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Theorem metustexhalf 20161
Description: For any element  A of the filter base generated by the metric  D, the half element (corresponding to half the distance) is also in this base. (Contributed by Thierry Arnoux, 28-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
Assertion
Ref Expression
metustexhalf  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  ->  E. v  e.  F  ( v  o.  v )  C_  A
)
Distinct variable groups:    D, a    X, a    A, a    F, a, v    v, A    v, D    v, F    v, X

Proof of Theorem metustexhalf
Dummy variables  b  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp-4r 766 . . . 4  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  ->  D  e.  (PsMet `  X
) )
2 simplr 754 . . . . . 6  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
a  e.  RR+ )
32rphalfcld 11060 . . . . 5  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( a  /  2
)  e.  RR+ )
4 eqidd 2444 . . . . 5  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
5 oveq2 6120 . . . . . . . 8  |-  ( b  =  ( a  / 
2 )  ->  (
0 [,) b )  =  ( 0 [,) ( a  /  2
) ) )
65imaeq2d 5190 . . . . . . 7  |-  ( b  =  ( a  / 
2 )  ->  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )
76eqeq2d 2454 . . . . . 6  |-  ( b  =  ( a  / 
2 )  ->  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) b ) )  <-> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )
87rspcev 3094 . . . . 5  |-  ( ( ( a  /  2
)  e.  RR+  /\  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  =  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  ->  E. b  e.  RR+  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  =  ( `' D " ( 0 [,) b
) ) )
93, 4, 8syl2anc 661 . . . 4  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  ->  E. b  e.  RR+  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  =  ( `' D " ( 0 [,) b
) ) )
10 metust.1 . . . . . . 7  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
11 oveq2 6120 . . . . . . . . . 10  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
1211imaeq2d 5190 . . . . . . . . 9  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
1312cbvmptv 4404 . . . . . . . 8  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
1413rneqi 5087 . . . . . . 7  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
1510, 14eqtri 2463 . . . . . 6  |-  F  =  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )
1615metustel 20149 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  e.  F  <->  E. b  e.  RR+  ( `' D " ( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) b ) ) ) )
1716biimpar 485 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  E. b  e.  RR+  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  =  ( `' D " ( 0 [,) b
) ) )  -> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  e.  F
)
181, 9, 17syl2anc 661 . . 3  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  e.  F
)
19 relco 5357 . . . . 5  |-  Rel  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
2019a1i 11 . . . 4  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  ->  Rel  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )
21 cossxp 5381 . . . . . . . . . 10  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  ( dom  ( `' D " ( 0 [,) (
a  /  2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
22 cnvimass 5210 . . . . . . . . . . . . . 14  |-  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  dom  D
23 psmetf 19904 . . . . . . . . . . . . . . 15  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
24 fdm 5584 . . . . . . . . . . . . . . 15  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
2523, 24syl 16 . . . . . . . . . . . . . 14  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
2622, 25syl5sseq 3425 . . . . . . . . . . . . 13  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) ( a  /  2
) ) )  C_  ( X  X.  X
) )
27 dmss 5060 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  dom  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  dom  ( X  X.  X ) )
28 rnss 5089 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ran  ( X  X.  X ) )
29 xpss12 4966 . . . . . . . . . . . . . 14  |-  ( ( dom  ( `' D " ( 0 [,) (
a  /  2 ) ) )  C_  dom  ( X  X.  X
)  /\  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ran  ( X  X.  X ) )  -> 
( dom  ( `' D " ( 0 [,) ( a  /  2
) ) )  X. 
