Step |
Hyp |
Ref |
Expression |
1 |
|
sge0fodjrnlem.k |
|- F/ k ph |
2 |
|
sge0fodjrnlem.n |
|- F/ n ph |
3 |
|
sge0fodjrnlem.bd |
|- ( k = G -> B = D ) |
4 |
|
sge0fodjrnlem.c |
|- ( ph -> C e. V ) |
5 |
|
sge0fodjrnlem.f |
|- ( ph -> F : C -onto-> A ) |
6 |
|
sge0fodjrnlem.dj |
|- ( ph -> Disj_ n e. C ( F ` n ) ) |
7 |
|
sge0fodjrnlem.fng |
|- ( ( ph /\ n e. C ) -> ( F ` n ) = G ) |
8 |
|
sge0fodjrnlem.b |
|- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) ) |
9 |
|
sge0fodjrnlem.b0 |
|- ( ( ph /\ k = (/) ) -> B = 0 ) |
10 |
|
sge0fodjrnlem.z |
|- Z = ( `' F " { (/) } ) |
11 |
|
fornex |
|- ( C e. V -> ( F : C -onto-> A -> A e. _V ) ) |
12 |
4 5 11
|
sylc |
|- ( ph -> A e. _V ) |
13 |
|
difssd |
|- ( ph -> ( A \ { (/) } ) C_ A ) |
14 |
|
simpl |
|- ( ( ph /\ k e. ( A \ { (/) } ) ) -> ph ) |
15 |
13
|
sselda |
|- ( ( ph /\ k e. ( A \ { (/) } ) ) -> k e. A ) |
16 |
14 15 8
|
syl2anc |
|- ( ( ph /\ k e. ( A \ { (/) } ) ) -> B e. ( 0 [,] +oo ) ) |
17 |
|
simpl |
|- ( ( ph /\ k e. ( A \ ( A \ { (/) } ) ) ) -> ph ) |
18 |
|
dfin4 |
|- ( A i^i { (/) } ) = ( A \ ( A \ { (/) } ) ) |
19 |
18
|
eqcomi |
|- ( A \ ( A \ { (/) } ) ) = ( A i^i { (/) } ) |
20 |
|
inss2 |
|- ( A i^i { (/) } ) C_ { (/) } |
21 |
19 20
|
eqsstri |
|- ( A \ ( A \ { (/) } ) ) C_ { (/) } |
22 |
|
id |
|- ( k e. ( A \ ( A \ { (/) } ) ) -> k e. ( A \ ( A \ { (/) } ) ) ) |
23 |
21 22
|
sselid |
|- ( k e. ( A \ ( A \ { (/) } ) ) -> k e. { (/) } ) |
24 |
|
elsni |
|- ( k e. { (/) } -> k = (/) ) |
25 |
23 24
|
syl |
|- ( k e. ( A \ ( A \ { (/) } ) ) -> k = (/) ) |
26 |
25
|
adantl |
|- ( ( ph /\ k e. ( A \ ( A \ { (/) } ) ) ) -> k = (/) ) |
27 |
17 26 9
|
syl2anc |
|- ( ( ph /\ k e. ( A \ ( A \ { (/) } ) ) ) -> B = 0 ) |
28 |
1 12 13 16 27
|
sge0ss |
|- ( ph -> ( sum^ ` ( k e. ( A \ { (/) } ) |-> B ) ) = ( sum^ ` ( k e. A |-> B ) ) ) |
29 |
28
|
eqcomd |
|- ( ph -> ( sum^ ` ( k e. A |-> B ) ) = ( sum^ ` ( k e. ( A \ { (/) } ) |-> B ) ) ) |
30 |
4
|
difexd |
|- ( ph -> ( C \ Z ) e. _V ) |
31 |
|
eqid |
|- ( n e. C |-> ( F ` n ) ) = ( n e. C |-> ( F ` n ) ) |
32 |
|
fof |
|- ( F : C -onto-> A -> F : C --> A ) |
33 |
5 32
|
syl |
|- ( ph -> F : C --> A ) |
34 |
33
|
ffvelrnda |
|- ( ( ph /\ n e. C ) -> ( F ` n ) e. A ) |
35 |
|
fveq2 |
|- ( m = n -> ( F ` m ) = ( F ` n ) ) |
36 |
35
|
neeq1d |
|- ( m = n -> ( ( F ` m ) =/= (/) <-> ( F ` n ) =/= (/) ) ) |
37 |
36
|
cbvrabv |
|- { m e. C | ( F ` m ) =/= (/) } = { n e. C | ( F ` n ) =/= (/) } |
