Step |
Hyp |
Ref |
Expression |
1 |
|
esum2d.0 |
|- F/_ k F |
2 |
|
esum2d.1 |
|- ( z = <. j , k >. -> F = C ) |
3 |
|
esum2d.2 |
|- ( ph -> A e. V ) |
4 |
|
esum2d.3 |
|- ( ( ph /\ j e. A ) -> B e. W ) |
5 |
|
esum2d.4 |
|- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. ( 0 [,] +oo ) ) |
6 |
|
xrltso |
|- < Or RR* |
7 |
6
|
a1i |
|- ( ph -> < Or RR* ) |
8 |
|
nfv |
|- F/ c ph |
9 |
|
nfcv |
|- F/_ c s |
10 |
|
nfmpt1 |
|- F/_ c ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
11 |
10
|
nfrn |
|- F/_ c ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
12 |
9 11
|
nfel |
|- F/ c s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
13 |
8 12
|
nfan |
|- F/ c ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) |
14 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
15 |
|
xrge0base |
|- ( 0 [,] +oo ) = ( Base ` ( RR*s |`s ( 0 [,] +oo ) ) ) |
16 |
|
xrge0cmn |
|- ( RR*s |`s ( 0 [,] +oo ) ) e. CMnd |
17 |
16
|
a1i |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( RR*s |`s ( 0 [,] +oo ) ) e. CMnd ) |
18 |
|
simpr |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) |
19 |
18
|
elin2d |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> c e. Fin ) |
20 |
|
simpll |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ z e. c ) -> ph ) |
21 |
18
|
elin1d |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> c e. ~P U_ j e. A ( { j } X. B ) ) |
22 |
21
|
adantr |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ z e. c ) -> c e. ~P U_ j e. A ( { j } X. B ) ) |
23 |
|
vex |
|- c e. _V |
24 |
23
|
elpw |
|- ( c e. ~P U_ j e. A ( { j } X. B ) <-> c C_ U_ j e. A ( { j } X. B ) ) |
25 |
22 24
|
sylib |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ z e. c ) -> c C_ U_ j e. A ( { j } X. B ) ) |
26 |
|
simpr |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ z e. c ) -> z e. c ) |
27 |
25 26
|
sseldd |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ z e. c ) -> z e. U_ j e. A ( { j } X. B ) ) |
28 |
|
nfv |
|- F/ j ph |
29 |
|
nfcv |
|- F/_ j z |
30 |
|
nfiu1 |
|- F/_ j U_ j e. A ( { j } X. B ) |
31 |
29 30
|
nfel |
|- F/ j z e. U_ j e. A ( { j } X. B ) |
32 |
28 31
|
nfan |
|- F/ j ( ph /\ z e. U_ j e. A ( { j } X. B ) ) |
33 |
|
nfv |
|- F/ k ( ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) /\ j e. A ) /\ z e. ( { j } X. B ) ) |
34 |
|
nfcv |
|- F/_ k ( 0 [,] +oo ) |
35 |
1 34
|
nfel |
|- F/ k F e. ( 0 [,] +oo ) |
36 |
2
|
adantl |
|- ( ( ( ( ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) /\ j e. A ) /\ z e. ( { j } X. B ) ) /\ k e. B ) /\ z = <. j , k >. ) -> F = C ) |
37 |
|
simp-5l |
|- ( ( ( ( ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) /\ j e. A ) /\ z e. ( { j } X. B ) ) /\ k e. B ) /\ z = <. j , k >. ) -> ph ) |
38 |
|
simp-4r |
|- ( ( ( ( ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) /\ j e. A ) /\ z e. ( { j } X. B ) ) /\ k e. B ) /\ z = <. j , k >. ) -> j e. A ) |
39 |
|
simplr |
|- ( ( ( ( ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) /\ j e. A ) /\ z e. ( { j } X. B ) ) /\ k e. B ) /\ z = <. j , k >. ) -> k e. B ) |
40 |
37 38 39 5
|
syl12anc |
|- ( ( ( ( ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) /\ j e. A ) /\ z e. ( { j } X. B ) ) /\ k e. B ) /\ z = <. j , k >. ) -> C e. ( 0 [,] +oo ) ) |
41 |
36 40
|
eqeltrd |
|- ( ( ( ( ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) /\ j e. A ) /\ z e. ( { j } X. B ) ) /\ k e. B ) /\ z = <. j , k >. ) -> F e. ( 0 [,] +oo ) ) |
42 |
|
elsnxp |
|- ( j e. A -> ( z e. ( { j } X. B ) <-> E. k e. B z = <. j , k >. ) ) |
43 |
42
|
biimpa |
|- ( ( j e. A /\ z e. ( { j } X. B ) ) -> E. k e. B z = <. j , k >. ) |
44 |
43
|
adantll |
|- ( ( ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) /\ j e. A ) /\ z e. ( { j } X. B ) ) -> E. k e. B z = <. j , k >. ) |
45 |
33 35 41 44
|
r19.29af2 |
|- ( ( ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) /\ j e. A ) /\ z e. ( { j } X. B ) ) -> F e. ( 0 [,] +oo ) ) |
46 |
|
simpr |
|- ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) -> z e. U_ j e. A ( { j } X. B ) ) |
47 |
|
eliun |
|- ( z e. U_ j e. A ( { j } X. B ) <-> E. j e. A z e. ( { j } X. B ) ) |
48 |
46 47
|
sylib |
|- ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) -> E. j e. A z e. ( { j } X. B ) ) |
49 |
32 45 48
|
r19.29af |
|- ( ( ph /\ z e. U_ j e. A ( { j } X. B ) ) -> F e. ( 0 [,] +oo ) ) |
50 |
20 27 49
|
syl2anc |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ z e. c ) -> F e. ( 0 [,] +oo ) ) |
51 |
50
|
ralrimiva |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> A. z e. c F e. ( 0 [,] +oo ) ) |
52 |
15 17 19 51
|
gsummptcl |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. ( 0 [,] +oo ) ) |
53 |
14 52
|
sselid |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. RR* ) |
54 |
53
|
ralrimiva |
|- ( ph -> A. c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. RR* ) |
55 |
|
eqid |
|- ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) = ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
56 |
55
|
rnmptss |
|- ( A. c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. RR* -> ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) C_ RR* ) |
