Step |
Hyp |
Ref |
Expression |
1 |
|
lgamgulm.r |
|- ( ph -> R e. NN ) |
2 |
|
lgamgulm.u |
|- U = { x e. CC | ( ( abs ` x ) <_ R /\ A. k e. NN0 ( 1 / R ) <_ ( abs ` ( x + k ) ) ) } |
3 |
|
lgamgulm.g |
|- G = ( m e. NN |-> ( z e. U |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) |
4 |
1 2
|
lgamgulmlem1 |
|- ( ph -> U C_ ( CC \ ( ZZ \ NN ) ) ) |
5 |
4
|
sselda |
|- ( ( ph /\ z e. U ) -> z e. ( CC \ ( ZZ \ NN ) ) ) |
6 |
|
ovex |
|- ( sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) - ( log ` z ) ) e. _V |
7 |
|
df-lgam |
|- log_G = ( z e. ( CC \ ( ZZ \ NN ) ) |-> ( sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) - ( log ` z ) ) ) |
8 |
7
|
fvmpt2 |
|- ( ( z e. ( CC \ ( ZZ \ NN ) ) /\ ( sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) - ( log ` z ) ) e. _V ) -> ( log_G ` z ) = ( sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) - ( log ` z ) ) ) |
9 |
5 6 8
|
sylancl |
|- ( ( ph /\ z e. U ) -> ( log_G ` z ) = ( sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) - ( log ` z ) ) ) |
10 |
|
nnuz |
|- NN = ( ZZ>= ` 1 ) |
11 |
|
1zzd |
|- ( ( ph /\ z e. U ) -> 1 e. ZZ ) |
12 |
|
oveq1 |
|- ( m = n -> ( m + 1 ) = ( n + 1 ) ) |
13 |
|
id |
|- ( m = n -> m = n ) |
14 |
12 13
|
oveq12d |
|- ( m = n -> ( ( m + 1 ) / m ) = ( ( n + 1 ) / n ) ) |
15 |
14
|
fveq2d |
|- ( m = n -> ( log ` ( ( m + 1 ) / m ) ) = ( log ` ( ( n + 1 ) / n ) ) ) |
16 |
15
|
oveq2d |
|- ( m = n -> ( z x. ( log ` ( ( m + 1 ) / m ) ) ) = ( z x. ( log ` ( ( n + 1 ) / n ) ) ) ) |
17 |
|
oveq2 |
|- ( m = n -> ( z / m ) = ( z / n ) ) |
18 |
17
|
fvoveq1d |
|- ( m = n -> ( log ` ( ( z / m ) + 1 ) ) = ( log ` ( ( z / n ) + 1 ) ) ) |
19 |
16 18
|
oveq12d |
|- ( m = n -> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) = ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) ) |
20 |
|
eqid |
|- ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) = ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) |
21 |
|
ovex |
|- ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) e. _V |
22 |
19 20 21
|
fvmpt |
|- ( n e. NN -> ( ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ` n ) = ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) ) |
23 |
22
|
adantl |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ` n ) = ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) ) |
24 |
5
|
eldifad |
|- ( ( ph /\ z e. U ) -> z e. CC ) |
25 |
24
|
adantr |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> z e. CC ) |
26 |
|
simpr |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> n e. NN ) |
27 |
26
|
peano2nnd |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( n + 1 ) e. NN ) |
28 |
27
|
nnrpd |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( n + 1 ) e. RR+ ) |
29 |
26
|
nnrpd |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> n e. RR+ ) |
30 |
28 29
|
rpdivcld |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( ( n + 1 ) / n ) e. RR+ ) |
31 |
30
|
relogcld |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( log ` ( ( n + 1 ) / n ) ) e. RR ) |
32 |
31
|
recnd |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( log ` ( ( n + 1 ) / n ) ) e. CC ) |
33 |
25 32
|
mulcld |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( z x. ( log ` ( ( n + 1 ) / n ) ) ) e. CC ) |
34 |
26
|
nncnd |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> n e. CC ) |
35 |
26
|
nnne0d |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> n =/= 0 ) |
36 |
25 34 35
|
divcld |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( z / n ) e. CC ) |
37 |
|
1cnd |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> 1 e. CC ) |
38 |
36 37
|
addcld |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( ( z / n ) + 1 ) e. CC ) |
39 |
5
|
adantr |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> z e. ( CC \ ( ZZ \ NN ) ) ) |
40 |
39 26
|
dmgmdivn0 |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( ( z / n ) + 1 ) =/= 0 ) |
41 |
38 40
|
