Step |
Hyp |
Ref |
Expression |
1 |
|
sge0split.a |
|- ( ph -> A e. V ) |
2 |
|
sge0split.b |
|- ( ph -> B e. W ) |
3 |
|
sge0split.u |
|- U = ( A u. B ) |
4 |
|
sge0split.in0 |
|- ( ph -> ( A i^i B ) = (/) ) |
5 |
|
sge0split.f |
|- ( ph -> F : U --> ( 0 [,] +oo ) ) |
6 |
1
|
adantr |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> A e. V ) |
7 |
2
|
adantr |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> B e. W ) |
8 |
4
|
adantr |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> ( A i^i B ) = (/) ) |
9 |
5
|
adantr |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> F : U --> ( 0 [,] +oo ) ) |
10 |
|
simpr |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> ( sum^ ` F ) e. RR ) |
11 |
6 7 3 8 9 10
|
sge0resplit |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> ( sum^ ` F ) = ( ( sum^ ` ( F |` A ) ) + ( sum^ ` ( F |` B ) ) ) ) |
12 |
|
unexg |
|- ( ( A e. V /\ B e. W ) -> ( A u. B ) e. _V ) |
13 |
1 2 12
|
syl2anc |
|- ( ph -> ( A u. B ) e. _V ) |
14 |
3 13
|
eqeltrid |
|- ( ph -> U e. _V ) |
15 |
14
|
adantr |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> U e. _V ) |
16 |
15 9 10
|
sge0ssre |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> ( sum^ ` ( F |` A ) ) e. RR ) |
17 |
15 9 10
|
sge0ssre |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> ( sum^ ` ( F |` B ) ) e. RR ) |
18 |
|
rexadd |
|- ( ( ( sum^ ` ( F |` A ) ) e. RR /\ ( sum^ ` ( F |` B ) ) e. RR ) -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) = ( ( sum^ ` ( F |` A ) ) + ( sum^ ` ( F |` B ) ) ) ) |
19 |
16 17 18
|
syl2anc |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) = ( ( sum^ ` ( F |` A ) ) + ( sum^ ` ( F |` B ) ) ) ) |
20 |
19
|
eqcomd |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> ( ( sum^ ` ( F |` A ) ) + ( sum^ ` ( F |` B ) ) ) = ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
21 |
11 20
|
eqtrd |
|- ( ( ph /\ ( sum^ ` F ) e. RR ) -> ( sum^ ` F ) = ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
22 |
|
simpl |
|- ( ( ph /\ -. ( sum^ ` F ) e. RR ) -> ph ) |
23 |
|
simpr |
|- ( ( ph /\ -. ( sum^ ` F ) e. RR ) -> -. ( sum^ ` F ) e. RR ) |
24 |
14 5
|
sge0repnf |
|- ( ph -> ( ( sum^ ` F ) e. RR <-> -. ( sum^ ` F ) = +oo ) ) |
25 |
24
|
adantr |
|- ( ( ph /\ -. ( sum^ ` F ) e. RR ) -> ( ( sum^ ` F ) e. RR <-> -. ( sum^ ` F ) = +oo ) ) |
26 |
23 25
|
mtbid |
|- ( ( ph /\ -. ( sum^ ` F ) e. RR ) -> -. -. ( sum^ ` F ) = +oo ) |
27 |
26
|
notnotrd |
|- ( ( ph /\ -. ( sum^ ` F ) e. RR ) -> ( sum^ ` F ) = +oo ) |
28 |
14 5
|
sge0xrcl |
|- ( ph -> ( sum^ ` F ) e. RR* ) |
29 |
28
|
adantr |
|- ( ( ph /\ ( sum^ ` F ) = +oo ) -> ( sum^ ` F ) e. RR* ) |
30 |
|
ssun1 |
|- A C_ ( A u. B ) |
31 |
30 3
|
sseqtrri |
|- A C_ U |
32 |
31
|
a1i |
|- ( ph -> A C_ U ) |
33 |
5 32
|
fssresd |
|- ( ph -> ( F |` A ) : A --> ( 0 [,] +oo ) ) |
34 |
1 33
|
sge0xrcl |
|- ( ph -> ( sum^ ` ( F |` A ) ) e. RR* ) |
35 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
36 |
|
ssun2 |
|- B C_ ( A u. B ) |
37 |
36 3
|
sseqtrri |
|- B C_ U |
38 |
37
|
a1i |
|- ( ph -> B C_ U ) |
39 |
5 38
|
fssresd |
|- ( ph -> ( F |` B ) : B --> ( 0 [,] +oo ) ) |
40 |
2 39
|
sge0cl |
|- ( ph -> ( sum^ ` ( F |` B ) ) e. ( 0 [,] +oo ) ) |
41 |
35 40
|
sseldi |
|- ( ph -> ( sum^ ` ( F |` B ) ) e. RR* ) |
42 |
34 41
|
xaddcld |
|- ( ph -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) e. RR* ) |
43 |
42
|
adantr |
|- ( ( ph /\ ( sum^ ` F ) = +oo ) -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) e. RR* ) |
44 |
|
pnfxr |
|- +oo e. RR* |
45 |
|
eqid |
|- +oo = +oo |
46 |
|
xreqle |
|- ( ( +oo e. RR* /\ +oo = +oo ) -> +oo <_ +oo ) |
47 |
44 45 46
|
mp2an |
|- +oo <_ +oo |
