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Theorem metustexhalfOLD 21044
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 768 . . . 4  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a ) ) )  ->  D  e.  ( *Met `  X
) )
2 simplr 755 . . . . . 6  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a ) ) )  ->  a  e.  RR+ )
32rphalfcld 11279 . . . . 5  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a ) ) )  ->  ( a  /  2 )  e.  RR+ )
4 eqidd 2444 . . . . 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 6289 . . . . . . . 8  |-  ( b  =  ( a  / 
2 )  ->  (
0 [,) b )  =  ( 0 [,) ( a  /  2
) ) )
65imaeq2d 5327 . . . . . . 7  |-  ( b  =  ( a  / 
2 )  ->  ( `' D " ( 0 [,) b ) )  =  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )
76eqeq2d 2457 . . . . . 6  |-  ( b  =  ( a  / 
2 )  ->  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) b ) )  <-> 
( `' D "
( 0 [,) (
a  /  2 ) ) )  =  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) ) )
87rspcev 3196 . . . . 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.  ( *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 6289 . . . . . . . . . 10  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
1211imaeq2d 5327 . . . . . . . . 9  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
1312cbvmptv 4528 . . . . . . . 8  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
1413rneqi 5219 . . . . . . 7  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
1510, 14eqtri 2472 . . . . . 6  |-  F  =  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )
1615metustelOLD 21032 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  (
( `' D "
( 0 [,) (
a  /  2 ) ) )  e.  F  <->  E. b  e.  RR+  ( `' D " ( 0 [,) ( a  / 
2 ) ) )  =  ( `' D " ( 0 [,) b
) ) ) )
1716biimpar 485 . . . 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 661 . . 3  |-  ( ( ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  A  e.  F )  /\  a  e.  RR+ )  /\  A  =  ( `' D " ( 0 [,) a ) ) )  ->  ( `' D " ( 0 [,) ( a  /  2
) ) )  e.  F )
19 relco 5495 . . . . 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 5520 . . . . . . . . . 10  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  ( dom  ( `' D " ( 0 [,) (
a  /  2 ) ) )  X.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
22 cnvimass 5347 . . . . . . . . . . . . . 14  |-  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  dom  D
23 xmetf 20810 . . . . . . . . . . . . . . 15  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
24 fdm 5725 . . . . . . . . . . . . . . 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 3537 . . . . . . . . . . . . 13  |-  ( D  e.  ( *Met `  X )  ->  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X ) )
27 dmss 5192 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  dom  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  dom  ( X  X.  X ) )
28 rnss 5221 . . . . . . . . . . . . . 14  |-  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ( X  X.  X )  ->  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) 
C_  ran  ( X  X.  X ) )
29 xpss12 5098 . . . . . . . . . . . . . 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.  ( *Met `  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.  ( *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 5211 . . . . . . . . . . . . 13  |-  ( X  =/=  (/)  ->  dom  ( X  X.  X )  =  X )
34 rnxp 5427 . . . . . . . . . . . . 13  |-  ( X  =/=  (/)  ->  ran  ( X  X.  X )  =  X )
3533, 34xpeq12d 5014 . . . . . . . . . . . 12  |-  ( X  =/=  (/)  ->  ( dom  ( X  X.  X
)  X.  ran  ( X  X.  X ) )  =  ( X  X.  X ) )
3635adantr 465 . . . . . . . . . . 11  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( dom  ( X  X.  X )  X.  ran  ( X  X.  X
) )  =  ( X  X.  X ) )
3732, 36sseqtrd 3525 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( dom  ( `' D " ( 0 [,) ( a  /  2
) ) )  X. 
ran  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  ( X  X.  X
) )
3821, 37syl5ss 3500 . . . . . . . . 9  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( ( `' D " ( 0 [,) (
a  /  2 ) ) )  o.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )  C_  ( X  X.  X ) )
3938ad3antrrr 729 . . . . . . . 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 3489 . . . . . . 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 5019 . . . . . . 7  |-  ( <.