57 |
54 56
|
syl |
|- ( ph -> ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) C_ RR* ) |
58 |
57
|
ad3antrrr |
|- ( ( ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) C_ RR* ) |
59 |
|
simpllr |
|- ( ( ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) |
60 |
58 59
|
sseldd |
|- ( ( ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> s e. RR* ) |
61 |
|
snex |
|- { j } e. _V |
62 |
|
xpexg |
|- ( ( { j } e. _V /\ B e. W ) -> ( { j } X. B ) e. _V ) |
63 |
61 4 62
|
sylancr |
|- ( ( ph /\ j e. A ) -> ( { j } X. B ) e. _V ) |
64 |
63
|
ralrimiva |
|- ( ph -> A. j e. A ( { j } X. B ) e. _V ) |
65 |
|
iunexg |
|- ( ( A e. V /\ A. j e. A ( { j } X. B ) e. _V ) -> U_ j e. A ( { j } X. B ) e. _V ) |
66 |
3 64 65
|
syl2anc |
|- ( ph -> U_ j e. A ( { j } X. B ) e. _V ) |
67 |
49
|
ralrimiva |
|- ( ph -> A. z e. U_ j e. A ( { j } X. B ) F e. ( 0 [,] +oo ) ) |
68 |
|
nfcv |
|- F/_ z U_ j e. A ( { j } X. B ) |
69 |
68
|
esumcl |
|- ( ( U_ j e. A ( { j } X. B ) e. _V /\ A. z e. U_ j e. A ( { j } X. B ) F e. ( 0 [,] +oo ) ) -> sum* z e. U_ j e. A ( { j } X. B ) F e. ( 0 [,] +oo ) ) |
70 |
66 67 69
|
syl2anc |
|- ( ph -> sum* z e. U_ j e. A ( { j } X. B ) F e. ( 0 [,] +oo ) ) |
71 |
14 70
|
sselid |
|- ( ph -> sum* z e. U_ j e. A ( { j } X. B ) F e. RR* ) |
72 |
71
|
ad3antrrr |
|- ( ( ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> sum* z e. U_ j e. A ( { j } X. B ) F e. RR* ) |
73 |
|
simpr |
|- ( ( ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
74 |
|
nfv |
|- F/ z ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) |
75 |
|
nfcv |
|- F/_ z c |
76 |
74 75 19 50
|
esumgsum |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> sum* z e. c F = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
77 |
66
|
adantr |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> U_ j e. A ( { j } X. B ) e. _V ) |
78 |
49
|
adantlr |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ z e. U_ j e. A ( { j } X. B ) ) -> F e. ( 0 [,] +oo ) ) |
79 |
21 24
|
sylib |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> c C_ U_ j e. A ( { j } X. B ) ) |
80 |
74 77 78 79
|
esummono |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> sum* z e. c F <_ sum* z e. U_ j e. A ( { j } X. B ) F ) |
81 |
76 80
|
eqbrtrrd |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) <_ sum* z e. U_ j e. A ( { j } X. B ) F ) |
82 |
81
|
adantlr |
|- ( ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) <_ sum* z e. U_ j e. A ( { j } X. B ) F ) |
83 |
82
|
adantr |
|- ( ( ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) <_ sum* z e. U_ j e. A ( { j } X. B ) F ) |
84 |
73 83
|
eqbrtrd |
|- ( ( ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> s <_ sum* z e. U_ j e. A ( { j } X. B ) F ) |
85 |
|
xrlenlt |
|- ( ( s e. RR* /\ sum* z e. U_ j e. A ( { j } X. B ) F e. RR* ) -> ( s <_ sum* z e. U_ j e. A ( { j } X. B ) F <-> -. sum* z e. U_ j e. A ( { j } X. B ) F < s ) ) |
86 |
85
|
biimpa |
|- ( ( ( s e. RR* /\ sum* z e. U_ j e. A ( { j } X. B ) F e. RR* ) /\ s <_ sum* z e. U_ j e. A ( { j } X. B ) F ) -> -. sum* z e. U_ j e. A ( { j } X. B ) F < s ) |
87 |
60 72 84 86
|
syl21anc |
|- ( ( ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> -. sum* z e. U_ j e. A ( { j } X. B ) F < s ) |
88 |
|
simpr |
|- ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) -> s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) |
89 |
|
ovex |
|- ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. _V |
90 |
55 89
|
elrnmpti |
|- ( s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) <-> E. c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
91 |
88 90
|
sylib |
|- ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) -> E. c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
92 |
13 87 91
|
r19.29af |
|- ( ( ph /\ s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) -> -. sum* z e. U_ j e. A ( { j } X. B ) F < s ) |
93 |
92
|
ralrimiva |
|- ( ph -> A. s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -. sum* z e. U_ j e. A ( { j } X. B ) F < s ) |
94 |
|
nfv |
|- F/ c ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) |
95 |
|
nfv |
|- F/ c s < t |
96 |
11 95
|
nfrex |
|- F/ c E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t |
97 |
76
|
adantlr |
|- ( ( ( ph /\ s e. RR* ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> sum* z e. c F = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
98 |
97
|
adantlr |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> sum* z e. c F = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
99 |
98
|
adantr |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s < sum* z e. c F ) -> sum* z e. c F = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
100 |
|
simplr |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s < sum* z e. c F ) -> c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) |
101 |
89
|
a1i |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s < sum* z e. c F ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. _V ) |
102 |
55
|
elrnmpt1 |
|- ( ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) /\ ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. _V ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) |
103 |
100 101 102
|
syl2anc |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s < sum* z e. c F ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) |
104 |
99 103
|
eqeltrd |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s < sum* z e. c F ) -> sum* z e. c F e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) |
105 |
|
simpr |
|- ( ( ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s < sum* z e. c F ) /\ t = sum* z e. c F ) -> t = sum* z e. c F ) |
106 |
105
|
breq2d |
|- ( ( ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s < sum* z e. c F ) /\ t = sum* z e. c F ) -> ( s < t <-> s < sum* z e. c F ) ) |
107 |
|
simpr |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s < sum* z e. c F ) -> s < sum* z e. c F ) |
108 |
104 106 107
|
rspcedvd |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ s < sum* z e. c F ) -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) |
109 |
|
nfv |
|- F/ z ( ph /\ s e. RR* ) |
110 |
|
nfcv |
|- F/_ z s |
111 |
|
nfcv |
|- F/_ z < |
112 |
68
|
nfesum1 |
|- F/_ z sum* z e. U_ j e. A ( { j } X. B ) F |
113 |
110 111 112
|
nfbr |
|- F/ z s < sum* z e. U_ j e. A ( { j } X. B ) F |
114 |
109 113
|
nfan |
|- F/ z ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) |
115 |
66
|
ad2antrr |
|- ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) -> U_ j e. A ( { j } X. B ) e. _V ) |
116 |
49
|
3ad2antr3 |
|- ( ( ph /\ ( s e. RR* /\ s < sum* z e. U_ j e. A ( { j } X. B ) F /\ z e. U_ j e. A ( { j } X. B ) ) ) -> F e. ( 0 [,] +oo ) ) |
117 |
116
|
3anassrs |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) /\ z e. U_ j e. A ( { j } X. B ) ) -> F e. ( 0 [,] +oo ) ) |
118 |
|
simplr |
|- ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) -> s e. RR* ) |
119 |
|
simpr |
|- ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) -> s < sum* z e. U_ j e. A ( { j } X. B ) F ) |
120 |
114 115 117 118 119
|
esumlub |
|- ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) -> E. c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) s < sum* z e. c F ) |
121 |
94 96 108 120
|
r19.29af2 |
|- ( ( ( ph /\ s e. RR* ) /\ s < sum* z e. U_ j e. A ( { j } X. B ) F ) -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) |
122 |
121
|
ex |
|- ( ( ph /\ s e. RR* ) -> ( s < sum* z e. U_ j e. A ( { j } X. B ) F -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) |
123 |
122
|
ralrimiva |
|- ( ph -> A. s e. RR* ( s < sum* z e. U_ j e. A ( { j } X. B ) F -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) |
124 |
93 123
|
jca |
|- ( ph -> ( A. s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -. sum* z e. U_ j e. A ( { j } X. B ) F < s /\ A. s e. RR* ( s < sum* z e. U_ j e. A ( { j } X. B ) F -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) ) |
125 |
|
simpr |
|- ( ( ph /\ r = sum* z e. U_ j e. A ( { j } X. B ) F ) -> r = sum* z e. U_ j e. A ( { j } X. B ) F ) |
126 |
125
|
breq1d |
|- ( ( ph /\ r = sum* z e. U_ j e. A ( { j } X. B ) F ) -> ( r < s <-> sum* z e. U_ j e. A ( { j } X. B ) F < s ) ) |
127 |
126
|
notbid |
|- ( ( ph /\ r = sum* z e. U_ j e. A ( { j } X. B ) F ) -> ( -. r < s <-> -. sum* z e. U_ j e. A ( { j } X. B ) F < s ) ) |
128 |
127
|
ralbidv |
|- ( ( ph /\ r = sum* z e. U_ j e. A ( { j } X. B ) F ) -> ( A. s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -. r < s <-> A. s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -. sum* z e. U_ j e. A ( { j } X. B ) F < s ) ) |
129 |
125
|
breq2d |
|- ( ( ph /\ r = sum* z e. U_ j e. A ( { j } X. B ) F ) -> ( s < r <-> s < sum* z e. U_ j e. A ( { j } X. B ) F ) ) |
130 |
129
|
imbi1d |
|- ( ( ph /\ r = sum* z e. U_ j e. A ( { j } X. B ) F ) -> ( ( s < r -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) <-> ( s < sum* z e. U_ j e. A ( { j } X. B ) F -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) ) |
131 |
130
|
ralbidv |
|- ( ( ph /\ r = sum* z e. U_ j e. A ( { j } X. B ) F ) -> ( A. s e. RR* ( s < r -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) <-> A. s e. RR* ( s < sum* z e. U_ j e. A ( { j } X. B ) F -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) ) |
132 |
128 131
|
anbi12d |
|- ( ( ph /\ r = sum* z e. U_ j e. A ( { j } X. B ) F ) -> ( ( A. s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -. r < s /\ A. s e. RR* ( s < r -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) <-> ( A. s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -. sum* z e. U_ j e. A ( { j } X. B ) F < s /\ A. s e. RR* ( s < sum* z e. U_ j e. A ( { j } X. B ) F -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) ) ) |
133 |
71 132
|
rspcedv |
|- ( ph -> ( ( A. s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -. sum* z e. U_ j e. A ( { j } X. B ) F < s /\ A. s e. RR* ( s < sum* z e. U_ j e. A ( { j } X. B ) F -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) -> E. r e. RR* ( A. s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -. r < s /\ A. s e. RR* ( s < r -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) ) ) |