48 |
47
|
a1i |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> +oo <_ +oo ) |
49 |
14
|
adantr |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> U e. _V ) |
50 |
5
|
adantr |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> F : U --> ( 0 [,] +oo ) ) |
51 |
|
rnresss |
|- ran ( F |` A ) C_ ran F |
52 |
51
|
sseli |
|- ( +oo e. ran ( F |` A ) -> +oo e. ran F ) |
53 |
52
|
adantl |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> +oo e. ran F ) |
54 |
49 50 53
|
sge0pnfval |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> ( sum^ ` F ) = +oo ) |
55 |
|
xrge0neqmnf |
|- ( ( sum^ ` ( F |` B ) ) e. ( 0 [,] +oo ) -> ( sum^ ` ( F |` B ) ) =/= -oo ) |
56 |
40 55
|
syl |
|- ( ph -> ( sum^ ` ( F |` B ) ) =/= -oo ) |
57 |
|
xaddpnf2 |
|- ( ( ( sum^ ` ( F |` B ) ) e. RR* /\ ( sum^ ` ( F |` B ) ) =/= -oo ) -> ( +oo +e ( sum^ ` ( F |` B ) ) ) = +oo ) |
58 |
41 56 57
|
syl2anc |
|- ( ph -> ( +oo +e ( sum^ ` ( F |` B ) ) ) = +oo ) |
59 |
58
|
eqcomd |
|- ( ph -> +oo = ( +oo +e ( sum^ ` ( F |` B ) ) ) ) |
60 |
59
|
adantr |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> +oo = ( +oo +e ( sum^ ` ( F |` B ) ) ) ) |
61 |
1
|
adantr |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> A e. V ) |
62 |
33
|
adantr |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> ( F |` A ) : A --> ( 0 [,] +oo ) ) |
63 |
|
simpr |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> +oo e. ran ( F |` A ) ) |
64 |
61 62 63
|
sge0pnfval |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> ( sum^ ` ( F |` A ) ) = +oo ) |
65 |
64
|
oveq1d |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) = ( +oo +e ( sum^ ` ( F |` B ) ) ) ) |
66 |
60 54 65
|
3eqtr4d |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> ( sum^ ` F ) = ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
67 |
66 54
|
eqtr3d |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) = +oo ) |
68 |
54 67
|
breq12d |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> ( ( sum^ ` F ) <_ ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) <-> +oo <_ +oo ) ) |
69 |
48 68
|
mpbird |
|- ( ( ph /\ +oo e. ran ( F |` A ) ) -> ( sum^ ` F ) <_ ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
70 |
47
|
a1i |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> +oo <_ +oo ) |
71 |
14
|
adantr |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> U e. _V ) |
72 |
5
|
adantr |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> F : U --> ( 0 [,] +oo ) ) |
73 |
|
rnresss |
|- ran ( F |` B ) C_ ran F |
74 |
73
|
sseli |
|- ( +oo e. ran ( F |` B ) -> +oo e. ran F ) |
75 |
74
|
adantl |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> +oo e. ran F ) |
76 |
71 72 75
|
sge0pnfval |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> ( sum^ ` F ) = +oo ) |
77 |
2
|
adantr |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> B e. W ) |
78 |
39
|
adantr |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> ( F |` B ) : B --> ( 0 [,] +oo ) ) |
79 |
|
simpr |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> +oo e. ran ( F |` B ) ) |
80 |
77 78 79
|
sge0pnfval |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> ( sum^ ` ( F |` B ) ) = +oo ) |
81 |
80
|
oveq2d |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) = ( ( sum^ ` ( F |` A ) ) +e +oo ) ) |
82 |
1 33
|
sge0cl |
|- ( ph -> ( sum^ ` ( F |` A ) ) e. ( 0 [,] +oo ) ) |
83 |
|
xrge0neqmnf |
|- ( ( sum^ ` ( F |` A ) ) e. ( 0 [,] +oo ) -> ( sum^ ` ( F |` A ) ) =/= -oo ) |
84 |
82 83
|
syl |
|- ( ph -> ( sum^ ` ( F |` A ) ) =/= -oo ) |
85 |
|
xaddpnf1 |
|- ( ( ( sum^ ` ( F |` A ) ) e. RR* /\ ( sum^ ` ( F |` A ) ) =/= -oo ) -> ( ( sum^ ` ( F |` A ) ) +e +oo ) = +oo ) |
