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Theorem metustexhalf 21571
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 777 . . . 4  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  ->  D  e.  (PsMet `  X
) )
2 simplr 762 . . . . . 6  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
a  e.  RR+ )
32rphalfcld 11353 . . . . 5  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( a  /  2
)  e.  RR+ )
4 eqidd 2452 . . . . 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 6298 . . . . . . . 8  |-  ( b  =  ( a  / 
2 )  ->  (
0 [,) b )  =  ( 0 [,) ( a  /  2
) ) )
65imaeq2d 5168 . . . . . . 7  |-  ( b  =  ( a  / 
2 )  ->  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )
76eqeq2d 2461 . . . . . 6  |-  ( b  =  ( a  / 
2 )  ->  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) b ) )  <-> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )
87rspcev 3150 . . . . 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 667 . . . 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 6298 . . . . . . . . . 10  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
1211imaeq2d 5168 . . . . . . . . 9  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
1312cbvmptv 4495 . . . . . . . 8  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
1413rneqi 5061 . . . . . . 7  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
1510, 14eqtri 2473 . . . . . 6  |-  F  =  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )
1615metustel 21565 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  e.  F  <->  E. b  e.  RR+  ( `' D " ( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) b ) ) ) )
1716biimpar 488 . . . 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 667 . . 3  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  e.  F
)
19 relco 5333 . . . . 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 5358 . . . . . . . . . 10  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  ( dom  ( `' D " ( 0 [,) (
a  /  2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
22 cnvimass 5188 . . . . . . . . . . . . . 14  |-  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  dom  D
23 psmetf 21322 . . . . . . . . . . . . . . 15  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
24 fdm 5733 . . . . . . . . . . . . . . 15  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
2523, 24syl 17 . . . . . . . . . . . . . 14  |-  ( D  e.  (PsMet `  X
)  ->  dom  D  =  ( X  X.  X
) )
2622, 25syl5sseq 3480 . . . . . . . . . . . . 13  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) ( a  /  2
) ) )  C_  ( X  X.  X
) )
27 dmss 5034 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  dom  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  dom  ( X  X.  X ) )
28 rnss 5063 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ran  ( X  X.  X ) )
29 xpss12 4940 . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . 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 17 . . . . . . . . . . . 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 468 . . . . . . . . . . 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 5053 . . . . . . . . . . . . 13  |-  ( X  =/=  (/)  ->  dom  ( X  X.  X )  =  X )
34 rnxp 5267 . . . . . . . . . . . . 13  |-  ( X  =/=  (/)  ->  ran  ( X  X.  X )  =  X )
3533, 34xpeq12d 4859 . . . . . . . . . . . 12  |-  ( X  =/=  (/)  ->  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) )  =  ( X  X.  X ) )
3635adantr 467 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) )  =  ( X  X.  X ) )
3732, 36sseqtrd 3468 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( dom  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( X  X.  X ) )
3821, 37syl5ss 3443 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  ( X  X.  X
) )
3938ad3antrrr 736 . . . . . . . 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 3432 . . . . . . 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 4864 . . . . . . 7  |-  ( <.
