Step |
Hyp |
Ref |
Expression |
1 |
|
hoicvr.2 |
|- I = ( j e. NN |-> ( x e. X |-> <. -u j , j >. ) ) |
2 |
|
hoicvr.3 |
|- ( ph -> X e. Fin ) |
3 |
|
reex |
|- RR e. _V |
4 |
|
mapdm0 |
|- ( RR e. _V -> ( RR ^m (/) ) = { (/) } ) |
5 |
3 4
|
ax-mp |
|- ( RR ^m (/) ) = { (/) } |
6 |
5
|
a1i |
|- ( X = (/) -> ( RR ^m (/) ) = { (/) } ) |
7 |
|
oveq2 |
|- ( X = (/) -> ( RR ^m X ) = ( RR ^m (/) ) ) |
8 |
|
ixpeq1 |
|- ( X = (/) -> X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) = X_ i e. (/) ( ( [,) o. ( I ` j ) ) ` i ) ) |
9 |
8
|
iuneq2d |
|- ( X = (/) -> U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) = U_ j e. NN X_ i e. (/) ( ( [,) o. ( I ` j ) ) ` i ) ) |
10 |
|
ixp0x |
|- X_ i e. (/) ( ( [,) o. ( I ` j ) ) ` i ) = { (/) } |
11 |
10
|
a1i |
|- ( j e. NN -> X_ i e. (/) ( ( [,) o. ( I ` j ) ) ` i ) = { (/) } ) |
12 |
11
|
iuneq2i |
|- U_ j e. NN X_ i e. (/) ( ( [,) o. ( I ` j ) ) ` i ) = U_ j e. NN { (/) } |
13 |
|
1nn |
|- 1 e. NN |
14 |
13
|
ne0ii |
|- NN =/= (/) |
15 |
|
iunconst |
|- ( NN =/= (/) -> U_ j e. NN { (/) } = { (/) } ) |
16 |
14 15
|
ax-mp |
|- U_ j e. NN { (/) } = { (/) } |
17 |
12 16
|
eqtri |
|- U_ j e. NN X_ i e. (/) ( ( [,) o. ( I ` j ) ) ` i ) = { (/) } |
18 |
17
|
a1i |
|- ( X = (/) -> U_ j e. NN X_ i e. (/) ( ( [,) o. ( I ` j ) ) ` i ) = { (/) } ) |
19 |
9 18
|
eqtrd |
|- ( X = (/) -> U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) = { (/) } ) |
20 |
6 7 19
|
3eqtr4d |
|- ( X = (/) -> ( RR ^m X ) = U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
21 |
|
eqimss |
|- ( ( RR ^m X ) = U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) -> ( RR ^m X ) C_ U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
22 |
20 21
|
syl |
|- ( X = (/) -> ( RR ^m X ) C_ U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
23 |
22
|
adantl |
|- ( ( ph /\ X = (/) ) -> ( RR ^m X ) C_ U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
24 |
|
simpll |
|- ( ( ( ph /\ -. X = (/) ) /\ f e. ( RR ^m X ) ) -> ph ) |
25 |
|
simpr |
|- ( ( ( ph /\ -. X = (/) ) /\ f e. ( RR ^m X ) ) -> f e. ( RR ^m X ) ) |
26 |
|
simplr |
|- ( ( ( ph /\ -. X = (/) ) /\ f e. ( RR ^m X ) ) -> -. X = (/) ) |
27 |
|
rncoss |
|- ran ( abs o. f ) C_ ran abs |
28 |
|
absf |
|- abs : CC --> RR |
29 |
|
frn |
|- ( abs : CC --> RR -> ran abs C_ RR ) |
30 |
28 29
|
ax-mp |
|- ran abs C_ RR |
31 |
27 30
|
sstri |
|- ran ( abs o. f ) C_ RR |
32 |
31
|
a1i |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) -> ran ( abs o. f ) C_ RR ) |
33 |
|
ltso |
|- < Or RR |
34 |
33
|
a1i |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) -> < Or RR ) |
35 |
28
|
a1i |
|- ( ( ph /\ f e. ( RR ^m X ) ) -> abs : CC --> RR ) |
36 |
|
elmapi |
|- ( f e. ( RR ^m X ) -> f : X --> RR ) |
37 |
36
|
adantl |
|- ( ( ph /\ f e. ( RR ^m X ) ) -> f : X --> RR ) |
38 |
|
ax-resscn |
|- RR C_ CC |
39 |
38
|
a1i |
|- ( ( ph /\ f e. ( RR ^m X ) ) -> RR C_ CC ) |
40 |
37 39
|
fssd |
|- ( ( ph /\ f e. ( RR ^m X ) ) -> f : X --> CC ) |
41 |
|
fco |
|- ( ( abs : CC --> RR /\ f : X --> CC ) -> ( abs o. f ) : X --> RR ) |
42 |
35 40 41
|
syl2anc |
|- ( ( ph /\ f e. ( RR ^m X ) ) -> ( abs o. f ) : X --> RR ) |