86 |
34 84 85
|
syl2anc |
|- ( ph -> ( ( sum^ ` ( F |` A ) ) +e +oo ) = +oo ) |
87 |
86
|
adantr |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> ( ( sum^ ` ( F |` A ) ) +e +oo ) = +oo ) |
88 |
81 87
|
eqtrd |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) = +oo ) |
89 |
76 88
|
breq12d |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> ( ( sum^ ` F ) <_ ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) <-> +oo <_ +oo ) ) |
90 |
70 89
|
mpbird |
|- ( ( ph /\ +oo e. ran ( F |` B ) ) -> ( sum^ ` F ) <_ ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
91 |
90
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ +oo e. ran ( F |` B ) ) -> ( sum^ ` F ) <_ ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
92 |
|
simpr |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ z e. ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) ) -> z e. ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) ) |
93 |
|
vex |
|- z e. _V |
94 |
|
eqid |
|- ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) = ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) |
95 |
94
|
elrnmpt |
|- ( z e. _V -> ( z e. ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) <-> E. x e. ( ~P U i^i Fin ) z = sum_ y e. x ( F ` y ) ) ) |
96 |
93 95
|
ax-mp |
|- ( z e. ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) <-> E. x e. ( ~P U i^i Fin ) z = sum_ y e. x ( F ` y ) ) |
97 |
92 96
|
sylib |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ z e. ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) ) -> E. x e. ( ~P U i^i Fin ) z = sum_ y e. x ( F ` y ) ) |
98 |
|
simp3 |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) /\ z = sum_ y e. x ( F ` y ) ) -> z = sum_ y e. x ( F ` y ) ) |
99 |
|
inss1 |
|- ( ( x i^i A ) i^i ( x i^i B ) ) C_ ( x i^i A ) |
100 |
|
inss2 |
|- ( x i^i A ) C_ A |
101 |
99 100
|
sstri |
|- ( ( x i^i A ) i^i ( x i^i B ) ) C_ A |
102 |
|
inss2 |
|- ( ( x i^i A ) i^i ( x i^i B ) ) C_ ( x i^i B ) |
103 |
|
inss2 |
|- ( x i^i B ) C_ B |
104 |
102 103
|
sstri |
|- ( ( x i^i A ) i^i ( x i^i B ) ) C_ B |
105 |
101 104
|
ssini |
|- ( ( x i^i A ) i^i ( x i^i B ) ) C_ ( A i^i B ) |
106 |
105
|
a1i |
|- ( ph -> ( ( x i^i A ) i^i ( x i^i B ) ) C_ ( A i^i B ) ) |
107 |
106 4
|
sseqtrd |
|- ( ph -> ( ( x i^i A ) i^i ( x i^i B ) ) C_ (/) ) |
108 |
|
ss0 |
|- ( ( ( x i^i A ) i^i ( x i^i B ) ) C_ (/) -> ( ( x i^i A ) i^i ( x i^i B ) ) = (/) ) |
109 |
107 108
|
syl |
|- ( ph -> ( ( x i^i A ) i^i ( x i^i B ) ) = (/) ) |
110 |
109
|
ad3antrrr |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( ( x i^i A ) i^i ( x i^i B ) ) = (/) ) |
111 |
|
indi |
|- ( x i^i ( A u. B ) ) = ( ( x i^i A ) u. ( x i^i B ) ) |
112 |
111
|
eqcomi |
|- ( ( x i^i A ) u. ( x i^i B ) ) = ( x i^i ( A u. B ) ) |
113 |
112
|
a1i |
|- ( x e. ( ~P U i^i Fin ) -> ( ( x i^i A ) u. ( x i^i B ) ) = ( x i^i ( A u. B ) ) ) |
114 |
3
|
eqcomi |
|- ( A u. B ) = U |
115 |
114
|
ineq2i |
|- ( x i^i ( A u. B ) ) = ( x i^i U ) |
116 |
115
|
a1i |
|- ( x e. ( ~P U i^i Fin ) -> ( x i^i ( A u. B ) ) = ( x i^i U ) ) |
117 |
|
elinel1 |
|- ( x e. ( ~P U i^i Fin ) -> x e. ~P U ) |
118 |
|
elpwi |
|- ( x e. ~P U -> x C_ U ) |
119 |
117 118
|
syl |
|- ( x e. ( ~P U i^i Fin ) -> x C_ U ) |
120 |
|
df-ss |
|- ( x C_ U <-> ( x i^i U ) = x ) |
121 |
120
|
biimpi |
|- ( x C_ U -> ( x i^i U ) = x ) |
122 |
119 121
|
syl |
|- ( x e. ( ~P U i^i Fin ) -> ( x i^i U ) = x ) |
123 |
113 116 122
|
3eqtrrd |
|- ( x e. ( ~P U i^i Fin ) -> x = ( ( x i^i A ) u. ( x i^i B ) ) ) |
124 |
123
|
adantl |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> x = ( ( x i^i A ) u. ( x i^i B ) ) ) |
125 |
|
elinel2 |
