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Theorem metustidOLD 20267
Description: The identity diagonal is included in all elements of the filter base generated by the metric  D. (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
metustidOLD  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  (  _I  |`  X )  C_  A
)
Distinct variable groups:    D, a    X, a    A, a    F, a

Proof of Theorem metustidOLD
Dummy variables  p  q are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5247 . . 3  |-  Rel  (  _I  |`  X )
21a1i 11 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  Rel  (  _I  |`  X ) )
3 vex 3081 . . . . . . . . . . . . . . 15  |-  q  e. 
_V
43brres 5226 . . . . . . . . . . . . . 14  |-  ( p (  _I  |`  X ) q  <->  ( p  _I  q  /\  p  e.  X ) )
5 df-br 4402 . . . . . . . . . . . . . 14  |-  ( p (  _I  |`  X ) q  <->  <. p ,  q
>.  e.  (  _I  |`  X ) )
63ideq 5101 . . . . . . . . . . . . . . 15  |-  ( p  _I  q  <->  p  =  q )
76anbi1i 695 . . . . . . . . . . . . . 14  |-  ( ( p  _I  q  /\  p  e.  X )  <->  ( p  =  q  /\  p  e.  X )
)
84, 5, 73bitr3i 275 . . . . . . . . . . . . 13  |-  ( <.
p ,  q >.  e.  (  _I  |`  X )  <-> 
( p  =  q  /\  p  e.  X
) )
98biimpi 194 . . . . . . . . . . . 12  |-  ( <.
p ,  q >.  e.  (  _I  |`  X )  ->  ( p  =  q  /\  p  e.  X ) )
109ad2antlr 726 . . . . . . . . . . 11  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  ( p  =  q  /\  p  e.  X ) )
1110simpld 459 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  p  =  q )
12 df-ov 6204 . . . . . . . . . . 11  |-  ( p D p )  =  ( D `  <. p ,  p >. )
13 opeq2 4169 . . . . . . . . . . . 12  |-  ( p  =  q  ->  <. p ,  p >.  =  <. p ,  q >. )
1413fveq2d 5804 . . . . . . . . . . 11  |-  ( p  =  q  ->  ( D `  <. p ,  p >. )  =  ( D `  <. p ,  q >. )
)
1512, 14syl5eq 2507 . . . . . . . . . 10  |-  ( p  =  q  ->  (
p D p )  =  ( D `  <. p ,  q >.
) )
1611, 15syl 16 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  ( p D p )  =  ( D `  <. p ,  q >. )
)
17 simplll 757 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  D  e.  ( *Met `  X
) )
1810simprd 463 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  p  e.  X )
19 xmet0 20050 . . . . . . . . . 10  |-  ( ( D  e.  ( *Met `  X )  /\  p  e.  X
)  ->  ( p D p )  =  0 )
2017, 18, 19syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  ( p D p )  =  0 )
2116, 20eqtr3d 2497 . . . . . . . 8  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  ( D `  <. p ,  q
>. )  =  0
)
22 0xr 9542 . . . . . . . . . . 11  |-  0  e.  RR*
2322a1i 11 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  0  e. 
RR* )
24 rpxr 11110 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  a  e. 