134 |
124 133
|
mpd |
|- ( ph -> E. r e. RR* ( A. s e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -. r < s /\ A. s e. RR* ( s < r -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) ) |
135 |
7 134
|
supcl |
|- ( ph -> sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) e. RR* ) |
136 |
|
nfv |
|- F/ a ph |
137 |
|
nfcv |
|- F/_ a s |
138 |
|
nfmpt1 |
|- F/_ a ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
139 |
138
|
nfrn |
|- F/_ a ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
140 |
137 139
|
nfel |
|- F/ a s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
141 |
136 140
|
nfan |
|- F/ a ( ph /\ s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) |
142 |
|
simpr |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> a e. ( ~P A i^i Fin ) ) |
143 |
|
simpll |
|- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ j e. a ) -> ph ) |
144 |
142
|
elin1d |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> a e. ~P A ) |
145 |
|
elpwi |
|- ( a e. ~P A -> a C_ A ) |
146 |
144 145
|
syl |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> a C_ A ) |
147 |
146
|
sselda |
|- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ j e. a ) -> j e. A ) |
148 |
143 147 4
|
syl2anc |
|- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ j e. a ) -> B e. W ) |
149 |
143
|
adantrr |
|- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ ( j e. a /\ k e. B ) ) -> ph ) |
150 |
147
|
adantrr |
|- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ ( j e. a /\ k e. B ) ) -> j e. A ) |
151 |
|
simprr |
|- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ ( j e. a /\ k e. B ) ) -> k e. B ) |
152 |
149 150 151 5
|
syl12anc |
|- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ ( j e. a /\ k e. B ) ) -> C e. ( 0 [,] +oo ) ) |
153 |
142
|
elin2d |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> a e. Fin ) |
154 |
1 2 142 148 152 153
|
esum2dlem |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> sum* j e. a sum* k e. B C = sum* z e. U_ j e. a ( { j } X. B ) F ) |
155 |
|
nfv |
|- F/ j ( ph /\ a e. ( ~P A i^i Fin ) ) |
156 |
|
nfcv |
|- F/_ j a |
157 |
5
|
anassrs |
|- ( ( ( ph /\ j e. A ) /\ k e. B ) -> C e. ( 0 [,] +oo ) ) |
158 |
157
|
ralrimiva |
|- ( ( ph /\ j e. A ) -> A. k e. B C e. ( 0 [,] +oo ) ) |
159 |
|
nfcv |
|- F/_ k B |
160 |
159
|
esumcl |
|- ( ( B e. W /\ A. k e. B C e. ( 0 [,] +oo ) ) -> sum* k e. B C e. ( 0 [,] +oo ) ) |
161 |
4 158 160
|
syl2anc |
|- ( ( ph /\ j e. A ) -> sum* k e. B C e. ( 0 [,] +oo ) ) |
162 |
143 147 161
|
syl2anc |
|- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ j e. a ) -> sum* k e. B C e. ( 0 [,] +oo ) ) |
163 |
155 156 153 162
|
esumgsum |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> sum* j e. a sum* k e. B C = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
164 |
154 163
|
eqtr3d |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> sum* z e. U_ j e. a ( { j } X. B ) F = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
165 |
|
nfv |
|- F/ z ( ph /\ a e. ( ~P A i^i Fin ) ) |
166 |
66
|
adantr |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> U_ j e. A ( { j } X. B ) e. _V ) |
167 |
49
|
adantlr |
|- ( ( ( ph /\ a e. ( ~P A i^i Fin ) ) /\ z e. U_ j e. A ( { j } X. B ) ) -> F e. ( 0 [,] +oo ) ) |
168 |
|
iunss1 |
|- ( a C_ A -> U_ j e. a ( { j } X. B ) C_ U_ j e. A ( { j } X. B ) ) |
169 |
146 168
|
syl |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> U_ j e. a ( { j } X. B ) C_ U_ j e. A ( { j } X. B ) ) |
170 |
165 166 167 169
|
esummono |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> sum* z e. U_ j e. a ( { j } X. B ) F <_ sum* z e. U_ j e. A ( { j } X. B ) F ) |
171 |
164 170
|
eqbrtrrd |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) <_ sum* z e. U_ j e. A ( { j } X. B ) F ) |
172 |
16
|
a1i |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( RR*s |`s ( 0 [,] +oo ) ) e. CMnd ) |
173 |
162
|
ralrimiva |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> A. j e. a sum* k e. B C e. ( 0 [,] +oo ) ) |
174 |
15 172 153 173
|
gsummptcl |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) e. ( 0 [,] +oo ) ) |
175 |
14 174
|
sselid |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) e. RR* ) |
176 |
71
|
adantr |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> sum* z e. U_ j e. A ( { j } X. B ) F e. RR* ) |
177 |
|
xrlenlt |
|- ( ( ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) e. RR* /\ sum* z e. U_ j e. A ( { j } X. B ) F e. RR* ) -> ( ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) <_ sum* z e. U_ j e. A ( { j } X. B ) F <-> -. sum* z e. U_ j e. A ( { j } X. B ) F < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) |
178 |
175 176 177
|
syl2anc |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) <_ sum* z e. U_ j e. A ( { j } X. B ) F <-> -. sum* z e. U_ j e. A ( { j } X. B ) F < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) |
179 |
171 178
|
mpbid |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> -. sum* z e. U_ j e. A ( { j } X. B ) F < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
180 |