43 |
2
|
adantr |
|- ( ( ph /\ f e. ( RR ^m X ) ) -> X e. Fin ) |
44 |
|
rnffi |
|- ( ( ( abs o. f ) : X --> RR /\ X e. Fin ) -> ran ( abs o. f ) e. Fin ) |
45 |
42 43 44
|
syl2anc |
|- ( ( ph /\ f e. ( RR ^m X ) ) -> ran ( abs o. f ) e. Fin ) |
46 |
45
|
adantr |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) -> ran ( abs o. f ) e. Fin ) |
47 |
36
|
frnd |
|- ( f e. ( RR ^m X ) -> ran f C_ RR ) |
48 |
28
|
fdmi |
|- dom abs = CC |
49 |
48
|
eqcomi |
|- CC = dom abs |
50 |
38 49
|
sseqtri |
|- RR C_ dom abs |
51 |
50
|
a1i |
|- ( f e. ( RR ^m X ) -> RR C_ dom abs ) |
52 |
47 51
|
sstrd |
|- ( f e. ( RR ^m X ) -> ran f C_ dom abs ) |
53 |
|
dmcosseq |
|- ( ran f C_ dom abs -> dom ( abs o. f ) = dom f ) |
54 |
52 53
|
syl |
|- ( f e. ( RR ^m X ) -> dom ( abs o. f ) = dom f ) |
55 |
36
|
fdmd |
|- ( f e. ( RR ^m X ) -> dom f = X ) |
56 |
54 55
|
eqtrd |
|- ( f e. ( RR ^m X ) -> dom ( abs o. f ) = X ) |
57 |
56
|
adantr |
|- ( ( f e. ( RR ^m X ) /\ -. X = (/) ) -> dom ( abs o. f ) = X ) |
58 |
|
neqne |
|- ( -. X = (/) -> X =/= (/) ) |
59 |
58
|
adantl |
|- ( ( f e. ( RR ^m X ) /\ -. X = (/) ) -> X =/= (/) ) |
60 |
57 59
|
eqnetrd |
|- ( ( f e. ( RR ^m X ) /\ -. X = (/) ) -> dom ( abs o. f ) =/= (/) ) |
61 |
60
|
neneqd |
|- ( ( f e. ( RR ^m X ) /\ -. X = (/) ) -> -. dom ( abs o. f ) = (/) ) |
62 |
|
dm0rn0 |
|- ( dom ( abs o. f ) = (/) <-> ran ( abs o. f ) = (/) ) |
63 |
61 62
|
sylnib |
|- ( ( f e. ( RR ^m X ) /\ -. X = (/) ) -> -. ran ( abs o. f ) = (/) ) |
64 |
63
|
neqned |
|- ( ( f e. ( RR ^m X ) /\ -. X = (/) ) -> ran ( abs o. f ) =/= (/) ) |
65 |
64
|
adantll |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) -> ran ( abs o. f ) =/= (/) ) |
66 |
|
fisupcl |
|- ( ( < Or RR /\ ( ran ( abs o. f ) e. Fin /\ ran ( abs o. f ) =/= (/) /\ ran ( abs o. f ) C_ RR ) ) -> sup ( ran ( abs o. f ) , RR , < ) e. ran ( abs o. f ) ) |
67 |
34 46 65 32 66
|
syl13anc |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) -> sup ( ran ( abs o. f ) , RR , < ) e. ran ( abs o. f ) ) |
68 |
32 67
|
sseldd |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) -> sup ( ran ( abs o. f ) , RR , < ) e. RR ) |
69 |
|
arch |
|- ( sup ( ran ( abs o. f ) , RR , < ) e. RR -> E. j e. NN sup ( ran ( abs o. f ) , RR , < ) < j ) |
70 |
68 69
|
syl |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) -> E. j e. NN sup ( ran ( abs o. f ) , RR , < ) < j ) |
71 |
37
|
ffnd |
|- ( ( ph /\ f e. ( RR ^m X ) ) -> f Fn X ) |
72 |
71
|
ad2antrr |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) -> f Fn X ) |
73 |
72
|
adantlr |
|- ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) -> f Fn X ) |
74 |
|
simplll |
|- ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( ph /\ f e. ( RR ^m X ) ) ) |
75 |
|
id |
|- ( j e. NN -> j e. NN ) |
76 |
75
|
ad3antlr |
|- ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> j e. NN ) |
77 |
|
simplr |
|- ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> sup ( ran ( abs o. f ) , RR , < ) < j ) |
78 |
|
simpr |
|- ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> i e. X ) |
79 |
|
