Step |
Hyp |
Ref |
Expression |
1 |
|
nfdisj1 |
|- F/ x Disj_ x e. A x |
2 |
|
nfv |
|- F/ x A e. Fin |
3 |
|
nfv |
|- F/ x A C_ Fin |
4 |
1 2 3
|
nf3an |
|- F/ x ( Disj_ x e. A x /\ A e. Fin /\ A C_ Fin ) |
5 |
|
simp2 |
|- ( ( Disj_ x e. A x /\ A e. Fin /\ A C_ Fin ) -> A e. Fin ) |
6 |
|
simp3 |
|- ( ( Disj_ x e. A x /\ A e. Fin /\ A C_ Fin ) -> A C_ Fin ) |
7 |
|
simp1 |
|- ( ( Disj_ x e. A x /\ A e. Fin /\ A C_ Fin ) -> Disj_ x e. A x ) |
8 |
4 5 6 7
|
hashunif |
|- ( ( Disj_ x e. A x /\ A e. Fin /\ A C_ Fin ) -> ( # ` U. A ) = sum_ x e. A ( # ` x ) ) |
9 |
|
simpl |
|- ( ( A e. Fin /\ A C_ Fin ) -> A e. Fin ) |
10 |
|
dfss3 |
|- ( A C_ Fin <-> A. x e. A x e. Fin ) |
11 |
|
hashcl |
|- ( x e. Fin -> ( # ` x ) e. NN0 ) |
12 |
|
nn0re |
|- ( ( # ` x ) e. NN0 -> ( # ` x ) e. RR ) |
13 |
|
nn0ge0 |
|- ( ( # ` x ) e. NN0 -> 0 <_ ( # ` x ) ) |
14 |
|
elrege0 |
|- ( ( # ` x ) e. ( 0 [,) +oo ) <-> ( ( # ` x ) e. RR /\ 0 <_ ( # ` x ) ) ) |
15 |
12 13 14
|
sylanbrc |
|- ( ( # ` x ) e. NN0 -> ( # ` x ) e. ( 0 [,) +oo ) ) |
16 |
11 15
|
syl |
|- ( x e. Fin -> ( # ` x ) e. ( 0 [,) +oo ) ) |
17 |
16
|
ralimi |
|- ( A. x e. A x e. Fin -> A. x e. A ( # ` x ) e. ( 0 [,) +oo ) ) |
18 |
10 17
|
sylbi |
|- ( A C_ Fin -> A. x e. A ( # ` x ) e. ( 0 [,) +oo ) ) |
19 |
18
|
r19.21bi |
|- ( ( A C_ Fin /\ x e. A ) -> ( # ` x ) e. ( 0 [,) +oo ) ) |
20 |
19
|
adantll |
|- ( ( ( A e. Fin /\ A C_ Fin ) /\ x e. A ) -> ( # ` x ) e. ( 0 [,) +oo ) ) |
21 |
9 20
|
esumpfinval |
|- ( ( A e. Fin /\ A C_ Fin ) -> sum* x e. A ( # ` x ) = sum_ x e. A ( # ` x ) ) |
22 |
21
|
3adant1 |
|- ( ( Disj_ x e. A x /\ A e. Fin /\ A C_ Fin ) -> sum* x e. A ( # ` x ) = sum_ x e. A ( # ` x ) ) |
23 |
8 22
|
eqtr4d |
|- ( ( Disj_ x e. A x /\ A e. Fin /\ A C_ Fin ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |
24 |
23
|
3adant1l |
|- ( ( ( A e. V /\ Disj_ x e. A x ) /\ A e. Fin /\ A C_ Fin ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |
25 |
24
|
3expa |
|- ( ( ( ( A e. V /\ Disj_ x e. A x ) /\ A e. Fin ) /\ A C_ Fin ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |
26 |
|
uniexg |
|- ( A e. V -> U. A e. _V ) |
27 |
10
|
notbii |
|- ( -. A C_ Fin <-> -. A. x e. A x e. Fin ) |
28 |
|
rexnal |
|- ( E. x e. A -. x e. Fin <-> -. A. x e. A x e. Fin ) |
29 |
27 28
|
bitr4i |
|- ( -. A C_ Fin <-> E. x e. A -. x e. Fin ) |
30 |
|
elssuni |
|- ( x e. A -> x C_ U. A ) |
31 |
|
ssfi |
|- ( ( U. A e. Fin /\ x C_ U. A ) -> x e. Fin ) |
32 |
31
|
expcom |
|- ( x C_ U. A -> ( U. A e. Fin -> x e. Fin ) ) |
33 |
32
|
con3d |
