Step |
Hyp |
Ref |
Expression |
1 |
|
iseralt.1 |
|- Z = ( ZZ>= ` M ) |
2 |
|
iseralt.2 |
|- ( ph -> M e. ZZ ) |
3 |
|
iseralt.3 |
|- ( ph -> G : Z --> RR ) |
4 |
|
iseralt.4 |
|- ( ( ph /\ k e. Z ) -> ( G ` ( k + 1 ) ) <_ ( G ` k ) ) |
5 |
|
iseralt.5 |
|- ( ph -> G ~~> 0 ) |
6 |
|
iseralt.6 |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) = ( ( -u 1 ^ k ) x. ( G ` k ) ) ) |
7 |
|
oveq2 |
|- ( x = 0 -> ( 2 x. x ) = ( 2 x. 0 ) ) |
8 |
|
2t0e0 |
|- ( 2 x. 0 ) = 0 |
9 |
7 8
|
eqtrdi |
|- ( x = 0 -> ( 2 x. x ) = 0 ) |
10 |
9
|
oveq2d |
|- ( x = 0 -> ( N + ( 2 x. x ) ) = ( N + 0 ) ) |
11 |
10
|
fveq2d |
|- ( x = 0 -> ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) = ( seq M ( + , F ) ` ( N + 0 ) ) ) |
12 |
11
|
oveq2d |
|- ( x = 0 -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) = ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + 0 ) ) ) ) |
13 |
12
|
breq1d |
|- ( x = 0 -> ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) <-> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + 0 ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) |
14 |
13
|
imbi2d |
|- ( x = 0 -> ( ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) <-> ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + 0 ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) ) |
15 |
|
oveq2 |
|- ( x = n -> ( 2 x. x ) = ( 2 x. n ) ) |
16 |
15
|
oveq2d |
|- ( x = n -> ( N + ( 2 x. x ) ) = ( N + ( 2 x. n ) ) ) |
17 |
16
|
fveq2d |
|- ( x = n -> ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) = ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) |
18 |
17
|
oveq2d |
|- ( x = n -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) = ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) ) |
19 |
18
|
breq1d |
|- ( x = n -> ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) <-> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) |
20 |
19
|
imbi2d |
|- ( x = n -> ( ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) <-> ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) ) |
21 |
|
oveq2 |
|- ( x = ( n + 1 ) -> ( 2 x. x ) = ( 2 x. ( n + 1 ) ) ) |
22 |
21
|
oveq2d |
|- ( x = ( n + 1 ) -> ( N + ( 2 x. x ) ) = ( N + ( 2 x. ( n + 1 ) ) ) ) |
23 |
22
|
fveq2d |
|- ( x = ( n + 1 ) -> ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) = ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) |
24 |
23
|
oveq2d |
|- ( x = ( n + 1 ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) = ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) ) |
25 |
24
|
breq1d |
|- ( x = ( n + 1 ) -> ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) <-> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) |
26 |
25
|
imbi2d |
|- ( x = ( n + 1 ) -> ( ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) <-> ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) ) |
27 |
|
oveq2 |
|- ( x = K -> ( 2 x. x ) = ( 2 x. K ) ) |
28 |
27
|
oveq2d |
|- ( x = K -> ( N + ( 2 x. x ) ) = ( N + ( 2 x. K ) ) ) |
29 |
28
|
fveq2d |
|- ( x = K -> ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) = ( seq M ( + , F ) ` ( N + ( 2 x. K ) ) ) ) |
30 |
29
|
oveq2d |
|- ( x = K -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) = ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. K ) ) ) ) ) |
31 |
30
|
breq1d |
|- ( x = K -> ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) <-> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. K ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) |
32 |
31
|
imbi2d |
|- ( x = K -> ( ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. x ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) <-> ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. K ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) ) |
33 |
|
uzssz |
|- ( ZZ>= ` M ) C_ ZZ |
34 |
1 33
|
eqsstri |
|- Z C_ ZZ |
35 |
34
|
a1i |
|- ( ph -> Z C_ ZZ ) |
36 |
35
|
sselda |
|- ( ( ph /\ N e. Z ) -> N e. ZZ ) |
37 |
36
|
zcnd |
|- ( ( ph /\ N e. Z ) -> N e. CC ) |
38 |
37
|
addid1d |
|- ( ( ph /\ N e. Z ) -> ( N + 0 ) = N ) |
39 |
38
|
fveq2d |
|- ( ( ph /\ N e. Z ) -> ( seq M ( + , F ) ` ( N + 0 ) ) = ( seq M ( + , F ) ` N ) ) |
40 |
39
|
oveq2d |
|- ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + 0 ) ) ) = ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) |
