| 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 |
|
focdmex |
|- ( 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 |
1 7
|
fmptd2f |
|- ( ph -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) ) |
| 14 |
13
|
adantr |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( k e. A |-> B ) : A --> ( 0 [,] +oo ) ) |
| 15 |
|
nfv |
|- F/ n +oo e. ran ( k e. A |-> B ) |
| 16 |
|
simp3 |
|- ( ( ph /\ n e. C /\ +oo = D ) -> +oo = D ) |
| 17 |
|
f1of |
|- ( F : C -1-1-onto-> A -> F : C --> A ) |
| 18 |
5 17
|
syl |
|- ( ph -> F : C --> A ) |
| 19 |
18
|
ffvelcdmda |
|- ( ( ph /\ n e. C ) -> ( F ` n ) e. A ) |
| 20 |
|
nfv |
|- F/ k ( F ` n ) = G |
| 21 |
|
nfcsb1v |
|- F/_ k [_ ( F ` n ) / k ]_ B |
| 22 |
21
|
nfeq1 |
|- F/ k [_ ( F ` n ) / k ]_ B = D |
| 23 |
20 22
|
nfim |
|- F/ k ( ( F ` n ) = G -> [_ ( F ` n ) / k ]_ B = D ) |
| 24 |
|
eqeq1 |
|- ( k = ( F ` n ) -> ( k = G <-> ( F ` n ) = G ) ) |
| 25 |
|
csbeq1a |
|- ( k = ( F ` n ) -> B = [_ ( F ` n ) / k ]_ B ) |
| 26 |
25
|
eqeq1d |
|- ( k = ( F ` n ) -> ( B = D <-> [_ ( F ` n ) / k ]_ B = D ) ) |
| 27 |
24 26
|
imbi12d |
|- ( k = ( F ` n ) -> ( ( k = G -> B = D ) <-> ( ( F ` n ) = G -> [_ ( F ` n ) / k ]_ B = D ) ) ) |
| 28 |
23 27 3
|
vtoclg1f |
|- ( ( F ` n ) e. A -> ( ( F ` n ) = G -> [_ ( F ` n ) / k ]_ B = D ) ) |
| 29 |
19 6 28
|
sylc |
|- ( ( ph /\ n e. C ) -> [_ ( F ` n ) / k ]_ B = D ) |
| 30 |
29
|
eqcomd |
|- ( ( ph /\ n e. C ) -> D = [_ ( F ` n ) / k ]_ B ) |
| 31 |
30
|
3adant3 |
|- ( ( ph /\ n e. C /\ +oo = D ) -> D = [_ ( F ` n ) / k ]_ B ) |
| 32 |
16 31
|
eqtrd |
|- ( ( ph /\ n e. C /\ +oo = D ) -> +oo = [_ ( F ` n ) / k ]_ B ) |
| 33 |
|
simpl |
|- ( ( ph /\ n e. C ) -> ph ) |
| 34 |
33 19
|
jca |
|- ( ( ph /\ n e. C ) -> ( ph /\ ( F ` n ) e. A ) ) |
| 35 |
|
nfv |
|- F/ k ( F ` n ) e. A |
| 36 |
1 35
|
nfan |
|- F/ k ( ph /\ ( F ` n ) e. A ) |
| 37 |
21
|
nfel1 |
|- F/ k [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) |
| 38 |
36 37
|
nfim |
|- F/ k ( ( ph /\ ( F ` n ) e. A ) -> [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) |
| 39 |
|
eleq1 |
|- ( k = ( F ` n ) -> ( k e. A <-> ( F ` n ) e. A ) ) |
| 40 |
39
|
anbi2d |
|- ( k = ( F ` n ) -> ( ( ph /\ k e. A ) <-> ( ph /\ ( F ` n ) e. A ) ) ) |
| 41 |
25
|
eleq1d |
|- ( k = ( F ` n ) -> ( B e. ( 0 [,] +oo ) <-> [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) ) |
| 42 |
40 41
|
imbi12d |
|- ( k = ( F ` n ) -> ( ( ( ph /\ k e. A ) -> B e. ( 0 [,] +oo ) ) <-> ( ( ph /\ ( F ` n ) e. A ) -> [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) ) ) |
| 43 |
38 42 7
|
vtoclg1f |
|- ( ( F ` n ) e. A -> ( ( ph /\ ( F ` n ) e. A ) -> [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) ) |
| 44 |
19 34 43
|
sylc |
|- ( ( ph /\ n e. C ) -> [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) |
| 45 |
|
eqid |
|- ( k e. A |-> B ) = ( k e. A |-> B ) |
| 46 |
21 45 25
|
elrnmpt1sf |
|- ( ( ( F ` n ) e. A /\ [_ ( F ` n ) / k ]_ B e. ( 0 [,] +oo ) ) -> [_ ( F ` n ) / k ]_ B e. ran ( k e. A |-> B ) ) |
| 47 |
19 44 46
|
syl2anc |
|- ( ( ph /\ n e. C ) -> [_ ( F ` n ) / k ]_ B e. ran ( k e. A |-> B ) ) |
| 48 |
47
|
3adant3 |
|- ( ( ph /\ n e. C /\ +oo = D ) -> [_ ( F ` n ) / k ]_ B e. ran ( k e. A |-> B ) ) |
| 49 |
32 48
|
eqeltrd |