|- ( x C_ U. A -> ( -. x e. Fin -> -. U. A e. Fin ) ) |
34 |
30 33
|
syl |
|- ( x e. A -> ( -. x e. Fin -> -. U. A e. Fin ) ) |
35 |
34
|
rexlimiv |
|- ( E. x e. A -. x e. Fin -> -. U. A e. Fin ) |
36 |
29 35
|
sylbi |
|- ( -. A C_ Fin -> -. U. A e. Fin ) |
37 |
|
hashinf |
|- ( ( U. A e. _V /\ -. U. A e. Fin ) -> ( # ` U. A ) = +oo ) |
38 |
26 36 37
|
syl2an |
|- ( ( A e. V /\ -. A C_ Fin ) -> ( # ` U. A ) = +oo ) |
39 |
|
vex |
|- x e. _V |
40 |
|
hashinf |
|- ( ( x e. _V /\ -. x e. Fin ) -> ( # ` x ) = +oo ) |
41 |
39 40
|
mpan |
|- ( -. x e. Fin -> ( # ` x ) = +oo ) |
42 |
41
|
reximi |
|- ( E. x e. A -. x e. Fin -> E. x e. A ( # ` x ) = +oo ) |
43 |
29 42
|
sylbi |
|- ( -. A C_ Fin -> E. x e. A ( # ` x ) = +oo ) |
44 |
|
nfv |
|- F/ x A e. V |
45 |
|
nfre1 |
|- F/ x E. x e. A ( # ` x ) = +oo |
46 |
44 45
|
nfan |
|- F/ x ( A e. V /\ E. x e. A ( # ` x ) = +oo ) |
47 |
|
simpl |
|- ( ( A e. V /\ E. x e. A ( # ` x ) = +oo ) -> A e. V ) |
48 |
|
hashf2 |
|- # : _V --> ( 0 [,] +oo ) |
49 |
|
ffvelrn |
|- ( ( # : _V --> ( 0 [,] +oo ) /\ x e. _V ) -> ( # ` x ) e. ( 0 [,] +oo ) ) |
50 |
48 39 49
|
mp2an |
|- ( # ` x ) e. ( 0 [,] +oo ) |
51 |
50
|
a1i |
|- ( ( ( A e. V /\ E. x e. A ( # ` x ) = +oo ) /\ x e. A ) -> ( # ` x ) e. ( 0 [,] +oo ) ) |
52 |
|
simpr |
|- ( ( A e. V /\ E. x e. A ( # ` x ) = +oo ) -> E. x e. A ( # ` x ) = +oo ) |
53 |
46 47 51 52
|
esumpinfval |
|- ( ( A e. V /\ E. x e. A ( # ` x ) = +oo ) -> sum* x e. A ( # ` x ) = +oo ) |
54 |
43 53
|
sylan2 |
|- ( ( A e. V /\ -. A C_ Fin ) -> sum* x e. A ( # ` x ) = +oo ) |
55 |
38 54
|
eqtr4d |
|- ( ( A e. V /\ -. A C_ Fin ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |
56 |
55
|
3adant2 |
|- ( ( A e. V /\ A e. Fin /\ -. A C_ Fin ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |
57 |
56
|
3adant1r |
|- ( ( ( A e. V /\ Disj_ x e. A x ) /\ A e. Fin /\ -. A C_ Fin ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |
58 |
57
|
3expa |
|- ( ( ( ( A e. V /\ Disj_ x e. A x ) /\ A e. Fin ) /\ -. A C_ Fin ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |
59 |
25 58
|
pm2.61dan |
|- ( ( ( A e. V /\ Disj_ x e. A x ) /\ A e. Fin ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |
60 |
|
pwfi |
|- ( U. A e. Fin <-> ~P U. A e. Fin ) |
61 |
|
pwuni |
|- A C_ ~P U. A |
62 |
|
ssfi |
|- ( ( ~P U. A e. Fin /\ A C_ ~P U. A ) -> A e. Fin ) |
63 |
61 62
|
mpan2 |
|- ( ~P U. A e. Fin -> A e. Fin ) |
64 |
60 63
|
sylbi |
|- ( U. A e. Fin -> A e. Fin ) |
65 |
64
|
con3i |
|- ( -. A e. Fin -> -. U. A e. Fin ) |
66 |
26 65 37
|
syl2an |
|- ( ( A e. V /\ -. A e. Fin ) -> ( # ` U. A ) = +oo ) |