41 |
|
neg1rr |
|- -u 1 e. RR |
42 |
|
neg1ne0 |
|- -u 1 =/= 0 |
43 |
|
reexpclz |
|- ( ( -u 1 e. RR /\ -u 1 =/= 0 /\ N e. ZZ ) -> ( -u 1 ^ N ) e. RR ) |
44 |
41 42 36 43
|
mp3an12i |
|- ( ( ph /\ N e. Z ) -> ( -u 1 ^ N ) e. RR ) |
45 |
35
|
sselda |
|- ( ( ph /\ k e. Z ) -> k e. ZZ ) |
46 |
|
reexpclz |
|- ( ( -u 1 e. RR /\ -u 1 =/= 0 /\ k e. ZZ ) -> ( -u 1 ^ k ) e. RR ) |
47 |
41 42 45 46
|
mp3an12i |
|- ( ( ph /\ k e. Z ) -> ( -u 1 ^ k ) e. RR ) |
48 |
3
|
ffvelrnda |
|- ( ( ph /\ k e. Z ) -> ( G ` k ) e. RR ) |
49 |
47 48
|
remulcld |
|- ( ( ph /\ k e. Z ) -> ( ( -u 1 ^ k ) x. ( G ` k ) ) e. RR ) |
50 |
6 49
|
eqeltrd |
|- ( ( ph /\ k e. Z ) -> ( F ` k ) e. RR ) |
51 |
1 2 50
|
serfre |
|- ( ph -> seq M ( + , F ) : Z --> RR ) |
52 |
51
|
ffvelrnda |
|- ( ( ph /\ N e. Z ) -> ( seq M ( + , F ) ` N ) e. RR ) |
53 |
44 52
|
remulcld |
|- ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) e. RR ) |
54 |
53
|
leidd |
|- ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) |
55 |
40 54
|
eqbrtrd |
|- ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + 0 ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) |
56 |
3
|
ad2antrr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> G : Z --> RR ) |
57 |
|
ax-1cn |
|- 1 e. CC |
58 |
57
|
2timesi |
|- ( 2 x. 1 ) = ( 1 + 1 ) |
59 |
58
|
oveq2i |
|- ( ( N + ( 2 x. n ) ) + ( 2 x. 1 ) ) = ( ( N + ( 2 x. n ) ) + ( 1 + 1 ) ) |
60 |
|
simpr |
|- ( ( ph /\ N e. Z ) -> N e. Z ) |
61 |
60 1
|
eleqtrdi |
|- ( ( ph /\ N e. Z ) -> N e. ( ZZ>= ` M ) ) |
62 |
61
|
adantr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> N e. ( ZZ>= ` M ) ) |
63 |
|
eluzelz |
|- ( N e. ( ZZ>= ` M ) -> N e. ZZ ) |
64 |
62 63
|
syl |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> N e. ZZ ) |
65 |
64
|
zcnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> N e. CC ) |
66 |
|
2cn |
|- 2 e. CC |
67 |
|
nn0cn |
|- ( n e. NN0 -> n e. CC ) |
68 |
67
|
adantl |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> n e. CC ) |
69 |
|
mulcl |
|- ( ( 2 e. CC /\ n e. CC ) -> ( 2 x. n ) e. CC ) |
70 |
66 68 69
|
sylancr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( 2 x. n ) e. CC ) |
71 |
66 57
|
mulcli |
|- ( 2 x. 1 ) e. CC |
72 |
71
|
a1i |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( 2 x. 1 ) e. CC ) |
73 |
65 70 72
|
addassd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( N + ( 2 x. n ) ) + ( 2 x. 1 ) ) = ( N + ( ( 2 x. n ) + ( 2 x. 1 ) ) ) ) |
74 |
59 73
|
eqtr3id |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( N + ( 2 x. n ) ) + ( 1 + 1 ) ) = ( N + ( ( 2 x. n ) + ( 2 x. 1 ) ) ) ) |
75 |
|
2nn0 |
|- 2 e. NN0 |
76 |
|
simpr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> n e. NN0 ) |
77 |
|
nn0mulcl |
|- ( ( 2 e. NN0 /\ n e. NN0 ) -> ( 2 x. n ) e. NN0 ) |
78 |
75 76 77
|
sylancr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( 2 x. n ) e. NN0 ) |
79 |
|
uzaddcl |
|- ( ( N e. ( ZZ>= ` M ) /\ ( 2 x. n ) e. NN0 ) -> ( N + ( 2 x. n ) ) e. ( ZZ>= ` M ) ) |
80 |
62 78 79
|
syl2anc |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( N + ( 2 x. n ) ) e. ( ZZ>= ` M ) ) |
81 |
33 80
|
sselid |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( N + ( 2 x. n ) ) e. ZZ ) |
82 |
81
|
zcnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( N + ( 2 x. n ) ) e. CC ) |
83 |
|
1cnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> 1 e. CC ) |
84 |
82 83 83
|
addassd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) = ( ( N + ( 2 x. n ) ) + ( 1 + 1 ) ) ) |
85 |
|
2cnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> 2 e. CC ) |
86 |
85 68 83
|
adddid |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( 2 x. ( n + 1 ) ) = ( ( 2 x. n ) + ( 2 x. 1 ) ) ) |
87 |
86
|
oveq2d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( N + ( 2 x. ( n + 1 ) ) ) = ( N + ( ( 2 x. n ) + ( 2 x. 1 ) ) ) ) |
88 |
74 84 87
|
3eqtr4d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) = ( N + ( 2 x. ( n + 1 ) ) ) ) |
89 |
|
peano2nn0 |
|- ( n e. NN0 -> ( n + 1 ) e. NN0 ) |
90 |
89
|
adantl |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( n + 1 ) e. NN0 ) |
91 |
|
nn0mulcl |
|- ( ( 2 e. NN0 /\ ( n + 1 ) e. NN0 ) -> ( 2 x. ( n + 1 ) ) e. NN0 ) |
92 |
75 90 91
|
