ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  bdxmet Unicode version

Theorem bdxmet 12581
Description: The standard bounded metric is an extended metric given an extended metric and a positive extended real cutoff. (Contributed by Mario Carneiro, 26-Aug-2015.) (Revised by Jim Kingdon, 9-May-2023.)
Hypothesis
Ref Expression
stdbdmet.1  |-  D  =  ( x  e.  X ,  y  e.  X  |-> inf ( { ( x C y ) ,  R } ,  RR* ,  <  ) )
Assertion
Ref Expression
bdxmet  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( *Met `  X
) )
Distinct variable groups:    x, y, C   
x, R, y    x, X, y
Allowed substitution hints:    D( x, y)

Proof of Theorem bdxmet
Dummy variables  a  b  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 966 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C  e.  ( *Met `  X
) )
2 xmetcl 12432 . . . . . . 7  |-  ( ( C  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x C y )  e. 
RR* )
3 xmetge0 12445 . . . . . . 7  |-  ( ( C  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  0  <_  ( x C y ) )
4 elxrge0 9716 . . . . . . 7  |-  ( ( x C y )  e.  ( 0 [,] +oo )  <->  ( ( x C y )  e. 
RR*  /\  0  <_  ( x C y ) ) )
52, 3, 4sylanbrc 413 . . . . . 6  |-  ( ( C  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x C y )  e.  ( 0 [,] +oo ) )
653expb 1167 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x C y )  e.  ( 0 [,] +oo ) )
71, 6sylan 281 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( x  e.  X  /\  y  e.  X ) )  -> 
( x C y )  e.  ( 0 [,] +oo ) )
8 xmetf 12430 . . . . . . 7  |-  ( C  e.  ( *Met `  X )  ->  C : ( X  X.  X ) --> RR* )
983ad2ant1 987 . . . . . 6  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C : ( X  X.  X ) -->
RR* )
109ffnd 5243 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C  Fn  ( X  X.  X ) )
11 fnovim 5847 . . . . 5  |-  ( C  Fn  ( X  X.  X )  ->  C  =  ( x  e.  X ,  y  e.  X  |->  ( x C y ) ) )
1210, 11syl 14 . . . 4  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C  =  ( x  e.  X , 
y  e.  X  |->  ( x C y ) ) )
13 eqidd 2118 . . . 4  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) )  =  ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) )
14 preq1 3570 . . . . 5  |-  ( z  =  ( x C y )  ->  { z ,  R }  =  { ( x C y ) ,  R } )
1514infeq1d 6867 . . . 4  |-  ( z  =  ( x C y )  -> inf ( { z ,  R } ,  RR* ,  <  )  = inf ( { ( x C y ) ,  R } ,  RR* ,  <  ) )
167, 12, 13, 15fmpoco 6081 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  ( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  )
)  o.  C )  =  ( x  e.  X ,  y  e.  X  |-> inf ( { ( x C y ) ,  R } ,  RR* ,  <  ) ) )
17 stdbdmet.1 . . 3  |-  D  =  ( x  e.  X ,  y  e.  X  |-> inf ( { ( x C y ) ,  R } ,  RR* ,  <  ) )
1816, 17syl6eqr 2168 . 2  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  ( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  )
)  o.  C )  =  D )
19 elxrge0 9716 . . . . . 6  |-  ( z  e.  ( 0 [,] +oo )  <->  ( z  e. 
RR*  /\  0  <_  z ) )
2019simplbi 272 . . . . 5  |-  ( z  e.  ( 0 [,] +oo )  ->  z  e. 
RR* )
21 simp2 967 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  R  e.  RR* )
22 xrmincl 10990 . . . . 5  |-  ( ( z  e.  RR*  /\  R  e.  RR* )  -> inf ( { z ,  R } ,  RR* ,  <  )  e.  RR* )
2320, 21, 22syl2anr 288 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  z  e.  ( 0 [,] +oo ) )  -> inf ( { z ,  R } ,  RR* ,  <  )  e.  RR* )
2423fmpttd 5543 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) : ( 0 [,] +oo ) --> RR* )
25 eqid 2117 . . . . . 6  |-  ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  )
)  =  ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  )
)
26 preq1 3570 . . . . . . 7  |-  ( z  =  a  ->  { z ,  R }  =  { a ,  R } )
2726infeq1d 6867 . . . . . 6  |-  ( z  =  a  -> inf ( { z ,  R } ,  RR* ,  <  )  = inf ( { a ,  R } ,  RR* ,  <  ) )
28 simpr 109 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  a  e.  ( 0 [,] +oo ) )
29 elxrge0 9716 . . . . . . . 8  |-  ( a  e.  ( 0 [,] +oo )  <->  ( a  e. 