67 |
|
nftru |
|- F/ x T. |
68 |
|
unrab |
|- ( { x e. A | ( # ` x ) = 0 } u. { x e. A | -. ( # ` x ) = 0 } ) = { x e. A | ( ( # ` x ) = 0 \/ -. ( # ` x ) = 0 ) } |
69 |
|
exmid |
|- ( ( # ` x ) = 0 \/ -. ( # ` x ) = 0 ) |
70 |
69
|
rgenw |
|- A. x e. A ( ( # ` x ) = 0 \/ -. ( # ` x ) = 0 ) |
71 |
|
rabid2 |
|- ( A = { x e. A | ( ( # ` x ) = 0 \/ -. ( # ` x ) = 0 ) } <-> A. x e. A ( ( # ` x ) = 0 \/ -. ( # ` x ) = 0 ) ) |
72 |
70 71
|
mpbir |
|- A = { x e. A | ( ( # ` x ) = 0 \/ -. ( # ` x ) = 0 ) } |
73 |
68 72
|
eqtr4i |
|- ( { x e. A | ( # ` x ) = 0 } u. { x e. A | -. ( # ` x ) = 0 } ) = A |
74 |
73
|
a1i |
|- ( T. -> ( { x e. A | ( # ` x ) = 0 } u. { x e. A | -. ( # ` x ) = 0 } ) = A ) |
75 |
67 74
|
esumeq1d |
|- ( T. -> sum* x e. ( { x e. A | ( # ` x ) = 0 } u. { x e. A | -. ( # ` x ) = 0 } ) ( # ` x ) = sum* x e. A ( # ` x ) ) |
76 |
75
|
mptru |
|- sum* x e. ( { x e. A | ( # ` x ) = 0 } u. { x e. A | -. ( # ` x ) = 0 } ) ( # ` x ) = sum* x e. A ( # ` x ) |
77 |
|
nfrab1 |
|- F/_ x { x e. A | ( # ` x ) = 0 } |
78 |
|
nfrab1 |
|- F/_ x { x e. A | -. ( # ` x ) = 0 } |
79 |
|
rabexg |
|- ( A e. V -> { x e. A | ( # ` x ) = 0 } e. _V ) |
80 |
|
rabexg |
|- ( A e. V -> { x e. A | -. ( # ` x ) = 0 } e. _V ) |
81 |
|
rabnc |
|- ( { x e. A | ( # ` x ) = 0 } i^i { x e. A | -. ( # ` x ) = 0 } ) = (/) |
82 |
81
|
a1i |
|- ( A e. V -> ( { x e. A | ( # ` x ) = 0 } i^i { x e. A | -. ( # ` x ) = 0 } ) = (/) ) |
83 |
50
|
a1i |
|- ( ( A e. V /\ x e. { x e. A | ( # ` x ) = 0 } ) -> ( # ` x ) e. ( 0 [,] +oo ) ) |
84 |
50
|
a1i |
|- ( ( A e. V /\ x e. { x e. A | -. ( # ` x ) = 0 } ) -> ( # ` x ) e. ( 0 [,] +oo ) ) |
85 |
44 77 78 79 80 82 83 84
|
esumsplit |
|- ( A e. V -> sum* x e. ( { x e. A | ( # ` x ) = 0 } u. { x e. A | -. ( # ` x ) = 0 } ) ( # ` x ) = ( sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) +e sum* x e. { x e. A | -. ( # ` x ) = 0 } ( # ` x ) ) ) |
86 |
76 85
|
eqtr3id |
|- ( A e. V -> sum* x e. A ( # ` x ) = ( sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) +e sum* x e. { x e. A | -. ( # ` x ) = 0 } ( # ` x ) ) ) |
87 |
86
|
adantr |
|- ( ( A e. V /\ -. A e. Fin ) -> sum* x e. A ( # ` x ) = ( sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) +e sum* x e. { x e. A | -. ( # ` x ) = 0 } ( # ` x ) ) ) |
88 |
|
nfv |
|- F/ x ( A e. V /\ -. A e. Fin ) |
89 |
80
|
adantr |
|- ( ( A e. V /\ -. A e. Fin ) -> { x e. A | -. ( # ` x ) = 0 } e. _V ) |
90 |
|
simpr |
|- ( ( A e. V /\ -. A e. Fin ) -> -. A e. Fin ) |
91 |
|
dfrab3 |
|- { x e. A | ( # ` x ) = 0 } = ( A i^i { x | ( # ` x ) = 0 } ) |