simp2 |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) -> j e. NN ) |
80 |
|
zssre |
|- ZZ C_ RR |
81 |
|
ressxr |
|- RR C_ RR* |
82 |
80 81
|
sstri |
|- ZZ C_ RR* |
83 |
|
nnnegz |
|- ( j e. NN -> -u j e. ZZ ) |
84 |
82 83
|
sseldi |
|- ( j e. NN -> -u j e. RR* ) |
85 |
84
|
adantr |
|- ( ( j e. NN /\ i e. X ) -> -u j e. RR* ) |
86 |
79 85
|
sylan |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> -u j e. RR* ) |
87 |
75
|
nnxrd |
|- ( j e. NN -> j e. RR* ) |
88 |
87
|
adantr |
|- ( ( j e. NN /\ i e. X ) -> j e. RR* ) |
89 |
79 88
|
sylan |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> j e. RR* ) |
90 |
36
|
3ad2ant1 |
|- ( ( f e. ( RR ^m X ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) -> f : X --> RR ) |
91 |
81
|
a1i |
|- ( ( f e. ( RR ^m X ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) -> RR C_ RR* ) |
92 |
90 91
|
fssd |
|- ( ( f e. ( RR ^m X ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) -> f : X --> RR* ) |
93 |
92
|
3adant1l |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) -> f : X --> RR* ) |
94 |
93
|
ffvelrnda |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) e. RR* ) |
95 |
|
nnre |
|- ( j e. NN -> j e. RR ) |
96 |
95
|
adantr |
|- ( ( j e. NN /\ i e. X ) -> j e. RR ) |
97 |
96
|
3ad2antl2 |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> j e. RR ) |
98 |
97
|
renegcld |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> -u j e. RR ) |
99 |
37
|
ffvelrnda |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ i e. X ) -> ( f ` i ) e. RR ) |
100 |
99
|
3ad2antl1 |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) e. RR ) |
101 |
100
|
renegcld |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> -u ( f ` i ) e. RR ) |
102 |
|
simpll |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ i e. X ) -> ph ) |
103 |
|
simplr |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ i e. X ) -> f e. ( RR ^m X ) ) |
104 |
|
n0i |
|- ( i e. X -> -. X = (/) ) |
105 |
104
|
adantl |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ i e. X ) -> -. X = (/) ) |
106 |
102 103 105 68
|
syl21anc |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ i e. X ) -> sup ( ran ( abs o. f ) , RR , < ) e. RR ) |
107 |
106
|
3ad2antl1 |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> sup ( ran ( abs o. f ) , RR , < ) e. RR ) |
108 |
36
|
ffvelrnda |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> ( f ` i ) e. RR ) |
109 |
38 108
|
sseldi |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> ( f ` i ) e. CC ) |
110 |
109
|
abscld |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> ( abs ` ( f ` i ) ) e. RR ) |
111 |
110
|
adantll |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ i e. X ) -> ( abs ` ( f ` i ) ) e. RR ) |
112 |
111
|
3ad2antl1 |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( abs ` ( f ` i ) ) e. RR ) |
113 |
108
|
renegcld |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> -u ( f ` i ) e. RR ) |
114 |
113
|
leabsd |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> -u ( f ` i ) <_ ( abs ` -u ( f ` i ) ) ) |
115 |
109
|
