Step |
Hyp |
Ref |
Expression |
1 |
|
xrtgioo.1 |
|- J = ( ( ordTop ` <_ ) |`t RR ) |
2 |
|
letop |
|- ( ordTop ` <_ ) e. Top |
3 |
|
ioof |
|- (,) : ( RR* X. RR* ) --> ~P RR |
4 |
|
ffn |
|- ( (,) : ( RR* X. RR* ) --> ~P RR -> (,) Fn ( RR* X. RR* ) ) |
5 |
3 4
|
ax-mp |
|- (,) Fn ( RR* X. RR* ) |
6 |
|
iooordt |
|- ( x (,) y ) e. ( ordTop ` <_ ) |
7 |
6
|
rgen2w |
|- A. x e. RR* A. y e. RR* ( x (,) y ) e. ( ordTop ` <_ ) |
8 |
|
ffnov |
|- ( (,) : ( RR* X. RR* ) --> ( ordTop ` <_ ) <-> ( (,) Fn ( RR* X. RR* ) /\ A. x e. RR* A. y e. RR* ( x (,) y ) e. ( ordTop ` <_ ) ) ) |
9 |
5 7 8
|
mpbir2an |
|- (,) : ( RR* X. RR* ) --> ( ordTop ` <_ ) |
10 |
|
frn |
|- ( (,) : ( RR* X. RR* ) --> ( ordTop ` <_ ) -> ran (,) C_ ( ordTop ` <_ ) ) |
11 |
9 10
|
ax-mp |
|- ran (,) C_ ( ordTop ` <_ ) |
12 |
|
tgss |
|- ( ( ( ordTop ` <_ ) e. Top /\ ran (,) C_ ( ordTop ` <_ ) ) -> ( topGen ` ran (,) ) C_ ( topGen ` ( ordTop ` <_ ) ) ) |
13 |
2 11 12
|
mp2an |
|- ( topGen ` ran (,) ) C_ ( topGen ` ( ordTop ` <_ ) ) |
14 |
|
tgtop |
|- ( ( ordTop ` <_ ) e. Top -> ( topGen ` ( ordTop ` <_ ) ) = ( ordTop ` <_ ) ) |
15 |
2 14
|
ax-mp |
|- ( topGen ` ( ordTop ` <_ ) ) = ( ordTop ` <_ ) |
16 |
13 15
|
sseqtri |
|- ( topGen ` ran (,) ) C_ ( ordTop ` <_ ) |
17 |
16
|
sseli |
|- ( x e. ( topGen ` ran (,) ) -> x e. ( ordTop ` <_ ) ) |
18 |
|
retopon |
|- ( topGen ` ran (,) ) e. ( TopOn ` RR ) |
19 |
|
toponss |
|- ( ( ( topGen ` ran (,) ) e. ( TopOn ` RR ) /\ x e. ( topGen ` ran (,) ) ) -> x C_ RR ) |
20 |
18 19
|
mpan |
|- ( x e. ( topGen ` ran (,) ) -> x C_ RR ) |
21 |
|
reordt |
|- RR e. ( ordTop ` <_ ) |
22 |
|
restopn2 |
|- ( ( ( ordTop ` <_ ) e. Top /\ RR e. ( ordTop ` <_ ) ) -> ( x e. ( ( ordTop ` <_ ) |`t RR ) <-> ( x e. ( ordTop ` <_ ) /\ x C_ RR ) ) ) |
23 |
2 21 22
|
mp2an |
|- ( x e. ( ( ordTop ` <_ ) |`t RR ) <-> ( x e. ( ordTop ` <_ ) /\ x C_ RR ) ) |
24 |
17 20 23
|
sylanbrc |
|- ( x e. ( topGen ` ran (,) ) -> x e. ( ( ordTop ` <_ ) |`t RR ) ) |
25 |
24
|
ssriv |
|- ( topGen ` ran (,) ) C_ ( ( ordTop ` <_ ) |`t RR ) |
26 |
|
eqid |
|- ran ( x e. RR* |-> ( x (,] +oo ) ) = ran ( x e. RR* |-> ( x (,] +oo ) ) |
27 |
|
eqid |
|- ran ( x e. RR* |-> ( -oo [,) x ) ) = ran ( x e. RR* |-> ( -oo [,) x ) ) |
28 |
|
eqid |
|- ran (,) = ran (,) |
29 |
26 27 28
|
leordtval |
|- ( ordTop ` <_ ) = ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) ) |
30 |
29
|
oveq1i |
|- ( ( ordTop ` <_ ) |`t RR ) = ( ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) ) |`t RR ) |
