MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  metustexhalfOLD Structured version   Unicode version

Theorem metustexhalfOLD 18624
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.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
metust.1  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
Assertion
Ref Expression
metustexhalfOLD  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 metustexhalfOLD
Dummy variables  b  p  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp-4r 745 . . . 4  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  ->  D  e.  ( * Met `  X ) )
2 simplr 733 . . . . . 6  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
a  e.  RR+ )
32rphalfcld 10691 . . . . 5  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( a  /  2
)  e.  RR+ )
4 eqidd 2443 . . . . 5  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
5 oveq2 6118 . . . . . . . 8  |-  ( b  =  ( a  / 
2 )  ->  (
0 [,) b )  =  ( 0 [,) ( a  /  2
) ) )
65imaeq2d 5232 . . . . . . 7  |-  ( b  =  ( a  / 
2 )  ->  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )
76eqeq2d 2453 . . . . . 6  |-  ( b  =  ( a  / 
2 )  ->  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) b ) )  <-> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )
87rspcev 3058 . . . . 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 644 . . . 4  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 6118 . . . . . . . . . 10  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
1211imaeq2d 5232 . . . . . . . . 9  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
1312cbvmptv 4325 . . . . . . . 8  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
1413rneqi 5125 . . . . . . 7  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
1510, 14eqtri 2462 . . . . . 6  |-  F  =  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )
1615metustelOLD 18612 . . . . 5  |-  ( D  e.  ( * Met `  X )  ->  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  e.  F  <->  E. b  e.  RR+  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  =  ( `' D " ( 0 [,) b
) ) ) )
1716biimpar 473 . . . 4  |-  ( ( D  e.  ( * Met `  X )  /\  E. b  e.  RR+  ( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) b ) ) )  ->  ( `' D " ( 0 [,) ( a  /  2
) ) )  e.  F )
181, 9, 17syl2anc 644 . . 3  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  e.  F
)
19 relco 5397 . . . . 5  |-  Rel  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
2019a1i 11 . . . 4  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  ->  Rel  ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )
21 cossxp 5421 . . . . . . . . . 10  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  ( dom  ( `' D " ( 0 [,) (
a  /  2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
22 cnvimass 5253 . . . . . . . . . . . . . 14  |-  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  dom  D
23 xmetf 18390 . . . . . . . . . . . . . . 15  |-  ( D  e.  ( * Met `  X )  ->  D : ( X  X.  X ) --> RR* )
24 fdm 5624 . . . . . . . . . . . . . . 15  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
2523, 24syl 16 . . . . . . . . . . . . . 14  |-  ( D  e.  ( * Met `  X )  ->  dom  D  =  ( X  X.  X ) )
2622, 25syl5sseq 3382 . . . . . . . . . . . . 13  |-  ( D  e.  ( * Met `  X )  ->  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X ) )
27 dmss 5098 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  dom  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  dom  ( X  X.  X ) )
28 rnss 5127 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ran  ( X  X.  X ) )
29 xpss12 5010 . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . 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.  ( * Met `  X )  ->  ( dom  ( `' D "
( 0 [,) (
a  /  2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) ) )
3231adantl 454 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  ( dom  ( `' D "
( 0 [,) (
a  /  2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) ) )
33 dmxp 5117 . . . . . . . . . . . . 13  |-  ( X  =/=  (/)  ->  dom  ( X  X.  X )  =  X )
34 rnxp 5328 . . . . . . . . . . . . 13  |-  ( X  =/=  (/)  ->  ran  ( X  X.  X )  =  X )
3533, 34xpeq12d 4932 . . . . . . . . . . . 12  |-  ( X  =/=  (/)  ->  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) )  =  ( X  X.  X ) )
3635adantr 453 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) )  =  ( X  X.  X ) )
3732, 36sseqtrd 3370 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  ( dom  ( `' D "
( 0 [,) (
a  /  2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( X  X.  X ) )
3821, 37syl5ss 3345 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( X  X.  X ) )
3938ad3antrrr 712 . . . . . . . 8  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 3334 . . . . . . 7  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 4937 . . . . . . 7  |-  ( <.
