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Theorem psmetxrge0 19889
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 19882 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> RR* )
2 ffn 5559 . . 3  |-  ( D : ( X  X.  X ) --> RR*  ->  D  Fn  ( X  X.  X ) )
31, 2syl 16 . 2  |-  ( D  e.  (PsMet `  X
)  ->  D  Fn  ( X  X.  X
) )
41ffvelrnda 5843 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  ( D `  a )  e.  RR* )
5 simpl 457 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  D  e.  (PsMet `  X )
)
6 elxp6 6608 . . . . . . . . . 10  |-  ( a  e.  ( X  X.  X )  <->  ( a  =  <. ( 1st `  a
) ,  ( 2nd `  a ) >.  /\  (
( 1st `  a
)  e.  X  /\  ( 2nd `  a )  e.  X ) ) )
76simprbi 464 . . . . . . . . 9  |-  ( a  e.  ( X  X.  X )  ->  (
( 1st `  a
)  e.  X  /\  ( 2nd `  a )  e.  X ) )
87adantl 466 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  (
( 1st `  a
)  e.  X  /\  ( 2nd `  a )  e.  X ) )
9 psmetge0 19888 . . . . . . . . 9  |-  ( ( D  e.  (PsMet `  X )  /\  ( 1st `  a )  e.  X  /\  ( 2nd `  a )  e.  X
)  ->  0  <_  ( ( 1st `  a
) D ( 2nd `  a ) ) )
1093expb 1188 . . . . . . . 8  |-  ( ( D  e.  (PsMet `  X )  /\  (
( 1st `  a
)  e.  X  /\  ( 2nd `  a )  e.  X ) )  ->  0  <_  (
( 1st `  a
) D ( 2nd `  a ) ) )
115, 8, 10syl2anc 661 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  0  <_  ( ( 1st `  a
) D ( 2nd `  a ) ) )
12 1st2nd2 6613 . . . . . . . . . 10  |-  ( a  e.  ( X  X.  X )  ->  a  =  <. ( 1st `  a
) ,  ( 2nd `  a ) >. )
1312fveq2d 5695 . . . . . . . . 9  |-  ( a  e.  ( X  X.  X )  ->  ( D `  a )  =  ( D `  <. ( 1st `  a
) ,  ( 2nd `  a ) >. )
)
14 df-ov 6094 . . . . . . . . 9  |-  ( ( 1st `  a ) D ( 2nd `  a
) )  =  ( D `  <. ( 1st `  a ) ,  ( 2nd `  a
) >. )
1513, 14syl6eqr 2493 . . . . . . . 8  |-  ( a  e.  ( X  X.  X )  ->  ( D `  a )  =  ( ( 1st `  a ) D ( 2nd `  a ) ) )
1615adantl 466 . . . . . . 7  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  ( D `  a )  =  ( ( 1st `  a ) D ( 2nd `  a ) ) )
1711, 16breqtrrd 4318 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  0  <_  ( D `  a
) )
184, 17jca 532 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  (
( D `  a
)  e.  RR*  /\  0  <_  ( D `  a
) ) )
19 elxrge0 11394 . . . . 5  |-  ( ( D `  a )  e.  ( 0 [,] +oo )  <->  ( ( D `
 a )  e. 
RR*  /\  0  <_  ( D `  a ) ) )
2018, 19sylibr 212 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  a  e.  ( X  X.  X
) )  ->  ( D `  a )  e.  ( 0 [,] +oo ) )
2120ralrimiva 2799 . . 3  |-  ( D  e.  (PsMet `  X
)  ->  A. a  e.  ( X  X.  X
) ( D `  a )  e.  ( 0 [,] +oo )
)
22 fnfvrnss 5871 . . 3  |-  ( ( D  Fn  ( X  X.  X )  /\  A. a  e.  ( X  X.  X ) ( D `  a )  e.  ( 0 [,] +oo ) )  ->  ran  D 
C_  ( 0 [,] +oo ) )
233, 21, 22syl2anc 661 . 2  |-  ( D  e.  (PsMet `  X
)  ->  ran  D  C_  ( 0 [,] +oo ) )
24 df-f 5422 . 2  |-  ( D : ( X  X.  X ) --> ( 0 [,] +oo )  <->  ( D  Fn  ( X  X.  X
)  /\  ran  D  C_  ( 0 [,] +oo ) ) )
253, 23, 24sylanbrc 664 1  |-  ( D  e.  (PsMet `  X
)  ->  D :
( X  X.  X
) --> ( 0 [,] +oo ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1369    e. wcel 1756   A.wral 2715    C_ wss 3328   <.cop 3883   class class class wbr 4292    X. cxp 4838   ran crn 4841    Fn wfn 5413   -->wf 5414   ` cfv 5418  (class class class)co 6091   1stc1st 6575   2ndc2nd 6576   0cc0 9282   +oocpnf 9415   RR*cxr 9417    <_ cle 9419   [,]cicc 11303  PsMetcpsmet 17800
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-po 4641  df-so 4642  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-1st 6577  df-2nd 6578  df-er 7101  df-map 7216  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-2 10380  df-rp 10992  df-xneg 11089  df-xadd 11090  df-xmul 11091  df-icc 11307  df-psmet 17809
This theorem is referenced by: (None)
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