p ,  q >.  e.  ( X  X.  X
)  <->  ( p  e.  X  /\  q  e.  X ) )
4240, 41sylib 196 . . . . . 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 753 . . . . . . 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 756 . . . . . . 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 757 . . . . . . 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 755 . . . . . . 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 759 . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . 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 532 . . . . . . . . . . . 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 1010 . . . . . . . . . . . 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 1011 . . . . . . . . . . . 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 1177 . . . . . . . . . . 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 755 . . . . . . . . . . 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 756 . . . . . . . . . . 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 757 . . . . . . . . . . 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 753 . . . . . . . . . . . . . . 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 1009 . . . . . . . . . . . . . 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 459 . . . . . . . . . . . . 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 5723 . . . . . . . . . . . . . 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 1010 . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . 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 664 . . . . . . . . . . . . 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 2534 . . . . . . . . . . . 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 9643 . . . . . . . . . . . . . 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 463 . . . . . . . . . . . . . 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 11268 . . . . . . . . . . . . 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 6017 . . . . . . . . . . . . 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 20825 . . . . . . . . . . . . . . 15  |-  ( ( D  e.  ( *Met `  X )  /\  p  e.  X  /\  q  e.  X
)  ->  0  <_  ( p D q ) )
7660, 64, 65, 75syl3anc 1229 . . . . . . . . . . . . . 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 6284 . . . . . . . . . . . . . 14  |-  ( p D q )  =  ( D `  <. p ,  q >. )
7876, 77syl6breq 4476 . . . . . . . . . . . . 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 2535 . . . . . . . . . . . . . . 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 9599 . . . . . . . . . . . . . . . . . . . 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 11267 . . . . . . . . . . . . . . . . . . . . 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 10792 . . . . . . . . . . . . . . . . . . . 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 9646 . . . . . . . . . . . . . . . . . . 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 6284 . . . . . . . . . . . . . . . . . . . 20  |-  ( p D r )  =  ( D `  <. p ,  r >. )
86 simplr 755 . . . . . . . . . . . . . . . . . . . . . . 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 5019 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
p ,  r >.  e.  ( X  X.  X
)  <->  ( p  e.  X  /\  r  e.  X ) )
8864, 86, 87sylanbrc 664 . . . . . . . . . . . . . . . . . . . . . 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 2534 . . . . . . . . . . . . . . . . . . . . 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 756 . . . . . . . . . . . . . . . . . . . . . 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 4438 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  <->  <. p ,  r >.  e.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
9290, 91sylib 196 . . . . . . . . . . . . . . . . . . . . 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 5987 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  D  /\  <. p ,  r >.  e.  dom  D )  ->  ( ( D `  <. p ,  r >. )  e.  ( 0 [,) ( a  /  2 ) )  <->  <. p ,  r >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) ) )
9493biimpar 485 . . . . . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . . . . . 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 2535 . . . . . . . . . . . . . . . . . . 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 11599 . . . . . . . . . . . . . . . . . . . . 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 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
) ) )
9998simp1d 1009 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( p D r )  e.  RR )
10081, 84, 96, 99syl21anc 1228 . . . . . . . . . . . . . . . . . 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 6284 . . . . . . . . . . . . . . . . . . . 20  |-  ( r D q )  =  ( D `  <. r ,  q >. )
102 opelxp 5019 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( <.
r ,  q >.  e.  ( X  X.  X
)  <->  ( r  e.  X  /\  q  e.  X ) )
10386, 65, 102sylanbrc 664 . . . . . . . . . . . . . . . . . . . . . 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 2534 . . . . . . . . . . . . . . . . . . . . 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 757 . . . . . . . . . . . . . . . . . . . . . 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 4438 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( r ( `' D "
( 0 [,) (
a  /  2 ) ) ) q  <->  <. r ,  q >.  e.  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
107105, 106sylib 196 . . . . . . . . . . . . . . . . . . . . 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 5987 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( Fun  D  /\  <. r ,  q >.  e.  dom  D )  ->  ( ( D `  <. r ,  q >. )  e.  ( 0 [,) ( a  /  2 ) )  <->  <. r ,  q >.  e.  ( `' D "
( 0 [,) (
a  /  2 ) ) ) ) )
109108biimpar 485 . . . . . . . . . . . . . . . . . . . . 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 1228 . . . . . . . . . . . . . . . . . . . 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 2535 . . . . . . . . . . . . . . . . . . 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 11599 . . . . . . . . . . . . . . . . . . . . 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 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
) ) )
114113simp1d 1009 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( r D q )  e.  RR )
11581, 84, 111, 114syl21anc 1228 . . . . . . . . . . . . . . . . . 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 11442 . . . . . . . . . . . . . . . . . 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 661 . . . . . . . . . . . . . . . . 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 9626 . . . . . . . . . . . . . . . . 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 2531 . . . . . . . . . . . . . . . 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 9646 . . . . . . . . . . . . . . 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 20832 . . . . . . . . . . . . . . . 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 1231 . . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( p D r )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( p D r )  <  (
a  /  2 ) )
12481, 84, 96, 123syl21anc 1228 . . . . . . . . . . . . . . . . 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 1011 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 0  e.  RR  /\  ( a  /  2
)  e.  RR* )  /\  ( r D q )  e.  ( 0 [,) ( a  / 
2 ) ) )  ->  ( r D q )  <  (
a  /  2 ) )
12681, 84, 111, 125syl21anc 1228 . . . . . . . . . . . . . . . . 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 10793 . . . . . . . . . . . . . . . 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 4457 . . . . . . . . . . . . . . 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 11374 . . . . . . . . . . . . . 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 4471 . . . . . . . . . . . . 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 11583 . . . . . . . . . . . . . 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 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
) )
13370, 72, 74, 78, 130, 132syl23anc 1236 . . . . . . . . . . . 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 5987 . . . . . . . . . . . . . 14  |-  ( ( Fun  D  /\  <. p ,  q >.  e.  dom  D )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
135134biimpa 484 . . . . . . . . . . . . 13  |-  ( ( ( Fun  D  /\  <.