|
nfv |
|- F/ z ph |
181 |
|
eqidd |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
182 |
180 68 66 49 181
|
esumval |
|- ( ph -> sum* z e. U_ j e. A ( { j } X. B ) F = sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) |
183 |
182
|
adantr |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> sum* z e. U_ j e. A ( { j } X. B ) F = sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) |
184 |
183
|
breq1d |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( sum* z e. U_ j e. A ( { j } X. B ) F < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) <-> sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) |
185 |
179 184
|
mtbid |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> -. sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
186 |
185
|
adantlr |
|- ( ( ( ph /\ s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) -> -. sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
187 |
186
|
adantr |
|- ( ( ( ( ph /\ s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) -> -. sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
188 |
|
simpr |
|- ( ( ( ( ph /\ s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) -> s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
189 |
188
|
breq2d |
|- ( ( ( ( ph /\ s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) -> ( sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) < s <-> sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) |
190 |
189
|
notbid |
|- ( ( ( ( ph /\ s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) -> ( -. sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) < s <-> -. sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) |
191 |
187 190
|
mpbird |
|- ( ( ( ( ph /\ s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) /\ a e. ( ~P A i^i Fin ) ) /\ s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) -> -. sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) < s ) |
192 |
|
eqid |
|- ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) = ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
193 |
|
ovex |
|- ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) e. _V |
194 |
192 193
|
elrnmpti |
|- ( s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) <-> E. a e. ( ~P A i^i Fin ) s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
195 |
194
|
biimpi |
|- ( s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) -> E. a e. ( ~P A i^i Fin ) s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
196 |
195
|
adantl |
|- ( ( ph /\ s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) -> E. a e. ( ~P A i^i Fin ) s = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
197 |
141 191 196
|
r19.29af |
|- ( ( ph /\ s e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) -> -. sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) < s ) |
198 |
9
|
nfel1 |
|- F/ c s e. RR* |
199 |
|
nfcv |
|- F/_ c < |
200 |
|
nfcv |
|- F/_ c RR* |
201 |
11 200 199
|
nfsup |
|- F/_ c sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) |
202 |
9 199 201
|
nfbr |
|- F/ c s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) |
203 |
198 202
|
nfan |
|- F/ c ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) |
204 |
8 203
|
nfan |
|- F/ c ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) |
205 |
|
nfcv |
|- F/_ c u |
206 |
205 11
|
nfel |
|- F/ c u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
207 |
204 206
|
nfan |
|- F/ c ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) |
208 |
|
nfv |
|- F/ c s < u |
209 |
207 208
|
nfan |
|- F/ c ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) |
210 |
|
simp-5l |
|- ( ( ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> ph ) |
211 |
|
simpr1l |
|- ( ( ph /\ ( ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ ( s < u /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) ) -> s e. RR* ) |
212 |
211
|
3anassrs |
|- ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ ( s < u /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) -> s e. RR* ) |
213 |
212
|
3anassrs |
|- ( ( ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> s e. RR* ) |
214 |
210 213
|
jca |
|- ( ( ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> ( ph /\ s e. RR* ) ) |
215 |
|
simpr1r |
|- ( ( ph /\ ( ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ ( s < u /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) ) -> s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) |
216 |
215
|
3anassrs |
|- ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ ( s < u /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) -> s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) |
217 |
216
|
3anassrs |
|- ( ( ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) |
218 |
214 217
|
jca |
|- ( ( ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> ( ( ph /\ s e. RR* ) /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) |
219 |
|
simpllr |
|- ( ( ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> s < u ) |
220 |
|
simpr |
|- ( ( ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
221 |
219 220
|
breqtrd |
|- ( ( ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
222 |
|
simplr |
|- ( ( ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) |
223 |
|
simpr |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) |
224 |
223
|
elin1d |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> c e. ~P U_ j e. A ( { j } X. B ) ) |
225 |
|
elpwi |
|- ( c e. ~P U_ j e. A ( { j } X. B ) -> c C_ U_ j e. A ( { j } X. B ) ) |
226 |
|
dmss |
|- ( c C_ U_ j e. A ( { j } X. B ) -> dom c C_ dom U_ j e. A ( { j } X. B ) ) |
227 |
|
dmiun |
|- dom U_ j e. A ( { j } X. B ) = U_ j e. A dom ( { j } X. B ) |
228 |
226 227
|
sseqtrdi |
|- ( c C_ U_ j e. A ( { j } X. B ) -> dom c C_ U_ j e. A dom ( { j } X. B ) ) |
229 |
|
dmxpss |
|- dom ( { j } X. B ) C_ { j } |
230 |
229
|
a1i |
|- ( j e. A -> dom ( { j } X. B ) C_ { j } ) |
231 |
|
snssi |
|- ( j e. A -> { j } C_ A ) |
232 |
230 231
|
sstrd |
|- ( j e. A -> dom ( { j } X. B ) C_ A ) |
233 |
232
|
rgen |
|- A. j e. A dom ( { j } X. B ) C_ A |
234 |
|
iunss |
|- ( U_ j e. A dom ( { j } X. B ) C_ A <-> A. j e. A dom ( { j } X. B ) C_ A ) |
235 |
233 234
|
mpbir |
|- U_ j e. A dom ( { j } X. B ) C_ A |
236 |
228 235
|
sstrdi |
|- ( c C_ U_ j e. A ( { j } X. B ) -> dom c C_ A ) |
237 |
23
|
dmex |
|- dom c e. _V |
238 |
237
|
elpw |
|- ( dom c e. ~P A <-> dom c C_ A ) |
239 |
236 238
|
sylibr |
|- ( c C_ U_ j e. A ( { j } X. B ) -> dom c e. ~P A ) |
240 |
224 225 239
|
3syl |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> dom c e. ~P A ) |
241 |
223
|
elin2d |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> c e. Fin ) |
242 |
|
dmfi |
|- ( c e. Fin -> dom c e. Fin ) |
243 |
241 242
|
syl |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> dom c e. Fin ) |
244 |
240 243
|
elind |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> dom c e. ( ~P A i^i Fin ) ) |
245 |
|
ovex |
|- ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) e. _V |
246 |
245
|
a1i |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) e. _V ) |
247 |
|
mpteq1 |
|- ( a = dom c -> ( j e. a |-> sum* k e. B C ) = ( j e. dom c |-> sum* k e. B C ) ) |
248 |
247
|
oveq2d |
|- ( a = dom c -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) |
249 |
192 248
|
elrnmpt1s |
|- ( ( dom c e. ( ~P A i^i Fin ) /\ ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) e. _V ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) |
250 |
244 246 249
|
syl2anc |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) ) |
251 |
|
simpr |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ t = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) -> t = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) |
252 |
251
|
breq2d |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ t = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) -> ( s < t <-> s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) ) |
253 |
|
simpllr |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> s e. RR* ) |
254 |
16
|
a1i |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( RR*s |`s ( 0 [,] +oo ) ) e. CMnd ) |
255 |
|
nfcv |
|- F/_ z ( RR*s |`s ( 0 [,] +oo ) ) |
256 |
|
nfcv |
|- F/_ z gsum |
257 |
|
nfmpt1 |
|- F/_ z ( z e. c |-> F ) |
258 |
255 256 257
|
nfov |
|- F/_ z ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) |
259 |
110 111 258
|
nfbr |
|- F/ z s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) |
260 |
109 259
|
nfan |
|- F/ z ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
261 |
|
nfv |
|- F/ z c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |
262 |
260 261
|
nfan |
|- F/ z ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) |
263 |
|
simp-4l |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ z e. c ) -> ph ) |
264 |
224 225
|
syl |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> c C_ U_ j e. A ( { j } X. B ) ) |
265 |
264
|
sselda |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ z e. c ) -> z e. U_ j e. A ( { j } X. B ) ) |
266 |
263 265 49
|
syl2anc |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ z e. c ) -> F e. ( 0 [,] +oo ) ) |
267 |
266
|
ex |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( z e. c -> F e. ( 0 [,] +oo ) ) ) |
268 |
262 267
|
ralrimi |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> A. z e. c F e. ( 0 [,] +oo ) ) |
269 |
15 254 241 268
|
gsummptcl |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. ( 0 [,] +oo ) ) |
270 |
14 269
|
sselid |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) e. RR* ) |
271 |
|
nfv |
|- F/ j ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
272 |
|
nfcv |
|- F/_ j c |
273 |
30
|
nfpw |
|- F/_ j ~P U_ j e. A ( { j } X. B ) |
274 |
|
nfcv |
|- F/_ j Fin |
275 |
273 274
|
nfin |
|- F/_ j ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |
276 |
272 275
|
nfel |
|- F/ j c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |
277 |
271 276
|
nfan |
|- F/ j ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) |
278 |
|
simpll |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> ph ) |
279 |
79 236
|
syl |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> dom c C_ A ) |
280 |
279
|
