Metamath Proof Explorer


Theorem esum2d

Description: Write a double extended sum as a sum over a two-dimensional region. Note that B ( j ) is a function of j . This can be seen as "slicing" the relation A . (Contributed by Thierry Arnoux, 17-May-2020)

Ref Expression
Hypotheses esum2d.0
|- F/_ k F
esum2d.1
|- ( z = <. j , k >. -> F = C )
esum2d.2
|- ( ph -> A e. V )
esum2d.3
|- ( ( ph /\ j e. A ) -> B e. W )
esum2d.4
|- ( ( ph /\ ( j e. A /\ k e. B ) ) -> C e. ( 0 [,] +oo ) )
Assertion esum2d
|- ( ph -> sum* j e. A sum* k e. B C = sum* z e. U_ j e. A ( { j } X. B ) F )

Proof

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 )