sylancr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( 2 x. ( n + 1 ) ) e. NN0 ) |
93 |
|
uzaddcl |
|- ( ( N e. ( ZZ>= ` M ) /\ ( 2 x. ( n + 1 ) ) e. NN0 ) -> ( N + ( 2 x. ( n + 1 ) ) ) e. ( ZZ>= ` M ) ) |
94 |
62 92 93
|
syl2anc |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( N + ( 2 x. ( n + 1 ) ) ) e. ( ZZ>= ` M ) ) |
95 |
94 1
|
eleqtrrdi |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( N + ( 2 x. ( n + 1 ) ) ) e. Z ) |
96 |
88 95
|
eqeltrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) e. Z ) |
97 |
56 96
|
ffvelrnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) e. RR ) |
98 |
|
peano2uz |
|- ( ( N + ( 2 x. n ) ) e. ( ZZ>= ` M ) -> ( ( N + ( 2 x. n ) ) + 1 ) e. ( ZZ>= ` M ) ) |
99 |
80 98
|
syl |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( N + ( 2 x. n ) ) + 1 ) e. ( ZZ>= ` M ) ) |
100 |
99 1
|
eleqtrrdi |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( N + ( 2 x. n ) ) + 1 ) e. Z ) |
101 |
56 100
|
ffvelrnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) e. RR ) |
102 |
97 101
|
resubcld |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) - ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) e. RR ) |
103 |
|
0red |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> 0 e. RR ) |
104 |
44
|
adantr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ N ) e. RR ) |
105 |
51
|
ad2antrr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> seq M ( + , F ) : Z --> RR ) |
106 |
80 1
|
eleqtrrdi |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( N + ( 2 x. n ) ) e. Z ) |
107 |
105 106
|
ffvelrnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) e. RR ) |
108 |
104 107
|
remulcld |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) e. RR ) |
109 |
|
fvoveq1 |
|- ( k = ( ( N + ( 2 x. n ) ) + 1 ) -> ( G ` ( k + 1 ) ) = ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) |
110 |
|
fveq2 |
|- ( k = ( ( N + ( 2 x. n ) ) + 1 ) -> ( G ` k ) = ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) |
111 |
109 110
|
breq12d |
|- ( k = ( ( N + ( 2 x. n ) ) + 1 ) -> ( ( G ` ( k + 1 ) ) <_ ( G ` k ) <-> ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) <_ ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
112 |
4
|
ralrimiva |
|- ( ph -> A. k e. Z ( G ` ( k + 1 ) ) <_ ( G ` k ) ) |
113 |
112
|
ad2antrr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> A. k e. Z ( G ` ( k + 1 ) ) <_ ( G ` k ) ) |
114 |
111 113 100
|
rspcdva |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) <_ ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) |
115 |
97 101
|
suble0d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) - ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) <_ 0 <-> ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) <_ ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
116 |
114 115
|
mpbird |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) - ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) <_ 0 ) |
117 |
102 103 108 116
|
leadd2dd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) + ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) - ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) <_ ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) + 0 ) ) |
118 |
|
seqp1 |
|- ( ( ( N + ( 2 x. n ) ) + 1 ) e. ( ZZ>= ` M ) -> ( seq M ( + , F ) ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) = ( ( seq M ( + , F ) ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
119 |
99 118
|
syl |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( seq M ( + , F ) ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) = ( ( seq M ( + , F ) ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
120 |
|
seqp1 |
|- ( ( N + ( 2 x. n ) ) e. ( ZZ>= ` M ) -> ( seq M ( + , F ) ` ( ( N + ( 2 x. n ) ) + 1 ) ) = ( ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) + ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
121 |
80 120
|
syl |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( seq M ( + , F ) ` ( ( N + ( 2 x. n ) ) + 1 ) ) = ( ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) + ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
122 |
121
|
oveq1d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( seq M ( + , F ) ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( ( ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) + ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
