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Theorem metustexhalf 20935
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 11280 . . . . 5  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( a  /  2
)  e.  RR+ )
4 eqidd 2468 . . . . 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 6303 . . . . . . . 8  |-  ( b  =  ( a  / 
2 )  ->  (
0 [,) b )  =  ( 0 [,) ( a  /  2
) ) )
65imaeq2d 5343 . . . . . . 7  |-  ( b  =  ( a  / 
2 )  ->  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )
76eqeq2d 2481 . . . . . 6  |-  ( b  =  ( a  / 
2 )  ->  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) b ) )  <-> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )
87rspcev 3219 . . . . 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 6303 . . . . . . . . . 10  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
1211imaeq2d 5343 . . . . . . . . 9  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
1312cbvmptv 4544 . . . . . . . 8  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
1413rneqi 5235 . . . . . . 7  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
1510, 14eqtri 2496 . . . . . 6  |-  F  =  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )
1615metustel 20923 . . . . 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 5511 . . . . 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 5536 . . . . . . . . . 10  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  ( dom  ( `' D " ( 0 [,) (
a  /  2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
22 cnvimass 5363 . . . . . . . . . . . . . 14  |-  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  dom  D
23 psmetf 20678 . . . . . . . . . . . . . . 15  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
24 fdm 5741 . . . . . . . . . . . . . . 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 3557 . . . . . . . . . . . . 13  |-  ( D  e.  (PsMet `  X
)  ->  ( `' D " ( 0 [,) ( a  /  2
) ) )  C_  ( X  X.  X
) )
27 dmss 5208 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  dom  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  dom  ( X  X.  X ) )
28 rnss 5237 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ran  ( X  X.  X ) )
29 xpss12 5114 . . . . . . . . . . . . . 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 5227 . . . . . . . . . . . . 13  |-  ( X  =/=  (/)  ->  dom  ( X  X.  X )  =  X )
34 rnxp 5443 . . . . . . . . . . . . 13  |-  ( X  =/=  (/)  ->  ran  ( X  X.  X )  =  X )
3533, 34xpeq12d 5030 . . . . . . . . . . . 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 3545 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  (PsMet `  X )
)  ->  ( dom  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( X  X.  X ) )
3821, 37syl5ss 3520 . . . . . . . . 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 3509 . . . . . . 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 5035 . . . . . . 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 1008 . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . 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 1010 . . . . . . . . . . . 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 1176 . . . . . . . . . . 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 1008 . . . . . . . . . . . . . 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 5739 . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . 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 2558 . . . . . . . . . . . 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 9652 . . . . . . . . . . . . . 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 11269 . . . . . . . . . . . . 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 6033 . . . . . . . . . . . . 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 20684 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  (PsMet `  X )  /\  p  e.  X  /\  q  e.  X )  ->  0  <_  ( p D q ) )
7660, 64, 65, 75syl3anc 1228 . . . . . . . . . . . . . 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 6298 . . . . . . . . . . . . . 14  |-  ( p D q )  =  ( D `  <. p ,  q >. )
7876, 77syl6breq 4492 . . . . . . . . . . . . 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 2559 . . . . . . . . . . . . . . 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 9609 . . . . . . . . . . . . . . . . . . 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 11268 . . . . . . . . . . . . . . . . . . . . 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 10797 . . . . . . . . . . . . . . . . . . . 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 9655 . . . . . . . . . . . . . . . . . . 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 6298 . . . . . . . . . . . . . . . . . . . 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 5035 . . . . . . . . . . . . . . . . . . . . . . 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 2558 . . . . . . . . . . . . . . . . . . . . 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 4454 . . . . . . . . . . . . . . . . . . . . . 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 6003 . . . . . . . . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . . . . . . . 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 2559 . . . . . . . . . . . . . . . . . . 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 11600 . . . . . . . . . . . . . . . . . . . . 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 1008 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( p D r )  e.  RR )
9980, 83, 95, 98syl21anc 1227 . . . . . . . . . . . . . . . . . 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 6298 . . . . . . . . . . . . . . . . . . . 20  |-  ( r D q )  =  ( D `  <. r ,  q >. )
101 opelxp 5035 . . . . . . . . . . . . . . . . . . . . . . 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 2558 . . . . . . . . . . . . . . . . . . . . 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 4454 . . . . . . . . . . . . . . . . . . . . . 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 6003 . . . . . . . . . . . . . . . . . . . . . 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 1227 . . . . . . . . . . . . . . . . . . . 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 2559 . . . . . . . . . . . . . . . . . . 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 11600 . . . . . . . . . . . . . . . . . . . . 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 1008 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( r D q )  e.  RR )
11480, 83, 110, 113syl21anc 1227 . . . . . . . . . . . . . . . . . 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 11443 . . . . . . . . . . . . . . . . . 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 9635 . . . . . . . . . . . . . . . . 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 2555 . . . . . . . . . . . . . . . 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 9655 . . . . . . . . . . . . . . 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 20683 . . . . . . . . . . . . . . . 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 1230 . . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( p D r )  <  (
a  /  2 ) )
12380, 83, 95, 122syl21anc 1227 . . . . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( r D q )  <  (
a  /  2 ) )
12580, 83, 110, 124syl21anc 1227 . . . . . . . . . . . . . . . . 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 10798 . . . . . . . . . . . . . . . 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 4473 . . . . . . . . . . . . . . 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 11375 . . . . . . . . . . . . . 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 4487 . . . . . . . . . . . . 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 11584 . . . . . . . . . . . . . 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 1235 . . . . . . . . . . . 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 6003 . . . . . . . . . . . . . 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 4454 . . . . . . . . . . . . 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 1227 . . . . . . . . . . 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 1229 . . . . . . . . . 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 4464 . . . . . . . . . 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 4454 . . . . . . . . . . . . 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 3121 . . . . . . . . . . . . 13  |-  p  e. 
_V
146 vex 3121 . . . . . . . . . . . . 13  |-  q  e. 
_V
147145, 146brco 5179 . . . . . . . . . . . 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 3545 . . . . . . . . . . . . . . . . . . . 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 3121 . . . . . . . . . . . . . . . . . . . . 21  |-  r  e. 
_V
155145, 154brelrn 5239 . . . . . . . . . . . . . . . . . . . 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 3510 . . . . . . . . . . . . . . . . . 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 1686 . . . . . . . . . . . . . 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 1017 . . . . . . . . . . . 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 2823 . . . . . . . . . 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 3008 . . . . . . . 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 4454 . . . . . . . 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 1231 . . . . . 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 5101 . . 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 5173 . . . . 5  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  (
v  o.  v )  =  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )
177176sseq1d 3536 . . . 4  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  (
( v  o.  v
)  C_  A  <->  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  A ) )
178177rspcev 3219 . . 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 20923 . . . 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 3008 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 973    = wceq 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2818    C_ wss 3481   (/)c0 3790   <.cop 4039   class class class wbr 4453    |-> cmpt 4511    X. cxp 5003   `'ccnv 5004   dom cdm 5005   ran crn 5006   "cima 5008    o. ccom 5009   Rel wrel 5010   Fun wfun 5588   -->wf 5590   ` cfv 5594  (class class class)co 6295   RRcr 9503   0cc0 9504    + caddc 9507   RR*cxr 9639    < clt 9640    <_ cle 9641    / cdiv 10218   2c2 10597   RR+crp 11232   +ecxad 11328   [,)cico 11543  PsMetcpsmet 18272
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-2 10606  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-ico 11547  df-psmet 18281
This theorem is referenced by:  metust  20939
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