92 |
|
hasheq0 |
|- ( x e. _V -> ( ( # ` x ) = 0 <-> x = (/) ) ) |
93 |
39 92
|
ax-mp |
|- ( ( # ` x ) = 0 <-> x = (/) ) |
94 |
93
|
abbii |
|- { x | ( # ` x ) = 0 } = { x | x = (/) } |
95 |
|
df-sn |
|- { (/) } = { x | x = (/) } |
96 |
94 95
|
eqtr4i |
|- { x | ( # ` x ) = 0 } = { (/) } |
97 |
96
|
ineq2i |
|- ( A i^i { x | ( # ` x ) = 0 } ) = ( A i^i { (/) } ) |
98 |
91 97
|
eqtri |
|- { x e. A | ( # ` x ) = 0 } = ( A i^i { (/) } ) |
99 |
|
snfi |
|- { (/) } e. Fin |
100 |
|
inss2 |
|- ( A i^i { (/) } ) C_ { (/) } |
101 |
|
ssfi |
|- ( ( { (/) } e. Fin /\ ( A i^i { (/) } ) C_ { (/) } ) -> ( A i^i { (/) } ) e. Fin ) |
102 |
99 100 101
|
mp2an |
|- ( A i^i { (/) } ) e. Fin |
103 |
98 102
|
eqeltri |
|- { x e. A | ( # ` x ) = 0 } e. Fin |
104 |
103
|
a1i |
|- ( ( A e. V /\ -. A e. Fin ) -> { x e. A | ( # ` x ) = 0 } e. Fin ) |
105 |
|
difinf |
|- ( ( -. A e. Fin /\ { x e. A | ( # ` x ) = 0 } e. Fin ) -> -. ( A \ { x e. A | ( # ` x ) = 0 } ) e. Fin ) |
106 |
90 104 105
|
syl2anc |
|- ( ( A e. V /\ -. A e. Fin ) -> -. ( A \ { x e. A | ( # ` x ) = 0 } ) e. Fin ) |
107 |
|
notrab |
|- ( A \ { x e. A | ( # ` x ) = 0 } ) = { x e. A | -. ( # ` x ) = 0 } |
108 |
107
|
eleq1i |
|- ( ( A \ { x e. A | ( # ` x ) = 0 } ) e. Fin <-> { x e. A | -. ( # ` x ) = 0 } e. Fin ) |
109 |
106 108
|
sylnib |
|- ( ( A e. V /\ -. A e. Fin ) -> -. { x e. A | -. ( # ` x ) = 0 } e. Fin ) |
110 |
50
|
a1i |
|- ( ( ( A e. V /\ -. A e. Fin ) /\ x e. { x e. A | -. ( # ` x ) = 0 } ) -> ( # ` x ) e. ( 0 [,] +oo ) ) |
111 |
39
|
a1i |
|- ( ( ( A e. V /\ -. A e. Fin ) /\ x e. { x e. A | -. ( # ` x ) = 0 } ) -> x e. _V ) |
112 |
|
simpr |
|- ( ( ( A e. V /\ -. A e. Fin ) /\ x e. { x e. A | -. ( # ` x ) = 0 } ) -> x e. { x e. A | -. ( # ` x ) = 0 } ) |
113 |
|
rabid |
|- ( x e. { x e. A | -. ( # ` x ) = 0 } <-> ( x e. A /\ -. ( # ` x ) = 0 ) ) |
114 |
112 113
|
sylib |
|- ( ( ( A e. V /\ -. A e. Fin ) /\ x e. { x e. A | -. ( # ` x ) = 0 } ) -> ( x e. A /\ -. ( # ` x ) = 0 ) ) |
115 |
114
|
simprd |
|- ( ( ( A e. V /\ -. A e. Fin ) /\ x e. { x e. A | -. ( # ` x ) = 0 } ) -> -. ( # ` x ) = 0 ) |
116 |
93
|
biimpri |
|- ( x = (/) -> ( # ` x ) = 0 ) |
117 |
116
|
necon3bi |
|- ( -. ( # ` x ) = 0 -> x =/= (/) ) |
118 |
115 117
|
syl |
|- ( ( ( A e. V /\ -. A e. Fin ) /\ x e. { x e. A | -. ( # ` x ) = 0 } ) -> x =/= (/) ) |
119 |
|
hashge1 |
|- ( ( x e. _V /\ x =/= (/) ) -> 1 <_ ( # ` x ) ) |
120 |
111 118 119
|
syl2anc |
|- ( ( ( A e. V /\ -. A e. Fin ) /\ x e. { x e. A | -. ( # ` x ) = 0 } ) -> 1 <_ ( # ` x ) ) |
