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Theorem bdxmet 12707
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 982 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C  e.  ( *Met `  X
) )
2 xmetcl 12558 . . . . . . 7  |-  ( ( C  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x C y )  e. 
RR* )
3 xmetge0 12571 . . . . . . 7  |-  ( ( C  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  0  <_  ( x C y ) )
4 elxrge0 9790 . . . . . . 7  |-  ( ( x C y )  e.  ( 0 [,] +oo )  <->  ( ( x C y )  e. 
RR*  /\  0  <_  ( x C y ) ) )
52, 3, 4sylanbrc 414 . . . . . 6  |-  ( ( C  e.  ( *Met `  X )  /\  x  e.  X  /\  y  e.  X
)  ->  ( x C y )  e.  ( 0 [,] +oo ) )
653expb 1183 . . . . 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 12556 . . . . . . 7  |-  ( C  e.  ( *Met `  X )  ->  C : ( X  X.  X ) --> RR* )
983ad2ant1 1003 . . . . . 6  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C : ( X  X.  X ) -->
RR* )
109ffnd 5280 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  C  Fn  ( X  X.  X ) )
11 fnovim 5886 . . . . 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 2141 . . . 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 3607 . . . . 5  |-  ( z  =  ( x C y )  ->  { z ,  R }  =  { ( x C y ) ,  R } )
1514infeq1d 6906 . . . 4  |-  ( z  =  ( x C y )  -> inf ( { z ,  R } ,  RR* ,  <  )  = inf ( { ( x C y ) ,  R } ,  RR* ,  <  ) )
167, 12, 13, 15fmpoco 6120 . . 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, 17eqtr4di 2191 . 2  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  ( ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  )
)  o.  C )  =  D )
19 elxrge0 9790 . . . . . 6  |-  ( z  e.  ( 0 [,] +oo )  <->  ( z  e. 
RR*  /\  0  <_  z ) )
2019simplbi 272 . . . . 5  |-  ( z  e.  ( 0 [,] +oo )  ->  z  e. 
RR* )
21 simp2 983 . . . . 5  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  R  e.  RR* )
22 xrmincl 11066 . . . . 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 5582 . . 3  |-  ( ( C  e.  ( *Met `  X )  /\  R  e.  RR*  /\  0  <  R )  ->  ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  ) ) : ( 0 [,] +oo ) --> RR* )
25 eqid 2140 . . . . . 6  |-  ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  )
)  =  ( z  e.  ( 0 [,] +oo )  |-> inf ( { z ,  R } ,  RR* ,  <  )
)
26 preq1 3607 . . . . . . 7  |-  ( z  =  a  ->  { z ,  R }  =  { a ,  R } )
2726infeq1d 6906 . . . . . 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 9790 . . . . . . . 8  |-  ( a  e.  ( 0 [,] +oo )  <->  ( a  e. 
RR*  /\  0  <_  a ) )
3029simplbi 272 . . . . . . 7  |-  ( a  e.  ( 0 [,] +oo )  ->  a  e. 
RR* )
31 xrmincl 11066 . . . . . . 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 5521 . . . . 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 2149 . . . 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 7835 . . . . . . . . 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 11070 . . . . . . . 8  |-  ( ( 0  e.  RR*  /\  a  e.  RR*  /\  R  e. 
RR* )  ->  (
0  < inf ( {
a ,  R } ,  RR* ,  <  )  <->  ( 0  <  a  /\  0  <  R ) ) )
4036, 37, 38, 39syl3anc 1217 . . . . . . 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 984 . . . . . . . . 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 657 . . . . 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 9613 . . . . . . . . . . . 12  |-  ( ( 0  e.  RR*  /\  R  e.  RR* )  ->  (
0  <  R  ->  0  <_  R ) )
4935, 21, 48sylancr 411 . . . . . . . . . . 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 11071 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  a  e.  RR*  /\  R  e. 
RR* )  ->  (
0  <_ inf ( {
a ,  R } ,  RR* ,  <  )  <->  ( 0  <_  a  /\  0  <_  R ) ) )
5336, 37, 38, 52syl3anc 1217 . . . . . . . . 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 929 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  a  e.  ( 0 [,] +oo ) )  ->  0  <_ inf ( { a ,  R } ,  RR* ,  <  ) )
55 xrlenlt 7852 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\ inf ( { a ,  R } ,  RR* ,  <  )  e.  RR* )  ->  (
0  <_ inf ( {
a ,  R } ,  RR* ,  <  )  <->  -. inf ( { a ,  R } ,  RR* ,  <  )  <  0
) )
5635, 32, 55sylancr 411 . . . . . . . 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 9612 . . . . . . 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 409 . . . . . 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 7852 . . . . . . . . 9  |-  ( ( 0  e.  RR*  /\  a  e.  RR* )  ->  (
0  <_  a  <->  -.  a  <  0 ) )
6335, 37, 62sylancr 411 . . . . . . . 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 9612 . . . . . . 7  |-  ( ( a  e.  RR*  /\  0  e.  RR* )  ->  (
a  =  0  <->  ( -.  a  <  0  /\  -.  0  <  a
) ) )
6737, 36, 66syl2anc 409 . . . . . 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 482 . . . . . . . 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 11067 . . . . . . . 8  |-  ( ( a  e.  RR*  /\  R  e.  RR* )  -> inf ( { a ,  R } ,  RR* ,  <  )  <_  a )
7471, 72, 73syl2anc 409 . . . . . . 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 409 . . . . . . . 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 9790 . . . . . . . . . 10  |-  ( b  e.  ( 0 [,] +oo )  <->  ( b  e. 