RR*  /\  0  <_  a ) )
3029simplbi 272 . . . . . . 7  |-  ( a  e.  ( 0 [,] +oo )  ->  a  e. 
RR* )
31 xrmincl 10990 . . . . . . 7  |-  ( ( a  e.  RR*  /\  R  e.  RR* )  -> inf ( { a ,  R } ,  RR* ,  <  )  e.  RR* )
3230, 21, 31syl2anr 288 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  -> inf ( { a ,  R } ,  RR* ,  <  )  e.  RR* )
3325, 27, 28, 32fvmptd3 5482 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `  a )  = inf ( { a ,  R } ,  RR* ,  <  ) )
3433eqeq1d 2126 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `
 a )  =  0  <-> inf ( { a ,  R } ,  RR* ,  <  )  =  0 ) )
35 0xr 7780 . . . . . . . . 9  |-  0  e.  RR*
3635a1i 9 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  0  e.  RR* )
3730adantl 275 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  a  e.  RR* )
3821adantr 274 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  R  e.  RR* )
39 xrltmininf 10994 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  a  e.  RR*  /\  R  e. 
RR* )  ->  (
0  < inf ( {
a ,  R } ,  RR* ,  <  )  <->  ( 0  <  a  /\  0  <  R ) ) )
4036, 37, 38, 39syl3anc 1201 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
0  < inf ( {
a ,  R } ,  RR* ,  <  )  <->  ( 0  <  a  /\  0  <  R ) ) )
41 simp3 968 . . . . . . . . 9  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  0  <  R
)
4241adantr 274 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  0  <  R )
4342biantrud 302 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
0  <  a  <->  ( 0  <  a  /\  0  <  R ) ) )
4440, 43bitr4d 190 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
0  < inf ( {
a ,  R } ,  RR* ,  <  )  <->  0  <  a ) )
4544notbid 641 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  ( -.  0  < inf ( { a ,  R } ,  RR* ,  <  )  <->  -.  0  <  a ) )
4628, 29sylib 121 . . . . . . . . . 10  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
a  e.  RR*  /\  0  <_  a ) )
4746simprd 113 . . . . . . . . 9  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  0  <_  a )
48 xrltle 9539 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR*  /\  R  e.  RR* )  ->  (
0  <  R  ->  0  <_  R ) )
4935, 21, 48sylancr 410 . . . . . . . . . . 11  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  ( 0  < 
R  ->  0  <_  R ) )
5041, 49mpd 13 . . . . . . . . . 10  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  0  <_  R
)
5150adantr 274 . . . . . . . . 9  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  0  <_  R )
52 xrlemininf 10995 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  a  e.  RR*  /\  R  e. 