ran  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  ( dom  ( X  X.  X )  X.  ran  ( X  X.  X
) ) )
3027, 28, 29syl2anc 661 . . . . . . . . . . . . 13  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  ( dom  ( `' D "
( 0 [,) (
a  /  2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) ) )
3126, 30syl 16 . . . . . . . . . . . 12  |-  ( D  e.  (PsMet `  X
)  ->  ( dom  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) ) )
3231adantl 466 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( dom  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) ) )
33 dmxp 5079 . . . . . . . . . . . . 13  |-  ( X  =/=  (/)  ->  dom  ( X  X.  X )  =  X )
34 rnxp 5289 . . . . . . . . . . . . 13  |-  ( X  =/=  (/)  ->  ran  ( X  X.  X )  =  X )
3533, 34xpeq12d 4886 . . . . . . . . . . . 12  |-  ( X  =/=  (/)  ->  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) )  =  ( X  X.  X ) )
3635adantr 465 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) )  =  ( X  X.  X ) )
3732, 36sseqtrd 3413 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( dom  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( X  X.  X ) )
3821, 37syl5ss 3388 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  ( X  X.  X
) )
3938ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( X  X.  X ) )
4039sselda 3377 . . . . . . 7  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  <.
p ,  q >.  e.  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )  ->  <. p ,  q >.  e.  ( X  X.  X ) )
41 opelxp 4890 . . . . . . 7  |-  ( <.
p ,  q >.  e.  ( X  X.  X
)  <->  ( p  e.  X  /\  q  e.  X ) )
4240, 41sylib 196 . . . . . 6  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  <.
p ,  q >.  e.  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )  ->  (
p  e.  X  /\  q  e.  X )
)
43 simpll 753 . . . . . . 7  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  <.
p ,  q >.  e.  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  ( (
( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a ) ) ) )
44 simprl 755 . . . . . . 7  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  <.
p ,  q >.  e.  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  p  e.  X )
45 simprr 756 . . . . . . 7  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  <.
p ,  q >.  e.  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  q  e.  X )
46 simplr 754 . . . . . . 7  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  <.
p ,  q >.  e.  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  <. p ,  q >.  e.  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )
47 simplll 757 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( (
( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )
)
4847simp1d 1000 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( (
( ( X  =/=  (/)  /\  D  e.  (PsMet `  X ) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a ) ) ) )
4948, 1syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  D  e.  (PsMet `  X ) )
5048, 2syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  a  e.  RR+ )
5149, 50jca 532 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( D  e.  (PsMet `  X )  /\  a  e.  RR+ )
)
5247simp2d 1001 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  p  e.  X )
5347simp3d 1002 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  q  e.  X )
5451, 52, 533jca 1168 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X ) )
55 simplr 754 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  r  e.  X )
56 simprl 755 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r )
57 simprr 756 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q )
58 simpll 753 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X ) )
5958simp1d 1000 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( D  e.  (PsMet `  X )  /\  a  e.  RR+ )
)
6059simpld 459 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  D  e.  (PsMet `  X ) )
61 ffun 5582 . . . . . . . . . . . . . 14  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
6223, 61syl 16 . . . . . . . . . . . . 13  |-  ( D  e.  (PsMet `  X
)  ->  Fun  D )
6360, 62syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  Fun  D )
6458simp2d 1001 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  p  e.  X )
6558simp3d 1002 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  q  e.  X )
6664, 65, 41sylanbrc 664 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  <. p ,  q >.  e.  ( X  X.  X ) )
6760, 25syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  dom  D  =  ( X  X.  X
) )
6866, 67eleqtrrd 2520 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  <. p ,  q >.  e.  dom  D )
69 0xr 9451 . . . . . . . . . . . . . 14  |-  0  e.  RR*
7069a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  0  e.  RR* )
7159simprd 463 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  a  e.  RR+ )
7271rpxrd 11049 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  a  e.  RR* )
7360, 23syl 16 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  D :
( X  X.  X
) --> RR* )
7473, 66ffvelrnd 5865 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( D `  <. p ,  q
>. )  e.  RR* )
75 psmetge0 19910 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  p  e.  X  /\  q  e.  X )  ->  0  <_  ( p D q ) )
7660, 64, 65, 75syl3anc 1218 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  0  <_  ( p D q ) )
77 df-ov 6115 . . . . . . . . . . . . . 14  |-  ( p D q )  =  ( D `  <. p ,  q >. )
7876, 77syl6breq 4352 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  0  <_  ( D `  <. p ,  q >. )
)
7977, 74syl5eqel 2527 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( p D q )  e. 