123 |
119 122
|
eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( seq M ( + , F ) ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) = ( ( ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) + ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
124 |
88
|
fveq2d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( seq M ( + , F ) ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) = ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) |
125 |
107
|
recnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) e. CC ) |
126 |
|
fveq2 |
|- ( k = ( ( N + ( 2 x. n ) ) + 1 ) -> ( F ` k ) = ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) |
127 |
|
oveq2 |
|- ( k = ( ( N + ( 2 x. n ) ) + 1 ) -> ( -u 1 ^ k ) = ( -u 1 ^ ( ( N + ( 2 x. n ) ) + 1 ) ) ) |
128 |
127 110
|
oveq12d |
|- ( k = ( ( N + ( 2 x. n ) ) + 1 ) -> ( ( -u 1 ^ k ) x. ( G ` k ) ) = ( ( -u 1 ^ ( ( N + ( 2 x. n ) ) + 1 ) ) x. ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
129 |
126 128
|
eqeq12d |
|- ( k = ( ( N + ( 2 x. n ) ) + 1 ) -> ( ( F ` k ) = ( ( -u 1 ^ k ) x. ( G ` k ) ) <-> ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) = ( ( -u 1 ^ ( ( N + ( 2 x. n ) ) + 1 ) ) x. ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) ) |
130 |
6
|
ralrimiva |
|- ( ph -> A. k e. Z ( F ` k ) = ( ( -u 1 ^ k ) x. ( G ` k ) ) ) |
131 |
130
|
ad2antrr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> A. k e. Z ( F ` k ) = ( ( -u 1 ^ k ) x. ( G ` k ) ) ) |
132 |
129 131 100
|
rspcdva |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) = ( ( -u 1 ^ ( ( N + ( 2 x. n ) ) + 1 ) ) x. ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
133 |
|
neg1cn |
|- -u 1 e. CC |
134 |
133
|
a1i |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> -u 1 e. CC ) |
135 |
42
|
a1i |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> -u 1 =/= 0 ) |
136 |
134 135 81
|
expp1zd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ ( ( N + ( 2 x. n ) ) + 1 ) ) = ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u 1 ) ) |
137 |
41
|
a1i |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> -u 1 e. RR ) |
138 |
137 135 81
|
reexpclzd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ ( N + ( 2 x. n ) ) ) e. RR ) |
139 |
138
|
recnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ ( N + ( 2 x. n ) ) ) e. CC ) |
140 |
|
mulcom |
|- ( ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) e. CC /\ -u 1 e. CC ) -> ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u 1 ) = ( -u 1 x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) ) |
141 |
139 133 140
|
sylancl |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u 1 ) = ( -u 1 x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) ) |
142 |
139
|
mulm1d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) = -u ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) |
143 |
136 141 142
|
3eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ ( ( N + ( 2 x. n ) ) + 1 ) ) = -u ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) |
144 |
143
|
oveq1d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ ( ( N + ( 2 x. n ) ) + 1 ) ) x. ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) = ( -u ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
145 |
101
|
recnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) e. CC ) |
146 |
|
mulneg12 |
|- ( ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) e. CC /\ ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) e. CC ) -> ( -u ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) = ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
147 |
139 145 146
|
syl2anc |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) = ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
148 |
132 144 147
|
3eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) = ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
149 |
101
|
renegcld |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) e. RR ) |
150 |
138 149
|
remulcld |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) e. RR ) |
151 |
148 150
|
eqeltrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) e. RR ) |
152 |
