Step |
Hyp |
Ref |
Expression |
1 |
|
sge0f1o.1 |
|- F/ k ph |
2 |
|
sge0f1o.2 |
|- F/ n ph |
3 |
|
sge0f1o.3 |
|- ( k = G -> B = D ) |
4 |
|
sge0f1o.4 |
|- ( ph -> C e. V ) |
5 |
|
sge0f1o.5 |
|- ( ph -> F : C -1-1-onto-> A ) |
6 |
|
sge0f1o.6 |
|- ( ( ph /\ n e. C ) -> ( F ` n ) = G ) |
7 |
|
sge0f1o.7 |
|- ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) ) |
8 |
|
f1ofo |
|- ( F : C -1-1-onto-> A -> F : C -onto-> A ) |
9 |
5 8
|
syl |
|- ( ph -> F : C -onto-> A ) |
10 |
|
fornex |
|- ( C e. V -> ( F : C -onto-> A -> A e. _V ) ) |
11 |
4 9 10
|
sylc |
|- ( ph -> A e. _V ) |
12 |
11
|
adantr |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> A e. _V ) |
13 |
|
eqid |
|- ( k e. A |-> B ) = ( k e. A |-> B ) |
14 |
1 7 13
|
fmptdf |
|- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) ) |
15 |
14
|
adantr |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) ) |
16 |
|
pnfex |
|- +oo e. _V |
17 |
|
eqid |
|- ( n e. C |-> D ) = ( n e. C |-> D ) |
18 |
17
|
elrnmpt |
|- ( +oo e. _V -> ( +oo e. ran ( n e. C |-> D ) <-> E. n e. C +oo = D ) ) |
19 |
16 18
|
ax-mp |
|- ( +oo e. ran ( n e. C |-> D ) <-> E. n e. C +oo = D ) |
20 |
19
|
biimpi |
|- ( +oo e. ran ( n e. C |-> D ) -> E. n e. C +oo = D ) |
21 |
20
|
adantl |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> E. n e. C +oo = D ) |
22 |
|
nfv |
|- F/ n +oo e. ran ( k e. A |-> B ) |
23 |
|
simp3 |
|- ( ( ph /\ n e. C /\ +oo = D ) -> +oo = D ) |
24 |
|
f1of |
|- ( F : C -1-1-onto-> A -> F : C --> A ) |
25 |
5 24
|
syl |
|- ( ph -> F : C --> A ) |
26 |
25
|
ffvelrnda |
|- ( ( ph /\ n e. C ) -> ( F ` n ) e. A ) |
27 |
|
nfcv |
|- F/_ k ( F ` n ) |
28 |
|
nfv |
|- F/ k ( F ` n ) = G |
29 |
27
|
nfcsb1 |
|- F/_ k [_ ( F ` n ) / k ]_ B |
30 |
|
nfcv |
|- F/_ k D |
31 |
29 30
|
nfeq |
|- F/ k [_ ( F ` n ) / k ]_ B = D |
32 |
28 31
|
nfim |
|- F/ k ( ( F ` n ) = G -> [_ ( F ` n ) / k ]_ B = D ) |
33 |
|
eqeq1 |
|- ( k = ( F ` n ) -> ( k = G <-> ( F ` n ) = G ) ) |
34 |
|
csbeq1a |
|- ( k = ( F ` n ) -> B = [_ ( F ` n ) / k ]_ B ) |
35 |
34
|
eqeq1d |
|- ( k = ( F ` n ) -> ( B = D <-> [_ ( F ` n ) / k ]_ B = D ) ) |
36 |
33 35
|
imbi12d |
|- ( k = ( F ` n ) -> ( ( k = G -> B = D ) <-> ( ( F ` n ) = G -> [_ ( F ` n ) / k ]_ B = D ) ) ) |
37 |
27 32 36 3
|
vtoclgf |
|- ( ( F ` n ) e. A -> ( ( F ` n ) = G -> [_ ( F ` n ) / k ]_ B = D ) ) |
38 |
26 6 37
|
sylc |
|- ( ( ph /\ n e. C ) -> [_ ( F ` n ) / k ]_ B = D ) |
39 |
38
|
eqcomd |
|- ( ( ph /\ n e. C ) -> D = [_ ( F ` n ) / k ]_ B ) |
40 |
39
|
3adant3 |
|- ( ( ph /\ n e. C /\ +oo = D ) -> D = [_ ( F ` n ) / k ]_ B ) |
41 |
23 40
|
eqtrd |
|- ( ( ph /\ n e. C /\ +oo = D ) -> +oo = [_ ( F ` n ) / k ]_ B ) |
42 |
|
simpl |
|- ( ( ph /\ n e. C ) -> ph ) |
43 |
42 26
|
jca |
|- ( ( ph /\ n e. C ) -> ( ph /\ ( F ` n ) e. A ) ) |
44 |
|
nfv |
|- F/ k ( F ` n ) e. A |
45 |
1 44
|
nfan |
|- F/ k ( ph /\ ( F ` n ) e. A ) |
46 |
29
|
nfel1 |
|- F/ k [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) |
47 |
45 46
|
nfim |
|- F/ k ( ( ph /\ ( F ` n ) e. A ) -> [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) |
48 |
|
eleq1 |
|- ( k = ( F ` n ) -> ( k e. A <-> ( F ` n ) e. A ) ) |
49 |
48
|
anbi2d |
|- ( k = ( F ` n ) -> ( ( ph /\ k e. A ) <-> ( ph /\ ( F ` n ) e. A ) ) ) |
50 |
34
|
eleq1d |
|- ( k = ( F ` n ) -> ( B e. ( 0 [,] +oo ) <-> [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) ) |
51 |
49 50
|
imbi12d |