absnegd |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> ( abs ` -u ( f ` i ) ) = ( abs ` ( f ` i ) ) ) |
116 |
114 115
|
breqtrd |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> -u ( f ` i ) <_ ( abs ` ( f ` i ) ) ) |
117 |
116
|
adantll |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ i e. X ) -> -u ( f ` i ) <_ ( abs ` ( f ` i ) ) ) |
118 |
117
|
3ad2antl1 |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> -u ( f ` i ) <_ ( abs ` ( f ` i ) ) ) |
119 |
31
|
a1i |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ran ( abs o. f ) C_ RR ) |
120 |
105 65
|
syldan |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ i e. X ) -> ran ( abs o. f ) =/= (/) ) |
121 |
120
|
3ad2antl1 |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ran ( abs o. f ) =/= (/) ) |
122 |
|
fimaxre2 |
|- ( ( ran ( abs o. f ) C_ RR /\ ran ( abs o. f ) e. Fin ) -> E. y e. RR A. z e. ran ( abs o. f ) z <_ y ) |
123 |
31 45 122
|
sylancr |
|- ( ( ph /\ f e. ( RR ^m X ) ) -> E. y e. RR A. z e. ran ( abs o. f ) z <_ y ) |
124 |
123
|
adantr |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ i e. X ) -> E. y e. RR A. z e. ran ( abs o. f ) z <_ y ) |
125 |
124
|
3ad2antl1 |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> E. y e. RR A. z e. ran ( abs o. f ) z <_ y ) |
126 |
|
elmapfun |
|- ( f e. ( RR ^m X ) -> Fun f ) |
127 |
126
|
adantr |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> Fun f ) |
128 |
|
simpr |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> i e. X ) |
129 |
55
|
eqcomd |
|- ( f e. ( RR ^m X ) -> X = dom f ) |
130 |
129
|
adantr |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> X = dom f ) |
131 |
128 130
|
eleqtrd |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> i e. dom f ) |
132 |
|
fvco |
|- ( ( Fun f /\ i e. dom f ) -> ( ( abs o. f ) ` i ) = ( abs ` ( f ` i ) ) ) |
133 |
127 131 132
|
syl2anc |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> ( ( abs o. f ) ` i ) = ( abs ` ( f ` i ) ) ) |
134 |
133
|
eqcomd |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> ( abs ` ( f ` i ) ) = ( ( abs o. f ) ` i ) ) |
135 |
|
absfun |
|- Fun abs |
136 |
135
|
a1i |
|- ( f e. ( RR ^m X ) -> Fun abs ) |
137 |
|
funco |
|- ( ( Fun abs /\ Fun f ) -> Fun ( abs o. f ) ) |
138 |
136 126 137
|
syl2anc |
|- ( f e. ( RR ^m X ) -> Fun ( abs o. f ) ) |
139 |
138
|
adantr |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> Fun ( abs o. f ) ) |
140 |
109 49
|
eleqtrdi |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> ( f ` i ) e. dom abs ) |
141 |
|
dmfco |
|- ( ( Fun f /\ i e. dom f ) -> ( i e. dom ( abs o. f ) <-> ( f ` i ) e. dom abs ) ) |
142 |
127 131 141
|
syl2anc |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> ( i e. dom ( abs o. f ) <-> ( f ` i ) e. dom abs ) ) |
143 |
140 142
|
mpbird |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> i e. dom ( abs o. f ) ) |
144 |
|
fvelrn |
|- ( ( Fun ( abs o. f ) /\ i e. dom ( abs o. f ) ) -> ( ( abs o. f ) ` i ) e. ran ( abs o. f ) ) |
145 |
139 143 144
|
syl2anc |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> ( ( abs o. f ) ` i ) e. ran ( abs o. f ) ) |
146 |
134 145
|
eqeltrd |
