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Theorem psmetsym 12498
Description: The distance function of a pseudometric is symmetrical. (Contributed by Thierry Arnoux, 7-Feb-2018.)
Assertion
Ref Expression
psmetsym |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) = ( B D A ) )
Proof of Theorem psmetsym
StepHypRef Expression
1 simp1 981 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> D e. (PsMet ` X ) )
2 simp3 983 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> B e. X )
3 simp2 982 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> A e. X )
4 psmettri2 12497 . . . 4 |- ( ( D e. (PsMet ` 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 1218 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( ( B D A ) +e ( B D B ) ) )
6 psmet0 12496 . . . . . 6 |- ( ( D e. (PsMet ` X ) /\ B e. X ) -> ( B D B ) = 0 )
763adant2 1000 . . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D B ) = 0 )
87oveq2d 5790 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e ( B D B ) ) = ( ( B D A ) +e 0 ) )
9 psmetcl 12495 . . . . . 6 |- ( ( D e. (PsMet ` X ) /\ B e. X /\ A e. X ) -> ( B D A ) e. RR* )
10 xaddid1 9645 . . . . . 6 |- ( ( B D A ) e. RR* -> ( ( B D A ) +e 0 ) = ( B D A ) )
119, 10syl 14 . . . . 5 |- ( ( D e. (PsMet ` X ) /\ B e. X /\ A e. X ) -> ( ( B D A ) +e 0 ) = ( B D A ) )
12113com23 1187 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e 0 ) = ( B D A ) )
138, 12eqtrd 2172 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e ( B D B ) ) = ( B D A ) )
145, 13breqtrd 3954 . 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( B D A ) )
15 psmettri2 12497 . . . 4 |- ( ( D e. (PsMet ` 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 1218 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( ( A D B ) +e ( A D A ) ) )
17 psmet0 12496 . . . . . 6 |- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )
18173adant3 1001 . . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D A ) = 0 )
1918oveq2d 5790 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e ( A D A ) ) = ( ( A D B ) +e 0 ) )
20 psmetcl 12495 . . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
21 xaddid1 9645 . . . . 5 |- ( ( A D B ) e. RR* -> ( ( A D B ) +e 0 ) = ( A D B ) )
2220, 21syl 14 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e 0 ) = ( A D B ) )
2319, 22eqtrd 2172 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e ( A D A ) ) = ( A D B ) )
2416, 23breqtrd 3954 . 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( A D B ) )
2593com23 1187 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) e. RR* )
26 xrletri3 9588 . . 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 408 . 2 |- ( ( D e. (PsMet ` 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 928 1 |- ( ( D e. (PsMet ` 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 962   = wceq 1331   e. wcel 1480   class class class wbr 3929   ` cfv 5123  (class class class)co 5774   0cc0 7620   RR*cxr 7799   <_ cle 7801   +ecxad 9557  PsMetcpsmet 12148
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-1re 7714  ax-addrcl 7717  ax-0id 7728  ax-rnegex 7729  ax-pre-ltirr 7732  ax-pre-apti 7735
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-if 3475  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-map 6544  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-xadd 9560  df-psmet 12156
This theorem is referenced by:  psmettri  12499  distspace  12504  elbl3ps  12563  blssps  12596
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