RR* )  ->  (
0  <_ inf ( {
a ,  R } ,  RR* ,  <  )  <->  ( 0  <_  a  /\  0  <_  R ) ) )
5336, 37, 38, 52syl3anc 1201 . . . . . . . . 9  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
0  <_ inf ( {
a ,  R } ,  RR* ,  <  )  <->  ( 0  <_  a  /\  0  <_  R ) ) )
5447, 51, 53mpbir2and 913 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  0  <_ inf ( { a ,  R } ,  RR* ,  <  ) )
55 xrlenlt 7797 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\ inf ( { a ,  R } ,  RR* ,  <  )  e.  RR* )  ->  (
0  <_ inf ( {
a ,  R } ,  RR* ,  <  )  <->  -. inf ( { a ,  R } ,  RR* ,  <  )  <  0
) )
5635, 32, 55sylancr 410 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
0  <_ inf ( {
a ,  R } ,  RR* ,  <  )  <->  -. inf ( { a ,  R } ,  RR* ,  <  )  <  0
) )
5754, 56mpbid 146 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  -. inf ( { a ,  R } ,  RR* ,  <  )  <  0 )
5857biantrurd 303 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  ( -.  0  < inf ( { a ,  R } ,  RR* ,  <  )  <->  ( -. inf ( { a ,  R } ,  RR* ,  <  )  <  0  /\  -.  0  < inf ( { a ,  R } ,  RR* ,  <  ) ) ) )
59 xrlttri3 9538 . . . . . . 7  |-  ( (inf ( { a ,  R } ,  RR* ,  <  )  e.  RR*  /\  0  e.  RR* )  ->  (inf ( { a ,  R } ,  RR* ,  <  )  =  0  <->  ( -. inf ( { a ,  R } ,  RR* ,  <  )  <  0  /\  -.  0  < inf ( { a ,  R } ,  RR* ,  <  ) ) ) )
6032, 36, 59syl2anc 408 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (inf ( { a ,  R } ,  RR* ,  <  )  =  0  <->  ( -. inf ( { a ,  R } ,  RR* ,  <  )  <  0  /\  -.  0  < inf ( { a ,  R } ,  RR* ,  <  ) ) ) )
6158, 60bitr4d 190 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  ( -.  0  < inf ( { a ,  R } ,  RR* ,  <  )  <-> inf ( { a ,  R } ,  RR* ,  <  )  =  0 ) )
62 xrlenlt 7797 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  a  e.  RR* )  ->  (
0  <_  a  <->  -.  a  <  0 ) )
6335, 37, 62sylancr 410 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
0  <_  a  <->  -.  a  <  0 ) )
6447, 63mpbid 146 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  -.  a  <  0 )
6564biantrurd 303 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  ( -.  0  <  a  <->  ( -.  a  <  0  /\  -.  0  <  a ) ) )
66 xrlttri3 9538 . . . . . . 7  |-  ( ( a  e.  RR*  /\  0  e.  RR* )  ->  (
a  =  0  <->  ( -.  a  <  0  /\  -.  0  <  a
) ) )
6737, 36, 66syl2anc 408 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
a  =  0  <->  ( -.  a  <  0  /\  -.  0  <  a
) ) )
6865, 67bitr4d 190 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  ( -.  0  <  a  <->  a  = 
0 ) )
6945, 61, 683bitr3d 217 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (inf ( { a ,  R } ,  RR* ,  <  )  =  0  <->  a  = 
0 ) )
7034, 69bitrd 187 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  (
( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `
 a )  =  0  <->  a  =  0 ) )
7130ad2antrl 481 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  a  e.  RR* )
7221adantr 274 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  R  e.  RR* )
73 xrmin1inf 10991 . . . . . . . 8  |-  ( ( a  e.  RR*  /\  R  e.  RR* )  -> inf ( { a ,  R } ,  RR* ,  <  )  <_  a )
7471, 72, 73syl2anc 408 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  -> inf ( { a ,  R } ,  RR* ,  <  )  <_  a )
7571, 72, 31syl2anc 408 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  -> inf ( { a ,  R } ,  RR* ,  <  )  e.  RR* )
76 elxrge0 9716 . . . . . . . . . 10  |-  ( b  e.  ( 0 [,] +oo )  <->  ( b  e. 
RR*  /\  0  <_  b ) )
7776simplbi 272 . . . . . . . . 9  |-  ( b  e.  ( 0 [,] +oo )  ->  b  e. 