RR* )
80 0red 9408 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  0  e.  RR )
8171rpred 11048 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  a  e.  RR )
8281rehalfcld 10592 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( a  /  2 )  e.  RR )
8382rexrd 9454 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( a  /  2 )  e. 
RR* )
84 df-ov 6115 . . . . . . . . . . . . . . . . . . . 20  |-  ( p D r )  =  ( D `  <. p ,  r >. )
85 simplr 754 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  r  e.  X )
86 opelxp 4890 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
p ,  r >.  e.  ( X  X.  X
)  <->  ( p  e.  X  /\  r  e.  X ) )
8764, 85, 86sylanbrc 664 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  <. p ,  r >.  e.  ( X  X.  X ) )
8887, 67eleqtrrd 2520 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  <. p ,  r >.  e.  dom  D )
89 simprl 755 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r )
90 df-br 4314 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  <->  <. p ,  r >.  e.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
9189, 90sylib 196 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  <. p ,  r >.  e.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
92 fvimacnv 5839 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  D  /\  <. p ,  r >.  e.  dom  D )  ->  ( ( D `  <. p ,  r >. )  e.  ( 0 [,) ( a  /  2 ) )  <->  <. p ,  r >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) ) )
9392biimpar 485 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( Fun  D  /\  <.
p ,  r >.  e.  dom  D )  /\  <.
p ,  r >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) )  -> 
( D `  <. p ,  r >. )  e.  ( 0 [,) (
a  /  2 ) ) )
9463, 88, 91, 93syl21anc 1217 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( D `  <. p ,  r
>. )  e.  (
0 [,) ( a  /  2 ) ) )
9584, 94syl5eqel 2527 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( p D r )  e.  ( 0 [,) (
a  /  2 ) ) )
96 elico2 11380 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  ->  ( ( p D r )  e.  ( 0 [,) ( a  /  2 ) )  <-> 
( ( p D r )  e.  RR  /\  0  <_  ( p D r )  /\  ( p D r )  <  ( a  /  2 ) ) ) )
9796biimpa 484 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( ( p D r )  e.  RR  /\  0  <_ 
( p D r )  /\  ( p D r )  < 
( a  /  2
) ) )
9897simp1d 1000 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( p D r )  e.  RR )
9980, 83, 95, 98syl21anc 1217 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( p D r )  e.  RR )
100 df-ov 6115 . . . . . . . . . . . . . . . . . . . 20  |-  ( r D q )  =  ( D `  <. r ,  q >. )
101 opelxp 4890 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
r ,  q >.  e.  ( X  X.  X
)  <->  ( r  e.  X  /\  q  e.  X ) )
10285, 65, 101sylanbrc 664 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  <. r ,  q >.  e.  ( X  X.  X ) )
103102, 67eleqtrrd 2520 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  <. r ,  q >.  e.  dom  D )
104 simprr 756 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q )
105 df-br 4314 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( r ( `' D "
( 0 [,) (
a  /  2 ) ) ) q  <->  <. r ,  q >.  e.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
106104, 105sylib 196 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  <. r ,  q >.  e.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
107 fvimacnv 5839 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  D  /\  <. r ,  q >.  e.  dom  D )  ->  ( ( D `  <. r ,  q >. )  e.  ( 0 [,) ( a  /  2 ) )  <->  <. r ,  q >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) ) )
108107biimpar 485 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( Fun  D  /\  <.
r ,  q >.  e.  dom  D )  /\  <.
