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Theorem xmetsym 20578
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 991 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  D  e.  ( *Met `  X
) )
2 simp3 993 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  B  e.  X )
3 simp2 992 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  A  e.  X )
4 xmettri2 20571 . . . 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 1225 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  <_  (
( B D A ) +e ( B D B ) ) )
6 xmet0 20573 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X
)  ->  ( B D B )  =  0 )
763adant2 1010 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D B )  =  0 )
87oveq2d 6291 . . . 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 20562 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  e.  RR* )
10 xaddid1 11427 . . . . . 6  |-  ( ( B D A )  e.  RR*  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
119, 10syl 16 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
12113com23 1197 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
138, 12eqtrd 2501 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( B D A ) +e ( B D B ) )  =  ( B D A ) )
145, 13breqtrd 4464 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  <_  ( B D A ) )
15 xmettri2 20571 . . . 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 1225 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  <_  (
( A D B ) +e ( A D A ) ) )
17 xmet0 20573 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X
)  ->  ( A D A )  =  0 )
18173adant3 1011 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D A )  =  0 )
1918oveq2d 6291 . . . 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 20562 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
21 xaddid1 11427 . . . . 5  |-  ( ( A D B )  e.  RR*  ->  ( ( A D B ) +e 0 )  =  ( A D B ) )
2220, 21syl 16 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B ) +e 0 )  =  ( A D B ) )
2319, 22eqtrd 2501 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B ) +e ( A D A ) )  =  ( A D B ) )
2416, 23breqtrd 4464 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  <_  ( A D B ) )
2593com23 1197 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  e.  RR* )
26 xrletri3 11347 . . 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 661 . 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 915 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 184    /\ wa 369    /\ w3a 968    = wceq 1374    e. wcel 1762   class class class wbr 4440   ` cfv 5579  (class class class)co 6275   0cc0 9481   RR*cxr 9616    <_ cle 9618   +ecxad 11305   *Metcxmt 18167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-po 4793  df-so 4794  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-er 7301  df-map 7412  df-en 7507  df-dom 7508  df-sdom 7509  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-xadd 11308  df-xmet 18176
This theorem is referenced by:  xmettpos  20580  metsym  20581  xmettri  20582  xmettri3  20584  xmetrtri2  20587  elbl3  20623  blss  20656  xmeter  20664  xmssym  20696  metcnp2  20773  metustsymOLD  20792  metdcnlem  21069  metdstri  21083  metdsle  21084  metdscn  21088  metnrmlem1  21091  metnrmlem3  21093  nmhmcn  21331  lmmbr2  21426  iscau2  21444  iscau3  21445  iscau4  21446  iscauf  21447  caucfil  21450  dvlip2  22124  nvlmle  25128  ubthlem1  25312  ubthlem2  25313  heicant  29477
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