|- ( k = ( F ` n ) -> ( ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) ) <-> ( ( ph /\ ( F ` n ) e. A ) -> [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) ) ) |
52 |
27 47 51 7
|
vtoclgf |
|- ( ( F ` n ) e. A -> ( ( ph /\ ( F ` n ) e. A ) -> [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) ) |
53 |
26 43 52
|
sylc |
|- ( ( ph /\ n e. C ) -> [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) |
54 |
29 13 34
|
elrnmpt1sf |
|- ( ( ( F ` n ) e. A /\ [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) -> [_ ( F ` n ) / k ]_ B e. ran ( k e. A |-> B ) ) |
55 |
26 53 54
|
syl2anc |
|- ( ( ph /\ n e. C ) -> [_ ( F ` n ) / k ]_ B e. ran ( k e. A |-> B ) ) |
56 |
55
|
3adant3 |
|- ( ( ph /\ n e. C /\ +oo = D ) -> [_ ( F ` n ) / k ]_ B e. ran ( k e. A |-> B ) ) |
57 |
41 56
|
eqeltrd |
|- ( ( ph /\ n e. C /\ +oo = D ) -> +oo e. ran ( k e. A |-> B ) ) |
58 |
57
|
3exp |
|- ( ph -> ( n e. C -> ( +oo = D -> +oo e. ran ( k e. A |-> B ) ) ) ) |
59 |
2 22 58
|
rexlimd |
|- ( ph -> ( E. n e. C +oo = D -> +oo e. ran ( k e. A |-> B ) ) ) |
60 |
59
|
adantr |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( E. n e. C +oo = D -> +oo e. ran ( k e. A |-> B ) ) ) |
61 |
21 60
|
mpd |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> +oo e. ran ( k e. A |-> B ) ) |
62 |
12 15 61
|
sge0pnfval |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( sum^ ` ( k e. A |-> B ) ) = +oo ) |
63 |
4
|
adantr |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> C e. V ) |
64 |
39 53
|
eqeltrd |
|- ( ( ph /\ n e. C ) -> D e. ( 0 [,] +oo ) ) |
65 |
2 64 17
|
fmptdf |
|- ( ph -> ( n e. C |-> D ) : C --> ( 0 [,] +oo ) ) |
66 |
65
|
adantr |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( n e. C |-> D ) : C --> ( 0 [,] +oo ) ) |
67 |
|
simpr |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> +oo e. ran ( n e. C |-> D ) ) |
68 |
63 66 67
|
sge0pnfval |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( sum^ ` ( n e. C |-> D ) ) = +oo ) |
69 |
62 68
|
eqtr4d |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( sum^ ` ( k e. A |-> B ) ) = ( sum^ ` ( n e. C |-> D ) ) ) |
70 |
|
sumex |
|- sum_ k e. y B e. _V |
71 |
70
|
a1i |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) -> sum_ k e. y B e. _V ) |
72 |
|
cnvimass |
|- ( `' F " y ) C_ dom F |
73 |
72 25
|
fssdm |
|- ( ph -> ( `' F " y ) C_ C ) |
74 |
|
fex |
|- ( ( F : C --> A /\ C e. V ) -> F e. _V ) |
75 |
25 4 74
|
syl2anc |
|- ( ph -> F e. _V ) |
76 |
|
cnvexg |
|- ( F e. _V -> `' F e. _V ) |
77 |
75 76
|
syl |
|- ( ph -> `' F e. _V ) |
78 |
|
imaexg |
|- ( `' F e. _V -> ( `' F " y ) e. _V ) |
79 |
77 78
|
syl |
|- ( ph -> ( `' F " y ) e. _V ) |
80 |
|
elpwg |
|- ( ( `' F " y ) e. _V -> ( ( `' F " y ) e. ~P C <-> ( `' F " y ) C_ C ) ) |
81 |
79 80
|
syl |
|- ( ph -> ( ( `' F " y ) e. ~P C <-> ( `' F " y ) C_ C ) ) |
82 |
73 81
|
mpbird |
|- ( ph -> ( `' F " y ) e. ~P C ) |
83 |
82
|
adantr |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) e. ~P C ) |
84 |
|
f1ocnv |
|- ( F : C -1-1-onto-> A -> `' F : A -1-1-onto-> C ) |
85 |
5 84
|
syl |
|- ( ph -> `' F : A -1-1-onto-> C ) |
86 |
|
f1ofun |
|- ( `' F : A -1-1-onto-> C -> Fun `' F ) |
87 |
85 86
|
syl |
|- ( ph -> Fun `' F ) |
88 |
87
|
adantr |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> Fun `' F ) |
89 |
|
elinel2 |
|- ( y e. ( ~P A i^i Fin ) -> y e. Fin ) |
90 |
89
|