r ,  q >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) )  -> 
( D `  <. r ,  q >. )  e.  ( 0 [,) (
a  /  2 ) ) )
10963, 103, 106, 108syl21anc 1217 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( D `  <. r ,  q
>. )  e.  (
0 [,) ( a  /  2 ) ) )
110100, 109syl5eqel 2527 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( r D q )  e.  ( 0 [,) (
a  /  2 ) ) )
111 elico2 11380 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  ->  ( ( r D q )  e.  ( 0 [,) ( a  /  2 ) )  <-> 
( ( r D q )  e.  RR  /\  0  <_  ( r D q )  /\  ( r D q )  <  ( a  /  2 ) ) ) )
112111biimpa 484 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( ( r D q )  e.  RR  /\  0  <_ 
( r D q )  /\  ( r D q )  < 
( a  /  2
) ) )
113112simp1d 1000 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( r D q )  e.  RR )
11480, 83, 110, 113syl21anc 1217 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( r D q )  e.  RR )
115 rexadd 11223 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( p D r )  e.  RR  /\  ( r D q )  e.  RR )  ->  ( ( p D r ) +e ( r D q ) )  =  ( ( p D r )  +  ( r D q ) ) )
11699, 114, 115syl2anc 661 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( (
p D r ) +e ( r D q ) )  =  ( ( p D r )  +  ( r D q ) ) )
11799, 114readdcld 9434 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( (
p D r )  +  ( r D q ) )  e.  RR )
118116, 117eqeltrd 2517 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( (
p D r ) +e ( r D q ) )  e.  RR )
119118rexrd 9454 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( (
p D r ) +e ( r D q ) )  e.  RR* )
120 psmettri 19909 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  (PsMet `  X )  /\  (
p  e.  X  /\  q  e.  X  /\  r  e.  X )
)  ->  ( p D q )  <_ 
( ( p D r ) +e
( r D q ) ) )
12160, 64, 65, 85, 120syl13anc 1220 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( p D q )  <_ 
( ( p D r ) +e
( r D q ) ) )
12297simp3d 1002 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( p D r )  <  (
a  /  2 ) )
12380, 83, 95, 122syl21anc 1217 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( p D r )  < 
( a  /  2
) )
124112simp3d 1002 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( r D q )  <  (
a  /  2 ) )
12580, 83, 110, 124syl21anc 1217 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( r D q )  < 
( a  /  2
) )
12699, 114, 81, 123, 125lt2halvesd 10593 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( (
p D r )  +  ( r D q ) )  < 
a )
127116, 126eqbrtrd 4333 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( (
p D r ) +e ( r D q ) )  <  a )
12879, 119, 72, 121, 127xrlelttrd 11155 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( p D q )  < 
a )
12977, 128syl5eqbrr 4347 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( D `  <. p ,  q
>. )  <  a )
130 elico1 11364 . . . . . . . . . . . . . 14  |-  ( ( 0  e.  RR*  /\  a  e.  RR* )  ->  (
( D `  <. p ,  q >. )  e.  ( 0 [,) a
)  <->  ( ( D `
 <. p ,  q
>. )  e.  RR*  /\  0  <_  ( D `  <. p ,  q >. )  /\  ( D `  <. p ,  q >. )  <  a ) ) )
131130biimpar 485 . . . . . . . . . . . . 13  |-  ( ( ( 0  e.  RR*  /\  a  e.  RR* )  /\  ( ( D `  <. p ,  q >.
)  e.  RR*  /\  0  <_  ( D `  <. p ,  q >. )  /\  ( D `  <. p ,  q >. )  <  a ) )  -> 
( D `  <. p ,  q >. )  e.  ( 0 [,) a
) )
13270, 72, 74, 78, 129, 131syl23anc 1225 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( D `  <. p ,  q
>. )  e.  (
0 [,) a ) )
133 fvimacnv 5839 . . . . . . . . . . . . . 14  |-  ( ( Fun  D  /\  <. p ,  q >.  e.  dom  D )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
134133biimpa 484 . . . . . . . . . . . . 13  |-  ( ( ( Fun  D  /\  <.
p ,  q >.  e.  dom  D )  /\  ( D `  <. p ,  q >. )  e.  ( 0 [,) a
) )  ->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) )
135 df-br 4314 . . . . . . . . . . . . 13  |-  ( p ( `' D "
( 0 [,) a
) ) q  <->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) )
136134, 135sylibr 212 . . . . . . . . . . . 12  |-  ( ( ( Fun  D  /\  <.
