( 5 < N -> N e. GoldbachOddW ) -> N e. GoldbachOddW ) ) |
21 |
19 20
|
syl |
|- ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> ( ( 5 < N -> N e. GoldbachOddW ) -> N e. GoldbachOddW ) ) |
22 |
|
isgbow |
|- ( N e. GoldbachOddW <-> ( N e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) ) |
23 |
|
1ex |
|- 1 e. _V |
24 |
|
2ex |
|- 2 e. _V |
25 |
|
3ex |
|- 3 e. _V |
26 |
|
vex |
|- p e. _V |
27 |
|
vex |
|- q e. _V |
28 |
|
vex |
|- r e. _V |
29 |
|
1ne2 |
|- 1 =/= 2 |
30 |
|
1re |
|- 1 e. RR |
31 |
|
1lt3 |
|- 1 < 3 |
32 |
30 31
|
ltneii |
|- 1 =/= 3 |
33 |
|
2re |
|- 2 e. RR |
34 |
|
2lt3 |
|- 2 < 3 |
35 |
33 34
|
ltneii |
|- 2 =/= 3 |
36 |
23 24 25 26 27 28 29 32 35
|
ftp |
|- { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : { 1 , 2 , 3 } --> { p , q , r } |
37 |
36
|
a1i |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : { 1 , 2 , 3 } --> { p , q , r } ) |
38 |
|
1p2e3 |
|- ( 1 + 2 ) = 3 |
39 |
38
|
eqcomi |
|- 3 = ( 1 + 2 ) |
40 |
39
|
oveq2i |
|- ( 1 ... 3 ) = ( 1 ... ( 1 + 2 ) ) |
41 |
|
1z |
|- 1 e. ZZ |
42 |
|
fztp |
|- ( 1 e. ZZ -> ( 1 ... ( 1 + 2 ) ) = { 1 , ( 1 + 1 ) , ( 1 + 2 ) } ) |
43 |
41 42
|
ax-mp |
|- ( 1 ... ( 1 + 2 ) ) = { 1 , ( 1 + 1 ) , ( 1 + 2 ) } |
44 |
|
eqid |
|- 1 = 1 |
45 |
|
id |
|- ( 1 = 1 -> 1 = 1 ) |
46 |
|
1p1e2 |
|- ( 1 + 1 ) = 2 |
47 |
46
|
a1i |
|- ( 1 = 1 -> ( 1 + 1 ) = 2 ) |
48 |
38
|
a1i |
|- ( 1 = 1 -> ( 1 + 2 ) = 3 ) |
49 |
45 47 48
|
tpeq123d |
|- ( 1 = 1 -> { 1 , ( 1 + 1 ) , ( 1 + 2 ) } = { 1 , 2 , 3 } ) |
50 |
44 49
|
ax-mp |
|- { 1 , ( 1 + 1 ) , ( 1 + 2 ) } = { 1 , 2 , 3 } |
51 |
40 43 50
|
3eqtri |
|- ( 1 ... 3 ) = { 1 , 2 , 3 } |
52 |
51
|
feq2i |
|- ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : ( 1 ... 3 ) --> { p , q , r } <-> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : { 1 , 2 , 3 } --> { p , q , r } ) |
53 |
37 52
|
sylibr |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : ( 1 ... 3 ) --> { p , q , r } ) |
54 |
|
df-3an |
|- ( ( p e. Prime /\ q e. Prime /\ r e. Prime ) <-> ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) ) |
55 |
26 27 28
|
tpss |
|- ( ( p e. Prime /\ q e. Prime /\ r e. Prime ) <-> { p , q , r } C_ Prime ) |
56 |
54 55
|
sylbb1 |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> { p , q , r } C_ Prime ) |
57 |
53 56
|
fssd |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : ( 1 ... 3 ) --> Prime ) |
58 |
|
prmex |
|- Prime e. _V |
59 |
|
ovex |
|- ( 1 ... 3 ) e. _V |
60 |
58 59
|
pm3.2i |
|- ( Prime e. _V /\ ( 1 ... 3 ) e. _V ) |
61 |
|
elmapg |
|- ( ( Prime e. _V /\ ( 1 ... 3 ) e. _V ) -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } e. ( Prime ^m ( 1 ... 3 ) ) <-> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : ( 1 ... 3 ) --> Prime ) ) |
62 |
60 61
|
mp1i |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } e. ( Prime ^m ( 1 ... 3 ) ) <-> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } : ( 1 ... 3 ) --> Prime ) ) |
63 |
57 62
|
mpbird |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } e. ( Prime ^m ( 1 ... 3 ) ) ) |
64 |
|
fveq1 |
|- ( f = { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } -> ( f ` k ) = ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) ) |
65 |
64
|
sumeq2sdv |
|- ( f = { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } -> sum_ k e. ( 1 ... 3 ) ( f ` k ) = sum_ k e. ( 1 ... 3 ) ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) ) |
66 |
65
|
eqeq2d |
|- ( f = { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } -> ( ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( f ` k ) <-> ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) ) ) |
67 |
66
|
adantl |
|- ( ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) /\ f = { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ) -> ( ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( f ` k ) <-> ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) ) ) |
68 |
51
|
a1i |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( 1 ... 3 ) = { 1 , 2 , 3 } ) |
69 |
68
|
sumeq1d |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> sum_ k e. ( 1 ... 3 ) ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = sum_ k e. { 1 , 2 , 3 } ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) ) |
