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Theorem xmetsym 15359
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 1024 . . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> D e. ( *Met ` X ) )
2 simp3 1026 . . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> B e. X )
3 simp2 1025 . . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> A e. X )
4 xmettri2 15352 . . . 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 1276 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( ( B D A ) +e ( B D B ) ) )
6 xmet0 15354 . . . . . 6 |- ( ( D e. ( *Met ` X ) /\ B e. X ) -> ( B D B ) = 0 )
763adant2 1043 . . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D B ) = 0 )
87oveq2d 6074 . . . 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 15343 . . . . . 6 |- ( ( D e. ( *Met ` X ) /\ B e. X /\ A e. X ) -> ( B D A ) e. RR* )
10 xaddid1 10214 . . . . . 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 1236 . . . 4 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e 0 ) = ( B D A ) )
138, 12eqtrd 2267 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e ( B D B ) ) = ( B D A ) )
145, 13breqtrd 4140 . 2 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( B D A ) )
15 xmettri2 15352 . . . 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 1276 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( ( A D B ) +e ( A D A ) ) )
17 xmet0 15354 . . . . . 6 |- ( ( D e. ( *Met ` X ) /\ A e. X ) -> ( A D A ) = 0 )
18173adant3 1044 . . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D A ) = 0 )
1918oveq2d 6074 . . . 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 15343 . . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
21 xaddid1 10214 . . . . 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 2267 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e ( A D A ) ) = ( A D B ) )
2416, 23breqtrd 4140 . 2 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( A D B ) )
2593com23 1236 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) e. RR* )
26 xrletri3 10156 . . 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 953 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 1005   = wceq 1398   e. wcel 2205   class class class wbr 4114   ` cfv 5357  (class class class)co 6058   0cc0 8143   RR*cxr 8323   <_ cle 8325   +ecxad 10122   *Metcxmet 14810
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1re 8237  ax-addrcl 8240  ax-0id 8251  ax-rnegex 8252  ax-pre-ltirr 8255  ax-pre-apti 8258
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-xadd 10125  df-xmet 14818
This theorem is referenced by:  xmettpos  15361  metsym  15362  xmettri  15363  xmettri3  15365  elbl3  15386  blss  15419  xmeter  15427  xmssym  15460  metcnp2  15504
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