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Theorem xmetsym 21349
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 1005 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  D  e.  ( *Met `  X
) )
2 simp3 1007 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  B  e.  X )
3 simp2 1006 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  A  e.  X )
4 xmettri2 21342 . . . 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 1266 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  <_  (
( B D A ) +e ( B D B ) ) )
6 xmet0 21344 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X
)  ->  ( B D B )  =  0 )
763adant2 1024 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D B )  =  0 )
87oveq2d 6318 . . . 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 21333 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  e.  RR* )
10 xaddid1 11533 . . . . . 6  |-  ( ( B D A )  e.  RR*  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
119, 10syl 17 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
12113com23 1211 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
138, 12eqtrd 2463 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( B D A ) +e ( B D B ) )  =  ( B D A ) )
145, 13breqtrd 4445 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  <_  ( B D A ) )
15 xmettri2 21342 . . . 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 1266 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  <_  (
( A D B ) +e ( A D A ) ) )
17 xmet0 21344 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X
)  ->  ( A D A )  =  0 )
18173adant3 1025 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D A )  =  0 )
1918oveq2d 6318 . . . 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 21333 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
21 xaddid1 11533 . . . . 5  |-  ( ( A D B )  e.  RR*  ->  ( ( A D B ) +e 0 )  =  ( A D B ) )
2220, 21syl 17 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B ) +e 0 )  =  ( A D B ) )
2319, 22eqtrd 2463 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B ) +e ( A D A ) )  =  ( A D B ) )
2416, 23breqtrd 4445 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  <_  ( A D B ) )
2593com23 1211 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  e.  RR* )
26 xrletri3 11452 . . 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 665 . 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 930 1  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  =  ( B D A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1868   class class class wbr 4420   ` cfv 5598  (class class class)co 6302   0cc0 9540   RR*cxr 9675    <_ cle 9677   +ecxad 11408   *Metcxmt 18943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-op 4003  df-uni 4217  df-br 4421  df-opab 4480  df-mpt 4481  df-id 4765  df-po 4771  df-so 4772  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-xadd 11411  df-xmet 18951
This theorem is referenced by:  xmettpos  21351  metsym  21352  xmettri  21353  xmettri3  21355  xmetrtri2  21358  elbl3  21394  blss  21427  xmeter  21435  xmssym  21467  metcnp2  21544  metdcnlem  21841  metdstri  21855  metdsle  21856  metdscn  21860  metnrmlem1  21863  metnrmlem3  21865  metdstriOLD  21870  metdsleOLD  21871  metdscnOLD  21875  metnrmlem1OLD  21878  metnrmlem3OLD  21880  nmhmcn  22121  lmmbr2  22216  iscau2  22234  iscau3  22235  iscau4  22236  iscauf  22237  caucfil  22240  dvlip2  22934  nvlmle  26314  ubthlem1  26498  ubthlem2  26499  heicant  31889
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