adantl |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> y e. Fin ) |
91 |
|
imafi |
|- ( ( Fun `' F /\ y e. Fin ) -> ( `' F " y ) e. Fin ) |
92 |
88 90 91
|
syl2anc |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) e. Fin ) |
93 |
83 92
|
elind |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) e. ( ~P C i^i Fin ) ) |
94 |
93
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) e. ( ~P C i^i Fin ) ) |
95 |
|
nfv |
|- F/ k -. +oo e. ran ( n e. C |-> D ) |
96 |
1 95
|
nfan |
|- F/ k ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) |
97 |
|
nfv |
|- F/ k y e. ( ~P A i^i Fin ) |
98 |
96 97
|
nfan |
|- F/ k ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) |
99 |
|
nfcv |
|- F/_ n +oo |
100 |
|
nfmpt1 |
|- F/_ n ( n e. C |-> D ) |
101 |
100
|
nfrn |
|- F/_ n ran ( n e. C |-> D ) |
102 |
99 101
|
nfel |
|- F/ n +oo e. ran ( n e. C |-> D ) |
103 |
102
|
nfn |
|- F/ n -. +oo e. ran ( n e. C |-> D ) |
104 |
2 103
|
nfan |
|- F/ n ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) |
105 |
|
nfv |
|- F/ n y e. ( ~P A i^i Fin ) |
106 |
104 105
|
nfan |
|- F/ n ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) |
107 |
92
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) e. Fin ) |
108 |
|
f1of1 |
|- ( F : C -1-1-onto-> A -> F : C -1-1-> A ) |
109 |
5 108
|
syl |
|- ( ph -> F : C -1-1-> A ) |
110 |
109
|
adantr |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> F : C -1-1-> A ) |
111 |
81
|
adantr |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( ( `' F " y ) e. ~P C <-> ( `' F " y ) C_ C ) ) |
112 |
83 111
|
mpbid |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) C_ C ) |
113 |
|
f1ores |
|- ( ( F : C -1-1-> A /\ ( `' F " y ) C_ C ) -> ( F |` ( `' F " y ) ) : ( `' F " y ) -1-1-onto-> ( F " ( `' F " y ) ) ) |
114 |
110 112 113
|
syl2anc |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( F |` ( `' F " y ) ) : ( `' F " y ) -1-1-onto-> ( F " ( `' F " y ) ) ) |
115 |
9
|
adantr |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> F : C -onto-> A ) |
116 |
|
elpwinss |
|- ( y e. ( ~P A i^i Fin ) -> y C_ A ) |
117 |
116
|
adantl |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> y C_ A ) |
118 |
|
foimacnv |
|- ( ( F : C -onto-> A /\ y C_ A ) -> ( F " ( `' F " y ) ) = y ) |
119 |
115 117 118
|
syl2anc |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( F " ( `' F " y ) ) = y ) |
120 |
119
|
f1oeq3d |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( ( F |` ( `' F " y ) ) : ( `' F " y ) -1-1-onto-> ( F " ( `' F " y ) ) <-> ( F |` ( `' F " y ) ) : ( `' F " y ) -1-1-onto-> y ) ) |
121 |
114 120
|
mpbid |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( F |` ( `' F " y ) ) : ( `' F " y ) -1-1-onto-> y ) |
122 |
121
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) -> ( F |` ( `' F " y ) ) : ( `' F " y ) -1-1-onto-> y ) |
123 |
79
|
ad2antrr |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( `' F " y ) e. _V ) |
124 |
|
simpll |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ph ) |
125 |
93
|
adantr |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( `' F " y ) e. ( ~P C i^i Fin ) ) |
126 |
|
simpr |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> n e. ( `' F " y ) ) |
127 |
124 125 126
|
jca31 |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( ( ph /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) /\ n e. ( `' F " y ) ) ) |
128 |
|
eleq1 |