31 |
29 2
|
eqeltrri |
|- ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) ) e. Top |
32 |
|
tgclb |
|- ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) e. TopBases <-> ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) ) e. Top ) |
33 |
31 32
|
mpbir |
|- ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) e. TopBases |
34 |
|
reex |
|- RR e. _V |
35 |
|
tgrest |
|- ( ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) e. TopBases /\ RR e. _V ) -> ( topGen ` ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) |`t RR ) ) = ( ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) ) |`t RR ) ) |
36 |
33 34 35
|
mp2an |
|- ( topGen ` ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) |`t RR ) ) = ( ( topGen ` ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) ) |`t RR ) |
37 |
30 36
|
eqtr4i |
|- ( ( ordTop ` <_ ) |`t RR ) = ( topGen ` ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) |`t RR ) ) |
38 |
|
retopbas |
|- ran (,) e. TopBases |
39 |
|
elrest |
|- ( ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) e. TopBases /\ RR e. _V ) -> ( u e. ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) |`t RR ) <-> E. v e. ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) u = ( v i^i RR ) ) ) |
40 |
33 34 39
|
mp2an |
|- ( u e. ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) |`t RR ) <-> E. v e. ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) u = ( v i^i RR ) ) |
41 |
|
elun |
|- ( v e. ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) <-> ( v e. ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) \/ v e. ran (,) ) ) |
42 |
|
elun |
|- ( v e. ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) <-> ( v e. ran ( x e. RR* |-> ( x (,] +oo ) ) \/ v e. ran ( x e. RR* |-> ( -oo [,) x ) ) ) ) |
43 |
|
eqid |
|- ( x e. RR* |-> ( x (,] +oo ) ) = ( x e. RR* |-> ( x (,] +oo ) ) |
44 |
43
|
elrnmpt |
|- ( v e. _V -> ( v e. ran ( x e. RR* |-> ( x (,] +oo ) ) <-> E. x e. RR* v = ( x (,] +oo ) ) ) |
45 |
44
|
elv |
|- ( v e. ran ( x e. RR* |-> ( x (,] +oo ) ) <-> E. x e. RR* v = ( x (,] +oo ) ) |
46 |
|
simpl |
|- ( ( x e. RR* /\ y e. RR ) -> x e. RR* ) |
47 |
|
pnfxr |
|- +oo e. RR* |
48 |
47
|
a1i |
|- ( ( x e. RR* /\ y e. RR ) -> +oo e. RR* ) |
49 |
|
rexr |
|- ( y e. RR -> y e. RR* ) |
50 |
49
|
adantl |
|- ( ( x e. RR* /\ y e. RR ) -> y e. RR* ) |
51 |
|
df-ioc |
|- (,] = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a < c /\ c <_ b ) } ) |
