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Theorem xmetsym 12909
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 986 . . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> D e. ( *Met ` X ) )
2 simp3 988 . . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> B e. X )
3 simp2 987 . . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> A e. X )
4 xmettri2 12902 . . . 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 1229 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( ( B D A ) +e ( B D B ) ) )
6 xmet0 12904 . . . . . 6 |- ( ( D e. ( *Met ` X ) /\ B e. X ) -> ( B D B ) = 0 )
763adant2 1005 . . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D B ) = 0 )
87oveq2d 5852 . . . 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 12893 . . . . . 6 |- ( ( D e. ( *Met ` X ) /\ B e. X /\ A e. X ) -> ( B D A ) e. RR* )
10 xaddid1 9789 . . . . . 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 1198 . . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e 0 ) = ( B D A ) )
138, 12eqtrd 2197 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e ( B D B ) ) = ( B D A ) )
145, 13breqtrd 4002 . 2 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( B D A ) )
15 xmettri2 12902 . . . 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 1229 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( ( A D B ) +e ( A D A ) ) )
17 xmet0 12904 . . . . . 6 |- ( ( D e. ( *Met ` X ) /\ A e. X ) -> ( A D A ) = 0 )
18173adant3 1006 . . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D A ) = 0 )
1918oveq2d 5852 . . . 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 12893 . . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
21 xaddid1 9789 . . . . 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 2197 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e ( A D A ) ) = ( A D B ) )
2416, 23breqtrd 4002 . 2 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( A D B ) )
2593com23 1198 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) e. RR* )
26 xrletri3 9731 . . 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 933 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 967   = wceq 1342   e. wcel 2135   class class class wbr 3976   ` cfv 5182  (class class class)co 5836   0cc0 7744   RR*cxr 7923   <_ cle 7925   +ecxad 9697   *Metcxmet 12521
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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-1re 7838  ax-addrcl 7841  ax-0id 7852  ax-rnegex 7853  ax-pre-ltirr 7856  ax-pre-apti 7859
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-sbc 2947  df-csb 3041  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-if 3516  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-iun 3862  df-br 3977  df-opab 4038  df-mpt 4039  df-id 4265  df-xp 4604  df-rel 4605  df-cnv 4606  df-co 4607  df-dm 4608  df-rn 4609  df-res 4610  df-ima 4611  df-iota 5147  df-fun 5184  df-fn 5185  df-f 5186  df-fv 5190  df-ov 5839  df-oprab 5840  df-mpo 5841  df-1st 6100  df-2nd 6101  df-map 6607  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929  df-le 7930  df-xadd 9700  df-xmet 12529
This theorem is referenced by:  xmettpos  12911  metsym  12912  xmettri  12913  xmettri3  12915  elbl3  12936  blss  12969  xmeter  12977  xmssym  13010  metcnp2  13054
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