|- ( x e. ( ~P U i^i Fin ) -> x e. Fin ) |
126 |
125
|
adantl |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> x e. Fin ) |
127 |
|
rge0ssre |
|- ( 0 [,) +oo ) C_ RR |
128 |
5
|
ad2antrr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> F : U --> ( 0 [,] +oo ) ) |
129 |
|
pm4.56 |
|- ( ( -. +oo e. ran ( F |` A ) /\ -. +oo e. ran ( F |` B ) ) <-> -. ( +oo e. ran ( F |` A ) \/ +oo e. ran ( F |` B ) ) ) |
130 |
129
|
biimpi |
|- ( ( -. +oo e. ran ( F |` A ) /\ -. +oo e. ran ( F |` B ) ) -> -. ( +oo e. ran ( F |` A ) \/ +oo e. ran ( F |` B ) ) ) |
131 |
|
elun |
|- ( +oo e. ( ran ( F |` A ) u. ran ( F |` B ) ) <-> ( +oo e. ran ( F |` A ) \/ +oo e. ran ( F |` B ) ) ) |
132 |
130 131
|
sylnibr |
|- ( ( -. +oo e. ran ( F |` A ) /\ -. +oo e. ran ( F |` B ) ) -> -. +oo e. ( ran ( F |` A ) u. ran ( F |` B ) ) ) |
133 |
132
|
adantll |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> -. +oo e. ( ran ( F |` A ) u. ran ( F |` B ) ) ) |
134 |
|
rnresun |
|- ran ( F |` ( A u. B ) ) = ( ran ( F |` A ) u. ran ( F |` B ) ) |
135 |
134
|
eqcomi |
|- ( ran ( F |` A ) u. ran ( F |` B ) ) = ran ( F |` ( A u. B ) ) |
136 |
135
|
a1i |
|- ( ph -> ( ran ( F |` A ) u. ran ( F |` B ) ) = ran ( F |` ( A u. B ) ) ) |
137 |
114
|
reseq2i |
|- ( F |` ( A u. B ) ) = ( F |` U ) |
138 |
137
|
rneqi |
|- ran ( F |` ( A u. B ) ) = ran ( F |` U ) |
139 |
138
|
a1i |
|- ( ph -> ran ( F |` ( A u. B ) ) = ran ( F |` U ) ) |
140 |
|
ffn |
|- ( F : U --> ( 0 [,] +oo ) -> F Fn U ) |
141 |
|
fnresdm |
|- ( F Fn U -> ( F |` U ) = F ) |
142 |
5 140 141
|
3syl |
|- ( ph -> ( F |` U ) = F ) |
143 |
142
|
rneqd |
|- ( ph -> ran ( F |` U ) = ran F ) |
144 |
136 139 143
|
3eqtrd |
|- ( ph -> ( ran ( F |` A ) u. ran ( F |` B ) ) = ran F ) |
145 |
144
|
ad2antrr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( ran ( F |` A ) u. ran ( F |` B ) ) = ran F ) |
146 |
133 145
|
neleqtrd |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> -. +oo e. ran F ) |
147 |
128 146
|
fge0iccico |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> F : U --> ( 0 [,) +oo ) ) |
148 |
147
|
ad2antrr |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. x ) -> F : U --> ( 0 [,) +oo ) ) |
149 |
119
|
adantr |
|- ( ( x e. ( ~P U i^i Fin ) /\ y e. x ) -> x C_ U ) |
150 |
|
simpr |
|- ( ( x e. ( ~P U i^i Fin ) /\ y e. x ) -> y e. x ) |
151 |
149 150
|
sseldd |
|- ( ( x e. ( ~P U i^i Fin ) /\ y e. x ) -> y e. U ) |
152 |
151
|
adantll |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. x ) -> y e. U ) |
153 |
148 152
|
ffvelrnd |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. ( 0 [,) +oo ) ) |
154 |
127 153
|
sseldi |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. RR ) |
155 |
154
|
recnd |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. x ) -> ( F ` y ) e. CC ) |
156 |
110 124 126 155
|
fsumsplit |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. x ( F ` y ) = ( sum_ y e. ( x i^i A ) ( F ` y ) + sum_ y e. ( x i^i B ) ( F ` y ) ) ) |
157 |
|
infi |
|- ( x e. Fin -> ( x i^i A ) e. Fin ) |
158 |
125 157
|
syl |
|- ( x e. ( ~P U i^i Fin ) -> ( x i^i A ) e. Fin ) |
159 |
158
|
adantl |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( x i^i A ) e. Fin ) |
160 |
|
simpl |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. ( x i^i A ) ) -> ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) ) |
161 |
|
elinel1 |
|- ( y e. ( x i^i A ) -> y e. x ) |
162 |
161
|
adantl |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. ( x i^i A ) ) -> y e. x ) |
163 |
160 162 154
|
syl2anc |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. ( x i^i A ) ) -> ( F ` y ) e. RR ) |
164 |
159 163
|