sselda |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> j e. A ) |
281 |
278 280 161
|
syl2anc |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> sum* k e. B C e. ( 0 [,] +oo ) ) |
282 |
281
|
adantllr |
|- ( ( ( ( ph /\ s e. RR* ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> sum* k e. B C e. ( 0 [,] +oo ) ) |
283 |
282
|
adantllr |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> sum* k e. B C e. ( 0 [,] +oo ) ) |
284 |
283
|
ex |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( j e. dom c -> sum* k e. B C e. ( 0 [,] +oo ) ) ) |
285 |
277 284
|
ralrimi |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> A. j e. dom c sum* k e. B C e. ( 0 [,] +oo ) ) |
286 |
15 254 243 285
|
gsummptcl |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) e. ( 0 [,] +oo ) ) |
287 |
14 286
|
sselid |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) e. RR* ) |
288 |
|
simplr |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
289 |
28 276
|
nfan |
|- F/ j ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) |
290 |
|
id |
|- ( c C_ U_ j e. A ( { j } X. B ) -> c C_ U_ j e. A ( { j } X. B ) ) |
291 |
|
xpss |
|- ( { j } X. B ) C_ ( _V X. _V ) |
292 |
291
|
rgenw |
|- A. j e. A ( { j } X. B ) C_ ( _V X. _V ) |
293 |
|
iunss |
|- ( U_ j e. A ( { j } X. B ) C_ ( _V X. _V ) <-> A. j e. A ( { j } X. B ) C_ ( _V X. _V ) ) |
294 |
292 293
|
mpbir |
|- U_ j e. A ( { j } X. B ) C_ ( _V X. _V ) |
295 |
294
|
a1i |
|- ( c C_ U_ j e. A ( { j } X. B ) -> U_ j e. A ( { j } X. B ) C_ ( _V X. _V ) ) |
296 |
290 295
|
sstrd |
|- ( c C_ U_ j e. A ( { j } X. B ) -> c C_ ( _V X. _V ) ) |
297 |
79 296
|
syl |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> c C_ ( _V X. _V ) ) |
298 |
|
df-rel |
|- ( Rel c <-> c C_ ( _V X. _V ) ) |
299 |
297 298
|
sylibr |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> Rel c ) |
300 |
1 289 15 2 299 19 17 50
|
gsummpt2d |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( k e. ( c " { j } ) |-> C ) ) ) ) ) |
301 |
|
nfcv |
|- F/_ j dom c |
302 |
237
|
a1i |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> dom c e. _V ) |
303 |
278
|
adantr |
|- ( ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) /\ k e. ( c " { j } ) ) -> ph ) |
304 |
280
|
adantr |
|- ( ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) /\ k e. ( c " { j } ) ) -> j e. A ) |
305 |
79
|
adantr |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> c C_ U_ j e. A ( { j } X. B ) ) |
306 |
|
imass1 |
|- ( c C_ U_ j e. A ( { j } X. B ) -> ( c " { j } ) C_ ( U_ j e. A ( { j } X. B ) " { j } ) ) |
307 |
305 306
|
syl |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> ( c " { j } ) C_ ( U_ j e. A ( { j } X. B ) " { j } ) ) |
308 |
3 4
|
iunsnima |
|- ( ( ph /\ j e. A ) -> ( U_ j e. A ( { j } X. B ) " { j } ) = B ) |
309 |
278 280 308
|
syl2anc |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> ( U_ j e. A ( { j } X. B ) " { j } ) = B ) |
310 |
307 309
|
sseqtrd |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> ( c " { j } ) C_ B ) |
311 |
310
|
sselda |
|- ( ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) /\ k e. ( c " { j } ) ) -> k e. B ) |
312 |
303 304 311 5
|
syl12anc |
|- ( ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) /\ k e. ( c " { j } ) ) -> C e. ( 0 [,] +oo ) ) |
313 |
312
|
ralrimiva |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> A. k e. ( c " { j } ) C e. ( 0 [,] +oo ) ) |
314 |
|
imaexg |
|- ( c e. _V -> ( c " { j } ) e. _V ) |
315 |
23 314
|
ax-mp |
|- ( c " { j } ) e. _V |
316 |
|
nfcv |
|- F/_ k ( c " { j } ) |
317 |
316
|
esumcl |
|- ( ( ( c " { j } ) e. _V /\ A. k e. ( c " { j } ) C e. ( 0 [,] +oo ) ) -> sum* k e. ( c " { j } ) C e. ( 0 [,] +oo ) ) |
318 |
315 317
|
mpan |
|- ( A. k e. ( c " { j } ) C e. ( 0 [,] +oo ) -> sum* k e. ( c " { j } ) C e. ( 0 [,] +oo ) ) |
319 |
313 318
|
syl |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> sum* k e. ( c " { j } ) C e. ( 0 [,] +oo ) ) |
320 |
|
nfv |
|- F/ k ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) |
321 |
278 280 4
|
syl2anc |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> B e. W ) |
322 |
278
|
adantr |
|- ( ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) /\ k e. B ) -> ph ) |
323 |
280
|
adantr |
|- ( ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) /\ k e. B ) -> j e. A ) |
324 |
|
simpr |
|- ( ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) /\ k e. B ) -> k e. B ) |
325 |
322 323 324 5
|
syl12anc |
|- ( ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) /\ k e. B ) -> C e. ( 0 [,] +oo ) ) |
326 |
320 321 325 310
|
esummono |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> sum* k e. ( c " { j } ) C <_ sum* k e. B C ) |
327 |
289 301 302 319 281 326
|
esumlef |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> sum* j e. dom c sum* k e. ( c " { j } ) C <_ sum* j e. dom c sum* k e. B C ) |
328 |
19 242
|
syl |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> dom c e. Fin ) |
329 |
289 301 328 319
|
esumgsum |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> sum* j e. dom c sum* k e. ( c " { j } ) C = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. ( c " { j } ) C ) ) ) |