logcld |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( log ` ( ( z / n ) + 1 ) ) e. CC ) |
42 |
33 41
|
subcld |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) e. CC ) |
43 |
|
1z |
|- 1 e. ZZ |
44 |
|
seqfn |
|- ( 1 e. ZZ -> seq 1 ( oF + , G ) Fn ( ZZ>= ` 1 ) ) |
45 |
43 44
|
ax-mp |
|- seq 1 ( oF + , G ) Fn ( ZZ>= ` 1 ) |
46 |
10
|
fneq2i |
|- ( seq 1 ( oF + , G ) Fn NN <-> seq 1 ( oF + , G ) Fn ( ZZ>= ` 1 ) ) |
47 |
45 46
|
mpbir |
|- seq 1 ( oF + , G ) Fn NN |
48 |
1 2 3
|
lgamgulm |
|- ( ph -> seq 1 ( oF + , G ) e. dom ( ~~>u ` U ) ) |
49 |
|
ulmdm |
|- ( seq 1 ( oF + , G ) e. dom ( ~~>u ` U ) <-> seq 1 ( oF + , G ) ( ~~>u ` U ) ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ) |
50 |
48 49
|
sylib |
|- ( ph -> seq 1 ( oF + , G ) ( ~~>u ` U ) ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ) |
51 |
|
ulmf2 |
|- ( ( seq 1 ( oF + , G ) Fn NN /\ seq 1 ( oF + , G ) ( ~~>u ` U ) ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ) -> seq 1 ( oF + , G ) : NN --> ( CC ^m U ) ) |
52 |
47 50 51
|
sylancr |
|- ( ph -> seq 1 ( oF + , G ) : NN --> ( CC ^m U ) ) |
53 |
52
|
adantr |
|- ( ( ph /\ z e. U ) -> seq 1 ( oF + , G ) : NN --> ( CC ^m U ) ) |
54 |
|
simpr |
|- ( ( ph /\ z e. U ) -> z e. U ) |
55 |
|
seqex |
|- seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) e. _V |
56 |
55
|
a1i |
|- ( ( ph /\ z e. U ) -> seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) e. _V ) |
57 |
3
|
a1i |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> G = ( m e. NN |-> ( z e. U |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ) |
58 |
57
|
seqeq3d |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> seq 1 ( oF + , G ) = seq 1 ( oF + , ( m e. NN |-> ( z e. U |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ) ) |
59 |
58
|
fveq1d |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( seq 1 ( oF + , G ) ` n ) = ( seq 1 ( oF + , ( m e. NN |-> ( z e. U |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ) ` n ) ) |
60 |
|
cnex |
|- CC e. _V |
61 |
2 60
|
rabex2 |
|- U e. _V |
62 |
61
|
a1i |
|- ( ( ph /\ n e. NN ) -> U e. _V ) |
63 |
|
simpr |
|- ( ( ph /\ n e. NN ) -> n e. NN ) |
64 |
63 10
|
eleqtrdi |
|- ( ( ph /\ n e. NN ) -> n e. ( ZZ>= ` 1 ) ) |
65 |
|
fz1ssnn |
|- ( 1 ... n ) C_ NN |
66 |
65
|
a1i |
|- ( ( ph /\ n e. NN ) -> ( 1 ... n ) C_ NN ) |
67 |
|
ovexd |
|- ( ( ( ph /\ n e. NN ) /\ ( m e. NN /\ z e. U ) ) -> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) e. _V ) |
68 |
62 64 66 67
|
seqof2 |
|- ( ( ph /\ n e. NN ) -> ( seq 1 ( oF + , ( m e. NN |-> ( z e. U |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ) ` n ) = ( z e. U |-> ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) ) |
69 |
68
|
adantlr |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( seq 1 ( oF + , ( m e. NN |-> ( z e. U |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ) ` n ) = ( z e. U |-> ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) ) |
70 |
59 69
|
eqtrd |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( seq 1 ( oF + , G ) ` n ) = ( z e. U |-> ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) ) |
71 |
70
|
fveq1d |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( ( seq 1 ( oF + , G ) ` n ) ` z ) = ( ( z e. U |-> ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) ` z ) ) |
72 |
54
|
adantr |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> z e. U ) |
73 |
|
fvex |
|- ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) e. _V |
74 |
|
eqid |
|- ( z e. U |-> ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) = ( z e. U |-> ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) |
75 |
74
|
fvmpt2 |
|- ( ( z e. U /\ ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) e. _V ) -> ( ( z e. U |-> ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) ` z ) = ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) |
76 |
72 73 75
|
sylancl |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( ( z e. U |-> ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) ` z ) = ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) |