fsumrecl |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i A ) ( F ` y ) e. RR ) |
165 |
|
infi |
|- ( x e. Fin -> ( x i^i B ) e. Fin ) |
166 |
125 165
|
syl |
|- ( x e. ( ~P U i^i Fin ) -> ( x i^i B ) e. Fin ) |
167 |
166
|
adantl |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( x i^i B ) e. Fin ) |
168 |
|
simpl |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. ( x i^i B ) ) -> ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) ) |
169 |
|
elinel1 |
|- ( y e. ( x i^i B ) -> y e. x ) |
170 |
169
|
adantl |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. ( x i^i B ) ) -> y e. x ) |
171 |
168 170 154
|
syl2anc |
|- ( ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. ( x i^i B ) ) -> ( F ` y ) e. RR ) |
172 |
167 171
|
fsumrecl |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i B ) ( F ` y ) e. RR ) |
173 |
|
rexadd |
|- ( ( sum_ y e. ( x i^i A ) ( F ` y ) e. RR /\ sum_ y e. ( x i^i B ) ( F ` y ) e. RR ) -> ( sum_ y e. ( x i^i A ) ( F ` y ) +e sum_ y e. ( x i^i B ) ( F ` y ) ) = ( sum_ y e. ( x i^i A ) ( F ` y ) + sum_ y e. ( x i^i B ) ( F ` y ) ) ) |
174 |
164 172 173
|
syl2anc |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( sum_ y e. ( x i^i A ) ( F ` y ) +e sum_ y e. ( x i^i B ) ( F ` y ) ) = ( sum_ y e. ( x i^i A ) ( F ` y ) + sum_ y e. ( x i^i B ) ( F ` y ) ) ) |
175 |
174
|
eqcomd |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( sum_ y e. ( x i^i A ) ( F ` y ) + sum_ y e. ( x i^i B ) ( F ` y ) ) = ( sum_ y e. ( x i^i A ) ( F ` y ) +e sum_ y e. ( x i^i B ) ( F ` y ) ) ) |
176 |
156 175
|
eqtrd |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. x ( F ` y ) = ( sum_ y e. ( x i^i A ) ( F ` y ) +e sum_ y e. ( x i^i B ) ( F ` y ) ) ) |
177 |
|
ressxr |
|- RR C_ RR* |
178 |
177 164
|
sseldi |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i A ) ( F ` y ) e. RR* ) |
179 |
177 172
|
sseldi |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i B ) ( F ` y ) e. RR* ) |
180 |
1
|
adantr |
|- ( ( ph /\ -. +oo e. ran ( F |` A ) ) -> A e. V ) |
181 |
33
|
adantr |
|- ( ( ph /\ -. +oo e. ran ( F |` A ) ) -> ( F |` A ) : A --> ( 0 [,] +oo ) ) |
182 |
|
simpr |
|- ( ( ph /\ -. +oo e. ran ( F |` A ) ) -> -. +oo e. ran ( F |` A ) ) |
183 |
181 182
|
fge0iccico |
|- ( ( ph /\ -. +oo e. ran ( F |` A ) ) -> ( F |` A ) : A --> ( 0 [,) +oo ) ) |
184 |
180 183
|
sge0reval |
|- ( ( ph /\ -. +oo e. ran ( F |` A ) ) -> ( sum^ ` ( F |` A ) ) = sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) ) |
185 |
184
|
eqcomd |
|- ( ( ph /\ -. +oo e. ran ( F |` A ) ) -> sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) = ( sum^ ` ( F |` A ) ) ) |
186 |
34
|
adantr |
|- ( ( ph /\ -. +oo e. ran ( F |` A ) ) -> ( sum^ ` ( F |` A ) ) e. RR* ) |
187 |
185 186
|
eqeltrd |
|- ( ( ph /\ -. +oo e. ran ( F |` A ) ) -> sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) e. RR* ) |
188 |
187
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) e. RR* ) |
189 |
2
|
adantr |
|- ( ( ph /\ -. +oo e. ran ( F |` B ) ) -> B e. W ) |
190 |
39
|
adantr |
|- ( ( ph /\ -. +oo e. ran ( F |` B ) ) -> ( F |` B ) : B --> ( 0 [,] +oo ) ) |
191 |
|
simpr |
|- ( ( ph /\ -. +oo e. ran ( F |` B ) ) -> -. +oo e. ran ( F |` B ) ) |
192 |
190 191
|
fge0iccico |
|- ( ( ph /\ -. +oo e. ran ( F |` B ) ) -> ( F |` B ) : B --> ( 0 [,) +oo ) ) |
193 |
189 192
|
sge0reval |
|- ( ( ph /\ -. +oo e. ran ( F |` B ) ) -> ( sum^ ` ( F |` B ) ) = sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) |
194 |
193
|
eqcomd |
|- ( ( ph /\ -. +oo e. ran ( F |` B ) ) -> sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) = ( sum^ ` ( F |` B ) ) ) |
195 |
41
|
adantr |
|- ( ( ph /\ -. +oo e. ran ( F |` B ) ) -> ( sum^ ` ( F |` B ) ) e. RR* ) |