38 |
35
|
cbvmptv |
|- ( m e. C |-> ( F ` m ) ) = ( n e. C |-> ( F ` n ) ) |
39 |
38
|
rneqi |
|- ran ( m e. C |-> ( F ` m ) ) = ran ( n e. C |-> ( F ` n ) ) |
40 |
39
|
difeq1i |
|- ( ran ( m e. C |-> ( F ` m ) ) \ { (/) } ) = ( ran ( n e. C |-> ( F ` n ) ) \ { (/) } ) |
41 |
2 31 34 6 37 40
|
disjf1o |
|- ( ph -> ( ( n e. C |-> ( F ` n ) ) |` { m e. C | ( F ` m ) =/= (/) } ) : { m e. C | ( F ` m ) =/= (/) } -1-1-onto-> ( ran ( m e. C |-> ( F ` m ) ) \ { (/) } ) ) |
42 |
33
|
feqmptd |
|- ( ph -> F = ( n e. C |-> ( F ` n ) ) ) |
43 |
|
difssd |
|- ( ph -> ( C \ Z ) C_ C ) |
44 |
43
|
sselda |
|- ( ( ph /\ n e. ( C \ Z ) ) -> n e. C ) |
45 |
|
eldifi |
|- ( n e. ( C \ Z ) -> n e. C ) |
46 |
45
|
adantr |
|- ( ( n e. ( C \ Z ) /\ ( F ` n ) = (/) ) -> n e. C ) |
47 |
|
id |
|- ( ( F ` n ) = (/) -> ( F ` n ) = (/) ) |
48 |
|
fvex |
|- ( F ` n ) e. _V |
49 |
48
|
elsn |
|- ( ( F ` n ) e. { (/) } <-> ( F ` n ) = (/) ) |
50 |
47 49
|
sylibr |
|- ( ( F ` n ) = (/) -> ( F ` n ) e. { (/) } ) |
51 |
50
|
adantl |
|- ( ( n e. ( C \ Z ) /\ ( F ` n ) = (/) ) -> ( F ` n ) e. { (/) } ) |
52 |
46 51
|
jca |
|- ( ( n e. ( C \ Z ) /\ ( F ` n ) = (/) ) -> ( n e. C /\ ( F ` n ) e. { (/) } ) ) |
53 |
52
|
adantll |
|- ( ( ( ph /\ n e. ( C \ Z ) ) /\ ( F ` n ) = (/) ) -> ( n e. C /\ ( F ` n ) e. { (/) } ) ) |
54 |
33
|
ffnd |
|- ( ph -> F Fn C ) |
55 |
|
elpreima |
|- ( F Fn C -> ( n e. ( `' F " { (/) } ) <-> ( n e. C /\ ( F ` n ) e. { (/) } ) ) ) |
56 |
54 55
|
syl |
|- ( ph -> ( n e. ( `' F " { (/) } ) <-> ( n e. C /\ ( F ` n ) e. { (/) } ) ) ) |
57 |
56
|
ad2antrr |
|- ( ( ( ph /\ n e. ( C \ Z ) ) /\ ( F ` n ) = (/) ) -> ( n e. ( `' F " { (/) } ) <-> ( n e. C /\ ( F ` n ) e. { (/) } ) ) ) |
58 |
53 57
|
mpbird |
|- ( ( ( ph /\ n e. ( C \ Z ) ) /\ ( F ` n ) = (/) ) -> n e. ( `' F " { (/) } ) ) |
59 |
58 10
|
eleqtrrdi |
|- ( ( ( ph /\ n e. ( C \ Z ) ) /\ ( F ` n ) = (/) ) -> n e. Z ) |
60 |
|
eldifn |
|- ( n e. ( C \ Z ) -> -. n e. Z ) |
61 |
60
|
ad2antlr |
|- ( ( ( ph /\ n e. ( C \ Z ) ) /\ ( F ` n ) = (/) ) -> -. n e. Z ) |
62 |
59 61
|
pm2.65da |
|- ( ( ph /\ n e. ( C \ Z ) ) -> -. ( F ` n ) = (/) ) |
63 |
62
|
neqned |
|- ( ( ph /\ n e. ( C \ Z ) ) -> ( F ` n ) =/= (/) ) |
64 |
44 63
|
jca |
|- ( ( ph /\ n e. ( C \ Z ) ) -> ( n e. C /\ ( F ` n ) =/= (/) ) ) |
65 |
36
|
elrab |
|- ( n e. { m e. C | ( F ` m ) =/= (/) } <-> ( n e. C /\ ( F ` n ) =/= (/) ) ) |
66 |
64 65
|
sylibr |
