Step |
Hyp |
Ref |
Expression |
1 |
|
bgoldbtbnd.m |
|- ( ph -> M e. ( ZZ>= ` ; 1 1 ) ) |
2 |
|
bgoldbtbnd.n |
|- ( ph -> N e. ( ZZ>= ` ; 1 1 ) ) |
3 |
|
bgoldbtbnd.b |
|- ( ph -> A. n e. Even ( ( 4 < n /\ n < N ) -> n e. GoldbachEven ) ) |
4 |
|
bgoldbtbnd.d |
|- ( ph -> D e. ( ZZ>= ` 3 ) ) |
5 |
|
bgoldbtbnd.f |
|- ( ph -> F e. ( RePart ` D ) ) |
6 |
|
bgoldbtbnd.i |
|- ( ph -> A. i e. ( 0 ..^ D ) ( ( F ` i ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( i + 1 ) ) - ( F ` i ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( i + 1 ) ) - ( F ` i ) ) ) ) |
7 |
|
bgoldbtbnd.0 |
|- ( ph -> ( F ` 0 ) = 7 ) |
8 |
|
bgoldbtbnd.1 |
|- ( ph -> ( F ` 1 ) = ; 1 3 ) |
9 |
|
bgoldbtbnd.l |
|- ( ph -> M < ( F ` D ) ) |
10 |
|
bgoldbtbnd.r |
|- ( ph -> ( F ` D ) e. RR ) |
11 |
|
bgoldbtbndlem3.s |
|- S = ( X - ( F ` I ) ) |
12 |
|
fzo0ss1 |
|- ( 1 ..^ D ) C_ ( 0 ..^ D ) |
13 |
12
|
sseli |
|- ( I e. ( 1 ..^ D ) -> I e. ( 0 ..^ D ) ) |
14 |
|
fveq2 |
|- ( i = I -> ( F ` i ) = ( F ` I ) ) |
15 |
14
|
eleq1d |
|- ( i = I -> ( ( F ` i ) e. ( Prime \ { 2 } ) <-> ( F ` I ) e. ( Prime \ { 2 } ) ) ) |
16 |
|
fvoveq1 |
|- ( i = I -> ( F ` ( i + 1 ) ) = ( F ` ( I + 1 ) ) ) |
17 |
16 14
|
oveq12d |
|- ( i = I -> ( ( F ` ( i + 1 ) ) - ( F ` i ) ) = ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) |
18 |
17
|
breq1d |
|- ( i = I -> ( ( ( F ` ( i + 1 ) ) - ( F ` i ) ) < ( N - 4 ) <-> ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) ) ) |
19 |
17
|
breq2d |
|- ( i = I -> ( 4 < ( ( F ` ( i + 1 ) ) - ( F ` i ) ) <-> 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) |
20 |
15 18 19
|
3anbi123d |
|- ( i = I -> ( ( ( F ` i ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( i + 1 ) ) - ( F ` i ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( i + 1 ) ) - ( F ` i ) ) ) <-> ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) ) |
21 |
20
|
rspcv |
|- ( I e. ( 0 ..^ D ) -> ( A. i e. ( 0 ..^ D ) ( ( F ` i ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( i + 1 ) ) - ( F ` i ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( i + 1 ) ) - ( F ` i ) ) ) -> ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) ) |
22 |
13 6 21
|
syl2imc |
|- ( ph -> ( I e. ( 1 ..^ D ) -> ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) ) |
23 |
22
|
a1d |
|- ( ph -> ( X e. Odd -> ( I e. ( 1 ..^ D ) -> ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) ) ) |
24 |
23
|
3imp |
|- ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) -> ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) |
25 |
|
simp2 |
|- ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) -> X e. Odd ) |
26 |
|
oddprmALTV |
|- ( ( F ` I ) e. ( Prime \ { 2 } ) -> ( F ` I ) e. Odd ) |
27 |
26
|
3ad2ant1 |
|- ( ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) -> ( F ` I ) e. Odd ) |
28 |
25 27
|
anim12i |
|- ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) -> ( X e. Odd /\ ( F ` I ) e. Odd ) ) |
29 |
28
|
adantr |
|- ( ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) /\ ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) /\ 4 < S ) ) -> ( X e. Odd /\ ( F ` I ) e. Odd ) ) |
30 |
|
omoeALTV |
|- ( ( X e. Odd /\ ( F ` I ) e. Odd ) -> ( X - ( F ` I ) ) e. Even ) |
31 |
29 30
|
syl |
|- ( ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) /\ ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) /\ 4 < S ) ) -> ( X - ( F ` I ) ) e. Even ) |