196 |
194 195
|
eqeltrd |
|- ( ( ph /\ -. +oo e. ran ( F |` B ) ) -> sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) e. RR* ) |
197 |
196
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) e. RR* ) |
198 |
188 197
|
jca |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) e. RR* /\ sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) e. RR* ) ) |
199 |
198
|
adantr |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) e. RR* /\ sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) e. RR* ) ) |
200 |
178 179 199
|
jca31 |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( ( sum_ y e. ( x i^i A ) ( F ` y ) e. RR* /\ sum_ y e. ( x i^i B ) ( F ` y ) e. RR* ) /\ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) e. RR* /\ sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) e. RR* ) ) ) |
201 |
180
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> A e. V ) |
202 |
181
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( F |` A ) : A --> ( 0 [,] +oo ) ) |
203 |
182
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> -. +oo e. ran ( F |` A ) ) |
204 |
202 203
|
fge0iccico |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( F |` A ) : A --> ( 0 [,) +oo ) ) |
205 |
100
|
a1i |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( x i^i A ) C_ A ) |
206 |
158
|
adantl |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( x i^i A ) e. Fin ) |
207 |
201 204 205 206
|
fsumlesge0 |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i A ) ( ( F |` A ) ` y ) <_ ( sum^ ` ( F |` A ) ) ) |
208 |
100
|
sseli |
|- ( y e. ( x i^i A ) -> y e. A ) |
209 |
|
fvres |
|- ( y e. A -> ( ( F |` A ) ` y ) = ( F ` y ) ) |
210 |
208 209
|
syl |
|- ( y e. ( x i^i A ) -> ( ( F |` A ) ` y ) = ( F ` y ) ) |
211 |
210
|
adantl |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. ( x i^i A ) ) -> ( ( F |` A ) ` y ) = ( F ` y ) ) |
212 |
211
|
sumeq2dv |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i A ) ( ( F |` A ) ` y ) = sum_ y e. ( x i^i A ) ( F ` y ) ) |
213 |
184
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( sum^ ` ( F |` A ) ) = sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) ) |
214 |
212 213
|
breq12d |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( sum_ y e. ( x i^i A ) ( ( F |` A ) ` y ) <_ ( sum^ ` ( F |` A ) ) <-> sum_ y e. ( x i^i A ) ( F ` y ) <_ sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) ) ) |
215 |
207 214
|
mpbid |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i A ) ( F ` y ) <_ sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) ) |
216 |
215
|
adantlr |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i A ) ( F ` y ) <_ sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) ) |
217 |
189
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> B e. W ) |
218 |
190
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( F |` B ) : B --> ( 0 [,] +oo ) ) |
219 |
191
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> -. +oo e. ran ( F |` B ) ) |
220 |
218 219
|
fge0iccico |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( F |` B ) : B --> ( 0 [,) +oo ) ) |
221 |
103
|
a1i |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( x i^i B ) C_ B ) |
222 |
166
|
adantl |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( x i^i B ) e. Fin ) |
223 |
217 220 221 222
|
fsumlesge0 |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i B ) ( ( F |` B ) ` y ) <_ ( sum^ ` ( F |` B ) ) ) |
224 |
103
|
sseli |
|- ( y e. ( x i^i B ) -> y e. B ) |
225 |
|
fvres |
|- ( y e. B -> ( ( F |` B ) ` y ) = ( F ` y ) ) |
226 |
224 225
|
syl |
|- ( y e. ( x i^i B ) -> ( ( F |` B ) ` y ) = ( F ` y ) ) |