RR*  /\  0  <_  b ) )
7776simplbi 272 . . . . . . . . 9  |-  ( b  e.  ( 0 [,] +oo )  ->  b  e. 
RR* )
7877ad2antll 483 . . . . . . . 8  |-  ( ( ( C  e.  ( *Met `  X
)  /\  R  e.  RR* 
/\  0  <  R
)  /\  ( a  e.  ( 0 [,] +oo )  /\  b  e.  ( 0 [,] +oo )
) )  ->  b  e.  RR* )
79 xrletr 9620 . . . . . . . 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 1217 . . . . . . 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 426 . . . . . 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 11068 . . . . . . 7  |-  ( ( a  e.  RR*  /\  R  e.  RR* )  -> inf ( { a ,  R } ,  RR* ,  <  )  <_  R )
8371, 72, 82syl2anc 409 . . . . . 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 11071 . . . . . 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 1217 . . . . 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 471 . . . . 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 3607 . . . . . . 7  |-  ( z  =  b  ->  { z ,  R }  =  { b ,  R } )
9089infeq1d 6906 . . . . . 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 11066 . . . . . . 7  |-  ( ( b  e.  RR*  /\  R  e.  RR* )  -> inf ( { b ,  R } ,  RR* ,  <  )  e.  RR* )
9478, 72, 93syl2anc 409 . . . . . 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 5521 . . . . 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 3949 . . . 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 482 . . . . 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 483 . . . . 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 11076 . . . . 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 1233 . . . 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 3607 . . . . . 6  |-  ( z  =  ( a +e b )  ->  { z ,  R }  =  { (
a +e b ) ,  R }
)
106105infeq1d 6906 . . . . 5  |-  ( z  =  ( a +e b )  -> inf ( { z ,  R } ,  RR* ,  <  )  = inf ( { ( a +e b ) ,  R } ,  RR* ,  <  )
)
107 ge0xaddcl 9795 . . . . . 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 9696 . . . . . 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 11066 . . . . . 6  |-  ( ( ( a +e
b )  e.  RR*  /\  R  e.  RR* )  -> inf ( { ( a +e b ) ,  R } ,  RR* ,  <  )  e. 
RR* )
111109, 72, 110syl2anc 409 . . . . 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 5521 . . . 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 5799 . . . 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 3967 . . 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 12705 . 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 2218 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 963    = wceq 1332    e. wcel 1481   {cpr 3532   class class class wbr 3936    |-> cmpt 3996    X. cxp 4544    o. ccom 4550    Fn wfn 5125   -->wf 5126   ` cfv 5130  (class class class)co 5781    e. cmpo 5783  infcinf 6877   0cc0 7643   +oocpnf 7820   RR*cxr 7822    < clt 7823    <_ cle 7824   +ecxad 9586   [,]cicc 9703   *Metcxmet 12186
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509  ax-cnex 7734  ax-resscn 7735  ax-1cn 7736  ax-1re 7737  ax-icn 7738  ax-addcl 7739  ax-addrcl 7740  ax-mulcl 7741  ax-mulrcl 7742  ax-addcom 7743  ax-mulcom 7744  ax-addass 7745  ax-mulass 7746  ax-distr 7747  ax-i2m1 7748  ax-0lt1 7749  ax-1rid 7750  ax-0id 7751  ax-rnegex 7752  ax-precex 7753  ax-cnre 7754  ax-pre-ltirr 7755  ax-pre-ltwlin 7756  ax-pre-lttrn 7757  ax-pre-apti 7758  ax-pre-ltadd 7759  ax-pre-mulgt0 7760  ax-pre-mulext 7761  ax-arch 7762  ax-caucvg 7763
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-reu 2424  df-rmo 2425  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-po 4225  df-iso 4226  df-iord 4295  df-on 4297  df-ilim 4298  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-isom 5139  df-riota 5737  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-recs 6209  df-frec 6295  df-map 6551  df-sup 6878  df-inf 6879  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-sub 7958  df-neg 7959  df-reap 8360  df-ap 8367  df-div 8456  df-inn 8744  df-2 8802  df-3 8803  df-4 8804  df-n0 9001  df-z 9078  df-uz 9350  df-rp 9470  df-xneg 9588  df-xadd 9589  df-icc 9707  df-seqfrec 10249  df-exp 10323  df-cj 10645  df-re 10646  df-im 10647  df-rsqrt 10801  df-abs 10802  df-xmet 12194
This theorem is referenced by:  bdmet  12708  bdbl  12709  bdmopn  12710
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