p ,  q >.  e.  ( X  X.  X
)  <->  ( p  e.  X  /\  q  e.  X ) )
4240, 41sylib 200 . . . . . 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 760 . . . . . . 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 764 . . . . . . 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 766 . . . . . . 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 762 . . . . . . 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 768 . . . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . . 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 17 . . . . . . . . . . . . 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 17 . . . . . . . . . . . . 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 535 . . . . . . . . . . . 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 1021 . . . . . . . . . . . 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 1022 . . . . . . . . . . . 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 1188 . . . . . . . . . . 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 762 . . . . . . . . . . 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 764 . . . . . . . . . . 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 766 . . . . . . . . . . 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 760 . . . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . . 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 461 . . . . . . . . . . . . 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 5731 . . . . . . . . . . . . . 14  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
6223, 61syl 17 . . . . . . . . . . . . 13  |-  ( D  e.  (PsMet `  X
)  ->  Fun  D )
6360, 62syl 17 . . . . . . . . . . . 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 1021 . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . 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 670 . . . . . . . . . . . . 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 17 . . . . . . . . . . . . 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 2532 . . . . . . . . . . . 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 9687 . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . 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 11342 . . . . . . . . . . . . 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 17 . . . . . . . . . . . . . 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 6023 . . . . . . . . . . . . 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 21328 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  p  e.  X  /\  q  e.  X )  ->  0  <_  ( p D q ) )
7660, 64, 65, 75syl3anc 1268 . . . . . . . . . . . . . 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 6293 . . . . . . . . . . . . . 14  |-  ( p D q )  =  ( D `  <. p ,  q >. )
7876, 77syl6breq 4442 . . . . . . . . . . . . 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 2533 . . . . . . . . . . . . . . 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 9644 . . . . . . . . . . . . . . . . . . 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 11341 . . . . . . . . . . . . . . . . . . . . 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 10859 . . . . . . . . . . . . . . . . . . . 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 9690 . . . . . . . . . . . . . . . . . . 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 6293 . . . . . . . . . . . . . . . . . . . 20  |-  ( p D r )  =  ( D `  <. p ,  r >. )
85 simplr 762 . . . . . . . . . . . . . . . . . . . . . . 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 4864 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
p ,  r >.  e.  ( X  X.  X
)  <->  ( p  e.  X  /\  r  e.  X ) )
8764, 85, 86sylanbrc 670 . . . . . . . . . . . . . . . . . . . . . 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 2532 . . . . . . . . . . . . . . . . . . . . 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 764 . . . . . . . . . . . . . . . . . . . . . 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 4403 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  <->  <. p ,  r >.  e.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
9189, 90sylib 200 . . . . . . . . . . . . . . . . . . . . 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 5997 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  D  /\  <. p ,  r >.  e.  dom  D )  ->  ( ( D `  <. p ,  r >. )  e.  ( 0 [,) ( a  /  2 ) )  <->  <. p ,  r >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) ) )
9392biimpar 488 . . . . . . . . . . . . . . . . . . . . 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 1267 . . . . . . . . . . . . . . . . . . . 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 2533 . . . . . . . . . . . . . . . . . . 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 11698 . . . . . . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( p D r )  e.  RR )
9980, 83, 95, 98syl21anc 1267 . . . . . . . . . . . . . . . . . 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 6293 . . . . . . . . . . . . . . . . . . . 20  |-  ( r D q )  =  ( D `  <. r ,  q >. )
101 opelxp 4864 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
r ,  q >.  e.  ( X  X.  X
)  <->  ( r  e.  X  /\  q  e.  X ) )
10285, 65, 101sylanbrc 670 . . . . . . . . . . . . . . . . . . . . . 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 2532 . . . . . . . . . . . . . . . . . . . . 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 766 . . . . . . . . . . . . . . . . . . . . . 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 4403 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( r ( `' D "
( 0 [,) (
a  /  2 ) ) ) q  <->  <. r ,  q >.  e.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
106104, 105sylib 200 . . . . . . . . . . . . . . . . . . . . 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 5997 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  D  /\  <. r ,  q >.  e.  dom  D )  ->  ( ( D `  <. r ,  q >. )  e.  ( 0 [,) ( a  /  2 ) )  <->  <. r ,  q >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) ) )
108107biimpar 488 . . . . . . . . . . . . . . . . . . . . 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 1267 . . . . . . . . . . . . . . . . . . . 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 2533 . . . . . . . . . . . . . . . . . . 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 11698 . . . . . . . . . . . . . . . . . . . . 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 487 . . . . . . . . . . . . . . . . . . . 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 1020 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( r D q )  e.  RR )
11480, 83, 110, 113syl21anc 1267 . . . . . . . . . . . . . . . . . 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 11525 . . . . . . . . . . . . . . . . . 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 667 . . . . . . . . . . . . . . . . 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 9670 . . . . . . . . . . . . . . . . 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 2529 . . . . . . . . . . . . . . . 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 9690 . . . . . . . . . . . . . . 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 21327 . . . . . . . . . . . . . . . 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 1270 . . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( p D r )  <  (
a  /  2 ) )
12380, 83, 95, 122syl21anc 1267 . . . . . . . . . . . . . . . . 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 1022 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( r D q )  <  (
a  /  2 ) )
12580, 83, 110, 124syl21anc 1267 . . . . . . . . . . . . . . . . 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 10860 . . . . . . . . . . . . . . . 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 4423 . . . . . . . . . . . . . . 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 11457 . . . . . . . . . . . . . 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 4437 . . . . . . . . . . . . 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 11679 . . . . . . . . . . . . . 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 488 . . . . . . . . . . . . 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 1275 . . . . . . . . . . . 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 5997 . . . . . . . . . . . . . 14  |-  ( ( Fun  D  /\  <. p ,  q >.  e.  dom  D )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
134133biimpa 487 . . . . . . . . . . . . 13  |-  ( ( ( Fun  D  /\  <.