121 |
|
1xr |
|- 1 e. RR* |
122 |
121
|
a1i |
|- ( ( A e. V /\ -. A e. Fin ) -> 1 e. RR* ) |
123 |
|
0lt1 |
|- 0 < 1 |
124 |
123
|
a1i |
|- ( ( A e. V /\ -. A e. Fin ) -> 0 < 1 ) |
125 |
88 78 89 109 110 120 122 124
|
esumpinfsum |
|- ( ( A e. V /\ -. A e. Fin ) -> sum* x e. { x e. A | -. ( # ` x ) = 0 } ( # ` x ) = +oo ) |
126 |
125
|
oveq2d |
|- ( ( A e. V /\ -. A e. Fin ) -> ( sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) +e sum* x e. { x e. A | -. ( # ` x ) = 0 } ( # ` x ) ) = ( sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) +e +oo ) ) |
127 |
|
iccssxr |
|- ( 0 [,] +oo ) C_ RR* |
128 |
79
|
adantr |
|- ( ( A e. V /\ -. A e. Fin ) -> { x e. A | ( # ` x ) = 0 } e. _V ) |
129 |
50
|
a1i |
|- ( ( ( A e. V /\ -. A e. Fin ) /\ x e. { x e. A | ( # ` x ) = 0 } ) -> ( # ` x ) e. ( 0 [,] +oo ) ) |
130 |
129
|
ralrimiva |
|- ( ( A e. V /\ -. A e. Fin ) -> A. x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) e. ( 0 [,] +oo ) ) |
131 |
77
|
esumcl |
|- ( ( { x e. A | ( # ` x ) = 0 } e. _V /\ A. x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) e. ( 0 [,] +oo ) ) -> sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) e. ( 0 [,] +oo ) ) |
132 |
128 130 131
|
syl2anc |
|- ( ( A e. V /\ -. A e. Fin ) -> sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) e. ( 0 [,] +oo ) ) |
133 |
127 132
|
sselid |
|- ( ( A e. V /\ -. A e. Fin ) -> sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) e. RR* ) |
134 |
|
xrge0neqmnf |
|- ( sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) e. ( 0 [,] +oo ) -> sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) =/= -oo ) |
135 |
132 134
|
syl |
|- ( ( A e. V /\ -. A e. Fin ) -> sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) =/= -oo ) |
136 |
|
xaddpnf1 |
|- ( ( sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) e. RR* /\ sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) =/= -oo ) -> ( sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) +e +oo ) = +oo ) |
137 |
133 135 136
|
syl2anc |
|- ( ( A e. V /\ -. A e. Fin ) -> ( sum* x e. { x e. A | ( # ` x ) = 0 } ( # ` x ) +e +oo ) = +oo ) |
138 |
87 126 137
|
3eqtrd |
|- ( ( A e. V /\ -. A e. Fin ) -> sum* x e. A ( # ` x ) = +oo ) |
139 |
66 138
|
eqtr4d |
|- ( ( A e. V /\ -. A e. Fin ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |
140 |
139
|
adantlr |
|- ( ( ( A e. V /\ Disj_ x e. A x ) /\ -. A e. Fin ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |
141 |
59 140
|
pm2.61dan |
|- ( ( A e. V /\ Disj_ x e. A x ) -> ( # ` U. A ) = sum* x e. A ( # ` x ) ) |