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Theorem metdsle 21224
Description: The distance from a point to a set is bounded by the distance to any member of the set. (Contributed by Mario Carneiro, 5-Sep-2015.)
Hypothesis
Ref Expression
metdscn.f  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
Assertion
Ref Expression
metdsle  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( F `  B
)  <_  ( A D B ) )
Distinct variable groups:    x, y, A    x, D, y    x, B, y    x, S, y   
x, X, y
Allowed substitution hints:    F( x, y)

Proof of Theorem metdsle
StepHypRef Expression
1 simprr 756 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  ->  B  e.  X )
2 simpr 461 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X
)  ->  S  C_  X
)
32sselda 3509 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  A  e.  S )  ->  A  e.  X )
43adantrr 716 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  ->  A  e.  X )
51, 4jca 532 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( B  e.  X  /\  A  e.  X
) )
6 metdscn.f . . . 4  |-  F  =  ( x  e.  X  |->  sup ( ran  (
y  e.  S  |->  ( x D y ) ) ,  RR* ,  `'  <  ) )
76metdstri 21223 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( B  e.  X  /\  A  e.  X ) )  -> 
( F `  B
)  <_  ( ( B D A ) +e ( F `  A ) ) )
85, 7syldan 470 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( F `  B
)  <_  ( ( B D A ) +e ( F `  A ) ) )
9 simpll 753 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  ->  D  e.  ( *Met `  X ) )
10 xmetsym 20718 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  =  ( A D B ) )
119, 1, 4, 10syl3anc 1228 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( B D A )  =  ( A D B ) )
126metds0 21222 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  =  0 )
13123expa 1196 . . . . 5  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  A  e.  S )  ->  ( F `  A )  =  0 )
1413adantrr 716 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( F `  A
)  =  0 )
1511, 14oveq12d 6313 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( ( B D A ) +e
( F `  A
) )  =  ( ( A D B ) +e 0 ) )
16 xmetcl 20702 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
179, 4, 1, 16syl3anc 1228 . . . 4  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( A D B )  e.  RR* )
18 xaddid1 11450 . . . 4  |-  ( ( A D B )  e.  RR*  ->  ( ( A D B ) +e 0 )  =  ( A D B ) )
1917, 18syl 16 . . 3  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( ( A D B ) +e 0 )  =  ( A D B ) )
2015, 19eqtrd 2508 . 2  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( ( B D A ) +e
( F `  A
) )  =  ( A D B ) )
218, 20breqtrd 4477 1  |-  ( ( ( D  e.  ( *Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( F `  B
)  <_  ( A D B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767    C_ wss 3481   class class class wbr 4453    |-> cmpt 4511   `'ccnv 5004   ran crn 5006   ` cfv 5594  (class class class)co 6295   supcsup 7912   0cc0 9504   RR*cxr 9639    < clt 9640    <_ cle 9641   +ecxad 11328   *Metcxmt 18273
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
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 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-po 4806  df-so 4807  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-1st 6795  df-2nd 6796  df-er 7323  df-ec 7325  df-map 7434  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-2 10606  df-rp 11233  df-xneg 11330  df-xadd 11331  df-xmul 11332  df-icc 11548  df-psmet 18281  df-xmet 18282  df-bl 18284
This theorem is referenced by:  metdsre  21225
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