|- ( ( ph /\ n e. C /\ +oo = D ) -> +oo e. ran ( k e. A |-> B ) ) |
| 50 |
49
|
3exp |
|- ( ph -> ( n e. C -> ( +oo = D -> +oo e. ran ( k e. A |-> B ) ) ) ) |
| 51 |
2 15 50
|
rexlimd |
|- ( ph -> ( E. n e. C +oo = D -> +oo e. ran ( k e. A |-> B ) ) ) |
| 52 |
|
pnfex |
|- +oo e. _V |
| 53 |
|
eqid |
|- ( n e. C |-> D ) = ( n e. C |-> D ) |
| 54 |
53
|
elrnmpt |
|- ( +oo e. _V -> ( +oo e. ran ( n e. C |-> D ) <-> E. n e. C +oo = D ) ) |
| 55 |
52 54
|
ax-mp |
|- ( +oo e. ran ( n e. C |-> D ) <-> E. n e. C +oo = D ) |
| 56 |
55
|
biimpi |
|- ( +oo e. ran ( n e. C |-> D ) -> E. n e. C +oo = D ) |
| 57 |
51 56
|
impel |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> +oo e. ran ( k e. A |-> B ) ) |
| 58 |
12 14 57
|
sge0pnfval |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( sum^ ` ( k e. A |-> B ) ) = +oo ) |
| 59 |
4
|
adantr |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> C e. V ) |
| 60 |
30 44
|
eqeltrd |
|- ( ( ph /\ n e. C ) -> D e. ( 0 [,] +oo ) ) |
| 61 |
2 60
|
fmptd2f |
|- ( ph -> ( n e. C |-> D ) : C --> ( 0 [,] +oo ) ) |
| 62 |
61
|
adantr |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( n e. C |-> D ) : C --> ( 0 [,] +oo ) ) |
| 63 |
|
simpr |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> +oo e. ran ( n e. C |-> D ) ) |
| 64 |
59 62 63
|
sge0pnfval |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( sum^ ` ( n e. C |-> D ) ) = +oo ) |
| 65 |
58 64
|
eqtr4d |
|- ( ( ph /\ +oo e. ran ( n e. C |-> D ) ) -> ( sum^ ` ( k e. A |-> B ) ) = ( sum^ ` ( n e. C |-> D ) ) ) |
| 66 |
|
sumex |
|- sum_ k e. y B e. _V |
| 67 |
66
|
a1i |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) -> sum_ k e. y B e. _V ) |
| 68 |
|
cnvimass |
|- ( `' F " y ) C_ dom F |
| 69 |
68 18
|
fssdm |
|- ( ph -> ( `' F " y ) C_ C ) |
| 70 |
4 69
|
sselpwd |
|- ( ph -> ( `' F " y ) e. ~P C ) |
| 71 |
70
|
adantr |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) e. ~P C ) |
| 72 |
|
f1ocnv |
|- ( F : C -1-1-onto-> A -> `' F : A -1-1-onto-> C ) |
| 73 |
5 72
|
syl |
|- ( ph -> `' F : A -1-1-onto-> C ) |
| 74 |
|
f1ofun |
|- ( `' F : A -1-1-onto-> C -> Fun `' F ) |
| 75 |
73 74
|
syl |
|- ( ph -> Fun `' F ) |
| 76 |
|
elinel2 |
|- ( y e. ( ~P A i^i Fin ) -> y e. Fin ) |
| 77 |
|
imafi |
|- ( ( Fun `' F /\ y e. Fin ) -> ( `' F " y ) e. Fin ) |
| 78 |
75 76 77
|
syl2an |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) e. Fin ) |
| 79 |
71 78
|
elind |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) e. ( ~P C i^i Fin ) ) |
| 80 |
79
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) e. ( ~P C i^i Fin ) ) |
| 81 |
|
nfv |
|- F/ k -. +oo e. ran ( n e. C |-> D ) |
| 82 |
1 81
|
nfan |
|- F/ k ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) |
| 83 |
|
nfv |
|- F/ k y e. ( ~P A i^i Fin ) |
| 84 |
82 83
|
nfan |
|- F/ k ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) |
| 85 |
|
nfmpt1 |
|- F/_ n ( n e. C |-> D ) |
| 86 |
85
|
nfrn |
|- F/_ n ran ( n e. C |-> D ) |
| 87 |
86
|
nfel2 |
|- F/ n +oo e. ran ( n e. C |-> D ) |
| 88 |
87
|
nfn |
|- F/ n -. +oo e. ran ( n e. C |-> D ) |
| 89 |
2 88
|
nfan |
|- F/ n ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) |
| 90 |
|
nfv |
|- F/ n y e. ( ~P A i^i Fin ) |
| 91 |
89 90
|
nfan |
|- F/ n ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) |