330 |
19
|
adantr |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> c e. Fin ) |
331 |
|
imafi2 |
|- ( c e. Fin -> ( c " { j } ) e. Fin ) |
332 |
330 331
|
syl |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> ( c " { j } ) e. Fin ) |
333 |
320 316 332 312
|
esumgsum |
|- ( ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ j e. dom c ) -> sum* k e. ( c " { j } ) C = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( k e. ( c " { j } ) |-> C ) ) ) |
334 |
289 333
|
mpteq2da |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( j e. dom c |-> sum* k e. ( c " { j } ) C ) = ( j e. dom c |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( k e. ( c " { j } ) |-> C ) ) ) ) |
335 |
334
|
oveq2d |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. ( c " { j } ) C ) ) = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( k e. ( c " { j } ) |-> C ) ) ) ) ) |
336 |
329 335
|
eqtrd |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> sum* j e. dom c sum* k e. ( c " { j } ) C = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( k e. ( c " { j } ) |-> C ) ) ) ) ) |
337 |
289 301 328 281
|
esumgsum |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> sum* j e. dom c sum* k e. B C = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) |
338 |
327 336 337
|
3brtr3d |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( k e. ( c " { j } ) |-> C ) ) ) ) <_ ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) |
339 |
300 338
|
eqbrtrd |
|- ( ( ph /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) <_ ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) |
340 |
339
|
adantlr |
|- ( ( ( ph /\ s e. RR* ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) <_ ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) |
341 |
340
|
adantlr |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) <_ ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) |
342 |
253 270 287 288 341
|
xrltletrd |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. dom c |-> sum* k e. B C ) ) ) |
343 |
250 252 342
|
rspcedvd |
|- ( ( ( ( ph /\ s e. RR* ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> E. t e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) s < t ) |
344 |
343
|
adantllr |
|- ( ( ( ( ( ph /\ s e. RR* ) /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) /\ s < ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) -> E. t e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) s < t ) |
345 |
218 221 222 344
|
syl21anc |
|- ( ( ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) /\ c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) ) /\ u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> E. t e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) s < t ) |
346 |
55 89
|
elrnmpti |
|- ( u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) <-> E. c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
347 |
346
|
biimpi |
|- ( u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) -> E. c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
348 |
347
|
ad2antlr |
|- ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) -> E. c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) u = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) |
349 |
209 345 348
|
r19.29af |
|- ( ( ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) /\ u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) ) /\ s < u ) -> E. t e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) s < t ) |
350 |
7 134
|
suplub |
|- ( ph -> ( ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) ) |
351 |
350
|
imp |
|- ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) -> E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t ) |
352 |
|
breq2 |
|- ( t = u -> ( s < t <-> s < u ) ) |
353 |
352
|
cbvrexvw |
|- ( E. t e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < t <-> E. u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < u ) |
354 |
351 353
|
sylib |
|- ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) -> E. u e. ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) s < u ) |
355 |
349 354
|
r19.29a |
|- ( ( ph /\ ( s e. RR* /\ s < sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) ) -> E. t e. ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) s < t ) |
356 |
7 135 197 355
|
eqsupd |
|- ( ph -> sup ( ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) , RR* , < ) = sup ( ran ( c e. ( ~P U_ j e. A ( { j } X. B ) i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( z e. c |-> F ) ) ) , RR* , < ) ) |
357 |
|
nfcv |
|- F/_ j A |
358 |
|
eqidd |
|- ( ( ph /\ a e. ( ~P A i^i Fin ) ) -> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) = ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) |
359 |
28 357 3 161 358
|
esumval |
|- ( ph -> sum* j e. A sum* k e. B C = sup ( ran ( a e. ( ~P A i^i Fin ) |-> ( ( RR*s |`s ( 0 [,] +oo ) ) gsum ( j e. a |-> sum* k e. B C ) ) ) , RR* , < ) ) |
360 |
356 359 182
|
3eqtr4d |
|- ( ph -> sum* j e. A sum* k e. B C = sum* z e. U_ j e. A ( { j } X. B ) F ) |