RR* )
7877ad2antll 482 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  b  e.  RR* )
79 xrletr 9546 . . . . . . . 8  |-  ( (inf ( { a ,  R } ,  RR* ,  <  )  e.  RR*  /\  a  e.  RR*  /\  b  e.  RR* )  ->  (
(inf ( { a ,  R } ,  RR* ,  <  )  <_ 
a  /\  a  <_  b )  -> inf ( {
a ,  R } ,  RR* ,  <  )  <_  b ) )
8075, 71, 78, 79syl3anc 1201 . . . . . . 7  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
(inf ( { a ,  R } ,  RR* ,  <  )  <_ 
a  /\  a  <_  b )  -> inf ( {
a ,  R } ,  RR* ,  <  )  <_  b ) )
8174, 80mpand 425 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
a  <_  b  -> inf ( { a ,  R } ,  RR* ,  <  )  <_  b ) )
82 xrmin2inf 10992 . . . . . . 7  |-  ( ( a  e.  RR*  /\  R  e.  RR* )  -> inf ( { a ,  R } ,  RR* ,  <  )  <_  R )
8371, 72, 82syl2anc 408 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  -> inf ( { a ,  R } ,  RR* ,  <  )  <_  R )
8481, 83jctird 315 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
a  <_  b  ->  (inf ( { a ,  R } ,  RR* ,  <  )  <_  b  /\ inf ( { a ,  R } ,  RR* ,  <  )  <_  R
) ) )
85 xrlemininf 10995 . . . . . 6  |-  ( (inf ( { a ,  R } ,  RR* ,  <  )  e.  RR*  /\  b  e.  RR*  /\  R  e.  RR* )  ->  (inf ( { a ,  R } ,  RR* ,  <  )  <_ inf ( { b ,  R } ,  RR* ,  <  )  <->  (inf ( { a ,  R } ,  RR* ,  <  )  <_  b  /\ inf ( { a ,  R } ,  RR* ,  <  )  <_  R ) ) )
8675, 78, 72, 85syl3anc 1201 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (inf ( { a ,  R } ,  RR* ,  <  )  <_ inf ( { b ,  R } ,  RR* ,  <  )  <->  (inf ( { a ,  R } ,  RR* ,  <  )  <_  b  /\ inf ( { a ,  R } ,  RR* ,  <  )  <_  R ) ) )
8784, 86sylibrd 168 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
a  <_  b  -> inf ( { a ,  R } ,  RR* ,  <  )  <_ inf ( { b ,  R } ,  RR* ,  <  ) ) )
8833adantrr 470 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `  a )  = inf ( { a ,  R } ,  RR* ,  <  ) )
89 preq1 3570 . . . . . . 7  |-  ( z  =  b  ->  { z ,  R }  =  { b ,  R } )
9089infeq1d 6867 . . . . . 6  |-  ( z  =  b  -> inf ( { z ,  R } ,  RR* ,  <  )  = inf ( { b ,  R } ,  RR* ,  <  ) )
91 simpr 109 . . . . . . 7  |-  ( ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo ) )  ->  b  e.  ( 0 [,] +oo )
)
9291adantl 275 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  b  e.  ( 0 [,] +oo ) )
93 xrmincl 10990 . . . . . . 7  |-  ( ( b  e.  RR*  /\  R  e.  RR* )  -> inf ( { b ,  R } ,  RR* ,  <  )  e.  RR* )
9478, 72, 93syl2anc 408 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  -> inf ( { b ,  R } ,  RR* ,  <  )  e.  RR* )
9525, 90, 92, 94fvmptd3 5482 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `  b )  = inf ( { b ,  R } ,  RR* ,  <  ) )
9688, 95breq12d 3912 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `
 a )  <_ 
( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `
 b )  <-> inf ( {
a ,  R } ,  RR* ,  <  )  <_ inf ( { b ,  R } ,  RR* ,  <  ) ) )
9787, 96sylibrd 168 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
a  <_  b  ->  ( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `  a )  <_  (
( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `  b ) ) )
9829simprbi 273 . . . . . 6  |-  ( a  e.  ( 0 [,] +oo )  ->  0  <_ 
a )
9998ad2antrl 481 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  0  <_  a )
10076simprbi 273 . . . . . 6  |-  ( b  e.  ( 0 [,] +oo )  ->  0  <_ 
b )
101100ad2antll 482 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  0  <_  b )
10241adantr 274 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  0  <  R )
103 xrbdtri 11000 . . . . 5  |-  ( ( ( a  e.  RR*  /\  0  <_  a )  /\  ( b  e.  RR*  /\  0  <_  b )  /\  ( R  e.  RR*  /\  0  <  R ) )  -> inf ( {
( a +e
b ) ,  R } ,  RR* ,  <  )  <_  (inf ( { a ,  R } ,  RR* ,  <  ) +einf ( { b ,  R } ,  RR* ,  <  ) ) )
