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Theorem metustidOLD 20889
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 5301 . . 3  |-  Rel  (  _I  |`  X )
21a1i 11 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  F
)  ->  Rel  (  _I  |`  X ) )
3 vex 3116 . . . . . . . . . . . . . . 15  |-  q  e. 
_V
43brres 5280 . . . . . . . . . . . . . 14  |-  ( p (  _I  |`  X ) q  <->  ( p  _I  q  /\  p  e.  X ) )
5 df-br 4448 . . . . . . . . . . . . . 14  |-  ( p (  _I  |`  X ) q  <->  <. p ,  q
>.  e.  (  _I  |`  X ) )
63ideq 5155 . . . . . . . . . . . . . . 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 6288 . . . . . . . . . . 11  |-  ( p D p )  =  ( D `  <. p ,  p >. )
13 opeq2 4214 . . . . . . . . . . . 12  |-  ( p  =  q  ->  <. p ,  p >.  =  <. p ,  q >. )
1413fveq2d 5870 . . . . . . . . . . 11  |-  ( p  =  q  ->  ( D `  <. p ,  p >. )  =  ( D `  <. p ,  q >. )
)
1512, 14syl5eq 2520 . . . . . . . . . 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 20672 . . . . . . . . . 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 2510 . . . . . . . 8  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  ( D `  <. p ,  q
>. )  =  0
)
22 0xr 9641 . . . . . . . . . . 11  |-  0  e.  RR*
2322a1i 11 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  0  e. 
RR* )
24 rpxr 11228 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  a  e. 
RR* )
25 rpgt0 11232 . . . . . . . . . 10  |-  ( a  e.  RR+  ->  0  < 
a )
26 lbico1 11580 . . . . . . . . . 10  |-  ( ( 0  e.  RR*  /\  a  e.  RR*  /\  0  < 
a )  ->  0  e.  ( 0 [,) a
) )
2723, 24, 25, 26syl3anc 1228 . . . . . . . . 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 2555 . . . . . . 7  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  ( D `  <. p ,  q
>. )  e.  (
0 [,) a ) )
30 xmetf 20659 . . . . . . . . . 10  |-  ( D  e.  ( *Met `  X )  ->  D : ( X  X.  X ) --> RR* )
31 ffun 5733 . . . . . . . . . 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 2556 . . . . . . . . . 10  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  q  e.  X )
35 opelxpi 5031 . . . . . . . . . 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 5735 . . . . . . . . . . 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 2558 . . . . . . . 8  |-  ( ( ( ( D  e.  ( *Met `  X )  /\  A  e.  F )  /\  <. p ,  q >.  e.  (  _I  |`  X )
)  /\  a  e.  RR+ )  ->  <. p ,  q >.  e.  dom  D )
41 fvimacnv 5997 . . . . . . . 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 2558 . . . 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 20881 . . . . . 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 3003 . . 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 5095 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 1379    e. wcel 1767   E.wrex 2815    C_ wss 3476   <.cop 4033   class class class wbr 4447    |-> cmpt 4505    _I cid 4790    X. cxp 4997   `'ccnv 4998   dom cdm 4999   ran crn 5000    |` cres 5001   "cima 5002   Rel wrel 5004   Fun wfun 5582   -->wf 5584   ` cfv 5588  (class class class)co 6285   0cc0 9493   RR*cxr 9628    < clt 9629   RR+crp 11221   [,)cico 11532   *Metcxmt 18214
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 6577  ax-cnex 9549  ax-resscn 9550  ax-1cn 9551  ax-icn 9552  ax-addcl 9553  ax-addrcl 9554  ax-mulcl 9555  ax-mulrcl 9556  ax-i2m1 9561  ax-1ne0 9562  ax-rnegex 9564  ax-rrecex 9565  ax-cnre 9566  ax-pre-lttri 9567  ax-pre-lttrn 9568
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-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 6288  df-oprab 6289  df-mpt2 6290  df-er 7312  df-map 7423  df-en 7518  df-dom 7519  df-sdom 7520  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635  df-rp 11222  df-ico 11536  df-xmet 18223
This theorem is referenced by:  metustfbasOLD  20895  metustOLD  20897
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