52 |
51
|
elixx3g |
|- ( y e. ( x (,] +oo ) <-> ( ( x e. RR* /\ +oo e. RR* /\ y e. RR* ) /\ ( x < y /\ y <_ +oo ) ) ) |
53 |
52
|
baib |
|- ( ( x e. RR* /\ +oo e. RR* /\ y e. RR* ) -> ( y e. ( x (,] +oo ) <-> ( x < y /\ y <_ +oo ) ) ) |
54 |
46 48 50 53
|
syl3anc |
|- ( ( x e. RR* /\ y e. RR ) -> ( y e. ( x (,] +oo ) <-> ( x < y /\ y <_ +oo ) ) ) |
55 |
|
pnfge |
|- ( y e. RR* -> y <_ +oo ) |
56 |
50 55
|
syl |
|- ( ( x e. RR* /\ y e. RR ) -> y <_ +oo ) |
57 |
56
|
biantrud |
|- ( ( x e. RR* /\ y e. RR ) -> ( x < y <-> ( x < y /\ y <_ +oo ) ) ) |
58 |
|
ltpnf |
|- ( y e. RR -> y < +oo ) |
59 |
58
|
adantl |
|- ( ( x e. RR* /\ y e. RR ) -> y < +oo ) |
60 |
59
|
biantrud |
|- ( ( x e. RR* /\ y e. RR ) -> ( x < y <-> ( x < y /\ y < +oo ) ) ) |
61 |
54 57 60
|
3bitr2d |
|- ( ( x e. RR* /\ y e. RR ) -> ( y e. ( x (,] +oo ) <-> ( x < y /\ y < +oo ) ) ) |
62 |
61
|
pm5.32da |
|- ( x e. RR* -> ( ( y e. RR /\ y e. ( x (,] +oo ) ) <-> ( y e. RR /\ ( x < y /\ y < +oo ) ) ) ) |
63 |
|
elin |
|- ( y e. ( ( x (,] +oo ) i^i RR ) <-> ( y e. ( x (,] +oo ) /\ y e. RR ) ) |
64 |
63
|
biancomi |
|- ( y e. ( ( x (,] +oo ) i^i RR ) <-> ( y e. RR /\ y e. ( x (,] +oo ) ) ) |
65 |
|
3anass |
|- ( ( y e. RR /\ x < y /\ y < +oo ) <-> ( y e. RR /\ ( x < y /\ y < +oo ) ) ) |
66 |
62 64 65
|
3bitr4g |
|- ( x e. RR* -> ( y e. ( ( x (,] +oo ) i^i RR ) <-> ( y e. RR /\ x < y /\ y < +oo ) ) ) |
67 |
|
elioo2 |
|- ( ( x e. RR* /\ +oo e. RR* ) -> ( y e. ( x (,) +oo ) <-> ( y e. RR /\ x < y /\ y < +oo ) ) ) |
68 |
47 67
|
mpan2 |
|- ( x e. RR* -> ( y e. ( x (,) +oo ) <-> ( y e. RR /\ x < y /\ y < +oo ) ) ) |
69 |
66 68
|
bitr4d |
|- ( x e. RR* -> ( y e. ( ( x (,] +oo ) i^i RR ) <-> y e. ( x (,) +oo ) ) ) |
70 |
69
|
eqrdv |
|- ( x e. RR* -> ( ( x (,] +oo ) i^i RR ) = ( x (,) +oo ) ) |
71 |
|
ioorebas |
|- ( x (,) +oo ) e. ran (,) |
72 |
70 71
|
eqeltrdi |
|- ( x e. RR* -> ( ( x (,] +oo ) i^i RR ) e. ran (,) ) |
73 |
|
ineq1 |
|- ( v = ( x (,] +oo ) -> ( v i^i RR ) = ( ( x (,] +oo ) i^i RR ) ) |
74 |
73
|
eleq1d |
|- ( v = ( x (,] +oo ) -> ( ( v i^i RR ) e. ran (,) <-> ( ( x (,] +oo ) i^i RR ) e. ran (,) ) ) |
75 |
72 74
|
syl5ibrcom |
|- ( x e. RR* -> ( v = ( x (,] +oo ) -> ( v i^i RR ) e. ran (,) ) ) |
76 |
75
|
rexlimiv |
|- ( E. x e. RR* v = ( x (,] +oo ) -> ( v i^i RR ) e. ran (,) ) |
77 |
45 76
|
sylbi |
|- ( v e. ran ( x e. RR* |-> ( x (,] +oo ) ) -> ( v i^i RR ) e. ran (,) ) |
78 |
|
eqid |
|- ( x e. RR* |-> ( -oo [,) x ) ) = ( x e. RR* |-> ( -oo [,) x ) ) |