10471, 99, 78, 101, 72, 102, 103syl222anc 1217 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  -> inf ( { ( a +e
b ) ,  R } ,  RR* ,  <  )  <_  (inf ( { a ,  R } ,  RR* ,  <  ) +einf ( { b ,  R } ,  RR* ,  <  ) ) )
105 preq1 3570 . . . . . 6  |-  ( z  =  ( a +e b )  ->  { z ,  R }  =  { (
a +e b ) ,  R }
)
106105infeq1d 6867 . . . . 5  |-  ( z  =  ( a +e b )  -> inf ( { z ,  R } ,  RR* ,  <  )  = inf ( { ( a +e b ) ,  R } ,  RR* ,  <  )
)
107 ge0xaddcl 9721 . . . . . 6  |-  ( ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo ) )  ->  ( a +e b )  e.  ( 0 [,] +oo ) )
108107adantl 275 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
a +e b )  e.  ( 0 [,] +oo ) )
10971, 78xaddcld 9622 . . . . . 6  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
a +e b )  e.  RR* )
110 xrmincl 10990 . . . . . 6  |-  ( ( ( a +e
b )  e.  RR*  /\  R  e.  RR* )  -> inf ( { ( a +e b ) ,  R } ,  RR* ,  <  )  e. 
RR* )
111109, 72, 110syl2anc 408 . . . . 5  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  -> inf ( { ( a +e
b ) ,  R } ,  RR* ,  <  )  e.  RR* )
11225, 106, 108, 111fvmptd3 5482 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `  ( a +e
b ) )  = inf ( { ( a +e b ) ,  R } ,  RR* ,  <  ) )
11388, 95oveq12d 5760 . . . 4  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `
 a ) +e ( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  )
) `  b )
)  =  (inf ( { a ,  R } ,  RR* ,  <  ) +einf ( { b ,  R } ,  RR* ,  <  )
) )
114104, 112, 1133brtr4d 3930 . . 3  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  (
( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `  ( a +e
b ) )  <_ 
( ( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  )
) `  a ) +e ( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) `  b ) ) )
1151, 24, 70, 97, 114comet 12579 . 2  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  ( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  )
)  o.  C )  e.  ( *Met `  X ) )
11618, 115eqeltrrd 2195 1  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  D  e.  ( *Met `  X
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 947    = wceq 1316    e. wcel 1465   {cpr 3498   class class class wbr 3899    |-> cmpt 3959    X. cxp 4507    o. ccom 4513    Fn wfn 5088   -->wf 5089   ` cfv 5093  (class class class)co 5742    e. cmpo 5744  infcinf 6838   0cc0 7588   +oocpnf 7765   RR*cxr 7767    < clt 7768    <_ cle 7769   +ecxad 9512   [,]cicc 9629   *Metcxmet 12060
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-coll 4013  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-iinf 4472  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-ltwlin 7701  ax-pre-lttrn 7702  ax-pre-apti 7703  ax-pre-ltadd 7704  ax-pre-mulgt0 7705  ax-pre-mulext 7706  ax-arch 7707  ax-caucvg 7708
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rmo 2401  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-int 3742  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-tr 3997  df-id 4185  df-po 4188  df-iso 4189  df-iord 4258  df-on 4260  df-ilim 4261  df-suc 4263  df-iom 4475  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-fv 5101  df-isom 5102  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-recs 6170  df-frec 6256  df-map 6512  df-sup 6839  df-inf 6840  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-reap 8304  df-ap 8311  df-div 8400  df-inn 8685  df-2 8743  df-3 8744  df-4 8745  df-n0 8936  df-z 9013  df-uz 9283  df-rp 9398  df-xneg 9514  df-xadd 9515  df-icc 9633  df-seqfrec 10174  df-exp 10248  df-cj 10569  df-re 10570  df-im 10571  df-rsqrt 10725  df-abs 10726  df-xmet 12068
This theorem is referenced by:  bdmet  12582  bdbl  12583  bdmopn  12584
  Copyright terms: Public domain W3C validator