RR* )
25 rpgt0 11114 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  0  < 
a )
26 lbico1 11462 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  a  e.  RR*  /\  0  < 
a )  ->  0  e.  ( 0 [,) a
) )
2723, 24, 25, 26syl3anc 1219 . . . . . . . . 9  |-  ( a  e.  RR+  ->  0  e.  ( 0 [,) a
) )
2827adantl 466 . . . . . . . 8  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  0  e.  ( 0 [,) a
) )
2921, 28eqeltrd 2542 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  ( D `  <. p ,  q
>. )  e.  (
0 [,) a ) )
30 xmetf 20037 . . . . . . . . . 10  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
31 ffun 5670 . . . . . . . . . 10  |-  ( D : ( X  X.  X ) --> RR*  ->  Fun 
D )
3230, 31syl 16 . . . . . . . . 9  |-  ( D  e.  ( *Met `  X )  ->  Fun  D )
3332ad3antrrr 729 . . . . . . . 8  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  Fun  D )
3411, 18eqeltrrd 2543 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  q  e.  X )
35 opelxpi 4980 . . . . . . . . . 10  |-  ( ( p  e.  X  /\  q  e.  X )  -> 
<. p ,  q >.  e.  ( X  X.  X
) )
3618, 34, 35syl2anc 661 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  <. p ,  q >.  e.  ( X  X.  X ) )
37 fdm 5672 . . . . . . . . . . 11  |-  ( D : ( X  X.  X ) --> RR*  ->  dom 
D  =  ( X  X.  X ) )
3830, 37syl 16 . . . . . . . . . 10  |-  ( D  e.  ( *Met `  X )  ->  dom  D  =  ( X  X.  X ) )
3938ad3antrrr 729 . . . . . . . . 9  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  dom  D  =  ( X  X.  X
) )
4036, 39eleqtrrd 2545 . . . . . . . 8  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  <. p ,  q >.  e.  dom  D )
41 fvimacnv 5928 . . . . . . . 8  |-  ( ( Fun  D  /\  <. p ,  q >.  e.  dom  D )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
4233, 40, 41syl2anc 661 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  ( ( D `  <. p ,  q >. )  e.  ( 0 [,) a )  <->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) ) )
4329, 42mpbid 210 . . . . . 6  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  <. p ,  q >.  e.  ( `' D " ( 0 [,) a ) ) )
4443adantr 465 . . . . 5  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  <. p ,  q >.  e.  ( `' D "
( 0 [,) a
) ) )
45 simpr 461 . . . . 5  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  A  =  ( `' D " ( 0 [,) a ) ) )
4644, 45eleqtrrd 2545 . . . 4  |-  ( ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  /\  A  =  ( `' D "
( 0 [,) a
) ) )  ->  <. p ,  q >.  e.  A )
47 simplr 754 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  A  e.  F )
48 metust.1 . . . . . . 7  |-  F  =  ran  ( a  e.  RR+  |->  ( `' D " ( 0 [,) a
) ) )
4948metustelOLD 20259 . . . . . 6  |-  ( D  e.  ( *Met `  X )  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) ) )
5049ad2antrr 725 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  ( A  e.  F  <->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a
) ) ) )
5147, 50mpbid 210 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  E. a  e.  RR+  A  =  ( `' D " ( 0 [,) a ) ) )
5246, 51r19.29a 2968 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  ->  <. p ,  q >.  e.  A
)
5352ex 434 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  ( <. p ,  q >.  e.  (  _I  |`  X )  -> 
<. p ,  q >.  e.  A ) )
542, 53relssdv 5041 1  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  (  _I  |`  X )  C_  A
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800    C_ wss 3437   <.cop 3992   class class class wbr 4401    |-> cmpt 4459    _I cid 4740    X. cxp 4947   `'ccnv 4948   dom cdm 4949   ran crn 4950    |` cres 4951   "cima 4952   Rel wrel 4954   Fun wfun 5521   -->wf 5523   ` cfv 5527  (class class class)co 6201   0cc0 9394   RR*cxr 9529    < clt 9530   RR+crp 11103   [,)cico 11414   *Metcxmt 17927
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483  ax-cnex 9450  ax-resscn 9451  ax-1cn 9452  ax-icn 9453  ax-addcl 9454  ax-addrcl 9455  ax-mulcl 9456  ax-mulrcl 9457  ax-i2m1 9462  ax-1ne0 9463  ax-rnegex 9465  ax-rrecex 9466  ax-cnre 9467  ax-pre-lttri 9468  ax-pre-lttrn 9469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-po 4750  df-so 4751  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-er 7212  df-map 7327  df-en 7422  df-dom 7423  df-sdom 7424  df-pnf 9532  df-mnf 9533  df-xr 9534  df-ltxr 9535  df-le 9536  df-rp 11104  df-ico 11418  df-xmet 17936
This theorem is referenced by:  metustfbasOLD  20273  metustOLD  20275
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