79 |
78
|
elrnmpt |
|- ( v e. _V -> ( v e. ran ( x e. RR* |-> ( -oo [,) x ) ) <-> E. x e. RR* v = ( -oo [,) x ) ) ) |
80 |
79
|
elv |
|- ( v e. ran ( x e. RR* |-> ( -oo [,) x ) ) <-> E. x e. RR* v = ( -oo [,) x ) ) |
81 |
|
mnfxr |
|- -oo e. RR* |
82 |
81
|
a1i |
|- ( ( x e. RR* /\ y e. RR ) -> -oo e. RR* ) |
83 |
|
df-ico |
|- [,) = ( a e. RR* , b e. RR* |-> { c e. RR* | ( a <_ c /\ c < b ) } ) |
84 |
83
|
elixx3g |
|- ( y e. ( -oo [,) x ) <-> ( ( -oo e. RR* /\ x e. RR* /\ y e. RR* ) /\ ( -oo <_ y /\ y < x ) ) ) |
85 |
84
|
baib |
|- ( ( -oo e. RR* /\ x e. RR* /\ y e. RR* ) -> ( y e. ( -oo [,) x ) <-> ( -oo <_ y /\ y < x ) ) ) |
86 |
82 46 50 85
|
syl3anc |
|- ( ( x e. RR* /\ y e. RR ) -> ( y e. ( -oo [,) x ) <-> ( -oo <_ y /\ y < x ) ) ) |
87 |
|
mnfle |
|- ( y e. RR* -> -oo <_ y ) |
88 |
50 87
|
syl |
|- ( ( x e. RR* /\ y e. RR ) -> -oo <_ y ) |
89 |
88
|
biantrurd |
|- ( ( x e. RR* /\ y e. RR ) -> ( y < x <-> ( -oo <_ y /\ y < x ) ) ) |
90 |
|
mnflt |
|- ( y e. RR -> -oo < y ) |
91 |
90
|
adantl |
|- ( ( x e. RR* /\ y e. RR ) -> -oo < y ) |
92 |
91
|
biantrurd |
|- ( ( x e. RR* /\ y e. RR ) -> ( y < x <-> ( -oo < y /\ y < x ) ) ) |
93 |
86 89 92
|
3bitr2d |
|- ( ( x e. RR* /\ y e. RR ) -> ( y e. ( -oo [,) x ) <-> ( -oo < y /\ y < x ) ) ) |
94 |
93
|
pm5.32da |
|- ( x e. RR* -> ( ( y e. RR /\ y e. ( -oo [,) x ) ) <-> ( y e. RR /\ ( -oo < y /\ y < x ) ) ) ) |
95 |
|
elin |
|- ( y e. ( ( -oo [,) x ) i^i RR ) <-> ( y e. ( -oo [,) x ) /\ y e. RR ) ) |
96 |
95
|
biancomi |
|- ( y e. ( ( -oo [,) x ) i^i RR ) <-> ( y e. RR /\ y e. ( -oo [,) x ) ) ) |
97 |
|
3anass |
|- ( ( y e. RR /\ -oo < y /\ y < x ) <-> ( y e. RR /\ ( -oo < y /\ y < x ) ) ) |
98 |
94 96 97
|
3bitr4g |
|- ( x e. RR* -> ( y e. ( ( -oo [,) x ) i^i RR ) <-> ( y e. RR /\ -oo < y /\ y < x ) ) ) |
99 |
|
elioo2 |
|- ( ( -oo e. RR* /\ x e. RR* ) -> ( y e. ( -oo (,) x ) <-> ( y e. RR /\ -oo < y /\ y < x ) ) ) |
100 |
81 99
|
mpan |
|- ( x e. RR* -> ( y e. ( -oo (,) x ) <-> ( y e. RR /\ -oo < y /\ y < x ) ) ) |
101 |
98 100
|
bitr4d |
|- ( x e. RR* -> ( y e. ( ( -oo [,) x ) i^i RR ) <-> y e. ( -oo (,) x ) ) ) |
102 |
101
|
eqrdv |
|- ( x e. RR* -> ( ( -oo [,) x ) i^i RR ) = ( -oo (,) x ) ) |
103 |
|
ioorebas |
|- ( -oo (,) x ) e. ran (,) |
104 |
102 103
|
eqeltrdi |
|- ( x e. RR* -> ( ( -oo [,) x ) i^i RR ) e. ran (,) ) |
105 |
|
ineq1 |
|- ( v = ( -oo [,) x ) -> ( v i^i RR ) = ( ( -oo [,) x ) i^i RR ) ) |
106 |
105
|
eleq1d |