p ,  q >.  e.  dom  D )  /\  ( D `  <. p ,  q >. )  e.  ( 0 [,) a
) )  ->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) )
135 df-br 4403 . . . . . . . . . . . . 13  |-  ( p ( `' D "
( 0 [,) a
) ) q  <->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) )
136134, 135sylibr 216 . . . . . . . . . . . 12  |-  ( ( ( Fun  D  /\  <.
p ,  q >.  e.  dom  D )  /\  ( D `  <. p ,  q >. )  e.  ( 0 [,) a
) )  ->  p
( `' D "
( 0 [,) a
) ) q )
13763, 68, 132, 136syl21anc 1267 . . . . . . . . . . 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 1269 . . . . . . . . . 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 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 ) ) ) ) )  /\  r  e.  X
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  A  =  ( `' D " ( 0 [,) a ) ) )
140139breqd 4413 . . . . . . . . . 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 236 . . . . . . . . 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 463 . . . . . . . . . . . . 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 4403 . . . . . . . . . . . . 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 216 . . . . . . . . . . . 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 3048 . . . . . . . . . . . . 13  |-  p  e. 
_V
146 vex 3048 . . . . . . . . . . . . 13  |-  q  e. 
_V
147145, 146brco 5005 . . . . . . . . . . . 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 200 . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( `' D " ( 0 [,) ( a  /  2
) ) )  C_  ( X  X.  X
) )
150149, 28syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ran  ( X  X.  X ) )
15134adantr 467 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( X  X.  X )  =  X )
152150, 151sseqtrd 3468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  X )
153152adantr 467 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r )  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  X )
154 vex 3048 . . . . . . . . . . . . . . . . . . . . 21  |-  r  e. 
_V
155145, 154brelrn 5065 . . . . . . . . . . . . . . . . . . . 20  |-  ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  -> 
r  e.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
156155adantl 468 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r )  ->  r  e.  ran  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )
157153, 156sseldd 3433 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r )  ->  r  e.  X )
158157adantrr 723 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  ( p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  r  e.  X )
159158ex 436 . . . . . . . . . . . . . . . 16  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q )  ->  r  e.  X
) )
160159ancrd 557 . . . . . . . . . . . . . . 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 1764 . . . . . . . . . . . . . 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 736 . . . . . . . . . . . . 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 1029 . . . . . . . . . . . 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 467 . . . . . . . . . . 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 2743 . . . . . . . . . 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 216 . . . . . . . . 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 2932 . . . . . . . 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 4403 . . . . . . . 8  |-  ( p A q  <->  <. p ,  q >.  e.  A
)
170168, 169sylib 200 . . . . . . 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 1271 . . . . . 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 674 . . . . 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 436 . . . 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 4927 . . 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 4999 . . . . 5  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  (
v  o.  v )  =  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )
177176sseq1d 3459 . . . 4  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  (
( v  o.  v
)  C_  A  <->  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  A ) )
178177rspcev 3150 . . 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 667 . 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 21565 . . . 4  |-  ( D  e.  (PsMet `  X
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
181180adantl 468 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
182181biimpa 487 . 2  |-  ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
183179, 182r19.29a 2932 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 188    /\ wa 371    /\ w3a 985    = wceq 1444   E.wex 1663    e. wcel 1887    =/= wne 2622   E.wrex 2738    C_ wss 3404   (/)c0 3731   <.cop 3974   class class class wbr 4402    |-> cmpt 4461    X. cxp 4832   `'ccnv 4833   dom cdm 4834   ran crn 4835   "cima 4837    o. ccom 4838   Rel wrel 4839   Fun wfun 5576   -->wf 5578   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539    + caddc 9542   RR*cxr 9674    < clt 9675    <_ cle 9676    / cdiv 10269   2c2 10659   RR+crp 11302   +ecxad 11407   [,)cico 11637  PsMetcpsmet 18954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rmo 2745  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-div 10270  df-2 10668  df-rp 11303  df-xneg 11409  df-xadd 11410  df-xmul 11411  df-ico 11641  df-psmet 18962
This theorem is referenced by:  metust  21573
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