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Theorem metdsle 18884
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 735 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  ->  B  e.  X )
2 simpr 449 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X
)  ->  S  C_  X
)
32sselda 3350 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  A  e.  S )  ->  A  e.  X )
43adantrr 699 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  ->  A  e.  X )
51, 4jca 520 . . 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 18883 . . 3  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( B  e.  X  /\  A  e.  X ) )  -> 
( F `  B
)  <_  ( ( B D A ) + e ( F `  A ) ) )
85, 7syldan 458 . 2  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( F `  B
)  <_  ( ( B D A ) + e ( F `  A ) ) )
9 simpll 732 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  ->  D  e.  ( * Met `  X ) )
10 xmetsym 18379 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  =  ( A D B ) )
119, 1, 4, 10syl3anc 1185 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( B D A )  =  ( A D B ) )
126metds0 18882 . . . . . 6  |-  ( ( D  e.  ( * Met `  X )  /\  S  C_  X  /\  A  e.  S
)  ->  ( F `  A )  =  0 )
13123expa 1154 . . . . 5  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  A  e.  S )  ->  ( F `  A )  =  0 )
1413adantrr 699 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( F `  A
)  =  0 )
1511, 14oveq12d 6101 . . 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 18363 . . . . 5  |-  ( ( D  e.  ( * Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
179, 4, 1, 16syl3anc 1185 . . . 4  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( A D B )  e.  RR* )
18 xaddid1 10827 . . . 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 2470 . 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 4238 1  |-  ( ( ( D  e.  ( * Met `  X
)  /\  S  C_  X
)  /\  ( A  e.  S  /\  B  e.  X ) )  -> 
( F `  B
)  <_  ( A D B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726    C_ wss 3322   class class class wbr 4214    e. cmpt 4268   `'ccnv 4879   ran crn 4881   ` cfv 5456  (class class class)co 6083   supcsup 7447   0cc0 8992   RR*cxr 9121    < clt 9122    <_ cle 9123   + ecxad 10710   * Metcxmt 16688
This theorem is referenced by:  metdsre  18885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-po 4505  df-so 4506  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-er 6907  df-ec 6909  df-map 7022  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-2 10060  df-rp 10615  df-xneg 10712  df-xadd 10713  df-xmul 10714  df-icc 10925  df-psmet 16696  df-xmet 16697  df-bl 16699
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