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Theorem xmetsym 13501
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 997 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  D  e.  ( *Met `  X
) )
2 simp3 999 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  B  e.  X )
3 simp2 998 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  A  e.  X )
4 xmettri2 13494 . . . 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 1240 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  <_  (
( B D A ) +e ( B D B ) ) )
6 xmet0 13496 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X
)  ->  ( B D B )  =  0 )
763adant2 1016 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D B )  =  0 )
87oveq2d 5884 . . . 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 13485 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  B  e.  X  /\  A  e.  X
)  ->  ( B D A )  e.  RR* )
10 xaddid1 9836 . . . . . 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 1209 . . . 4  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
138, 12eqtrd 2210 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( B D A ) +e ( B D B ) )  =  ( B D A ) )
145, 13breqtrd 4026 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  <_  ( B D A ) )
15 xmettri2 13494 . . . 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 1240 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  <_  (
( A D B ) +e ( A D A ) ) )
17 xmet0 13496 . . . . . 6  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X
)  ->  ( A D A )  =  0 )
18173adant3 1017 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D A )  =  0 )
1918oveq2d 5884 . . . 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 13485 . . . . 5  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( A D B )  e.  RR* )
21 xaddid1 9836 . . . . 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 2210 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( ( A D B ) +e ( A D A ) )  =  ( A D B ) )
2416, 23breqtrd 4026 . 2  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  <_  ( A D B ) )
2593com23 1209 . . 3  |-  ( ( D  e.  ( *Met `  X )  /\  A  e.  X  /\  B  e.  X
)  ->  ( B D A )  e.  RR* )
26 xrletri3 9778 . . 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 411 . 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 944 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 104    <-> wb 105    /\ w3a 978    = wceq 1353    e. wcel 2148   class class class wbr 4000   ` cfv 5211  (class class class)co 5868   0cc0 7789   RR*cxr 7968    <_ cle 7970   +ecxad 9744   *Metcxmet 13113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429  ax-setind 4532  ax-cnex 7880  ax-resscn 7881  ax-1re 7883  ax-addrcl 7886  ax-0id 7897  ax-rnegex 7898  ax-pre-ltirr 7901  ax-pre-apti 7904
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4289  df-xp 4628  df-rel 4629  df-cnv 4630  df-co 4631  df-dm 4632  df-rn 4633  df-res 4634  df-ima 4635  df-iota 5173  df-fun 5213  df-fn 5214  df-f 5215  df-fv 5219  df-ov 5871  df-oprab 5872  df-mpo 5873  df-1st 6134  df-2nd 6135  df-map 6643  df-pnf 7971  df-mnf 7972  df-xr 7973  df-ltxr 7974  df-le 7975  df-xadd 9747  df-xmet 13121
This theorem is referenced by:  xmettpos  13503  metsym  13504  xmettri  13505  xmettri3  13507  elbl3  13528  blss  13561  xmeter  13569  xmssym  13602  metcnp2  13646
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