p ,  q >.  e.  ( X  X.  X
)  <->  ( p  e.  X  /\  q  e.  X ) )
4240, 41sylib 190 . . . . . 6  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 732 . . . . . . 7  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) ) )
44 simprl 734 . . . . . . 7  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 735 . . . . . . 7  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 733 . . . . . . 7  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 736 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  /\  p  e.  X  /\  q  e.  X )
)
4847simp1d 970 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) ) )
4948, 1syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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.  ( * Met `  X
) )
5048, 2syl 16 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 520 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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.  ( * Met `  X
)  /\  a  e.  RR+ ) )
5247simp2d 971 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 972 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 1135 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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.  ( * Met `  X )  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )
)
55 simplr 733 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 734 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 735 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 732 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( * Met `  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.  ( * Met `  X )  /\  a  e.  RR+ )  /\  p  e.  X  /\  q  e.  X )
)
5958simp1d 970 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  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.  ( * Met `  X
)  /\  a  e.  RR+ ) )
6059simpld 447 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( * Met `  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.  ( * Met `  X
) )
61 ffun 5622 . . . . . . . . . . . . . 14  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
6223, 61syl 16 . . . . . . . . . . . . 13  |-  ( D  e.  ( * Met `  X )  ->  Fun  D )
6360, 62syl 16 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( * Met `  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 971 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  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 972 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  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 647 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( * Met `  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.  ( * Met `  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 2519 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( * Met `  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 9162 . . . . . . . . . . . . . 14  |-  0  e.  RR*
7069a1i 11 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( * Met `  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 451 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  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 10680 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( * Met `  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.  ( * Met `  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 5900 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( * Met `  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 xmetge0 18405 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( * Met `  X )  /\  p  e.  X  /\  q  e.  X
)  ->  0  <_  ( p D q ) )
7660, 64, 65, 75syl3anc 1185 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  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 6113 . . . . . . . . . . . . . 14  |-  ( p D q )  =  ( D `  <. p ,  q >. )
7876, 77syl6breq 4276 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( * Met `  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 2526 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( * Met `  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 0re 9122 . . . . . . . . . . . . . . . . . . . 20  |-  0  e.  RR
8180a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
8271rpred 10679 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
8382rehalfcld 10245 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
8483rexrd 9165 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  ( * Met `  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* )
85 df-ov 6113 . . . . . . . . . . . . . . . . . . . 20  |-  ( p D r )  =  ( D `  <. p ,  r >. )
86 simplr 733 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
87 opelxp 4937 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
p ,  r >.  e.  ( X  X.  X
)  <->  ( p  e.  X  /\  r  e.  X ) )
8864, 86, 87sylanbrc 647 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) )
8988, 67eleqtrrd 2519 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
90 simprl 734 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
91 df-br 4238 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  <->  <. p ,  r >.  e.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
9290, 91sylib 190 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) ) ) )
93 fvimacnv 5874 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  D  /\  <. p ,  r >.  e.  dom  D )  ->  ( ( D `  <. p ,  r >. )  e.  ( 0 [,) ( a  /  2 ) )  <->  <. p ,  r >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) ) )
9493biimpar 473 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( Fun  D  /\  <.
p ,  r >.  e.  dom  D )  /\  <.
p ,  r >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) )  -> 
( D `  <. p ,  r >. )  e.  ( 0 [,) (
a  /  2 ) ) )
9563, 89, 92, 94syl21anc 1184 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) ) )
9685, 95syl5eqel 2526 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) ) )
97 elico2 11005 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
9897biimpa 472 . . . . . . . . . . . . . . . . . . . 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
) ) )
9998simp1d 970 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( p D r )  e.  RR )
10081, 84, 96, 99syl21anc 1184 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
101 df-ov 6113 . . . . . . . . . . . . . . . . . . . 20  |-  ( r D q )  =  ( D `  <. r ,  q >. )
102 opelxp 4937 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
r ,  q >.  e.  ( X  X.  X
)  <->  ( r  e.  X  /\  q  e.  X ) )
10386, 65, 102sylanbrc 647 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) )
104103, 67eleqtrrd 2519 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
105 simprr 735 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
106 df-br 4238 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( r ( `' D "
( 0 [,) (
a  /  2 ) ) ) q  <->  <. r ,  q >.  e.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
107105, 106sylib 190 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) ) ) )
108 fvimacnv 5874 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  D  /\  <. r ,  q >.  e.  dom  D )  ->  ( ( D `  <. r ,  q >. )  e.  ( 0 [,) ( a  /  2 ) )  <->  <. r ,  q >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) ) )
109108biimpar 473 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( ( Fun  D  /\  <.
r ,  q >.  e.  dom  D )  /\  <.
