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Theorem psmetsym 12257
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 949 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> D e. (PsMet ` X ) )
2 simp3 951 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> B e. X )
3 simp2 950 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> A e. X )
4 psmettri2 12256 . . . 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 1186 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( ( B D A ) +e ( B D B ) ) )
6 psmet0 12255 . . . . . 6 |- ( ( D e. (PsMet ` X ) /\ B e. X ) -> ( B D B ) = 0 )
763adant2 968 . . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D B ) = 0 )
87oveq2d 5722 . . . 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 12254 . . . . . 6 |- ( ( D e. (PsMet ` X ) /\ B e. X /\ A e. X ) -> ( B D A ) e. RR* )
10 xaddid1 9486 . . . . . 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 1155 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e 0 ) = ( B D A ) )
138, 12eqtrd 2132 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e ( B D B ) ) = ( B D A ) )
145, 13breqtrd 3899 . 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( B D A ) )
15 psmettri2 12256 . . . 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 1186 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( ( A D B ) +e ( A D A ) ) )
17 psmet0 12255 . . . . . 6 |- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )
18173adant3 969 . . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D A ) = 0 )
1918oveq2d 5722 . . . 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 12254 . . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
21 xaddid1 9486 . . . . 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 2132 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e ( A D A ) ) = ( A D B ) )
2416, 23breqtrd 3899 . 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( A D B ) )
2593com23 1155 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) e. RR* )
26 xrletri3 9429 . . 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 406 . 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 896 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 930   = wceq 1299   e. wcel 1448   class class class wbr 3875   ` cfv 5059  (class class class)co 5706   0cc0 7500   RR*cxr 7671   <_ cle 7673   +ecxad 9398  PsMetcpsmet 11930
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1re 7589  ax-addrcl 7592  ax-0id 7603  ax-rnegex 7604  ax-pre-ltirr 7607  ax-pre-apti 7610
This theorem depends on definitions:  df-bi 116  df-dc 787  df-3or 931  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-rab 2384  df-v 2643  df-sbc 2863  df-csb 2956  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-if 3422  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-mpt 3931  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-rn 4488  df-res 4489  df-ima 4490  df-iota 5024  df-fun 5061  df-fn 5062  df-f 5063  df-fv 5067  df-ov 5709  df-oprab 5710  df-mpo 5711  df-map 6474  df-pnf 7674  df-mnf 7675  df-xr 7676  df-ltxr 7677  df-le 7678  df-xadd 9401  df-psmet 11938
This theorem is referenced by:  psmettri  12258  distspace  12263  elbl3ps  12322  blssps  12355
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