|- ( x = ( `' F " y ) -> ( x e. ( ~P C i^i Fin ) <-> ( `' F " y ) e. ( ~P C i^i Fin ) ) ) |
129 |
128
|
anbi2d |
|- ( x = ( `' F " y ) -> ( ( ph /\ x e. ( ~P C i^i Fin ) ) <-> ( ph /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) ) ) |
130 |
|
eleq2 |
|- ( x = ( `' F " y ) -> ( n e. x <-> n e. ( `' F " y ) ) ) |
131 |
129 130
|
anbi12d |
|- ( x = ( `' F " y ) -> ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) <-> ( ( ph /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) /\ n e. ( `' F " y ) ) ) ) |
132 |
|
reseq2 |
|- ( x = ( `' F " y ) -> ( F |` x ) = ( F |` ( `' F " y ) ) ) |
133 |
132
|
fveq1d |
|- ( x = ( `' F " y ) -> ( ( F |` x ) ` n ) = ( ( F |` ( `' F " y ) ) ` n ) ) |
134 |
133
|
eqeq1d |
|- ( x = ( `' F " y ) -> ( ( ( F |` x ) ` n ) = G <-> ( ( F |` ( `' F " y ) ) ` n ) = G ) ) |
135 |
131 134
|
imbi12d |
|- ( x = ( `' F " y ) -> ( ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ( ( F |` x ) ` n ) = G ) <-> ( ( ( ph /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( ( F |` ( `' F " y ) ) ` n ) = G ) ) ) |
136 |
|
fvres |
|- ( n e. x -> ( ( F |` x ) ` n ) = ( F ` n ) ) |
137 |
136
|
adantl |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ( ( F |` x ) ` n ) = ( F ` n ) ) |
138 |
|
simpll |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ph ) |
139 |
|
elpwinss |
|- ( x e. ( ~P C i^i Fin ) -> x C_ C ) |
140 |
139
|
adantl |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> x C_ C ) |
141 |
140
|
sselda |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> n e. C ) |
142 |
138 141 6
|
syl2anc |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ( F ` n ) = G ) |
143 |
137 142
|
eqtrd |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ( ( F |` x ) ` n ) = G ) |
144 |
135 143
|
vtoclg |
|- ( ( `' F " y ) e. _V -> ( ( ( ph /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( ( F |` ( `' F " y ) ) ` n ) = G ) ) |
145 |
123 127 144
|
sylc |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( ( F |` ( `' F " y ) ) ` n ) = G ) |
146 |
145
|
adantllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( ( F |` ( `' F " y ) ) ` n ) = G ) |
147 |
79
|
ad3antrrr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> ( `' F " y ) e. _V ) |
148 |
|
simpll |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) ) |
149 |
82
|
ad3antrrr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> ( `' F " y ) e. ~P C ) |
150 |
107
|
adantr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> ( `' F " y ) e. Fin ) |
151 |
149 150
|
elind |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> ( `' F " y ) e. ( ~P C i^i Fin ) ) |
152 |
|
simpr |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> k e. y ) |
153 |
119
|
eqcomd |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> y = ( F " ( `' F " y ) ) ) |
154 |
153
|
adantr |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> y = ( F " ( `' F " y ) ) ) |
155 |
152 154
|
eleqtrd |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> k e. ( F " ( `' F " y ) ) ) |
156 |
155
|
adantllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> k e. ( F " ( `' F " y ) ) ) |
157 |
148 151 156
|
jca31 |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) /\ k e. ( F " ( `' F " y ) ) ) ) |
158 |
128
|
anbi2d |
|- ( x = ( `' F " y ) -> ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) <-> ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) ) ) |