|- ( ( f e. ( RR ^m X ) /\ i e. X ) -> ( abs ` ( f ` i ) ) e. ran ( abs o. f ) ) |
147 |
146
|
adantll |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ i e. X ) -> ( abs ` ( f ` i ) ) e. ran ( abs o. f ) ) |
148 |
147
|
3ad2antl1 |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( abs ` ( f ` i ) ) e. ran ( abs o. f ) ) |
149 |
|
suprub |
|- ( ( ( ran ( abs o. f ) C_ RR /\ ran ( abs o. f ) =/= (/) /\ E. y e. RR A. z e. ran ( abs o. f ) z <_ y ) /\ ( abs ` ( f ` i ) ) e. ran ( abs o. f ) ) -> ( abs ` ( f ` i ) ) <_ sup ( ran ( abs o. f ) , RR , < ) ) |
150 |
119 121 125 148 149
|
syl31anc |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( abs ` ( f ` i ) ) <_ sup ( ran ( abs o. f ) , RR , < ) ) |
151 |
101 112 107 118 150
|
letrd |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> -u ( f ` i ) <_ sup ( ran ( abs o. f ) , RR , < ) ) |
152 |
|
simpl3 |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> sup ( ran ( abs o. f ) , RR , < ) < j ) |
153 |
101 107 97 151 152
|
lelttrd |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> -u ( f ` i ) < j ) |
154 |
101 97
|
ltnegd |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( -u ( f ` i ) < j <-> -u j < -u -u ( f ` i ) ) ) |
155 |
153 154
|
mpbid |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> -u j < -u -u ( f ` i ) ) |
156 |
38 100
|
sseldi |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) e. CC ) |
157 |
156
|
negnegd |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> -u -u ( f ` i ) = ( f ` i ) ) |
158 |
155 157
|
breqtrd |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> -u j < ( f ` i ) ) |
159 |
98 100 158
|
ltled |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> -u j <_ ( f ` i ) ) |
160 |
100
|
leabsd |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) <_ ( abs ` ( f ` i ) ) ) |
161 |
100 112 107 160 150
|
letrd |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) <_ sup ( ran ( abs o. f ) , RR , < ) ) |
162 |
100 107 97 161 152
|
lelttrd |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) < j ) |
163 |
86 89 94 159 162
|
elicod |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) e. ( -u j [,) j ) ) |
164 |
74 76 77 78 163
|
syl31anc |
|- ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) e. ( -u j [,) j ) ) |
165 |
164
|
adantl3r |
|- ( ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) e. ( -u j [,) j ) ) |
166 |
|
simpr |
|- ( ( ph /\ j e. NN ) -> j e. NN ) |
167 |
|
mptexg |
|- ( X e. Fin -> ( x e. X |-> <. -u j , j >. ) e. _V ) |
168 |
2 167
|
syl |
|- ( ph -> ( x e. X |-> <. -u j , j >. ) e. _V ) |
169 |
168
|
adantr |
|- ( ( ph /\ j e. NN ) -> ( x e. X |-> <. -u j , j >. ) e. _V ) |
170 |
1
|
fvmpt2 |
|- ( ( j e. NN /\ ( x e. X |-> <. -u j , j >. ) e. _V ) -> ( I ` j ) = ( x e. X |-> <. -u j , j >. ) ) |
171 |
166 169 170
|
syl2anc |
|- ( ( ph /\ j e. NN ) -> ( I ` j ) = ( x e. X |-> <. -u j , j >. ) ) |
172 |
171
|
fveq1d |
|- ( ( ph /\ j e. NN ) -> ( ( I ` j ) ` i ) = ( ( x e. X |-> <. -u j , j >. ) ` i ) ) |
173 |
172
|
3adant3 |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( ( I ` j ) ` i ) = ( ( x e. X |-> <. -u j , j >. ) ` i ) ) |
