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Theorem xmetsym 15179
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 15172 . . . 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 15174 . . . . . 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 6044 . . . 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 15163 . . . . . 6 |- ( ( D e. ( *Met ` X ) /\ B e. X /\ A e. X ) -> ( B D A ) e. RR* )
10 xaddid1 10158 . . . . . 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 2264 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e ( B D B ) ) = ( B D A ) )
145, 13breqtrd 4119 . 2 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( B D A ) )
15 xmettri2 15172 . . . 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 15174 . . . . . 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 6044 . . . 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 15163 . . . . 5 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
21 xaddid1 10158 . . . . 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 2264 . . 3 |- ( ( D e. ( *Met ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e ( A D A ) ) = ( A D B ) )
2416, 23breqtrd 4119 . 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 10100 . . 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 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   0cc0 8092   RR*cxr 8272   <_ cle 8274   +ecxad 10066   *Metcxmet 14632
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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8183  ax-resscn 8184  ax-1re 8186  ax-addrcl 8189  ax-0id 8200  ax-rnegex 8201  ax-pre-ltirr 8204  ax-pre-apti 8207
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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fv 5341  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-map 6862  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-xadd 10069  df-xmet 14640
This theorem is referenced by:  xmettpos  15181  metsym  15182  xmettri  15183  xmettri3  15185  elbl3  15206  blss  15239  xmeter  15247  xmssym  15280  metcnp2  15324
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