32 |
11 31
|
eqeltrid |
|- ( ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) /\ ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) /\ 4 < S ) ) -> S e. Even ) |
33 |
|
eldifi |
|- ( ( F ` I ) e. ( Prime \ { 2 } ) -> ( F ` I ) e. Prime ) |
34 |
|
prmz |
|- ( ( F ` I ) e. Prime -> ( F ` I ) e. ZZ ) |
35 |
34
|
zred |
|- ( ( F ` I ) e. Prime -> ( F ` I ) e. RR ) |
36 |
|
fzofzp1 |
|- ( I e. ( 1 ..^ D ) -> ( I + 1 ) e. ( 1 ... D ) ) |
37 |
|
elfzo2 |
|- ( I e. ( 1 ..^ D ) <-> ( I e. ( ZZ>= ` 1 ) /\ D e. ZZ /\ I < D ) ) |
38 |
|
1zzd |
|- ( ( I e. ( ZZ>= ` 1 ) /\ D e. ZZ /\ I < D ) -> 1 e. ZZ ) |
39 |
|
simp2 |
|- ( ( I e. ( ZZ>= ` 1 ) /\ D e. ZZ /\ I < D ) -> D e. ZZ ) |
40 |
|
eluz2 |
|- ( I e. ( ZZ>= ` 1 ) <-> ( 1 e. ZZ /\ I e. ZZ /\ 1 <_ I ) ) |
41 |
|
zre |
|- ( 1 e. ZZ -> 1 e. RR ) |
42 |
|
zre |
|- ( I e. ZZ -> I e. RR ) |
43 |
|
zre |
|- ( D e. ZZ -> D e. RR ) |
44 |
|
leltletr |
|- ( ( 1 e. RR /\ I e. RR /\ D e. RR ) -> ( ( 1 <_ I /\ I < D ) -> 1 <_ D ) ) |
45 |
41 42 43 44
|
syl3an |
|- ( ( 1 e. ZZ /\ I e. ZZ /\ D e. ZZ ) -> ( ( 1 <_ I /\ I < D ) -> 1 <_ D ) ) |
46 |
45
|
exp5o |
|- ( 1 e. ZZ -> ( I e. ZZ -> ( D e. ZZ -> ( 1 <_ I -> ( I < D -> 1 <_ D ) ) ) ) ) |
47 |
46
|
com34 |
|- ( 1 e. ZZ -> ( I e. ZZ -> ( 1 <_ I -> ( D e. ZZ -> ( I < D -> 1 <_ D ) ) ) ) ) |
48 |
47
|
3imp |
|- ( ( 1 e. ZZ /\ I e. ZZ /\ 1 <_ I ) -> ( D e. ZZ -> ( I < D -> 1 <_ D ) ) ) |
49 |
40 48
|
sylbi |
|- ( I e. ( ZZ>= ` 1 ) -> ( D e. ZZ -> ( I < D -> 1 <_ D ) ) ) |
50 |
49
|
3imp |
|- ( ( I e. ( ZZ>= ` 1 ) /\ D e. ZZ /\ I < D ) -> 1 <_ D ) |
51 |
|
eluz2 |
|- ( D e. ( ZZ>= ` 1 ) <-> ( 1 e. ZZ /\ D e. ZZ /\ 1 <_ D ) ) |
52 |
38 39 50 51
|
syl3anbrc |
|- ( ( I e. ( ZZ>= ` 1 ) /\ D e. ZZ /\ I < D ) -> D e. ( ZZ>= ` 1 ) ) |
53 |
37 52
|
sylbi |
|- ( I e. ( 1 ..^ D ) -> D e. ( ZZ>= ` 1 ) ) |
54 |
|
fzisfzounsn |
|- ( D e. ( ZZ>= ` 1 ) -> ( 1 ... D ) = ( ( 1 ..^ D ) u. { D } ) ) |
55 |
53 54
|
syl |
|- ( I e. ( 1 ..^ D ) -> ( 1 ... D ) = ( ( 1 ..^ D ) u. { D } ) ) |
56 |
55
|
eleq2d |
|- ( I e. ( 1 ..^ D ) -> ( ( I + 1 ) e. ( 1 ... D ) <-> ( I + 1 ) e. ( ( 1 ..^ D ) u. { D } ) ) ) |
57 |
|
elun |
|- ( ( I + 1 ) e. ( ( 1 ..^ D ) u. { D } ) <-> ( ( I + 1 ) e. ( 1 ..^ D ) \/ ( I + 1 ) e. { D } ) ) |
58 |
56 57
|
bitrdi |
|- ( I e. ( 1 ..^ D ) -> ( ( I + 1 ) e. ( 1 ... D ) <-> ( ( I + 1 ) e. ( 1 ..^ D ) \/ ( I + 1 ) e. { D } ) ) ) |
59 |
|
eluzge3nn |
|- ( D e. ( ZZ>= ` 3 ) -> D e. NN ) |
60 |
4 59
|
syl |
|- ( ph -> D e. NN ) |
61 |
60
|
ad2antrl |
|- ( ( ( I e. ( 1 ..^ D ) /\ ( I + 1 ) e. ( 1 ..^ D ) ) /\ ( ph /\ X e. Odd ) ) -> D e. NN ) |
62 |
5
|
ad2antrl |
|- ( ( ( I e. ( 1 ..^ D ) /\ ( I + 1 ) e. ( 1 ..^ D ) ) /\ ( ph /\ X e. Odd ) ) -> F e. ( RePart ` D ) ) |
63 |
|
simplr |
|- ( ( ( I e. ( 1 ..^ D ) /\ ( I + 1 ) e. ( 1 ..^ D ) ) /\ ( ph /\ X e. Odd ) ) -> ( I + 1 ) e. ( 1 ..^ D ) ) |
64 |
61 62 63
|
iccpartipre |
|- ( ( ( I e. ( 1 ..^ D ) /\ ( I + 1 ) e. ( 1 ..^ D ) ) /\ ( ph /\ X e. Odd ) ) -> ( F ` ( I + 1 ) ) e. RR ) |
65 |
64
|
exp31 |
|- ( I e. ( 1 ..^ D ) -> ( ( I + 1 ) e. ( 1 ..^ D ) -> ( ( ph /\ X e. Odd ) -> ( F ` ( I + 1 ) ) e. RR ) ) ) |
66 |
|
elsni |
|- ( ( I + 1 ) e. { D } -> ( I + 1 ) = D ) |
67 |
10
|
ad2antrl |
|- ( ( ( I + 1 ) = D /\ ( ph /\ X e. Odd ) ) -> ( F ` D ) e. RR ) |