174 |
|
eqidd |
|- ( i e. X -> ( x e. X |-> <. -u j , j >. ) = ( x e. X |-> <. -u j , j >. ) ) |
175 |
|
eqid |
|- <. -u j , j >. = <. -u j , j >. |
176 |
175
|
a1i |
|- ( ( i e. X /\ x = i ) -> <. -u j , j >. = <. -u j , j >. ) |
177 |
|
id |
|- ( i e. X -> i e. X ) |
178 |
|
opex |
|- <. -u j , j >. e. _V |
179 |
178
|
a1i |
|- ( i e. X -> <. -u j , j >. e. _V ) |
180 |
174 176 177 179
|
fvmptd |
|- ( i e. X -> ( ( x e. X |-> <. -u j , j >. ) ` i ) = <. -u j , j >. ) |
181 |
180
|
3ad2ant3 |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( ( x e. X |-> <. -u j , j >. ) ` i ) = <. -u j , j >. ) |
182 |
173 181
|
eqtrd |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( ( I ` j ) ` i ) = <. -u j , j >. ) |
183 |
182
|
fveq2d |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( 1st ` ( ( I ` j ) ` i ) ) = ( 1st ` <. -u j , j >. ) ) |
184 |
|
negex |
|- -u j e. _V |
185 |
|
vex |
|- j e. _V |
186 |
184 185
|
op1st |
|- ( 1st ` <. -u j , j >. ) = -u j |
187 |
186
|
a1i |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( 1st ` <. -u j , j >. ) = -u j ) |
188 |
183 187
|
eqtrd |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( 1st ` ( ( I ` j ) ` i ) ) = -u j ) |
189 |
182
|
fveq2d |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( 2nd ` ( ( I ` j ) ` i ) ) = ( 2nd ` <. -u j , j >. ) ) |
190 |
184 185
|
op2nd |
|- ( 2nd ` <. -u j , j >. ) = j |
191 |
190
|
a1i |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( 2nd ` <. -u j , j >. ) = j ) |
192 |
189 191
|
eqtrd |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( 2nd ` ( ( I ` j ) ` i ) ) = j ) |
193 |
188 192
|
oveq12d |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( ( 1st ` ( ( I ` j ) ` i ) ) [,) ( 2nd ` ( ( I ` j ) ` i ) ) ) = ( -u j [,) j ) ) |
194 |
193
|
eqcomd |
|- ( ( ph /\ j e. NN /\ i e. X ) -> ( -u j [,) j ) = ( ( 1st ` ( ( I ` j ) ` i ) ) [,) ( 2nd ` ( ( I ` j ) ` i ) ) ) ) |
195 |
194
|
3adant1r |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ j e. NN /\ i e. X ) -> ( -u j [,) j ) = ( ( 1st ` ( ( I ` j ) ` i ) ) [,) ( 2nd ` ( ( I ` j ) ` i ) ) ) ) |
196 |
195
|
ad5ant135 |
|- ( ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( -u j [,) j ) = ( ( 1st ` ( ( I ` j ) ` i ) ) [,) ( 2nd ` ( ( I ` j ) ` i ) ) ) ) |
197 |
165 196
|
eleqtrd |
|- ( ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) e. ( ( 1st ` ( ( I ` j ) ` i ) ) [,) ( 2nd ` ( ( I ` j ) ` i ) ) ) ) |
198 |
80 83
|
sseldi |
|- ( j e. NN -> -u j e. RR ) |
199 |
|
opelxpi |
|- ( ( -u j e. RR /\ j e. RR ) -> <. -u j , j >. e. ( RR X. RR ) ) |
200 |
198 95 199
|
syl2anc |
|- ( j e. NN -> <. -u j , j >. e. ( RR X. RR ) ) |
201 |
200
|
ad2antlr |
|- ( ( ( ph /\ j e. NN ) /\ x e. X ) -> <. -u j , j >. e. ( RR X. RR ) ) |
202 |
|
eqid |
|- ( x e. X |-> <. -u j , j >. ) = ( x e. X |-> <. -u j , j >. ) |
203 |
201 202
|
fmptd |
|- ( ( ph /\ j e. NN ) -> ( x e. X |-> <. -u j , j >. ) : X --> ( RR X. RR ) ) |
204 |
171
|
feq1d |
|- ( ( ph /\ j e. NN ) -> ( ( I ` j ) : X --> ( RR X. RR ) <-> ( x e. X |-> <. -u j , j >. ) : X --> ( RR X. RR ) ) ) |
205 |
203 204
|