r ,  q >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) )  -> 
( D `  <. r ,  q >. )  e.  ( 0 [,) (
a  /  2 ) ) )
11063, 104, 107, 109syl21anc 1184 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) ) )
111101, 110syl5eqel 2526 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) ) )
112 elico2 11005 . . . . . . . . . . . . . . . . . . . . 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 ) ) ) )
113112biimpa 472 . . . . . . . . . . . . . . . . . . . 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
) ) )
114113simp1d 970 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( r D q )  e.  RR )
11581, 84, 111, 114syl21anc 1184 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
116 rexadd 10849 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( p D r )  e.  RR  /\  ( r D q )  e.  RR )  ->  ( ( p D r ) + e ( r D q ) )  =  ( ( p D r )  +  ( r D q ) ) )
117100, 115, 116syl2anc 644 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) ) )
118100, 115readdcld 9146 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
119117, 118eqeltrd 2516 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
120119rexrd 9165 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( * Met `  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* )
121 xmettri 18412 . . . . . . . . . . . . . . . 16  |-  ( ( D  e.  ( * Met `  X )  /\  ( p  e.  X  /\  q  e.  X  /\  r  e.  X ) )  -> 
( p D q )  <_  ( (
p D r ) + e ( r D q ) ) )
12260, 64, 65, 86, 121syl13anc 1187 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) ) )
12398simp3d 972 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( p D r )  <  (
a  /  2 ) )
12481, 84, 96, 123syl21anc 1184 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  ( * Met `  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
) )
125113simp3d 972 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( r D q )  <  (
a  /  2 ) )
12681, 84, 111, 125syl21anc 1184 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( D  e.  ( * Met `  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
) )
127100, 115, 82, 124, 126lt2halvesd 10246 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
128117, 127eqbrtrd 4257 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
12979, 120, 72, 122, 128xrlelttrd 10781 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
13077, 129syl5eqbrr 4271 . . . . . . . . . . . . 13  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
131 elico1 10990 . . . . . . . . . . . . . 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 ) ) )
132131biimpar 473 . . . . . . . . . . . . 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
) )
13370, 72, 74, 78, 130, 132syl23anc 1192 . . . . . . . . . . . 12  |-  ( ( ( ( ( D  e.  ( * Met `  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 ) )
134 fvimacnv 5874 . . . . . . . . . . . . . 14  |-  ( ( Fun  D  /\  <. p ,  q >.  e.  dom  D )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
135134biimpa 472 . . . . . . . . . . . . 13  |-  ( ( ( Fun  D  /\  <.
p ,  q >.  e.  dom  D )  /\  ( D `  <. p ,  q >. )  e.  ( 0 [,) a
) )  ->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) )
136 df-br 4238 . . . . . . . . . . . . 13  |-  ( p ( `' D "
( 0 [,) a
) ) q  <->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) )
137135, 136sylibr 205 . . . . . . . . . . . 12  |-  ( ( ( Fun  D  /\  <.
p ,  q >.  e.  dom  D )  /\  ( D `  <. p ,  q >. )  e.  ( 0 [,) a
) )  ->  p
( `' D "
( 0 [,) a
) ) q )
13863, 68, 133, 137syl21anc 1184 . . . . . . . . . . 11  |-  ( ( ( ( ( D  e.  ( * Met `  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 )
13954, 55, 56, 57, 138syl22anc 1186 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 )
14048simprd 451 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) ) )
141140breqd 4248 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) )
142139, 141mpbird 225 . . . . . . . . 9  |-  ( ( ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 )
143 simpr 449 . . . . . . . . . . . . 13  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) ) ) ) )
144 df-br 4238 . . . . . . . . . . . . 13  |-  ( p ( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) q  <->  <. p ,  q >.  e.  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )
145143, 144sylibr 205 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 )
146 vex 2965 . . . . . . . . . . . . 13  |-  p  e. 
_V
147 vex 2965 . . . . . . . . . . . . 13  |-  q  e. 
_V
148146, 147brco 5072 . . . . . . . . . . . 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 ) )
149145, 148sylib 190 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) )
15026adantl 454 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X ) )
151150, 28syl 16 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ran  ( X  X.  X ) )
15234adantr 453 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  ran  ( X  X.  X
)  =  X )
153151, 152sseqtrd 3370 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  X )
154153adantr 453 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r )  ->  ran  ( `' D " ( 0 [,) ( a  /  2
) ) )  C_  X )
155 vex 2965 . . . . . . . . . . . . . . . . . . . . 21  |-  r  e. 
