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Theorem metusttoOLD 21186
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.) (Proof modification is discouraged.) (New usage is discouraged.)
Hypothesis
Ref Expression
metust.1  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
Assertion
Ref Expression
metusttoOLD  |-  ( ( D  e.  ( *Met `  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 metusttoOLD
Dummy variable  b is distinct from all other variables.
StepHypRef Expression
1 simplr 755 . . . . 5  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  b  e.  RR+ )
21rpred 11281 . . . 4  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  b  e.  RR )
3 simpll 753 . . . . 5  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  a  e.  RR+ )
43rpred 11281 . . . 4  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  a  e.  RR )
5 simplll 759 . . . . . . . 8  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  a  e.  RR+ )
65rpred 11281 . . . . . . 7  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  a  e.  RR )
7 0xr 9657 . . . . . . . . . 10  |-  0  e.  RR*
87a1i 11 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
0  e.  RR* )
9 simpl 457 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
a  e.  RR )
109rexrd 9660 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
a  e.  RR* )
11 0le0 10646 . . . . . . . . . 10  |-  0  <_  0
1211a1i 11 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
0  <_  0 )
13 simpr 461 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
b  <_  a )
14 icossico 11619 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\  a  e.  RR* )  /\  ( 0  <_  0  /\  b  <_  a ) )  ->  ( 0 [,) b )  C_  ( 0 [,) a
) )
158, 10, 12, 13, 14syl22anc 1229 . . . . . . . 8  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
( 0 [,) b
)  C_  ( 0 [,) a ) )
16 imass2 5382 . . . . . . . 8  |-  ( ( 0 [,) b ) 
C_  ( 0 [,) a )  ->  ( `' D " ( 0 [,) b ) ) 
C_  ( `' D " ( 0 [,) a
) ) )
1715, 16syl 16 . . . . . . 7  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
( `' D "
( 0 [,) b
) )  C_  ( `' D " ( 0 [,) a ) ) )
186, 17sylancom 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 ) ) )
19 simplrr 762 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  B  =  ( `' D " ( 0 [,) b ) ) )
20 simplrl 761 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  A  =  ( `' D " ( 0 [,) a ) ) )
2118, 19, 203sstr4d 3542 . . . . 5  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  B  C_  A
)
2221olcd 393 . . . 4  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  ( A  C_  B  \/  B  C_  A ) )
23 simpllr 760 . . . . . . . 8  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  b  e.  RR+ )
2423rpred 11281 . . . . . . 7  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  b  e.  RR )
257a1i 11 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
0  e.  RR* )
26 simpl 457 . . . . . . . . . 10  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
b  e.  RR )
2726rexrd 9660 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
b  e.  RR* )
2811a1i 11 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
0  <_  0 )
29 simpr 461 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
a  <_  b )
30 icossico 11619 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\  b  e.  RR* )  /\  ( 0  <_  0  /\  a  <_  b ) )  ->  ( 0 [,) a )  C_  ( 0 [,) b
) )
3125, 27, 28, 29, 30syl22anc 1229 . . . . . . . 8  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
( 0 [,) a
)  C_  ( 0 [,) b ) )
32 imass2 5382 . . . . . . . 8  |-  ( ( 0 [,) a ) 
C_  ( 0 [,) b )  ->  ( `' D " ( 0 [,) a ) ) 
C_  ( `' D " ( 0 [,) b
) ) )
3331, 32syl 16 . . . . . . 7  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
( `' D "
( 0 [,) a
) )  C_  ( `' D " ( 0 [,) b ) ) )
3424, 33sylancom 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 ) ) )
35 simplrl 761 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  A  =  ( `' D " ( 0 [,) a ) ) )
36 simplrr 762 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  B  =  ( `' D " ( 0 [,) b ) ) )
3734, 35, 363sstr4d 3542 . . . . 5  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  A  C_  B
)
3837orcd 392 . . . 4  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  ( A  C_  B  \/  B  C_  A ) )
392, 4, 22, 38lecasei 9707 . . 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.  ( *Met `  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
) ) )
4241metustelOLD 21180 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) ) )
4342biimpa 484 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
44433adant3 1016 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F  /\  B  e.  F
)  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
45 oveq2 6304 . . . . . . . . . 10  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
4645imaeq2d 5347 . . . . . . . . 9  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
4746cbvmptv 4548 . . . . . . . 8  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
4847rneqi 5239 . . . . . . 7  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
4941, 48eqtri 2486 . . . . . 6  |-  F  =  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )
5049metustelOLD 21180 . . . . 5  |-  ( D  e.  ( *Met `  X )  ->  ( B  e.  F  <->  E. b  e.  RR+  B  =  ( `' D " ( 0 [,) b ) ) ) )
5150biimpa 484 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  F
)  ->  E. b  e.  RR+  B  =  ( `' D " ( 0 [,) b ) ) )
52513adant2 1015 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F  /\  B  e.  F
)  ->  E. b  e.  RR+  B  =  ( `' D " ( 0 [,) b ) ) )
53 reeanv 3025 . . 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.  ( *Met `  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 3001 1  |-  ( ( D  e.  ( *Met `  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 1395    e. wcel 1819   E.wrex 2808    C_ wss 3471   class class class wbr 4456    |-> cmpt 4515   `'ccnv 5007   ran crn 5009   "cima 5011   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   RR*cxr 9644    <_ cle 9646   RR+crp 11245   [,)cico 11556   *Metcxmt 18530
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-i2m1 9577  ax-1ne0 9578  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-po 4809  df-so 4810  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-1st 6799  df-2nd 6800  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-rp 11246  df-ico 11560
This theorem is referenced by:  metustfbasOLD  21194
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