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Theorem metustto 21568
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 760 . . . . 5  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  a  e.  RR+ )
21rpred 11341 . . . 4  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  a  e.  RR )
3 simplr 762 . . . . 5  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  b  e.  RR+ )
43rpred 11341 . . . 4  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  b  e.  RR )
5 simpllr 769 . . . . . . . 8  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  b  e.  RR+ )
65rpred 11341 . . . . . . 7  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  b  e.  RR )
7 0xr 9687 . . . . . . . . . 10  |-  0  e.  RR*
87a1i 11 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
0  e.  RR* )
9 simpl 459 . . . . . . . . . 10  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
b  e.  RR )
109rexrd 9690 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
b  e.  RR* )
11 0le0 10699 . . . . . . . . . 10  |-  0  <_  0
1211a1i 11 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
0  <_  0 )
13 simpr 463 . . . . . . . . 9  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
a  <_  b )
14 icossico 11704 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\  b  e.  RR* )  /\  ( 0  <_  0  /\  a  <_  b ) )  ->  ( 0 [,) a )  C_  ( 0 [,) b
) )
158, 10, 12, 13, 14syl22anc 1269 . . . . . . . 8  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
( 0 [,) a
)  C_  ( 0 [,) b ) )
16 imass2 5204 . . . . . . . 8  |-  ( ( 0 [,) a ) 
C_  ( 0 [,) b )  ->  ( `' D " ( 0 [,) a ) ) 
C_  ( `' D " ( 0 [,) b
) ) )
1715, 16syl 17 . . . . . . 7  |-  ( ( b  e.  RR  /\  a  <_  b )  -> 
( `' D "
( 0 [,) a
) )  C_  ( `' D " ( 0 [,) b ) ) )
186, 17sylancom 673 . . . . . 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 770 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  A  =  ( `' D " ( 0 [,) a ) ) )
20 simplrr 771 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  B  =  ( `' D " ( 0 [,) b ) ) )
2118, 19, 203sstr4d 3475 . . . . 5  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  a  <_  b
)  ->  A  C_  B
)
2221orcd 394 . . . 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 768 . . . . . . . 8  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  a  e.  RR+ )
2423rpred 11341 . . . . . . 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 459 . . . . . . . . . 10  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
a  e.  RR )
2726rexrd 9690 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
a  e.  RR* )
2811a1i 11 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
0  <_  0 )
29 simpr 463 . . . . . . . . 9  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
b  <_  a )
30 icossico 11704 . . . . . . . . 9  |-  ( ( ( 0  e.  RR*  /\  a  e.  RR* )  /\  ( 0  <_  0  /\  b  <_  a ) )  ->  ( 0 [,) b )  C_  ( 0 [,) a
) )
3125, 27, 28, 29, 30syl22anc 1269 . . . . . . . 8  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
( 0 [,) b
)  C_  ( 0 [,) a ) )
32 imass2 5204 . . . . . . . 8  |-  ( ( 0 [,) b ) 
C_  ( 0 [,) a )  ->  ( `' D " ( 0 [,) b ) ) 
C_  ( `' D " ( 0 [,) a
) ) )
3331, 32syl 17 . . . . . . 7  |-  ( ( a  e.  RR  /\  b  <_  a )  -> 
( `' D "
( 0 [,) b
) )  C_  ( `' D " ( 0 [,) a ) ) )
3424, 33sylancom 673 . . . . . 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 771 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  B  =  ( `' D " ( 0 [,) b ) ) )
36 simplrl 770 . . . . . 6  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  A  =  ( `' D " ( 0 [,) a ) ) )
3734, 35, 363sstr4d 3475 . . . . 5  |-  ( ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D "
( 0 [,) a
) )  /\  B  =  ( `' D " ( 0 [,) b
) ) ) )  /\  b  <_  a
)  ->  B  C_  A
)
3837olcd 395 . . . 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 9740 . . 3  |-  ( ( ( a  e.  RR+  /\  b  e.  RR+ )  /\  ( A  =  ( `' D " ( 0 [,) a ) )  /\  B  =  ( `' D " ( 0 [,) b ) ) ) )  ->  ( A  C_  B  \/  B  C_  A ) )
4039adantlll 724 . 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 21565 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
4342biimpa 487 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
44433adant3 1028 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
45 oveq2 6298 . . . . . . . . . 10  |-  ( a  =  b  ->  (
0 [,) a )  =  ( 0 [,) b ) )
4645imaeq2d 5168 . . . . . . . . 9  |-  ( a  =  b  ->  ( `' D " ( 0 [,) a ) )  =  ( `' D " ( 0 [,) b
) ) )
4746cbvmptv 4495 . . . . . . . 8  |-  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
4847rneqi 5061 . . . . . . 7  |-  ran  (
a  e.  RR+  |->  ( `' D " ( 0 [,) a ) ) )  =  ran  (
b  e.  RR+  |->  ( `' D " ( 0 [,) b ) ) )
4941, 48eqtri 2473 . . . . . 6  |-  F  =  ran  ( b  e.  RR+  |->  ( `' D " ( 0 [,) b
) ) )
5049metustel 21565 . . . . 5  |-  ( D  e.  (PsMet `  X
)  ->  ( B  e.  F  <->  E. b  e.  RR+  B  =  ( `' D " ( 0 [,) b
) ) ) )
5150biimpa 487 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  B  e.  F )  ->  E. b  e.  RR+  B  =  ( `' D " ( 0 [,) b ) ) )
52513adant2 1027 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  F  /\  B  e.  F )  ->  E. b  e.  RR+  B  =  ( `' D " ( 0 [,) b ) ) )
53 reeanv 2958 . . 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 670 . 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, 54r19.29vva 2934 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 370    /\ wa 371    /\ w3a 985    = wceq 1444    e. wcel 1887   E.wrex 2738    C_ wss 3404   class class class wbr 4402    |-> cmpt 4461   `'ccnv 4833   ran crn 4835   "cima 4837   ` cfv 5582  (class class class)co 6290   RRcr 9538   0cc0 9539   RR*cxr 9674    <_ cle 9676   RR+crp 11302   [,)cico 11637  PsMetcpsmet 18954
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-i2m1 9607  ax-1ne0 9608  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-po 4755  df-so 4756  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-1st 6793  df-2nd 6794  df-er 7363  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-rp 11303  df-ico 11641
This theorem is referenced by:  metustfbas  21572
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