p ,  q >.  e.  dom  D )  /\  ( D `  <. p ,  q >. )  e.  ( 0 [,) a
) )  ->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) )
136 df-br 4438 . . . . . . . . . . . . 13  |-  ( p ( `' D "
( 0 [,) a
) ) q  <->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) )
137135, 136sylibr 212 . . . . . . . . . . . 12  |-  ( ( ( Fun  D  /\  <.
p ,  q >.  e.  dom  D )  /\  ( D `  <. p ,  q >. )  e.  ( 0 [,) a
) )  ->  p
( `' D "
( 0 [,) a
) ) q )
13863, 68, 133, 137syl21anc 1228 . . . . . . . . . . 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 1230 . . . . . . . . . 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 463 . . . . . . . . . . 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 4448 . . . . . . . . . 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 232 . . . . . . . . 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 461 . . . . . . . . . . . . 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 4438 . . . . . . . . . . . . 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 212 . . . . . . . . . . . 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 3098 . . . . . . . . . . . . 13  |-  p  e. 
_V
147 vex 3098 . . . . . . . . . . . . 13  |-  q  e. 
_V
148146, 147brco 5163 . . . . . . . . . . . 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 196 . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  ran  ( X  X.  X
)  =  X )
153151, 152sseqtrd 3525 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  ->  ran  ( `' D "
( 0 [,) (
a  /  2 ) ) )  C_  X
)
154153adantr 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r )  ->  ran  ( `' D " ( 0 [,) ( a  /  2
) ) )  C_  X )
155 vex 3098 . . . . . . . . . . . . . . . . . . . . 21  |-  r  e. 
_V
156146, 155brelrn 5223 . . . . . . . . . . . . . . . . . . . 20  |-  ( p ( `' D "
( 0 [,) (
a  /  2 ) ) ) r  -> 
r  e.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
157156adantl 466 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r )  ->  r  e.  ran  ( `' D " ( 0 [,) ( a  / 
2 ) ) ) )
158154, 157sseldd 3490 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  p ( `' D " ( 0 [,) (
a  /  2 ) ) ) r )  ->  r  e.  X
)
159158adantrr 716 . . . . . . . . . . . . . . . . 17  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  ( p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r  /\  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q ) )  -> 
r  e.  X )
160159ex 434 . . . . . . . . . . . . . . . 16  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( ( p ( `' D " ( 0 [,) ( a  / 
2 ) ) ) r  /\  r ( `' D " ( 0 [,) ( a  / 
2 ) ) ) q )  ->  r  e.  X ) )
161160ancrd 554 . . . . . . . . . . . . . . 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 1697 . . . . . . . . . . . . . 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 729 . . . . . . . . . . . . 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 1018 . . . . . . . . . . . 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 465 . . . . . . . . . . 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 2799 . . . . . . . . . 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 212 . . . . . . . . 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 2985 . . . . . . . 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 4438 . . . . . . . 8  |-  ( p A q  <->  <. p ,  q >.  e.  A
)
171169, 170sylib 196 . . . . . . 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 1232 . . . . . 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 668 . . . . 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 434 . . . 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 5085 . . 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 22 . . . . . 6  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )
177176, 176coeq12d 5157 . . . . 5  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  (
v  o.  v )  =  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) ) )
178177sseq1d 3516 . . . 4  |-  ( v  =  ( `' D " ( 0 [,) (
a  /  2 ) ) )  ->  (
( v  o.  v
)  C_  A  <->  ( ( `' D " ( 0 [,) ( a  / 
2 ) ) )  o.  ( `' D " ( 0 [,) (
a  /  2 ) ) ) )  C_  A ) )
179178rspcev 3196 . . 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 661 . 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 21032 . . . 4  |-  ( D  e.  ( *Met `  X )  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) ) )
182181adantl 466 . . 3  |-  ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  -> 
( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D "
( 0 [,) a
) ) ) )
183182biimpa 484 . 2  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  X ) )  /\  A  e.  F )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) )
184180, 183r19.29a 2985 1  |-  ( ( ( X  =/=  (/)  /\  D  e.  ( *Met `  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 974    = wceq 1383   E.wex 1599    e. wcel 1804    =/= wne 2638   E.wrex 2794    C_ wss 3461   (/)c0 3770   <.cop 4020   class class class wbr 4437    |-> cmpt 4495    X. cxp 4987   `'ccnv 4988   dom cdm 4989   ran crn 4990   "cima 4992    o. ccom 4993   Rel wrel 4994   Fun wfun 5572   -->wf 5574   ` cfv 5578  (class class class)co 6281   RRcr 9494   0cc0 9495    + caddc 9498   RR*cxr 9630    < clt 9631    <_ cle 9632    / cdiv 10213   2c2 10592   RR+crp 11231   +ecxad 11327   [,)cico 11542   *Metcxmt 18382
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-po 4790  df-so 4791  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-2 10601  df-rp 11232  df-xneg 11329  df-xadd 11330  df-xmul 11331  df-ico 11546  df-xmet 18391
This theorem is referenced by:  metustOLD  21048
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