159 |
|
imaeq2 |
|- ( x = ( `' F " y ) -> ( F " x ) = ( F " ( `' F " y ) ) ) |
160 |
159
|
eleq2d |
|- ( x = ( `' F " y ) -> ( k e. ( F " x ) <-> k e. ( F " ( `' F " y ) ) ) ) |
161 |
158 160
|
anbi12d |
|- ( x = ( `' F " y ) -> ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) <-> ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) /\ k e. ( F " ( `' F " y ) ) ) ) ) |
162 |
161
|
imbi1d |
|- ( x = ( `' F " y ) -> ( ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> B e. CC ) <-> ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) /\ k e. ( F " ( `' F " y ) ) ) -> B e. CC ) ) ) |
163 |
|
rge0ssre |
|- ( 0 [,) +oo ) C_ RR |
164 |
|
ax-resscn |
|- RR C_ CC |
165 |
163 164
|
sstri |
|- ( 0 [,) +oo ) C_ CC |
166 |
|
simplll |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> ph ) |
167 |
|
simpllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> -. +oo e. ran ( n e. C |-> D ) ) |
168 |
|
fimass |
|- ( F : C --> A -> ( F " x ) C_ A ) |
169 |
25 168
|
syl |
|- ( ph -> ( F " x ) C_ A ) |
170 |
169
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> ( F " x ) C_ A ) |
171 |
|
simpr |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> k e. ( F " x ) ) |
172 |
170 171
|
sseldd |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> k e. A ) |
173 |
172
|
adantllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> k e. A ) |
174 |
|
foelrni |
|- ( ( F : C -onto-> A /\ k e. A ) -> E. n e. C ( F ` n ) = k ) |
175 |
9 174
|
sylan |
|- ( ( ph /\ k e. A ) -> E. n e. C ( F ` n ) = k ) |
176 |
175
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ k e. A ) -> E. n e. C ( F ` n ) = k ) |
177 |
|
nfv |
|- F/ n k e. A |
178 |
104 177
|
nfan |
|- F/ n ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ k e. A ) |
179 |
|
nfv |
|- F/ n B e. ( 0 [,) +oo ) |
180 |
|
csbid |
|- [_ k / k ]_ B = B |
181 |
180
|
eqcomi |
|- B = [_ k / k ]_ B |
182 |
181
|
a1i |
|- ( ( ph /\ n e. C /\ ( F ` n ) = k ) -> B = [_ k / k ]_ B ) |
183 |
|
id |
|- ( ( F ` n ) = k -> ( F ` n ) = k ) |
184 |
183
|
eqcomd |
|- ( ( F ` n ) = k -> k = ( F ` n ) ) |
185 |
184
|
csbeq1d |
|- ( ( F ` n ) = k -> [_ k / k ]_ B = [_ ( F ` n ) / k ]_ B ) |
186 |
185
|
3ad2ant3 |
|- ( ( ph /\ n e. C /\ ( F ` n ) = k ) -> [_ k / k ]_ B = [_ ( F ` n ) / k ]_ B ) |
187 |
38
|
idi |
|- ( ( ph /\ n e. C ) -> [_ ( F ` n ) / k ]_ B = D ) |
188 |
187
|
3adant3 |
|- ( ( ph /\ n e. C /\ ( F ` n ) = k ) -> [_ ( F ` n ) / k ]_ B = D ) |
189 |
182 186 188
|
3eqtrd |
|- ( ( ph /\ n e. C /\ ( F ` n ) = k ) -> B = D ) |
190 |
189
|
3adant1r |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C /\ ( F ` n ) = k ) -> B = D ) |
191 |
|
0xr |
|- 0 e. RR* |
192 |
191
|
a1i |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> 0 e. RR* ) |
193 |
|
pnfxr |
|- +oo e. RR* |
194 |
193
|
a1i |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> +oo e. RR* ) |
195 |
64
|
adantr |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> D e. ( 0 [,] +oo ) ) |
196 |
|
simpr |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> -. D e. ( 0 [,) +oo ) ) |
197 |
192 194 195 196
|
eliccnelico |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> D = +oo ) |
198 |
197
|
eqcomd |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> +oo = D ) |
199 |
|
simpr |
|- ( ( ph /\ n e. C ) -> n e. C ) |
200 |
64
|
idi |
|- ( ( ph /\ n e. C ) -> D e. ( 0 [,] +oo ) ) |
201 |
17
|
elrnmpt1 |