151
|
recnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) e. CC ) |
153 |
|
fveq2 |
|- ( k = ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) -> ( F ` k ) = ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) |
154 |
|
oveq2 |
|- ( k = ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) -> ( -u 1 ^ k ) = ( -u 1 ^ ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) |
155 |
|
fveq2 |
|- ( k = ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) -> ( G ` k ) = ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) |
156 |
154 155
|
oveq12d |
|- ( k = ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) -> ( ( -u 1 ^ k ) x. ( G ` k ) ) = ( ( -u 1 ^ ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
157 |
153 156
|
eqeq12d |
|- ( k = ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) -> ( ( F ` k ) = ( ( -u 1 ^ k ) x. ( G ` k ) ) <-> ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) = ( ( -u 1 ^ ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) ) |
158 |
157 131 96
|
rspcdva |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) = ( ( -u 1 ^ ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
159 |
81
|
peano2zd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( N + ( 2 x. n ) ) + 1 ) e. ZZ ) |
160 |
134 135 159
|
expp1zd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) = ( ( -u 1 ^ ( ( N + ( 2 x. n ) ) + 1 ) ) x. -u 1 ) ) |
161 |
143
|
oveq1d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ ( ( N + ( 2 x. n ) ) + 1 ) ) x. -u 1 ) = ( -u ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u 1 ) ) |
162 |
|
mul2neg |
|- ( ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) e. CC /\ 1 e. CC ) -> ( -u ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u 1 ) = ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. 1 ) ) |
163 |
139 57 162
|
sylancl |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u 1 ) = ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. 1 ) ) |
164 |
139
|
mulid1d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. 1 ) = ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) |
165 |
163 164
|
eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u 1 ) = ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) |
166 |
160 161 165
|
3eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) = ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) |
167 |
166
|
oveq1d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
168 |
158 167
|
eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) = ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
169 |
138 97
|
remulcld |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) e. RR ) |
170 |
168 169
|
eqeltrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) e. RR ) |
171 |
170
|
recnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) e. CC ) |
172 |
125 152 171
|
addassd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) + ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) + ( ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) ) |
173 |
123 124 172
|
3eqtr3d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) = ( ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) + ( ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) ) |
174 |
173
|
oveq2d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) = ( ( -u 1 ^ N ) x. ( ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) + ( ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) ) ) |
175 |
104
|
recnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ N ) e. CC ) |
176 |
151 170
|
readdcld |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) e. RR ) |
177 |
176
|
recnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) e. CC ) |
178 |
175 125 177
|
adddid |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) + ( ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) ) = ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) + ( ( -u 1 ^ N ) x. ( ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) ) ) |
179 |
175 152 171
|
adddid |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) = ( ( ( -u 1 ^ N ) x. ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) + ( ( -u 1 ^ N ) x. ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) ) |
180 |
148
|