| 92 |
78
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) e. Fin ) |
| 93 |
|
f1of1 |
|- ( F : C -1-1-onto-> A -> F : C -1-1-> A ) |
| 94 |
5 93
|
syl |
|- ( ph -> F : C -1-1-> A ) |
| 95 |
94
|
adantr |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> F : C -1-1-> A ) |
| 96 |
69
|
adantr |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( `' F " y ) C_ C ) |
| 97 |
|
f1ores |
|- ( ( F : C -1-1-> A /\ ( `' F " y ) C_ C ) -> ( F |` ( `' F " y ) ) : ( `' F " y ) -1-1-onto-> ( F " ( `' F " y ) ) ) |
| 98 |
95 96 97
|
syl2anc |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( F |` ( `' F " y ) ) : ( `' F " y ) -1-1-onto-> ( F " ( `' F " y ) ) ) |
| 99 |
|
elpwinss |
|- ( y e. ( ~P A i^i Fin ) -> y C_ A ) |
| 100 |
|
foimacnv |
|- ( ( F : C -onto-> A /\ y C_ A ) -> ( F " ( `' F " y ) ) = y ) |
| 101 |
9 99 100
|
syl2an |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( F " ( `' F " y ) ) = y ) |
| 102 |
101
|
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 ) ) |
| 103 |
98 102
|
mpbid |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> ( F |` ( `' F " y ) ) : ( `' F " y ) -1-1-onto-> y ) |
| 104 |
103
|
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 ) |
| 105 |
18 4
|
fexd |
|- ( ph -> F e. _V ) |
| 106 |
|
cnvexg |
|- ( F e. _V -> `' F e. _V ) |
| 107 |
105 106
|
syl |
|- ( ph -> `' F e. _V ) |
| 108 |
107
|
imaexd |
|- ( ph -> ( `' F " y ) e. _V ) |
| 109 |
108
|
ad2antrr |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( `' F " y ) e. _V ) |
| 110 |
|
simpll |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ph ) |
| 111 |
79
|
adantr |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( `' F " y ) e. ( ~P C i^i Fin ) ) |
| 112 |
|
simpr |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> n e. ( `' F " y ) ) |
| 113 |
110 111 112
|
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 ) ) ) |
| 114 |
|
eleq1 |
|- ( x = ( `' F " y ) -> ( x e. ( ~P C i^i Fin ) <-> ( `' F " y ) e. ( ~P C i^i Fin ) ) ) |
| 115 |
114
|
anbi2d |
|- ( x = ( `' F " y ) -> ( ( ph /\ x e. ( ~P C i^i Fin ) ) <-> ( ph /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) ) ) |
| 116 |
|
eleq2w2 |
|- ( x = ( `' F " y ) -> ( n e. x <-> n e. ( `' F " y ) ) ) |
| 117 |
115 116
|
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 ) ) ) ) |
| 118 |
|
reseq2 |
|- ( x = ( `' F " y ) -> ( F |` x ) = ( F |` ( `' F " y ) ) ) |
| 119 |
118
|
fveq1d |
|- ( x = ( `' F " y ) -> ( ( F |` x ) ` n ) = ( ( F |` ( `' F " y ) ) ` n ) ) |
| 120 |
119
|
eqeq1d |
|- ( x = ( `' F " y ) -> ( ( ( F |` x ) ` n ) = G <-> ( ( F |` ( `' F " y ) ) ` n ) = G ) ) |
| 121 |
117 120
|
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 ) ) ) |
| 122 |
|
fvres |
|- ( n e. x -> ( ( F |` x ) ` n ) = ( F ` n ) ) |
| 123 |
122
|
adantl |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ( ( F |` x ) ` n ) = ( F ` n ) ) |
| 124 |
|
simpll |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ph ) |
| 125 |
|
elpwinss |
|- ( x e. ( ~P C i^i Fin ) -> x C_ C ) |
| 126 |
125
|
adantl |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> x C_ C ) |
| 127 |
126
|
sselda |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> n e. C ) |
| 128 |
124 127 6
|
syl2anc |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ( F ` n ) = G ) |