|- ( ( n e. C /\ D e. ( 0 [,] +oo ) ) -> D e. ran ( n e. C |-> D ) ) |
202 |
199 200 201
|
syl2anc |
|- ( ( ph /\ n e. C ) -> D e. ran ( n e. C |-> D ) ) |
203 |
202
|
adantr |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> D e. ran ( n e. C |-> D ) ) |
204 |
198 203
|
eqeltrd |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> +oo e. ran ( n e. C |-> D ) ) |
205 |
204
|
adantllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> +oo e. ran ( n e. C |-> D ) ) |
206 |
|
simpllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> -. +oo e. ran ( n e. C |-> D ) ) |
207 |
205 206
|
condan |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C ) -> D e. ( 0 [,) +oo ) ) |
208 |
207
|
3adant3 |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C /\ ( F ` n ) = k ) -> D e. ( 0 [,) +oo ) ) |
209 |
190 208
|
eqeltrd |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C /\ ( F ` n ) = k ) -> B e. ( 0 [,) +oo ) ) |
210 |
209
|
3exp |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> ( n e. C -> ( ( F ` n ) = k -> B e. ( 0 [,) +oo ) ) ) ) |
211 |
210
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ k e. A ) -> ( n e. C -> ( ( F ` n ) = k -> B e. ( 0 [,) +oo ) ) ) ) |
212 |
178 179 211
|
rexlimd |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ k e. A ) -> ( E. n e. C ( F ` n ) = k -> B e. ( 0 [,) +oo ) ) ) |
213 |
176 212
|
mpd |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ k e. A ) -> B e. ( 0 [,) +oo ) ) |
214 |
166 167 173 213
|
syl21anc |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> B e. ( 0 [,) +oo ) ) |
215 |
165 214
|
sseldi |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> B e. CC ) |
216 |
215
|
idi |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> B e. CC ) |
217 |
162 216
|
vtoclg |
|- ( ( `' F " y ) e. _V -> ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) /\ k e. ( F " ( `' F " y ) ) ) -> B e. CC ) ) |
218 |
147 157 217
|
sylc |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> B e. CC ) |
219 |
98 106 3 107 122 146 218
|
fsumf1of |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) -> sum_ k e. y B = sum_ n e. ( `' F " y ) D ) |
220 |
|
sumeq1 |
|- ( x = ( `' F " y ) -> sum_ n e. x D = sum_ n e. ( `' F " y ) D ) |
221 |
220
|
rspceeqv |
|- ( ( ( `' F " y ) e. ( ~P C i^i Fin ) /\ sum_ k e. y B = sum_ n e. ( `' F " y ) D ) -> E. x e. ( ~P C i^i Fin ) sum_ k e. y B = sum_ n e. x D ) |
222 |
94 219 221
|
syl2anc |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) -> E. x e. ( ~P C i^i Fin ) sum_ k e. y B = sum_ n e. x D ) |
223 |
71 222
|
rnmptssrn |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> ran ( y e. ( ~P A i^i Fin ) |-> sum_ k e. y B ) C_ ran ( x e. ( ~P C i^i Fin ) |-> sum_ n e. x D ) ) |
224 |
|
sumex |
|- sum_ n e. x D e. _V |
225 |
224
|
a1i |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> sum_ n e. x D e. _V ) |
226 |
11 169
|
ssexd |
|- ( ph -> ( F " x ) e. _V ) |
227 |
|
elpwg |
|- ( ( F " x ) e. _V -> ( ( F " x ) e. ~P A <-> ( F " x ) C_ A ) ) |
228 |
226 227
|
syl |
|- ( ph -> ( ( F " x ) e. ~P A <-> ( F " x ) C_ A ) ) |
229 |
169 228
|
mpbird |
|- ( ph -> ( F " x ) e. ~P A ) |
230 |
229
|
adantr |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> ( F " x ) e. ~P A ) |
231 |
25
|
ffund |
|- ( ph -> Fun F ) |
232 |
231
|
adantr |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> Fun F ) |
233 |
|
elinel2 |