|- ( ( ph /\ n e. ( C \ Z ) ) -> n e. { m e. C | ( F ` m ) =/= (/) } ) |
67 |
66
|
ex |
|- ( ph -> ( n e. ( C \ Z ) -> n e. { m e. C | ( F ` m ) =/= (/) } ) ) |
68 |
65
|
simplbi |
|- ( n e. { m e. C | ( F ` m ) =/= (/) } -> n e. C ) |
69 |
68
|
adantl |
|- ( ( ph /\ n e. { m e. C | ( F ` m ) =/= (/) } ) -> n e. C ) |
70 |
10
|
eleq2i |
|- ( n e. Z <-> n e. ( `' F " { (/) } ) ) |
71 |
70
|
biimpi |
|- ( n e. Z -> n e. ( `' F " { (/) } ) ) |
72 |
71
|
adantl |
|- ( ( ph /\ n e. Z ) -> n e. ( `' F " { (/) } ) ) |
73 |
56
|
adantr |
|- ( ( ph /\ n e. Z ) -> ( n e. ( `' F " { (/) } ) <-> ( n e. C /\ ( F ` n ) e. { (/) } ) ) ) |
74 |
72 73
|
mpbid |
|- ( ( ph /\ n e. Z ) -> ( n e. C /\ ( F ` n ) e. { (/) } ) ) |
75 |
74
|
simprd |
|- ( ( ph /\ n e. Z ) -> ( F ` n ) e. { (/) } ) |
76 |
|
elsni |
|- ( ( F ` n ) e. { (/) } -> ( F ` n ) = (/) ) |
77 |
75 76
|
syl |
|- ( ( ph /\ n e. Z ) -> ( F ` n ) = (/) ) |
78 |
77
|
adantlr |
|- ( ( ( ph /\ n e. { m e. C | ( F ` m ) =/= (/) } ) /\ n e. Z ) -> ( F ` n ) = (/) ) |
79 |
65
|
simprbi |
|- ( n e. { m e. C | ( F ` m ) =/= (/) } -> ( F ` n ) =/= (/) ) |
80 |
79
|
ad2antlr |
|- ( ( ( ph /\ n e. { m e. C | ( F ` m ) =/= (/) } ) /\ n e. Z ) -> ( F ` n ) =/= (/) ) |
81 |
80
|
neneqd |
|- ( ( ( ph /\ n e. { m e. C | ( F ` m ) =/= (/) } ) /\ n e. Z ) -> -. ( F ` n ) = (/) ) |
82 |
78 81
|
pm2.65da |
|- ( ( ph /\ n e. { m e. C | ( F ` m ) =/= (/) } ) -> -. n e. Z ) |
83 |
69 82
|
eldifd |
|- ( ( ph /\ n e. { m e. C | ( F ` m ) =/= (/) } ) -> n e. ( C \ Z ) ) |
84 |
83
|
ex |
|- ( ph -> ( n e. { m e. C | ( F ` m ) =/= (/) } -> n e. ( C \ Z ) ) ) |
85 |
2 84
|
ralrimi |
|- ( ph -> A. n e. { m e. C | ( F ` m ) =/= (/) } n e. ( C \ Z ) ) |
86 |
|
dfss3 |
|- ( { m e. C | ( F ` m ) =/= (/) } C_ ( C \ Z ) <-> A. n e. { m e. C | ( F ` m ) =/= (/) } n e. ( C \ Z ) ) |
87 |
85 86
|
sylibr |
|- ( ph -> { m e. C | ( F ` m ) =/= (/) } C_ ( C \ Z ) ) |
88 |
87
|
sseld |
|- ( ph -> ( n e. { m e. C | ( F ` m ) =/= (/) } -> n e. ( C \ Z ) ) ) |
89 |
67 88
|
impbid |
|- ( ph -> ( n e. ( C \ Z ) <-> n e. { m e. C | ( F ` m ) =/= (/) } ) ) |
90 |
2 89
|
alrimi |
|- ( ph -> A. n ( n e. ( C \ Z ) <-> n e. { m e. C | ( F ` m ) =/= (/) } ) ) |
91 |
|
dfcleq |
|- ( ( C \ Z ) = { m e. C | ( F ` m ) =/= (/) } <-> A. n ( n e. ( C \ Z ) <-> n e. { m e. C | ( F ` m ) =/= (/) } ) ) |
92 |
90 91
|
sylibr |
|- ( ph -> ( C \ Z ) = { m e. C | ( F ` m ) =/= (/) } ) |
93 |
42 92
|
reseq12d |