| 129 |
123 128
|
eqtrd |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ( ( F |` x ) ` n ) = G ) |
| 130 |
121 129
|
vtoclg |
|- ( ( `' F " y ) e. _V -> ( ( ( ph /\ ( `' F " y ) e. ( ~P C i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( ( F |` ( `' F " y ) ) ` n ) = G ) ) |
| 131 |
109 113 130
|
sylc |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ n e. ( `' F " y ) ) -> ( ( F |` ( `' F " y ) ) ` n ) = G ) |
| 132 |
131
|
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 ) |
| 133 |
108
|
ad3antrrr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> ( `' F " y ) e. _V ) |
| 134 |
|
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 ) ) ) |
| 135 |
70
|
ad3antrrr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> ( `' F " y ) e. ~P C ) |
| 136 |
92
|
adantr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> ( `' F " y ) e. Fin ) |
| 137 |
135 136
|
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 ) ) |
| 138 |
|
simpr |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> k e. y ) |
| 139 |
101
|
eqcomd |
|- ( ( ph /\ y e. ( ~P A i^i Fin ) ) -> y = ( F " ( `' F " y ) ) ) |
| 140 |
139
|
adantr |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> y = ( F " ( `' F " y ) ) ) |
| 141 |
138 140
|
eleqtrd |
|- ( ( ( ph /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> k e. ( F " ( `' F " y ) ) ) |
| 142 |
141
|
adantllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> k e. ( F " ( `' F " y ) ) ) |
| 143 |
134 137 142
|
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 ) ) ) ) |
| 144 |
114
|
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 ) ) ) ) |
| 145 |
|
imaeq2 |
|- ( x = ( `' F " y ) -> ( F " x ) = ( F " ( `' F " y ) ) ) |
| 146 |
145
|
eleq2d |
|- ( x = ( `' F " y ) -> ( k e. ( F " x ) <-> k e. ( F " ( `' F " y ) ) ) ) |
| 147 |
144 146
|
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 ) ) ) ) ) |
| 148 |
147
|
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 ) ) ) |
| 149 |
|
rge0ssre |
|- ( 0 [,) +oo ) C_ RR |
| 150 |
|
ax-resscn |
|- RR C_ CC |
| 151 |
149 150
|
sstri |
|- ( 0 [,) +oo ) C_ CC |
| 152 |
|
simplll |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> ph ) |
| 153 |
|
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 ) ) |
| 154 |
18
|
fimassd |
|- ( ph -> ( F " x ) C_ A ) |
| 155 |
154
|
ad2antrr |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> ( F " x ) C_ A ) |
| 156 |
|
simpr |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> k e. ( F " x ) ) |
| 157 |
155 156
|
sseldd |
|- ( ( ( ph /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> k e. A ) |
| 158 |
157
|
adantllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> k e. A ) |
| 159 |
|
foelcdmi |
|- ( ( F : C -onto-> A /\ k e. A ) -> E. n e. C ( F ` n ) = k ) |
| 160 |
9 159
|
sylan |
|- ( ( ph /\ k e. A ) -> E. n e. C ( F ` n ) = k ) |
| 161 |
160
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ k e. A ) -> E. n e. C ( F ` n ) = k ) |
| 162 |
|
nfv |
|- F/ n k e. A |
| 163 |
89 162
|
nfan |
|- F/ n ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ k e. A ) |
| 164 |
|
nfv |
|- F/ n B e. ( 0 [,) +oo ) |
| 165 |
|
csbid |
|- [_ k / k ]_ B = B |
| 166 |
165
|
eqcomi |
|- B = [_ k / k ]_ B |
| 167 |