oveq2d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) = ( ( -u 1 ^ N ) x. ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) ) |
181 |
149
|
recnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) e. CC ) |
182 |
175 139 181
|
mulassd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( -u 1 ^ N ) x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) = ( ( -u 1 ^ N ) x. ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) ) |
183 |
180 182
|
eqtr4d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) = ( ( ( -u 1 ^ N ) x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
184 |
85 65 68
|
adddid |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( 2 x. ( N + n ) ) = ( ( 2 x. N ) + ( 2 x. n ) ) ) |
185 |
65
|
2timesd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( 2 x. N ) = ( N + N ) ) |
186 |
185
|
oveq1d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( 2 x. N ) + ( 2 x. n ) ) = ( ( N + N ) + ( 2 x. n ) ) ) |
187 |
65 65 70
|
addassd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( N + N ) + ( 2 x. n ) ) = ( N + ( N + ( 2 x. n ) ) ) ) |
188 |
184 186 187
|
3eqtrrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( N + ( N + ( 2 x. n ) ) ) = ( 2 x. ( N + n ) ) ) |
189 |
188
|
oveq2d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ ( N + ( N + ( 2 x. n ) ) ) ) = ( -u 1 ^ ( 2 x. ( N + n ) ) ) ) |
190 |
|
expaddz |
|- ( ( ( -u 1 e. CC /\ -u 1 =/= 0 ) /\ ( N e. ZZ /\ ( N + ( 2 x. n ) ) e. ZZ ) ) -> ( -u 1 ^ ( N + ( N + ( 2 x. n ) ) ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) ) |
191 |
134 135 64 81 190
|
syl22anc |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ ( N + ( N + ( 2 x. n ) ) ) ) = ( ( -u 1 ^ N ) x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) ) |
192 |
|
2z |
|- 2 e. ZZ |
193 |
192
|
a1i |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> 2 e. ZZ ) |
194 |
|
nn0z |
|- ( n e. NN0 -> n e. ZZ ) |
195 |
|
zaddcl |
|- ( ( N e. ZZ /\ n e. ZZ ) -> ( N + n ) e. ZZ ) |
196 |
36 194 195
|
syl2an |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( N + n ) e. ZZ ) |
197 |
|
expmulz |
|- ( ( ( -u 1 e. CC /\ -u 1 =/= 0 ) /\ ( 2 e. ZZ /\ ( N + n ) e. ZZ ) ) -> ( -u 1 ^ ( 2 x. ( N + n ) ) ) = ( ( -u 1 ^ 2 ) ^ ( N + n ) ) ) |
198 |
134 135 193 196 197
|
syl22anc |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ ( 2 x. ( N + n ) ) ) = ( ( -u 1 ^ 2 ) ^ ( N + n ) ) ) |
199 |
|
neg1sqe1 |
|- ( -u 1 ^ 2 ) = 1 |
200 |
199
|
oveq1i |
|- ( ( -u 1 ^ 2 ) ^ ( N + n ) ) = ( 1 ^ ( N + n ) ) |
201 |
|
1exp |
|- ( ( N + n ) e. ZZ -> ( 1 ^ ( N + n ) ) = 1 ) |
202 |
196 201
|
syl |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( 1 ^ ( N + n ) ) = 1 ) |
203 |
200 202
|
eqtrid |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ 2 ) ^ ( N + n ) ) = 1 ) |
204 |
198 203
|
eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u 1 ^ ( 2 x. ( N + n ) ) ) = 1 ) |
205 |
189 191 204
|
3eqtr3d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) = 1 ) |
206 |
205
|
oveq1d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( -u 1 ^ N ) x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) = ( 1 x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
207 |
181
|
mulid2d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( 1 x. -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) = -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) |
208 |
183 206 207
|
3eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) = -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) |
209 |
168
|
oveq2d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( ( -u 1 ^ N ) x. ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) ) |
210 |
97
|
recnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) e. CC ) |
211 |
175 139 210
|
mulassd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( -u 1 ^ N ) x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( ( -u 1 ^ N ) x. ( ( -u 1 ^ ( N + ( 2 x. n ) ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) ) |
212 |
209 211
|
eqtr4d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( ( ( -u 1 ^ N ) x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
213 |
205
|