|- ( v = ( -oo [,) x ) -> ( ( v i^i RR ) e. ran (,) <-> ( ( -oo [,) x ) i^i RR ) e. ran (,) ) ) |
107 |
104 106
|
syl5ibrcom |
|- ( x e. RR* -> ( v = ( -oo [,) x ) -> ( v i^i RR ) e. ran (,) ) ) |
108 |
107
|
rexlimiv |
|- ( E. x e. RR* v = ( -oo [,) x ) -> ( v i^i RR ) e. ran (,) ) |
109 |
80 108
|
sylbi |
|- ( v e. ran ( x e. RR* |-> ( -oo [,) x ) ) -> ( v i^i RR ) e. ran (,) ) |
110 |
77 109
|
jaoi |
|- ( ( v e. ran ( x e. RR* |-> ( x (,] +oo ) ) \/ v e. ran ( x e. RR* |-> ( -oo [,) x ) ) ) -> ( v i^i RR ) e. ran (,) ) |
111 |
42 110
|
sylbi |
|- ( v e. ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) -> ( v i^i RR ) e. ran (,) ) |
112 |
|
elssuni |
|- ( v e. ran (,) -> v C_ U. ran (,) ) |
113 |
|
unirnioo |
|- RR = U. ran (,) |
114 |
112 113
|
sseqtrrdi |
|- ( v e. ran (,) -> v C_ RR ) |
115 |
|
df-ss |
|- ( v C_ RR <-> ( v i^i RR ) = v ) |
116 |
114 115
|
sylib |
|- ( v e. ran (,) -> ( v i^i RR ) = v ) |
117 |
|
id |
|- ( v e. ran (,) -> v e. ran (,) ) |
118 |
116 117
|
eqeltrd |
|- ( v e. ran (,) -> ( v i^i RR ) e. ran (,) ) |
119 |
111 118
|
jaoi |
|- ( ( v e. ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) \/ v e. ran (,) ) -> ( v i^i RR ) e. ran (,) ) |
120 |
41 119
|
sylbi |
|- ( v e. ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) -> ( v i^i RR ) e. ran (,) ) |
121 |
|
eleq1 |
|- ( u = ( v i^i RR ) -> ( u e. ran (,) <-> ( v i^i RR ) e. ran (,) ) ) |
122 |
120 121
|
syl5ibrcom |
|- ( v e. ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) -> ( u = ( v i^i RR ) -> u e. ran (,) ) ) |
123 |
122
|
rexlimiv |
|- ( E. v e. ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) u = ( v i^i RR ) -> u e. ran (,) ) |
124 |
40 123
|
sylbi |
|- ( u e. ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) |`t RR ) -> u e. ran (,) ) |
125 |
124
|
ssriv |
|- ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) |`t RR ) C_ ran (,) |
126 |
|
tgss |
|- ( ( ran (,) e. TopBases /\ ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) |`t RR ) C_ ran (,) ) -> ( topGen ` ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) |`t RR ) ) C_ ( topGen ` ran (,) ) ) |
127 |
38 125 126
|
mp2an |
|- ( topGen ` ( ( ( ran ( x e. RR* |-> ( x (,] +oo ) ) u. ran ( x e. RR* |-> ( -oo [,) x ) ) ) u. ran (,) ) |`t RR ) ) C_ ( topGen ` ran (,) ) |
128 |
37 127
|
eqsstri |
|- ( ( ordTop ` <_ ) |`t RR ) C_ ( topGen ` ran (,) ) |
129 |
25 128
|
eqssi |
|- ( topGen ` ran (,) ) = ( ( ordTop ` <_ ) |`t RR ) |
130 |
129 1
|
eqtr4i |
|- ( topGen ` ran (,) ) = J |