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Theorem psmetxrge0 12428
Description: The distance function of a pseudometric space is a function into the nonnegative extended real numbers. (Contributed by Thierry Arnoux, 24-Feb-2018.)
Assertion
Ref Expression
psmetxrge0  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> ( 0 [,] +oo ) )

Proof of Theorem psmetxrge0
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 psmetf 12421 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
21ffnd 5243 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  Fn  ( X  X.  X
) )
31ffvelrnda 5523 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  ( D `  a )  e.  RR* )
4 elxp6 6035 . . . . . . . 8  |-  ( a  e.  ( X  X.  X )  <->  ( a  =  <. ( 1st `  a
) ,  ( 2nd `  a ) >.  /\  (
( 1st `  a
)  e.  X  /\  ( 2nd `  a )  e.  X ) ) )
54simprbi 273 . . . . . . 7  |-  ( a  e.  ( X  X.  X )  ->  (
( 1st `  a
)  e.  X  /\  ( 2nd `  a )  e.  X ) )
6 psmetge0 12427 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  ( 1st `  a )  e.  X  /\  ( 2nd `  a )  e.  X
)  ->  0  <_  ( ( 1st `  a
) D ( 2nd `  a ) ) )
763expb 1167 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  (
( 1st `  a
)  e.  X  /\  ( 2nd `  a )  e.  X ) )  ->  0  <_  (
( 1st `  a
) D ( 2nd `  a ) ) )
85, 7sylan2 284 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  0  <_  ( ( 1st `  a
) D ( 2nd `  a ) ) )
9 1st2nd2 6041 . . . . . . . . 9  |-  ( a  e.  ( X  X.  X )  ->  a  =  <. ( 1st `  a
) ,  ( 2nd `  a ) >. )
109fveq2d 5393 . . . . . . . 8  |-  ( a  e.  ( X  X.  X )  ->  ( D `  a )  =  ( D `  <. ( 1st `  a
) ,  ( 2nd `  a ) >. )
)
11 df-ov 5745 . . . . . . . 8  |-  ( ( 1st `  a ) D ( 2nd `  a
) )  =  ( D `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
1210, 11syl6eqr 2168 . . . . . . 7  |-  ( a  e.  ( X  X.  X )  ->  ( D `  a )  =  ( ( 1st `  a ) D ( 2nd `  a ) ) )
1312adantl 275 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  ( D `  a )  =  ( ( 1st `  a ) D ( 2nd `  a ) ) )
148, 13breqtrrd 3926 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  0  <_  ( D `  a
) )
15 elxrge0 9729 . . . . 5  |-  ( ( D `  a )  e.  ( 0 [,] +oo )  <->  ( ( D `
 a )  e. 
RR*  /\  0  <_  ( D `  a ) ) )
163, 14, 15sylanbrc 413 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  ( D `  a )  e.  ( 0 [,] +oo ) )
1716ralrimiva 2482 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  A. a  e.  ( X  X.  X
) ( D `  a )  e.  ( 0 [,] +oo )
)
18 fnfvrnss 5548 . . 3  |-  ( ( D  Fn  ( X  X.  X )  /\  A. a  e.  ( X  X.  X ) ( D `  a )  e.  ( 0 [,] +oo ) )  ->  ran  D 
C_  ( 0 [,] +oo ) )
192, 17, 18syl2anc 408 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ran  D  C_  ( 0 [,] +oo ) )
20 df-f 5097 . 2  |-  ( D : ( X  X.  X ) --> ( 0 [,] +oo )  <->  ( D  Fn  ( X  X.  X
)  /\  ran  D  C_  ( 0 [,] +oo ) ) )
212, 19, 20sylanbrc 413 1  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> ( 0 [,] +oo ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    = wceq 1316    e. wcel 1465   A.wral 2393    C_ wss 3041   <.cop 3500   class class class wbr 3899    X. cxp 4507   ran crn 4510    Fn wfn 5088   -->wf 5089   ` cfv 5093  (class class class)co 5742   1stc1st 6004   2ndc2nd 6005   0cc0 7588   +oocpnf 7765   RR*cxr 7767    <_ cle 7769   [,]cicc 9642  PsMetcpsmet 12075
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-un 4325  ax-setind 4422  ax-cnex 7679  ax-resscn 7680  ax-1cn 7681  ax-1re 7682  ax-icn 7683  ax-addcl 7684  ax-addrcl 7685  ax-mulcl 7686  ax-mulrcl 7687  ax-addcom 7688  ax-mulcom 7689  ax-addass 7690  ax-mulass 7691  ax-distr 7692  ax-i2m1 7693  ax-0lt1 7694  ax-1rid 7695  ax-0id 7696  ax-rnegex 7697  ax-precex 7698  ax-cnre 7699  ax-pre-ltirr 7700  ax-pre-lttrn 7702  ax-pre-ltadd 7704  ax-pre-mulgt0 7705
This theorem depends on definitions:  df-bi 116  df-dc 805  df-3or 948  df-3an 949  df-tru 1319  df-fal 1322  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ne 2286  df-nel 2381  df-ral 2398  df-rex 2399  df-reu 2400  df-rab 2402  df-v 2662  df-sbc 2883  df-csb 2976  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-if 3445  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-iun 3785  df-br 3900  df-opab 3960  df-mpt 3961  df-id 4185  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-res 4521  df-ima 4522  df-iota 5058  df-fun 5095  df-fn 5096  df-f 5097  df-fv 5101  df-riota 5698  df-ov 5745  df-oprab 5746  df-mpo 5747  df-1st 6006  df-2nd 6007  df-map 6512  df-pnf 7770  df-mnf 7771  df-xr 7772  df-ltxr 7773  df-le 7774  df-sub 7903  df-neg 7904  df-2 8747  df-xadd 9528  df-icc 9646  df-psmet 12083
This theorem is referenced by: (None)
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