oveq1d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( -u 1 ^ N ) x. ( -u 1 ^ ( N + ( 2 x. n ) ) ) ) x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( 1 x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
214 |
210
|
mulid2d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( 1 x. ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) |
215 |
212 213 214
|
3eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) |
216 |
208 215
|
oveq12d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( -u 1 ^ N ) x. ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) + ( ( -u 1 ^ N ) x. ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) = ( -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) |
217 |
145
|
negcld |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) e. CC ) |
218 |
217 210
|
addcomd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) + -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
219 |
210 145
|
negsubd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) + -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) = ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) - ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
220 |
218 219
|
eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( -u ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) = ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) - ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
221 |
179 216 220
|
3eqtrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) = ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) - ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) |
222 |
221
|
oveq2d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) + ( ( -u 1 ^ N ) x. ( ( F ` ( ( N + ( 2 x. n ) ) + 1 ) ) + ( F ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) ) ) ) = ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) + ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) - ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) ) |
223 |
174 178 222
|
3eqtrrd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) + ( ( G ` ( ( ( N + ( 2 x. n ) ) + 1 ) + 1 ) ) - ( G ` ( ( N + ( 2 x. n ) ) + 1 ) ) ) ) = ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) ) |
224 |
108
|
recnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) e. CC ) |
225 |
224
|
addid1d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) + 0 ) = ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) ) |
226 |
117 223 225
|
3brtr3d |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) ) |
227 |
105 95
|
ffvelrnd |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) e. RR ) |
228 |
104 227
|
remulcld |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) e. RR ) |
229 |
53
|
adantr |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) e. RR ) |
230 |
|
letr |
|- ( ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) e. RR /\ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) e. RR /\ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) e. RR ) -> ( ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) /\ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) |
231 |
228 108 229 230
|
syl3anc |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) /\ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) |
232 |
226 231
|
mpand |
|- ( ( ( ph /\ N e. Z ) /\ n e. NN0 ) -> ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) |
233 |
232
|
expcom |
|- ( n e. NN0 -> ( ( ph /\ N e. Z ) -> ( ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) ) |
234 |
233
|
a2d |
|- ( n e. NN0 -> ( ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. n ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) -> ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. ( n + 1 ) ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) ) |
235 |
14 20 26 32 55 234
|
nn0ind |
|- ( K e. NN0 -> ( ( ph /\ N e. Z ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. K ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) |
236 |
235
|
com12 |
|- ( ( ph /\ N e. Z ) -> ( K e. NN0 -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. K ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) ) |
237 |
236
|
3impia |
|- ( ( ph /\ N e. Z /\ K e. NN0 ) -> ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` ( N + ( 2 x. K ) ) ) ) <_ ( ( -u 1 ^ N ) x. ( seq M ( + , F ) ` N ) ) ) |