|- ( x e. ( ~P C i^i Fin ) -> x e. Fin ) |
234 |
233
|
adantl |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> x e. Fin ) |
235 |
|
imafi |
|- ( ( Fun F /\ x e. Fin ) -> ( F " x ) e. Fin ) |
236 |
232 234 235
|
syl2anc |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> ( F " x ) e. Fin ) |
237 |
230 236
|
elind |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> ( F " x ) e. ( ~P A i^i Fin ) ) |
238 |
237
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> ( F " x ) e. ( ~P A i^i Fin ) ) |
239 |
|
nfv |
|- F/ k x e. ( ~P C i^i Fin ) |
240 |
96 239
|
nfan |
|- F/ k ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) |
241 |
|
nfv |
|- F/ n x e. ( ~P C i^i Fin ) |
242 |
104 241
|
nfan |
|- F/ n ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) |
243 |
233
|
adantl |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> x e. Fin ) |
244 |
109
|
adantr |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> F : C -1-1-> A ) |
245 |
|
f1ores |
|- ( ( F : C -1-1-> A /\ x C_ C ) -> ( F |` x ) : x -1-1-onto-> ( F " x ) ) |
246 |
244 140 245
|
syl2anc |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> ( F |` x ) : x -1-1-onto-> ( F " x ) ) |
247 |
246
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> ( F |` x ) : x -1-1-onto-> ( F " x ) ) |
248 |
143
|
adantllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ( ( F |` x ) ` n ) = G ) |
249 |
240 242 3 243 247 248 215
|
fsumf1of |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> sum_ k e. ( F " x ) B = sum_ n e. x D ) |
250 |
249
|
eqcomd |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> sum_ n e. x D = sum_ k e. ( F " x ) B ) |
251 |
|
sumeq1 |
|- ( y = ( F " x ) -> sum_ k e. y B = sum_ k e. ( F " x ) B ) |
252 |
251
|
rspceeqv |
|- ( ( ( F " x ) e. ( ~P A i^i Fin ) /\ sum_ n e. x D = sum_ k e. ( F " x ) B ) -> E. y e. ( ~P A i^i Fin ) sum_ n e. x D = sum_ k e. y B ) |
253 |
238 250 252
|
syl2anc |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> E. y e. ( ~P A i^i Fin ) sum_ n e. x D = sum_ k e. y B ) |
254 |
225 253
|
rnmptssrn |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> ran ( x e. ( ~P C i^i Fin ) |-> sum_ n e. x D ) C_ ran ( y e. ( ~P A i^i Fin ) |-> sum_ k e. y B ) ) |
255 |
223 254
|
eqssd |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> ran ( y e. ( ~P A i^i Fin ) |-> sum_ k e. y B ) = ran ( x e. ( ~P C i^i Fin ) |-> sum_ n e. x D ) ) |
256 |
255
|
supeq1d |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> sup ( ran ( y e. ( ~P A i^i Fin ) |-> sum_ k e. y B ) , RR* , < ) = sup ( ran ( x e. ( ~P C i^i Fin ) |-> sum_ n e. x D ) , RR* , < ) ) |
257 |
11
|
adantr |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> A e. _V ) |
258 |
96 257 213
|
sge0revalmpt |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> ( sum^ ` ( k e. A |-> B ) ) = sup ( ran ( y e. ( ~P A i^i Fin ) |-> sum_ k e. y B ) , RR* , < ) ) |
259 |
4
|
adantr |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> C e. V ) |
260 |
104 259 207
|
sge0revalmpt |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> ( sum^ ` ( n e. C |-> D ) ) = sup ( ran ( x e. ( ~P C i^i Fin ) |-> sum_ n e. x D ) , RR* , < ) ) |
261 |
256 258 260
|
3eqtr4d |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> ( sum^ ` ( k e. A |-> B ) ) = ( sum^ ` ( n e. C |-> D ) ) ) |
262 |
69 261
|
pm2.61dan |
|- ( ph -> ( sum^ ` ( k e. A |-> B ) ) = ( sum^ ` ( n e. C |-> D ) ) ) |