77 |
71 76
|
eqtrd |
|- ( ( ( ph /\ z e. U ) /\ n e. NN ) -> ( ( seq 1 ( oF + , G ) ` n ) ` z ) = ( seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ` n ) ) |
78 |
50
|
adantr |
|- ( ( ph /\ z e. U ) -> seq 1 ( oF + , G ) ( ~~>u ` U ) ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ) |
79 |
10 11 53 54 56 77 78
|
ulmclm |
|- ( ( ph /\ z e. U ) -> seq 1 ( + , ( m e. NN |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) ~~> ( ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ` z ) ) |
80 |
10 11 23 42 79
|
isumclim |
|- ( ( ph /\ z e. U ) -> sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) = ( ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ` z ) ) |
81 |
|
ulmcl |
|- ( seq 1 ( oF + , G ) ( ~~>u ` U ) ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) -> ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) : U --> CC ) |
82 |
50 81
|
syl |
|- ( ph -> ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) : U --> CC ) |
83 |
82
|
ffvelrnda |
|- ( ( ph /\ z e. U ) -> ( ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ` z ) e. CC ) |
84 |
80 83
|
eqeltrd |
|- ( ( ph /\ z e. U ) -> sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) e. CC ) |
85 |
5
|
dmgmn0 |
|- ( ( ph /\ z e. U ) -> z =/= 0 ) |
86 |
24 85
|
logcld |
|- ( ( ph /\ z e. U ) -> ( log ` z ) e. CC ) |
87 |
84 86
|
subcld |
|- ( ( ph /\ z e. U ) -> ( sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) - ( log ` z ) ) e. CC ) |
88 |
9 87
|
eqeltrd |
|- ( ( ph /\ z e. U ) -> ( log_G ` z ) e. CC ) |
89 |
88
|
ralrimiva |
|- ( ph -> A. z e. U ( log_G ` z ) e. CC ) |
90 |
|
ffn |
|- ( ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) : U --> CC -> ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) Fn U ) |
91 |
50 81 90
|
3syl |
|- ( ph -> ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) Fn U ) |
92 |
|
nfcv |
|- F/_ z ( ~~>u ` U ) |
93 |
|
nfcv |
|- F/_ z 1 |
94 |
|
nfcv |
|- F/_ z oF + |
95 |
|
nfcv |
|- F/_ z NN |
96 |
|
nfmpt1 |
|- F/_ z ( z e. U |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) |
97 |
95 96
|
nfmpt |
|- F/_ z ( m e. NN |-> ( z e. U |-> ( ( z x. ( log ` ( ( m + 1 ) / m ) ) ) - ( log ` ( ( z / m ) + 1 ) ) ) ) ) |
98 |
3 97
|
nfcxfr |
|- F/_ z G |
99 |
93 94 98
|
nfseq |
|- F/_ z seq 1 ( oF + , G ) |
100 |
92 99
|
nffv |
|- F/_ z ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) |
101 |
100
|
dffn5f |
|- ( ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) Fn U <-> ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) = ( z e. U |-> ( ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ` z ) ) ) |
102 |
91 101
|
sylib |
|- ( ph -> ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) = ( z e. U |-> ( ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ` z ) ) ) |
103 |
9
|
oveq1d |
|- ( ( ph /\ z e. U ) -> ( ( log_G ` z ) + ( log ` z ) ) = ( ( sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) - ( log ` z ) ) + ( log ` z ) ) ) |
104 |
84 86
|
npcand |
|- ( ( ph /\ z e. U ) -> ( ( sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) - ( log ` z ) ) + ( log ` z ) ) = sum_ n e. NN ( ( z x. ( log ` ( ( n + 1 ) / n ) ) ) - ( log ` ( ( z / n ) + 1 ) ) ) ) |
105 |
103 104 80
|
3eqtrrd |
|- ( ( ph /\ z e. U ) -> ( ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ` z ) = ( ( log_G ` z ) + ( log ` z ) ) ) |
106 |
105
|
mpteq2dva |
|- ( ph -> ( z e. U |-> ( ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) ` z ) ) = ( z e. U |-> ( ( log_G ` z ) + ( log ` z ) ) ) ) |
107 |
102 106
|
eqtrd |
|- ( ph -> ( ( ~~>u ` U ) ` seq 1 ( oF + , G ) ) = ( z e. U |-> ( ( log_G ` z ) + ( log ` z ) ) ) ) |
108 |
50 107
|
breqtrd |
|- ( ph -> seq 1 ( oF + , G ) ( ~~>u ` U ) ( z e. U |-> ( ( log_G ` z ) + ( log ` z ) ) ) ) |
109 |
89 108
|
jca |
|- ( ph -> ( A. z e. U ( log_G ` z ) e. CC /\ seq 1 ( oF + , G ) ( ~~>u ` U ) ( z e. U |-> ( ( log_G ` z ) + ( log ` z ) ) ) ) ) |