p ,  q >.  e.  dom  D )  /\  ( D `  <. p ,  q >. )  e.  ( 0 [,) a
) )  ->  p
( `' D "
( 0 [,) a
) ) q )
13763, 68, 132, 136syl21anc 1217 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )  /\  r  e.  X )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  p ( `' D " ( 0 [,) a ) ) q )
13854, 55, 56, 57, 137syl22anc 1219 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  p ( `' D " ( 0 [,) a ) ) q )
13948simprd 463 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  A  =  ( `' D " ( 0 [,) a ) ) )
140139breqd 4324 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  ( p A q  <->  p ( `' D " ( 0 [,) a ) ) q ) )
141138, 140mpbird 232 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  p A
q )
142 simpr 461 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  ->  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )
143 df-br 4314 . . . . . . . . . . . . 13  |-  ( p ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) q  <->  <. p ,  q >.  e.  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )
144142, 143sylibr 212 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  ->  p ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) q )
145 vex 2996 . . . . . . . . . . . . 13  |-  p  e. 
_V
146 vex 2996 . . . . . . . . . . . . 13  |-  q  e. 
_V
147145, 146brco 5031 . . . . . . . . . . . 12  |-  ( p ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) q  <->  E. r
( p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r  /\  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q ) )
148144, 147sylib 196 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  ->  E. r ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )
14926adantl 466 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( `' D " ( 0 [,) ( a  /  2
) ) )  C_  ( X  X.  X
) )
150149, 28syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ran  ( X  X.  X ) )
15134adantr 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( X  X.  X )  =  X )
152150, 151sseqtrd 3413 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  X )
153152adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r )  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  X )
154 vex 2996 . . . . . . . . . . . . . . . . . . . . 21  |-  r  e. 
_V
155145, 154brelrn 5091 . . . . . . . . . . . . . . . . . . . 20  |-  ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  -> 
r  e.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
156155adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r )  ->  r  e.  ran  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )
157153, 156sseldd 3378 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r )  ->  r  e.  X )
158157adantrr 716 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  r  e.  X )
159158ex 434 . . . . . . . . . . . . . . . 16  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q )  ->  r  e.  X
) )
160159ancrd 554 . . . . . . . . . . . . . . 15  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q )  ->  ( r  e.  X  /\  ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) ) ) )
161160eximdv 1676 . . . . . . . . . . . . . 14  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( E. r ( p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r  /\  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q )  ->  E. r
( r  e.  X  /\  ( p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r  /\  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q ) ) ) )
162161ad3antrrr 729 . . . . . . . . . . . . 13  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( E. r ( p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q )  ->  E. r ( r  e.  X  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) ) ) )
1631623ad2ant1 1009 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  ->  ( E. r ( p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q )  ->  E. r ( r  e.  X  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) ) ) )
164163adantr 465 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  ->  ( E. r
( p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r  /\  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q )  ->  E. r
( r  e.  X  /\  ( p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r  /\  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q ) ) ) )
165148, 164mpd 15 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  ->  E. r ( r  e.  X  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) ) )
166 df-rex 2742 . . . . . . . . . 10  |-  ( E. r  e.  X  ( p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q )  <->  E. r ( r  e.  X  /\  ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) ) )
167165, 166sylibr 212 . . . . . . . . 9  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  ->  E. r  e.  X  ( p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r  /\  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q ) )
168141, 167r19.29a 2883 . . . . . . . 8  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  ->  p A q )
169 df-br 4314 . . . . . . . 8  |-  ( p A q  <->  <. p ,  q >.  e.  A
)
170168, 169sylib 196 . . . . . . 7  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )  /\  <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )  ->  <. p ,  q
>.  e.  A )
17143, 44, 45, 46, 170syl31anc 1221 . . . . . 6  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  <.