|- ( ph -> ( F |` ( C \ Z ) ) = ( ( n e. C |-> ( F ` n ) ) |` { m e. C | ( F ` m ) =/= (/) } ) ) |
94 |
42 38
|
eqtr4di |
|- ( ph -> F = ( m e. C |-> ( F ` m ) ) ) |
95 |
94
|
eqcomd |
|- ( ph -> ( m e. C |-> ( F ` m ) ) = F ) |
96 |
95
|
rneqd |
|- ( ph -> ran ( m e. C |-> ( F ` m ) ) = ran F ) |
97 |
|
forn |
|- ( F : C -onto-> A -> ran F = A ) |
98 |
5 97
|
syl |
|- ( ph -> ran F = A ) |
99 |
96 98
|
eqtr2d |
|- ( ph -> A = ran ( m e. C |-> ( F ` m ) ) ) |
100 |
99
|
difeq1d |
|- ( ph -> ( A \ { (/) } ) = ( ran ( m e. C |-> ( F ` m ) ) \ { (/) } ) ) |
101 |
93 92 100
|
f1oeq123d |
|- ( ph -> ( ( F |` ( C \ Z ) ) : ( C \ Z ) -1-1-onto-> ( A \ { (/) } ) <-> ( ( n e. C |-> ( F ` n ) ) |` { m e. C | ( F ` m ) =/= (/) } ) : { m e. C | ( F ` m ) =/= (/) } -1-1-onto-> ( ran ( m e. C |-> ( F ` m ) ) \ { (/) } ) ) ) |
102 |
41 101
|
mpbird |
|- ( ph -> ( F |` ( C \ Z ) ) : ( C \ Z ) -1-1-onto-> ( A \ { (/) } ) ) |
103 |
|
fvres |
|- ( n e. ( C \ Z ) -> ( ( F |` ( C \ Z ) ) ` n ) = ( F ` n ) ) |
104 |
103
|
adantl |
|- ( ( ph /\ n e. ( C \ Z ) ) -> ( ( F |` ( C \ Z ) ) ` n ) = ( F ` n ) ) |
105 |
|
simpl |
|- ( ( ph /\ n e. ( C \ Z ) ) -> ph ) |
106 |
105 44 7
|
syl2anc |
|- ( ( ph /\ n e. ( C \ Z ) ) -> ( F ` n ) = G ) |
107 |
104 106
|
eqtrd |
|- ( ( ph /\ n e. ( C \ Z ) ) -> ( ( F |` ( C \ Z ) ) ` n ) = G ) |
108 |
1 2 3 30 102 107 16
|
sge0f1o |
|- ( ph -> ( sum^ ` ( k e. ( A \ { (/) } ) |-> B ) ) = ( sum^ ` ( n e. ( C \ Z ) |-> D ) ) ) |
109 |
7
|
eqcomd |
|- ( ( ph /\ n e. C ) -> G = ( F ` n ) ) |
110 |
109 34
|
eqeltrd |
|- ( ( ph /\ n e. C ) -> G e. A ) |
111 |
105 44 110
|
syl2anc |
|- ( ( ph /\ n e. ( C \ Z ) ) -> G e. A ) |
112 |
111
|
ex |
|- ( ph -> ( n e. ( C \ Z ) -> G e. A ) ) |
113 |
112
|
imdistani |
|- ( ( ph /\ n e. ( C \ Z ) ) -> ( ph /\ G e. A ) ) |
114 |
|
nfcv |
|- F/_ k G |
115 |
|
nfv |
|- F/ k G e. A |
116 |
1 115
|
nfan |
|- F/ k ( ph /\ G e. A ) |
117 |
|
nfv |
|- F/ k D e. ( 0 [,] +oo ) |
118 |
116 117
|
nfim |
|- F/ k ( ( ph /\ G e. A ) -> D e. ( 0 [,] +oo ) ) |
119 |
|
eleq1 |
|- ( k = G -> ( k e. A <-> G e. A ) ) |
120 |
119
|
anbi2d |
|- ( k = G -> ( ( ph /\ k e. A ) <-> ( ph /\ G e. A ) ) ) |
121 |
3
|
eleq1d |
|- ( k = G -> ( B e. ( 0 [,] +oo ) <-> D e. ( 0 [,] +oo ) ) ) |
122 |
120 121
|
imbi12d |
|- ( k = G -> ( ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) ) <-> ( ( ph /\ G e. A ) -> D e. ( 0 [,] +oo ) ) ) ) |
123 |
114 118 122 8
|
vtoclgf |
|- ( G e. A -> ( ( ph /\ G e. A ) -> D e. ( 0 [,] +oo ) ) ) |