70 |
|
fveq2 |
|- ( k = 1 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 1 ) ) |
71 |
23 26
|
fvtp1 |
|- ( ( 1 =/= 2 /\ 1 =/= 3 ) -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 1 ) = p ) |
72 |
29 32 71
|
mp2an |
|- ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 1 ) = p |
73 |
70 72
|
eqtrdi |
|- ( k = 1 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = p ) |
74 |
|
fveq2 |
|- ( k = 2 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 2 ) ) |
75 |
24 27
|
fvtp2 |
|- ( ( 1 =/= 2 /\ 2 =/= 3 ) -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 2 ) = q ) |
76 |
29 35 75
|
mp2an |
|- ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 2 ) = q |
77 |
74 76
|
eqtrdi |
|- ( k = 2 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = q ) |
78 |
|
fveq2 |
|- ( k = 3 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 3 ) ) |
79 |
25 28
|
fvtp3 |
|- ( ( 1 =/= 3 /\ 2 =/= 3 ) -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 3 ) = r ) |
80 |
32 35 79
|
mp2an |
|- ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` 3 ) = r |
81 |
78 80
|
eqtrdi |
|- ( k = 3 -> ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = r ) |
82 |
|
prmz |
|- ( p e. Prime -> p e. ZZ ) |
83 |
82
|
zcnd |
|- ( p e. Prime -> p e. CC ) |
84 |
|
prmz |
|- ( q e. Prime -> q e. ZZ ) |
85 |
84
|
zcnd |
|- ( q e. Prime -> q e. CC ) |
86 |
|
prmz |
|- ( r e. Prime -> r e. ZZ ) |
87 |
86
|
zcnd |
|- ( r e. Prime -> r e. CC ) |
88 |
83 85 87
|
3anim123i |
|- ( ( p e. Prime /\ q e. Prime /\ r e. Prime ) -> ( p e. CC /\ q e. CC /\ r e. CC ) ) |
89 |
88
|
3expa |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( p e. CC /\ q e. CC /\ r e. CC ) ) |
90 |
|
2z |
|- 2 e. ZZ |
91 |
|
3z |
|- 3 e. ZZ |
92 |
41 90 91
|
3pm3.2i |
|- ( 1 e. ZZ /\ 2 e. ZZ /\ 3 e. ZZ ) |
93 |
92
|
a1i |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( 1 e. ZZ /\ 2 e. ZZ /\ 3 e. ZZ ) ) |
94 |
29
|
a1i |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> 1 =/= 2 ) |
95 |
32
|
a1i |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> 1 =/= 3 ) |
96 |
35
|
a1i |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> 2 =/= 3 ) |
97 |
73 77 81 89 93 94 95 96
|
sumtp |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> sum_ k e. { 1 , 2 , 3 } ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) = ( ( p + q ) + r ) ) |
98 |
69 97
|
eqtr2d |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( { <. 1 , p >. , <. 2 , q >. , <. 3 , r >. } ` k ) ) |
99 |
63 67 98
|
rspcedvd |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) |
100 |
|
eqeq1 |
|- ( N = ( ( p + q ) + r ) -> ( N = sum_ k e. ( 1 ... 3 ) ( f ` k ) <-> ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) ) |
101 |
100
|
rexbidv |
|- ( N = ( ( p + q ) + r ) -> ( E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) <-> E. f e. ( Prime ^m ( 1 ... 3 ) ) ( ( p + q ) + r ) = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) ) |
102 |
99 101
|
syl5ibrcom |
|- ( ( ( p e. Prime /\ q e. Prime ) /\ r e. Prime ) -> ( N = ( ( p + q ) + r ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) ) |
103 |
102
|
rexlimdva |
|- ( ( p e. Prime /\ q e. Prime ) -> ( E. r e. Prime N = ( ( p + q ) + r ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) ) |
104 |
103
|
rexlimivv |
|- ( E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) |
105 |
104
|
adantl |
|- ( ( N e. Odd /\ E. p e. Prime E. q e. Prime E. r e. Prime N = ( ( p + q ) + r ) ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) |
106 |
22 105
|
sylbi |
|- ( N e. GoldbachOddW -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) |
107 |
106
|
a1i |
|- ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> ( N e. GoldbachOddW -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) ) |
108 |
5 21 107
|
3syld |
|- ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) ) |
109 |
108
|
com12 |
|- ( A. m e. Odd ( 5 < m -> m e. GoldbachOddW ) -> ( ( N e. ( ZZ>= ` 6 ) /\ N e. Odd ) -> E. f e. ( Prime ^m ( 1 ... 3 ) ) N = sum_ k e. ( 1 ... 3 ) ( f ` k ) ) ) |