p ,  q >.  e.  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )  /\  (
p  e.  X  /\  q  e.  X )
)  ->  <. p ,  q >.  e.  A
)
17242, 171mpdan 668 . . . . 5  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  <.
p ,  q >.  e.  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )  ->  <. p ,  q >.  e.  A
)
173172ex 434 . . . 4  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( <. p ,  q
>.  e.  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  ->  <. p ,  q >.  e.  A ) )
17420, 173relssdv 4953 . . 3  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  A )
175 id 22 . . . . . 6  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )
176175, 175coeq12d 5025 . . . . 5  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  (
v  o.  v )  =  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )
177176sseq1d 3404 . . . 4  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  (
( v  o.  v
)  C_  A  <->  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  A ) )
178177rspcev 3094 . . 3  |-  ( ( ( `' D "
( 0 [,) (
a  /  2 ) ) )  e.  F  /\  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  A )  ->  E. v  e.  F  ( v  o.  v
)  C_  A )
17918, 174, 178syl2anc 661 . 2  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  ->  E. v  e.  F  ( v  o.  v
)  C_  A )
18010metustel 20149 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
181180adantl 466 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
182181biimpa 484 . 2  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
183179, 182r19.29a 2883 1  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  ->  E. v  e.  F  ( v  o.  v )  C_  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2620   E.wrex 2737    C_ wss 3349   (/)c0 3658   <.cop 3904   class class class wbr 4313    e. cmpt 4371    X. cxp 4859   `'ccnv 4860   dom cdm 4861   ran crn 4862   "cima 4864    o. ccom 4865   Rel wrel 4866   Fun wfun 5433   -->wf 5435   ` cfv 5439  (class class class)co 6112   RRcr 9302   0cc0 9303    + caddc 9306   RR*cxr 9438    < clt 9439    <_ cle 9440    / cdiv 10014   2c2 10392   RR+crp 11012   +ecxad 11108   [,)cico 11323  PsMetcpsmet 17822
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4434  ax-nul 4442  ax-pow 4491  ax-pr 4552  ax-un 6393  ax-cnex 9359  ax-resscn 9360  ax-1cn 9361  ax-icn 9362  ax-addcl 9363  ax-addrcl 9364  ax-mulcl 9365  ax-mulrcl 9366  ax-mulcom 9367  ax-addass 9368  ax-mulass 9369  ax-distr 9370  ax-i2m1 9371  ax-1ne0 9372  ax-1rid 9373  ax-rnegex 9374  ax-rrecex 9375  ax-cnre 9376  ax-pre-lttri 9377  ax-pre-lttrn 9378  ax-pre-ltadd 9379  ax-pre-mulgt0 9380
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-nel 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 2995  df-sbc 3208  df-csb 3310  df-dif 3352  df-un 3354  df-in 3356  df-ss 3363  df-nul 3659  df-if 3813  df-pw 3883  df-sn 3899  df-pr 3901  df-op 3905  df-uni 4113  df-iun 4194  df-br 4314  df-opab 4372  df-mpt 4373  df-id 4657  df-po 4662  df-so 4663  df-xp 4867  df-rel 4868  df-cnv 4869  df-co 4870  df-dm 4871  df-rn 4872  df-res 4873  df-ima 4874  df-iota 5402  df-fun 5441  df-fn 5442  df-f 5443  df-f1 5444  df-fo 5445  df-f1o 5446  df-fv 5447  df-riota 6073  df-ov 6115  df-oprab 6116  df-mpt2 6117  df-1st 6598  df-2nd 6599  df-er 7122  df-map 7237  df-en 7332  df-dom 7333  df-sdom 7334  df-pnf 9441  df-mnf 9442  df-xr 9443  df-ltxr 9444  df-le 9445  df-sub 9618  df-neg 9619  df-div 10015  df-2 10401  df-rp 11013  df-xneg 11110  df-xadd 11111  df-xmul 11112  df-ico 11327  df-psmet 17831
This theorem is referenced by:  metust  20165
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