_V
156146, 155brelrn 5129 . . . . . . . . . . . . . . . . . . . 20  |-  ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  -> 
r  e.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
157156adantl 454 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r )  ->  r  e.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
158154, 157sseldd 3335 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  p
( `' D "
( 0 [,) (
a  /  2 ) ) ) r )  ->  r  e.  X
)
159158adantrr 699 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  (
p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r  /\  r ( `' D " ( 0 [,) (
a  /  2 ) ) ) q ) )  ->  r  e.  X )
160159ex 425 . . . . . . . . . . . . . . . 16  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  (
( p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r  /\  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q )  ->  r  e.  X ) )
161160ancrd 539 . . . . . . . . . . . . . . 15  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) ) ) )
162161eximdv 1633 . . . . . . . . . . . . . 14  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) ) ) )
163162ad3antrrr 712 . . . . . . . . . . . . 13  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) ) ) )
1641633ad2ant1 979 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) ) ) )
165164adantr 453 . . . . . . . . . . 11  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) ) ) )
166149, 165mpd 15 . . . . . . . . . 10  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) ) )
167 df-rex 2717 . . . . . . . . . 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 ) ) )
168166, 167sylibr 205 . . . . . . . . 9  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) )
169142, 168r19.29a 2856 . . . . . . . 8  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 )
170 df-br 4238 . . . . . . . 8  |-  ( p A q  <->  <. p ,  q >.  e.  A
)
171169, 170sylib 190 . . . . . . 7  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 )
17243, 44, 45, 46, 171syl31anc 1188 . . . . . 6  |-  ( ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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
)
17342, 172mpdan 651 . . . . 5  |-  ( ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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
)
174173ex 425 . . . 4  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  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 ) )
17520, 174relssdv 4997 . . 3  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  -> 
( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  A )
176 id 21 . . . . . 6  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )
177176, 176coeq12d 5066 . . . . 5  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  (
v  o.  v )  =  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )
178177sseq1d 3361 . . . 4  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  (
( v  o.  v
)  C_  A  <->  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  A ) )
179178rspcev 3058 . . 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 )
18018, 175, 179syl2anc 644 . 2  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a
) ) )  ->  E. v  e.  F  ( v  o.  v
)  C_  A )
18110metustelOLD 18612 . . . 4  |-  ( D  e.  ( * Met `  X )  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) ) )
182181adantl 454 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) ) )
183182biimpa 472 . 2  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  A  e.  F )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
184180, 183r19.29a 2856 1  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( * Met `  X
) )  /\  A  e.  F )  ->  E. v  e.  F  ( v  o.  v )  C_  A
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937   E.wex 1551    = wceq 1653    e. wcel 1727    =/= wne 2605   E.wrex 2712    C_ wss 3306   (/)c0 3613   <.cop 3841   class class class wbr 4237    e. cmpt 4291    X. cxp 4905   `'ccnv 4906   dom cdm 4907   ran crn 4908   "cima 4910    o. ccom 4911   Rel wrel 4912   Fun wfun 5477   -->wf 5479   ` cfv 5483  (class class class)co 6110   RRcr 9020   0cc0 9021    + caddc 9024   RR*cxr 9150    < clt 9151    <_ cle 9152    / cdiv 9708   2c2 10080   RR+crp 10643   + ecxad 10739   [,)cico 10949   * Metcxmt 16717
This theorem is referenced by:  metustOLD  18628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730  ax-cnex 9077  ax-resscn 9078  ax-1cn 9079  ax-icn 9080  ax-addcl 9081  ax-addrcl 9082  ax-mulcl 9083  ax-mulrcl 9084  ax-mulcom 9085  ax-addass 9086  ax-mulass 9087  ax-distr 9088  ax-i2m1 9089  ax-1ne0 9090  ax-1rid 9091  ax-rnegex 9092  ax-rrecex 9093  ax-cnre 9094  ax-pre-lttri 9095  ax-pre-lttrn 9096  ax-pre-ltadd 9097  ax-pre-mulgt0 9098
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2716  df-rex 2717  df-reu 2718  df-rmo 2719  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-pw 3825  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-po 4532  df-so 4533  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-f1 5488  df-fo 5489  df-f1o 5490  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-er 6934  df-map 7049  df-en 7139  df-dom 7140  df-sdom 7141  df-pnf 9153  df-mnf 9154  df-xr 9155  df-ltxr 9156  df-le 9157  df-sub 9324  df-neg 9325  df-div 9709  df-2 10089  df-rp 10644  df-xneg 10741  df-xadd 10742  df-xmul 10743  df-ico 10953  df-xmet 16726
  Copyright terms: Public domain W3C validator