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Theorem xmetsym 12574
Description: The distance function of an extended metric space is symmetric. (Contributed by Mario Carneiro, 20-Aug-2015.)
Assertion
Ref Expression
xmetsym  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  =  ( B D A ) )

Proof of Theorem xmetsym
StepHypRef Expression
1 simp1 982 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  D  e.  ( *Met `  X
) )
2 simp3 984 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  B  e.  X )
3 simp2 983 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  A  e.  X )
4 xmettri2 12567 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( B  e.  X  /\  A  e.  X  /\  B  e.  X ) )  -> 
( A D B )  <_  ( ( B D A ) +e ( B D B ) ) )
51, 2, 3, 2, 4syl13anc 1219 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  <_  (
( B D A ) +e ( B D B ) ) )
6 xmet0 12569 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X
)  ->  ( B D B )  =  0 )
763adant2 1001 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D B )  =  0 )
87oveq2d 5797 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( B D A ) +e ( B D B ) )  =  ( ( B D A ) +e 0 ) )
9 xmetcl 12558 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  e.  RR* )
10 xaddid1 9674 . . . . . 6  |-  ( ( B D A )  e.  RR*  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
119, 10syl 14 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
12113com23 1188 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
138, 12eqtrd 2173 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( B D A ) +e ( B D B ) )  =  ( B D A ) )
145, 13breqtrd 3961 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  <_  ( B D A ) )
15 xmettri2 12567 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  A  e.  X ) )  -> 
( B D A )  <_  ( ( A D B ) +e ( A D A ) ) )
161, 3, 2, 3, 15syl13anc 1219 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  <_  (
( A D B ) +e ( A D A ) ) )
17 xmet0 12569 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X
)  ->  ( A D A )  =  0 )
18173adant3 1002 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D A )  =  0 )
1918oveq2d 5797 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B ) +e ( A D A ) )  =  ( ( A D B ) +e 0 ) )
20 xmetcl 12558 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
21 xaddid1 9674 . . . . 5  |-  ( ( A D B )  e.  RR*  ->  ( ( A D B ) +e 0 )  =  ( A D B ) )
2220, 21syl 14 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B ) +e 0 )  =  ( A D B ) )
2319, 22eqtrd 2173 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B ) +e ( A D A ) )  =  ( A D B ) )
2416, 23breqtrd 3961 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  <_  ( A D B ) )
2593com23 1188 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  e.  RR* )
26 xrletri3 9617 . . 3  |-  ( ( ( A D B )  e.  RR*  /\  ( B D A )  e. 
RR* )  ->  (
( A D B )  =  ( B D A )  <->  ( ( A D B )  <_ 
( B D A )  /\  ( B D A )  <_ 
( A D B ) ) ) )
2720, 25, 26syl2anc 409 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B )  =  ( B D A )  <->  ( ( A D B )  <_ 
( B D A )  /\  ( B D A )  <_ 
( A D B ) ) ) )
2814, 24, 27mpbir2and 929 1  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  =  ( B D A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    /\ w3a 963    = wceq 1332    e. wcel 1481   class class class wbr 3936   ` cfv 5130  (class class class)co 5781   0cc0 7643   RR*cxr 7822    <_ cle 7824   +ecxad 9586   *Metcxmet 12186
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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-cnex 7734  ax-resscn 7735  ax-1re 7737  ax-addrcl 7740  ax-0id 7751  ax-rnegex 7752  ax-pre-ltirr 7755  ax-pre-apti 7758
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-if 3479  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fv 5138  df-ov 5784  df-oprab 5785  df-mpo 5786  df-1st 6045  df-2nd 6046  df-map 6551  df-pnf 7825  df-mnf 7826  df-xr 7827  df-ltxr 7828  df-le 7829  df-xadd 9589  df-xmet 12194
This theorem is referenced by:  xmettpos  12576  metsym  12577  xmettri  12578  xmettri3  12580  elbl3  12601  blss  12634  xmeter  12642  xmssym  12675  metcnp2  12719
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