68 |
|
fveq2 |
|- ( ( I + 1 ) = D -> ( F ` ( I + 1 ) ) = ( F ` D ) ) |
69 |
68
|
eleq1d |
|- ( ( I + 1 ) = D -> ( ( F ` ( I + 1 ) ) e. RR <-> ( F ` D ) e. RR ) ) |
70 |
69
|
adantr |
|- ( ( ( I + 1 ) = D /\ ( ph /\ X e. Odd ) ) -> ( ( F ` ( I + 1 ) ) e. RR <-> ( F ` D ) e. RR ) ) |
71 |
67 70
|
mpbird |
|- ( ( ( I + 1 ) = D /\ ( ph /\ X e. Odd ) ) -> ( F ` ( I + 1 ) ) e. RR ) |
72 |
71
|
ex |
|- ( ( I + 1 ) = D -> ( ( ph /\ X e. Odd ) -> ( F ` ( I + 1 ) ) e. RR ) ) |
73 |
66 72
|
syl |
|- ( ( I + 1 ) e. { D } -> ( ( ph /\ X e. Odd ) -> ( F ` ( I + 1 ) ) e. RR ) ) |
74 |
73
|
a1i |
|- ( I e. ( 1 ..^ D ) -> ( ( I + 1 ) e. { D } -> ( ( ph /\ X e. Odd ) -> ( F ` ( I + 1 ) ) e. RR ) ) ) |
75 |
65 74
|
jaod |
|- ( I e. ( 1 ..^ D ) -> ( ( ( I + 1 ) e. ( 1 ..^ D ) \/ ( I + 1 ) e. { D } ) -> ( ( ph /\ X e. Odd ) -> ( F ` ( I + 1 ) ) e. RR ) ) ) |
76 |
58 75
|
sylbid |
|- ( I e. ( 1 ..^ D ) -> ( ( I + 1 ) e. ( 1 ... D ) -> ( ( ph /\ X e. Odd ) -> ( F ` ( I + 1 ) ) e. RR ) ) ) |
77 |
36 76
|
mpd |
|- ( I e. ( 1 ..^ D ) -> ( ( ph /\ X e. Odd ) -> ( F ` ( I + 1 ) ) e. RR ) ) |
78 |
77
|
com12 |
|- ( ( ph /\ X e. Odd ) -> ( I e. ( 1 ..^ D ) -> ( F ` ( I + 1 ) ) e. RR ) ) |
79 |
78
|
3impia |
|- ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) -> ( F ` ( I + 1 ) ) e. RR ) |
80 |
|
eluzelre |
|- ( N e. ( ZZ>= ` ; 1 1 ) -> N e. RR ) |
81 |
2 80
|
syl |
|- ( ph -> N e. RR ) |
82 |
|
oddz |
|- ( X e. Odd -> X e. ZZ ) |
83 |
82
|
zred |
|- ( X e. Odd -> X e. RR ) |
84 |
|
rexr |
|- ( ( F ` ( I + 1 ) ) e. RR -> ( F ` ( I + 1 ) ) e. RR* ) |
85 |
|
rexr |
|- ( ( F ` I ) e. RR -> ( F ` I ) e. RR* ) |
86 |
84 85
|
anim12ci |
|- ( ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) -> ( ( F ` I ) e. RR* /\ ( F ` ( I + 1 ) ) e. RR* ) ) |
87 |
86
|
adantl |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( ( F ` I ) e. RR* /\ ( F ` ( I + 1 ) ) e. RR* ) ) |
88 |
|
elico1 |
|- ( ( ( F ` I ) e. RR* /\ ( F ` ( I + 1 ) ) e. RR* ) -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) <-> ( X e. RR* /\ ( F ` I ) <_ X /\ X < ( F ` ( I + 1 ) ) ) ) ) |
89 |
87 88
|
syl |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) <-> ( X e. RR* /\ ( F ` I ) <_ X /\ X < ( F ` ( I + 1 ) ) ) ) ) |
90 |
|
simpllr |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> X e. RR ) |
91 |
|
simplrl |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> ( F ` ( I + 1 ) ) e. RR ) |
92 |
|
simplrr |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> ( F ` I ) e. RR ) |
93 |
|
simpr |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> X < ( F ` ( I + 1 ) ) ) |
94 |
90 91 92 93
|
ltsub1dd |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) |
95 |
|
simplr |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> X e. RR ) |
96 |
|
simprr |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( F ` I ) e. RR ) |
97 |
95 96
|
resubcld |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( X - ( F ` I ) ) e. RR ) |
98 |
97
|
adantr |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) e. RR ) |
99 |
91 92
|
resubcld |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> ( ( F ` ( I + 1 ) ) - ( F ` I ) ) e. RR ) |
100 |
|
simplll |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> N e. RR ) |
101 |
|
4re |
|- 4 e. RR |
102 |
101
|
a1i |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> 4 e. RR ) |
