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Theorem metustto 20132
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 11026 . . . 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 11026 . . . 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 11026 . . . . . . 7  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  b  e.  RR )
7 0xr 9429 . . . . . . . . . 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 9432 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
b  e.  RR* )
11 0le0 10410 . . . . . . . . . 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 11364 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\  b  e.  RR* )  /\  ( 0  <_  0  /\  a  <_  b ) )  ->  ( 0 [,) a )  C_  ( 0 [,) b
) )
158, 10, 12, 13, 14syl22anc 1219 . . . . . . . 8  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
( 0 [,) a
)  C_  ( 0 [,) b ) )
16 imass2 5203 . . . . . . . 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 3398 . . . . 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 11026 . . . . . . 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 9432 . . . . . . . . 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 11364 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\  a  e.  RR* )  /\  ( 0  <_  0  /\  b  <_  a ) )  ->  ( 0 [,) b )  C_  ( 0 [,) a
) )
3125, 27, 28, 29, 30syl22anc 1219 . . . . . . . 8  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
( 0 [,) b
)  C_  ( 0 [,) a ) )
32 imass2 5203 . . . . . . . 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 3398 . . . . 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 9479 . . 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 20126 . . . . 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 1008 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
45 oveq2 6098 . . . . . . . . . 10  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
4645imaeq2d 5168 . . . . . . . . 9  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
4746cbvmptv 4382 . . . . . . . 8  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
4847rneqi 5065 . . . . . . 7  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
4941, 48eqtri 2462 . . . . . 6  |-  F  =  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )
5049metustel 20126 . . . . 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 1007 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. b  e.  RR+  B  =  ( `' D " ( 0 [,) b ) ) )
53 reeanv 2887 . . 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 2863 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 965    = wceq 1369    e. wcel 1756   E.wrex 2715    C_ wss 3327   class class class wbr 4291    e. cmpt 4349   `'ccnv 4838   ran crn 4840   "cima 4842   ` cfv 5417  (class class class)co 6090   RRcr 9280   0cc0 9281   RR*cxr 9416    <_ cle 9418   RR+crp 10990   [,)cico 11301  PsMetcpsmet 17799
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530  ax-un 6371  ax-cnex 9337  ax-resscn 9338  ax-1cn 9339  ax-icn 9340  ax-addcl 9341  ax-addrcl 9342  ax-mulcl 9343  ax-mulrcl 9344  ax-i2m1 9349  ax-1ne0 9350  ax-rnegex 9352  ax-rrecex 9353  ax-cnre 9354  ax-pre-lttri 9355  ax-pre-lttrn 9356
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-nel 2608  df-ral 2719  df-rex 2720  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-pw 3861  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-po 4640  df-so 4641  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-1st 6576  df-2nd 6577  df-er 7100  df-en 7310  df-dom 7311  df-sdom 7312  df-pnf 9419  df-mnf 9420  df-xr 9421  df-ltxr 9422  df-le 9423  df-rp 10991  df-ico 11305
This theorem is referenced by:  metustfbas  20140
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