227 |
226
|
adantl |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) /\ y e. ( x i^i B ) ) -> ( ( F |` B ) ` y ) = ( F ` y ) ) |
228 |
227
|
sumeq2dv |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i B ) ( ( F |` B ) ` y ) = sum_ y e. ( x i^i B ) ( F ` y ) ) |
229 |
193
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( sum^ ` ( F |` B ) ) = sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) |
230 |
228 229
|
breq12d |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( sum_ y e. ( x i^i B ) ( ( F |` B ) ` y ) <_ ( sum^ ` ( F |` B ) ) <-> sum_ y e. ( x i^i B ) ( F ` y ) <_ sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) |
231 |
223 230
|
mpbid |
|- ( ( ( ph /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i B ) ( F ` y ) <_ sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) |
232 |
231
|
adantllr |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. ( x i^i B ) ( F ` y ) <_ sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) |
233 |
216 232
|
jca |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( sum_ y e. ( x i^i A ) ( F ` y ) <_ sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) /\ sum_ y e. ( x i^i B ) ( F ` y ) <_ sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) |
234 |
|
xle2add |
|- ( ( ( sum_ y e. ( x i^i A ) ( F ` y ) e. RR* /\ sum_ y e. ( x i^i B ) ( F ` y ) e. RR* ) /\ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) e. RR* /\ sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) e. RR* ) ) -> ( ( sum_ y e. ( x i^i A ) ( F ` y ) <_ sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) /\ sum_ y e. ( x i^i B ) ( F ` y ) <_ sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) -> ( sum_ y e. ( x i^i A ) ( F ` y ) +e sum_ y e. ( x i^i B ) ( F ` y ) ) <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) ) |
235 |
200 233 234
|
sylc |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> ( sum_ y e. ( x i^i A ) ( F ` y ) +e sum_ y e. ( x i^i B ) ( F ` y ) ) <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) |
236 |
176 235
|
eqbrtrd |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) ) -> sum_ y e. x ( F ` y ) <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) |
237 |
236
|
3adant3 |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) /\ z = sum_ y e. x ( F ` y ) ) -> sum_ y e. x ( F ` y ) <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) |
238 |
98 237
|
eqbrtrd |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ x e. ( ~P U i^i Fin ) /\ z = sum_ y e. x ( F ` y ) ) -> z <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) |
239 |
238
|
3exp |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( x e. ( ~P U i^i Fin ) -> ( z = sum_ y e. x ( F ` y ) -> z <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) ) ) |
240 |
239
|
rexlimdv |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( E. x e. ( ~P U i^i Fin ) z = sum_ y e. x ( F ` y ) -> z <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) ) |
241 |
240
|
adantr |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ z e. ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) ) -> ( E. x e. ( ~P U i^i Fin ) z = sum_ y e. x ( F ` y ) -> z <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) ) |
242 |
97 241
|
mpd |
|- ( ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) /\ z e. ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) ) -> z <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) |
243 |
242
|
ralrimiva |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> A. z e. ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) z <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) |
244 |
147