103 |
100 102
|
resubcld |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> ( N - 4 ) e. RR ) |
104 |
|
lttr |
|- ( ( ( X - ( F ` I ) ) e. RR /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) e. RR /\ ( N - 4 ) e. RR ) -> ( ( ( X - ( F ` I ) ) < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) ) -> ( X - ( F ` I ) ) < ( N - 4 ) ) ) |
105 |
98 99 103 104
|
syl3anc |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> ( ( ( X - ( F ` I ) ) < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) ) -> ( X - ( F ` I ) ) < ( N - 4 ) ) ) |
106 |
94 105
|
mpand |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ X < ( F ` ( I + 1 ) ) ) -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( X - ( F ` I ) ) < ( N - 4 ) ) ) |
107 |
106
|
impr |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ ( X < ( F ` ( I + 1 ) ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) ) ) -> ( X - ( F ` I ) ) < ( N - 4 ) ) |
108 |
|
4pos |
|- 0 < 4 |
109 |
101
|
a1i |
|- ( ( N e. RR /\ X e. RR ) -> 4 e. RR ) |
110 |
|
simpl |
|- ( ( N e. RR /\ X e. RR ) -> N e. RR ) |
111 |
109 110
|
ltsubposd |
|- ( ( N e. RR /\ X e. RR ) -> ( 0 < 4 <-> ( N - 4 ) < N ) ) |
112 |
108 111
|
mpbii |
|- ( ( N e. RR /\ X e. RR ) -> ( N - 4 ) < N ) |
113 |
112
|
adantr |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( N - 4 ) < N ) |
114 |
113
|
adantr |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ ( X < ( F ` ( I + 1 ) ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) ) ) -> ( N - 4 ) < N ) |
115 |
|
simpll |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> N e. RR ) |
116 |
101
|
a1i |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> 4 e. RR ) |
117 |
115 116
|
resubcld |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( N - 4 ) e. RR ) |
118 |
|
lttr |
|- ( ( ( X - ( F ` I ) ) e. RR /\ ( N - 4 ) e. RR /\ N e. RR ) -> ( ( ( X - ( F ` I ) ) < ( N - 4 ) /\ ( N - 4 ) < N ) -> ( X - ( F ` I ) ) < N ) ) |
119 |
97 117 115 118
|
syl3anc |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( ( ( X - ( F ` I ) ) < ( N - 4 ) /\ ( N - 4 ) < N ) -> ( X - ( F ` I ) ) < N ) ) |
120 |
119
|
adantr |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ ( X < ( F ` ( I + 1 ) ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) ) ) -> ( ( ( X - ( F ` I ) ) < ( N - 4 ) /\ ( N - 4 ) < N ) -> ( X - ( F ` I ) ) < N ) ) |
121 |
107 114 120
|
mp2and |
|- ( ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) /\ ( X < ( F ` ( I + 1 ) ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) ) ) -> ( X - ( F ` I ) ) < N ) |
122 |
121
|
exp32 |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( X < ( F ` ( I + 1 ) ) -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( X - ( F ` I ) ) < N ) ) ) |
123 |
122
|
com12 |
|- ( X < ( F ` ( I + 1 ) ) -> ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( X - ( F ` I ) ) < N ) ) ) |
124 |
123
|
3ad2ant3 |
|- ( ( X e. RR* /\ ( F ` I ) <_ X /\ X < ( F ` ( I + 1 ) ) ) -> ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( X - ( F ` I ) ) < N ) ) ) |
125 |
124
|
com12 |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( ( X e. RR* /\ ( F ` I ) <_ X /\ X < ( F ` ( I + 1 ) ) ) -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( X - ( F ` I ) ) < N ) ) ) |
126 |
89 125
|