mpbird |
|- ( ( ph /\ j e. NN ) -> ( I ` j ) : X --> ( RR X. RR ) ) |
206 |
205
|
ad4ant14 |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) -> ( I ` j ) : X --> ( RR X. RR ) ) |
207 |
206
|
ad2antrr |
|- ( ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( I ` j ) : X --> ( RR X. RR ) ) |
208 |
|
simpr |
|- ( ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> i e. X ) |
209 |
207 208
|
fvovco |
|- ( ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( ( [,) o. ( I ` j ) ) ` i ) = ( ( 1st ` ( ( I ` j ) ` i ) ) [,) ( 2nd ` ( ( I ` j ) ` i ) ) ) ) |
210 |
209
|
eqcomd |
|- ( ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( ( 1st ` ( ( I ` j ) ` i ) ) [,) ( 2nd ` ( ( I ` j ) ` i ) ) ) = ( ( [,) o. ( I ` j ) ) ` i ) ) |
211 |
197 210
|
eleqtrd |
|- ( ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) /\ i e. X ) -> ( f ` i ) e. ( ( [,) o. ( I ` j ) ) ` i ) ) |
212 |
211
|
ralrimiva |
|- ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) -> A. i e. X ( f ` i ) e. ( ( [,) o. ( I ` j ) ) ` i ) ) |
213 |
73 212
|
jca |
|- ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) -> ( f Fn X /\ A. i e. X ( f ` i ) e. ( ( [,) o. ( I ` j ) ) ` i ) ) ) |
214 |
|
vex |
|- f e. _V |
215 |
214
|
elixp |
|- ( f e. X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) <-> ( f Fn X /\ A. i e. X ( f ` i ) e. ( ( [,) o. ( I ` j ) ) ` i ) ) ) |
216 |
213 215
|
sylibr |
|- ( ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) /\ sup ( ran ( abs o. f ) , RR , < ) < j ) -> f e. X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
217 |
216
|
ex |
|- ( ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) /\ j e. NN ) -> ( sup ( ran ( abs o. f ) , RR , < ) < j -> f e. X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) ) |
218 |
217
|
reximdva |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) -> ( E. j e. NN sup ( ran ( abs o. f ) , RR , < ) < j -> E. j e. NN f e. X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) ) |
219 |
70 218
|
mpd |
|- ( ( ( ph /\ f e. ( RR ^m X ) ) /\ -. X = (/) ) -> E. j e. NN f e. X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
220 |
24 25 26 219
|
syl21anc |
|- ( ( ( ph /\ -. X = (/) ) /\ f e. ( RR ^m X ) ) -> E. j e. NN f e. X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
221 |
|
eliun |
|- ( f e. U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) <-> E. j e. NN f e. X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
222 |
220 221
|
sylibr |
|- ( ( ( ph /\ -. X = (/) ) /\ f e. ( RR ^m X ) ) -> f e. U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
223 |
222
|
ralrimiva |
|- ( ( ph /\ -. X = (/) ) -> A. f e. ( RR ^m X ) f e. U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
224 |
|
dfss3 |
|- ( ( RR ^m X ) C_ U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) <-> A. f e. ( RR ^m X ) f e. U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
225 |
223 224
|
sylibr |
|- ( ( ph /\ -. X = (/) ) -> ( RR ^m X ) C_ U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |
226 |
23 225
|
pm2.61dan |
|- ( ph -> ( RR ^m X ) C_ U_ j e. NN X_ i e. X ( ( [,) o. ( I ` j ) ) ` i ) ) |