|
sge0rnre |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) C_ RR ) |
245 |
177
|
a1i |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> RR C_ RR* ) |
246 |
244 245
|
sstrd |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) C_ RR* ) |
247 |
188 197
|
xaddcld |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) e. RR* ) |
248 |
|
supxrleub |
|- ( ( ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) C_ RR* /\ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) e. RR* ) -> ( sup ( ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) <-> A. z e. ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) z <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) ) |
249 |
246 247 248
|
syl2anc |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( sup ( ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) <-> A. z e. ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) z <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) ) |
250 |
243 249
|
mpbird |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> sup ( ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) <_ ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) |
251 |
14
|
ad2antrr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> U e. _V ) |
252 |
251 147
|
sge0reval |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( sum^ ` F ) = sup ( ran ( x e. ( ~P U i^i Fin ) |-> sum_ y e. x ( F ` y ) ) , RR* , < ) ) |
253 |
184
|
adantr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( sum^ ` ( F |` A ) ) = sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) ) |
254 |
193
|
adantlr |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( sum^ ` ( F |` B ) ) = sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) |
255 |
253 254
|
oveq12d |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) = ( sup ( ran ( a e. ( ~P A i^i Fin ) |-> sum_ b e. a ( ( F |` A ) ` b ) ) , RR* , < ) +e sup ( ran ( c e. ( ~P B i^i Fin ) |-> sum_ d e. c ( ( F |` B ) ` d ) ) , RR* , < ) ) ) |
256 |
250 252 255
|
3brtr4d |
|- ( ( ( ph /\ -. +oo e. ran ( F |` A ) ) /\ -. +oo e. ran ( F |` B ) ) -> ( sum^ ` F ) <_ ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
257 |
91 256
|
pm2.61dan |
|- ( ( ph /\ -. +oo e. ran ( F |` A ) ) -> ( sum^ ` F ) <_ ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
258 |
69 257
|
pm2.61dan |
|- ( ph -> ( sum^ ` F ) <_ ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
259 |
258
|
adantr |
|- ( ( ph /\ ( sum^ ` F ) = +oo ) -> ( sum^ ` F ) <_ ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
260 |
|
pnfge |
|- ( ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) e. RR* -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) <_ +oo ) |
261 |
42 260
|
syl |
|- ( ph -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) <_ +oo ) |
262 |
261
|
adantr |
|- ( ( ph /\ ( sum^ ` F ) = +oo ) -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) <_ +oo ) |
263 |
|
id |
|- ( ( sum^ ` F ) = +oo -> ( sum^ ` F ) = +oo ) |
264 |
263
|
eqcomd |
|- ( ( sum^ ` F ) = +oo -> +oo = ( sum^ ` F ) ) |
265 |
264
|
adantl |
|- ( ( ph /\ ( sum^ ` F ) = +oo ) -> +oo = ( sum^ ` F ) ) |
266 |
262 265
|
breqtrd |
|- ( ( ph /\ ( sum^ ` F ) = +oo ) -> ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) <_ ( sum^ ` F ) ) |
267 |
29 43 259 266
|
xrletrid |
|- ( ( ph /\ ( sum^ ` F ) = +oo ) -> ( sum^ ` F ) = ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
268 |
22 27 267
|
syl2anc |
|- ( ( ph /\ -. ( sum^ ` F ) e. RR ) -> ( sum^ ` F ) = ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |
269 |
21 268
|
pm2.61dan |
|- ( ph -> ( sum^ ` F ) = ( ( sum^ ` ( F |` A ) ) +e ( sum^ ` ( F |` B ) ) ) ) |