sylbid |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( X - ( F ` I ) ) < N ) ) ) |
127 |
126
|
com23 |
|- ( ( ( N e. RR /\ X e. RR ) /\ ( ( F ` ( I + 1 ) ) e. RR /\ ( F ` I ) e. RR ) ) -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) ) |
128 |
127
|
exp32 |
|- ( ( N e. RR /\ X e. RR ) -> ( ( F ` ( I + 1 ) ) e. RR -> ( ( F ` I ) e. RR -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) ) ) ) |
129 |
128
|
com34 |
|- ( ( N e. RR /\ X e. RR ) -> ( ( F ` ( I + 1 ) ) e. RR -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( ( F ` I ) e. RR -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) ) ) ) |
130 |
81 83 129
|
syl2an |
|- ( ( ph /\ X e. Odd ) -> ( ( F ` ( I + 1 ) ) e. RR -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( ( F ` I ) e. RR -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) ) ) ) |
131 |
130
|
3adant3 |
|- ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) -> ( ( F ` ( I + 1 ) ) e. RR -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( ( F ` I ) e. RR -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) ) ) ) |
132 |
79 131
|
mpd |
|- ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( ( F ` I ) e. RR -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) ) ) |
133 |
132
|
com13 |
|- ( ( F ` I ) e. RR -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) ) ) |
134 |
33 35 133
|
3syl |
|- ( ( F ` I ) e. ( Prime \ { 2 } ) -> ( ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) -> ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) ) ) |
135 |
134
|
imp |
|- ( ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) ) -> ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) ) |
136 |
135
|
3adant3 |
|- ( ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) -> ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) ) |
137 |
136
|
impcom |
|- ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) -> ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) -> ( X - ( F ` I ) ) < N ) ) |
138 |
137
|
imp |
|- ( ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) /\ X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) ) -> ( X - ( F ` I ) ) < N ) |
139 |
138
|
adantrr |
|- ( ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) /\ ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) /\ 4 < S ) ) -> ( X - ( F ` I ) ) < N ) |
140 |
11 139
|
eqbrtrid |
|- ( ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) /\ ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) /\ 4 < S ) ) -> S < N ) |
141 |
|
simprr |
|- ( ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) /\ ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) /\ 4 < S ) ) -> 4 < S ) |
142 |
32 140 141
|
3jca |
|- ( ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) /\ ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) /\ 4 < S ) ) -> ( S e. Even /\ S < N /\ 4 < S ) ) |
143 |
142
|
ex |
|- ( ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) /\ ( ( F ` I ) e. ( Prime \ { 2 } ) /\ ( ( F ` ( I + 1 ) ) - ( F ` I ) ) < ( N - 4 ) /\ 4 < ( ( F ` ( I + 1 ) ) - ( F ` I ) ) ) ) -> ( ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) /\ 4 < S ) -> ( S e. Even /\ S < N /\ 4 < S ) ) ) |
144 |
24 143
|
mpdan |
|- ( ( ph /\ X e. Odd /\ I e. ( 1 ..^ D ) ) -> ( ( X e. ( ( F ` I ) [,) ( F ` ( I + 1 ) ) ) /\ 4 < S ) -> ( S e. Even /\ S < N /\ 4 < S ) ) ) |