124 |
111 113 123
|
sylc |
|- ( ( ph /\ n e. ( C \ Z ) ) -> D e. ( 0 [,] +oo ) ) |
125 |
|
simpl |
|- ( ( ph /\ n e. ( C \ ( C \ Z ) ) ) -> ph ) |
126 |
|
eldifi |
|- ( n e. ( C \ ( C \ Z ) ) -> n e. C ) |
127 |
126
|
adantl |
|- ( ( ph /\ n e. ( C \ ( C \ Z ) ) ) -> n e. C ) |
128 |
125 127 110
|
syl2anc |
|- ( ( ph /\ n e. ( C \ ( C \ Z ) ) ) -> G e. A ) |
129 |
|
dfin4 |
|- ( Z i^i C ) = ( Z \ ( Z \ C ) ) |
130 |
|
difss |
|- ( Z \ ( Z \ C ) ) C_ Z |
131 |
129 130
|
eqsstri |
|- ( Z i^i C ) C_ Z |
132 |
|
inss2 |
|- ( C i^i Z ) C_ Z |
133 |
|
id |
|- ( n e. ( C \ ( C \ Z ) ) -> n e. ( C \ ( C \ Z ) ) ) |
134 |
|
dfin4 |
|- ( C i^i Z ) = ( C \ ( C \ Z ) ) |
135 |
134
|
eqcomi |
|- ( C \ ( C \ Z ) ) = ( C i^i Z ) |
136 |
133 135
|
eleqtrdi |
|- ( n e. ( C \ ( C \ Z ) ) -> n e. ( C i^i Z ) ) |
137 |
132 136
|
sselid |
|- ( n e. ( C \ ( C \ Z ) ) -> n e. Z ) |
138 |
137 126
|
elind |
|- ( n e. ( C \ ( C \ Z ) ) -> n e. ( Z i^i C ) ) |
139 |
131 138
|
sselid |
|- ( n e. ( C \ ( C \ Z ) ) -> n e. Z ) |
140 |
139
|
adantl |
|- ( ( ph /\ n e. ( C \ ( C \ Z ) ) ) -> n e. Z ) |
141 |
77
|
eqcomd |
|- ( ( ph /\ n e. Z ) -> (/) = ( F ` n ) ) |
142 |
|
simpl |
|- ( ( ph /\ n e. Z ) -> ph ) |
143 |
74
|
simpld |
|- ( ( ph /\ n e. Z ) -> n e. C ) |
144 |
142 143 7
|
syl2anc |
|- ( ( ph /\ n e. Z ) -> ( F ` n ) = G ) |
145 |
141 144
|
eqtr2d |
|- ( ( ph /\ n e. Z ) -> G = (/) ) |
146 |
125 140 145
|
syl2anc |
|- ( ( ph /\ n e. ( C \ ( C \ Z ) ) ) -> G = (/) ) |
147 |
125 146
|
jca |
|- ( ( ph /\ n e. ( C \ ( C \ Z ) ) ) -> ( ph /\ G = (/) ) ) |
148 |
|
nfv |
|- F/ k G = (/) |
149 |
1 148
|
nfan |
|- F/ k ( ph /\ G = (/) ) |
150 |
|
nfv |
|- F/ k D = 0 |
151 |
149 150
|
nfim |
|- F/ k ( ( ph /\ G = (/) ) -> D = 0 ) |
152 |
|
eqeq1 |
|- ( k = G -> ( k = (/) <-> G = (/) ) ) |
153 |
152
|
anbi2d |
|- ( k = G -> ( ( ph /\ k = (/) ) <-> ( ph /\ G = (/) ) ) ) |
154 |
3
|
eqeq1d |
|- ( k = G -> ( B = 0 <-> D = 0 ) ) |
155 |
153 154
|
imbi12d |
|- ( k = G -> ( ( ( ph /\ k = (/) ) -> B = 0 ) <-> ( ( ph /\ G = (/) ) -> D = 0 ) ) ) |
156 |
114 151 155 9
|
vtoclgf |
|- ( G e. A -> ( ( ph /\ G = (/) ) -> D = 0 ) ) |
157 |
128 147 156
|
sylc |
|- ( ( ph /\ n e. ( C \ ( C \ Z ) ) ) -> D = 0 ) |
158 |
2 4 43 124 157
|
sge0ss |
|- ( ph -> ( sum^ ` ( n e. ( C \ Z ) |-> D ) ) = ( sum^ ` ( n e. C |-> D ) ) ) |
159 |
29 108 158
|
3eqtrd |
|- ( ph -> ( sum^ ` ( k e. A |-> B ) ) = ( sum^ ` ( n e. C |-> D ) ) ) |