166
|
a1i |
|- ( ( ph /\ n e. C /\ ( F ` n ) = k ) -> B = [_ k / k ]_ B ) |
| 168 |
|
id |
|- ( ( F ` n ) = k -> ( F ` n ) = k ) |
| 169 |
168
|
eqcomd |
|- ( ( F ` n ) = k -> k = ( F ` n ) ) |
| 170 |
169
|
csbeq1d |
|- ( ( F ` n ) = k -> [_ k / k ]_ B = [_ ( F ` n ) / k ]_ B ) |
| 171 |
170
|
3ad2ant3 |
|- ( ( ph /\ n e. C /\ ( F ` n ) = k ) -> [_ k / k ]_ B = [_ ( F ` n ) / k ]_ B ) |
| 172 |
29
|
3adant3 |
|- ( ( ph /\ n e. C /\ ( F ` n ) = k ) -> [_ ( F ` n ) / k ]_ B = D ) |
| 173 |
167 171 172
|
3eqtrd |
|- ( ( ph /\ n e. C /\ ( F ` n ) = k ) -> B = D ) |
| 174 |
173
|
3adant1r |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C /\ ( F ` n ) = k ) -> B = D ) |
| 175 |
|
0xr |
|- 0 e. RR* |
| 176 |
175
|
a1i |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> 0 e. RR* ) |
| 177 |
|
pnfxr |
|- +oo e. RR* |
| 178 |
177
|
a1i |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> +oo e. RR* ) |
| 179 |
60
|
adantr |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> D e. ( 0 [,] +oo ) ) |
| 180 |
|
simpr |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> -. D e. ( 0 [,) +oo ) ) |
| 181 |
176 178 179 180
|
eliccnelico |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> D = +oo ) |
| 182 |
181
|
eqcomd |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> +oo = D ) |
| 183 |
|
simpr |
|- ( ( ph /\ n e. C ) -> n e. C ) |
| 184 |
53 183 60
|
elrnmpt1d |
|- ( ( ph /\ n e. C ) -> D e. ran ( n e. C |-> D ) ) |
| 185 |
184
|
adantr |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> D e. ran ( n e. C |-> D ) ) |
| 186 |
182 185
|
eqeltrd |
|- ( ( ( ph /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> +oo e. ran ( n e. C |-> D ) ) |
| 187 |
186
|
adantllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> +oo e. ran ( n e. C |-> D ) ) |
| 188 |
|
simpllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C ) /\ -. D e. ( 0 [,) +oo ) ) -> -. +oo e. ran ( n e. C |-> D ) ) |
| 189 |
187 188
|
condan |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C ) -> D e. ( 0 [,) +oo ) ) |
| 190 |
189
|
3adant3 |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C /\ ( F ` n ) = k ) -> D e. ( 0 [,) +oo ) ) |
| 191 |
174 190
|
eqeltrd |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ n e. C /\ ( F ` n ) = k ) -> B e. ( 0 [,) +oo ) ) |
| 192 |
191
|
3exp |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> ( n e. C -> ( ( F ` n ) = k -> B e. ( 0 [,) +oo ) ) ) ) |
| 193 |
192
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ k e. A ) -> ( n e. C -> ( ( F ` n ) = k -> B e. ( 0 [,) +oo ) ) ) ) |
| 194 |
163 164 193
|
rexlimd |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ k e. A ) -> ( E. n e. C ( F ` n ) = k -> B e. ( 0 [,) +oo ) ) ) |
| 195 |
161 194
|
mpd |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ k e. A ) -> B e. ( 0 [,) +oo ) ) |
| 196 |
152 153 158 195
|
syl21anc |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> B e. ( 0 [,) +oo ) ) |
| 197 |
151 196
|
sselid |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ k e. ( F " x ) ) -> B e. CC ) |
| 198 |
148 197
|
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 ) ) |
| 199 |
133 143 198
|
sylc |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ y e. ( ~P A i^i Fin ) ) /\ k e. y ) -> B e. CC ) |
