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Theorem metustto 20824
Description: Any two elements of the filter base generated by the metric 
D can be compared, like for RR+ (i.e. it's totally ordered). (Contributed by Thierry Arnoux, 22-Nov-2017.) (Revised by Thierry Arnoux, 11-Feb-2018.)
Hypothesis
Ref Expression
metust.1  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
Assertion
Ref Expression
metustto  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  C_  B  \/  B  C_  A ) )
Distinct variable groups:    B, a    D, a    X, a    A, a    F, a

Proof of Theorem metustto
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 simpll 753 . . . . 5  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  a  e.  RR+ )
21rpred 11256 . . . 4  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  a  e.  RR )
3 simplr 754 . . . . 5  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  b  e.  RR+ )
43rpred 11256 . . . 4  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  b  e.  RR )
5 simpllr 758 . . . . . . . 8  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  b  e.  RR+ )
65rpred 11256 . . . . . . 7  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  b  e.  RR )
7 0xr 9640 . . . . . . . . . 10  |-  0  e.  RR*
87a1i 11 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
0  e.  RR* )
9 simpl 457 . . . . . . . . . 10  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
b  e.  RR )
109rexrd 9643 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
b  e.  RR* )
11 0le0 10625 . . . . . . . . . 10  |-  0  <_  0
1211a1i 11 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
0  <_  0 )
13 simpr 461 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
a  <_  b )
14 icossico 11594 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\  b  e.  RR* )  /\  ( 0  <_  0  /\  a  <_  b ) )  ->  ( 0 [,) a )  C_  ( 0 [,) b
) )
158, 10, 12, 13, 14syl22anc 1229 . . . . . . . 8  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
( 0 [,) a
)  C_  ( 0 [,) b ) )
16 imass2 5372 . . . . . . . 8  |-  ( ( 0 [,) a ) 
C_  ( 0 [,) b )  ->  ( `' D " ( 0 [,) a ) ) 
C_  ( `' D " ( 0 [,) b
) ) )
1715, 16syl 16 . . . . . . 7  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
( `' D "
( 0 [,) a
) )  C_  ( `' D " ( 0 [,) b ) ) )
186, 17sylancom 667 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  ( `' D " ( 0 [,) a ) )  C_  ( `' D " ( 0 [,) b ) ) )
19 simplrl 759 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  A  =  ( `' D " ( 0 [,) a ) ) )
20 simplrr 760 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  B  =  ( `' D " ( 0 [,) b ) ) )
2118, 19, 203sstr4d 3547 . . . . 5  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  A  C_  B
)
2221orcd 392 . . . 4  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  ( A  C_  B  \/  B  C_  A ) )
23 simplll 757 . . . . . . . 8  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  a  e.  RR+ )
2423rpred 11256 . . . . . . 7  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  a  e.  RR )
257a1i 11 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
0  e.  RR* )
26 simpl 457 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
a  e.  RR )
2726rexrd 9643 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
a  e.  RR* )
2811a1i 11 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
0  <_  0 )
29 simpr 461 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
b  <_  a )
30 icossico 11594 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\  a  e.  RR* )  /\  ( 0  <_  0  /\  b  <_  a ) )  ->  ( 0 [,) b )  C_  ( 0 [,) a
) )
3125, 27, 28, 29, 30syl22anc 1229 . . . . . . . 8  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
( 0 [,) b
)  C_  ( 0 [,) a ) )
32 imass2 5372 . . . . . . . 8  |-  ( ( 0 [,) b ) 
C_  ( 0 [,) a )  ->  ( `' D " ( 0 [,) b ) ) 
C_  ( `' D " ( 0 [,) a
) ) )
3331, 32syl 16 . . . . . . 7  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
( `' D "
( 0 [,) b
) )  C_  ( `' D " ( 0 [,) a ) ) )
3424, 33sylancom 667 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  ( `' D " ( 0 [,) b ) )  C_  ( `' D " ( 0 [,) a ) ) )
35 simplrr 760 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  B  =  ( `' D " ( 0 [,) b ) ) )
36 simplrl 759 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  A  =  ( `' D " ( 0 [,) a ) ) )
3734, 35, 363sstr4d 3547 . . . . 5  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  B  C_  A
)
3837olcd 393 . . . 4  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  ( A  C_  B  \/  B  C_  A ) )
392, 4, 22, 38lecasei 9690 . . 3  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  ( A  C_  B  \/  B  C_  A ) )
4039adantlll 717 . 2  |-  ( ( ( ( ( D  e.  (PsMet `  X
)  /\  A  e.  F  /\  B  e.  F
)  /\  a  e.  RR+ )  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  ( A  C_  B  \/  B  C_  A ) )
41 metust.1 . . . . . 6  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
4241metustel 20818 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
4342biimpa 484 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
44433adant3 1016 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
45 oveq2 6292 . . . . . . . . . 10  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
4645imaeq2d 5337 . . . . . . . . 9  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
4746cbvmptv 4538 . . . . . . . 8  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
4847rneqi 5229 . . . . . . 7  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
4941, 48eqtri 2496 . . . . . 6  |-  F  =  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )
5049metustel 20818 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( B  e.  F  <->  E. b  e.  RR+  B  =  ( `' D " ( 0 [,) b
) ) ) )
5150biimpa 484 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  B  e.  F )  ->  E. b  e.  RR+  B  =  ( `' D " ( 0 [,) b ) ) )
52513adant2 1015 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. b  e.  RR+  B  =  ( `' D " ( 0 [,) b ) ) )
53 reeanv 3029 . . 3  |-  ( E. a  e.  RR+  E. b  e.  RR+  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) )  <->  ( E. a  e.  RR+  A  =  ( `' D "
( 0 [,) a
) )  /\  E. b  e.  RR+  B  =  ( `' D "
( 0 [,) b
) ) ) )
5444, 52, 53sylanbrc 664 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. a  e.  RR+  E. b  e.  RR+  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )
5540, 54r19.29_2a 3005 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F  /\  B  e.  F )  ->  ( A  C_  B  \/  B  C_  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   class class class wbr 4447    |-> cmpt 4505   `'ccnv 4998   ran crn 5000   "cima 5002   ` cfv 5588  (class class class)co 6284   RRcr 9491   0cc0 9492   RR*cxr 9627    <_ cle 9629   RR+crp 11220   [,)cico 11531  PsMetcpsmet 18201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-i2m1 9560  ax-1ne0 9561  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-1st 6784  df-2nd 6785  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-rp 11221  df-ico 11535
This theorem is referenced by:  metustfbas  20832
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