| 200 |
84 91 3 92 104 132 199
|
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 ) |
| 201 |
|
sumeq1 |
|- ( x = ( `' F " y ) -> sum_ n e. x D = sum_ n e. ( `' F " y ) D ) |
| 202 |
201
|
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 ) |
| 203 |
80 200 202
|
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 ) |
| 204 |
67 203
|
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 ) ) |
| 205 |
|
sumex |
|- sum_ n e. x D e. _V |
| 206 |
205
|
a1i |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> sum_ n e. x D e. _V ) |
| 207 |
11 154
|
sselpwd |
|- ( ph -> ( F " x ) e. ~P A ) |
| 208 |
207
|
adantr |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> ( F " x ) e. ~P A ) |
| 209 |
18
|
ffund |
|- ( ph -> Fun F ) |
| 210 |
|
elinel2 |
|- ( x e. ( ~P C i^i Fin ) -> x e. Fin ) |
| 211 |
|
imafi |
|- ( ( Fun F /\ x e. Fin ) -> ( F " x ) e. Fin ) |
| 212 |
209 210 211
|
syl2an |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> ( F " x ) e. Fin ) |
| 213 |
208 212
|
elind |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> ( F " x ) e. ( ~P A i^i Fin ) ) |
| 214 |
213
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> ( F " x ) e. ( ~P A i^i Fin ) ) |
| 215 |
|
nfv |
|- F/ k x e. ( ~P C i^i Fin ) |
| 216 |
82 215
|
nfan |
|- F/ k ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) |
| 217 |
|
nfv |
|- F/ n x e. ( ~P C i^i Fin ) |
| 218 |
89 217
|
nfan |
|- F/ n ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) |
| 219 |
210
|
adantl |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> x e. Fin ) |
| 220 |
|
f1ores |
|- ( ( F : C -1-1-> A /\ x C_ C ) -> ( F |` x ) : x -1-1-onto-> ( F " x ) ) |
| 221 |
94 125 220
|
syl2an |
|- ( ( ph /\ x e. ( ~P C i^i Fin ) ) -> ( F |` x ) : x -1-1-onto-> ( F " x ) ) |
| 222 |
221
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) -> ( F |` x ) : x -1-1-onto-> ( F " x ) ) |
| 223 |
129
|
adantllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) /\ x e. ( ~P C i^i Fin ) ) /\ n e. x ) -> ( ( F |` x ) ` n ) = G ) |
| 224 |
216 218 3 219 222 223 197
|
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 ) |
| 225 |
224
|
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 ) |
| 226 |
|
sumeq1 |
|- ( y = ( F " x ) -> sum_ k e. y B = sum_ k e. ( F " x ) B ) |
| 227 |
226
|
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 ) |
| 228 |
214 225 227
|
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 ) |
| 229 |
206 228
|
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 ) ) |
| 230 |
204 229
|
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 ) ) |
| 231 |
230
|
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* , < ) ) |
| 232 |
11
|
adantr |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> A e. _V ) |
| 233 |
82 232 195
|
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* , < ) ) |
| 234 |
4
|
adantr |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> C e. V ) |
| 235 |
89 234 189
|
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* , < ) ) |
| 236 |
231 233 235
|
3eqtr4d |
|- ( ( ph /\ -. +oo e. ran ( n e. C |-> D ) ) -> ( sum^ ` ( k e. A |-> B ) ) = ( sum^ ` ( n e. C |-> D ) ) ) |
| 237 |
65 236
|
pm2.61dan |
|- ( ph -> ( sum^ ` ( k e. A |-> B ) ) = ( sum^ ` ( n e. C |-> D ) ) ) |