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Theorem psmettri 12258
Description: Triangle inequality for the distance function of a pseudometric space. (Contributed by Thierry Arnoux, 11-Feb-2018.)
Assertion
Ref Expression
psmettri  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  <_  (
( A D C ) +e ( C D B ) ) )

Proof of Theorem psmettri
StepHypRef Expression
1 simpl 108 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  D  e.  (PsMet `  X ) )
2 simpr3 957 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  C  e.  X )
3 simpr1 955 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  A  e.  X )
4 simpr2 956 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  B  e.  X )
5 psmettri2 12256 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( C  e.  X  /\  A  e.  X  /\  B  e.  X )
)  ->  ( A D B )  <_  (
( C D A ) +e ( C D B ) ) )
61, 2, 3, 4, 5syl13anc 1186 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  <_  (
( C D A ) +e ( C D B ) ) )
7 psmetsym 12257 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  C  e.  X  /\  A  e.  X )  ->  ( C D A )  =  ( A D C ) )
81, 2, 3, 7syl3anc 1184 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( C D A )  =  ( A D C ) )
98oveq1d 5721 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( ( C D A ) +e ( C D B ) )  =  ( ( A D C ) +e
( C D B ) ) )
106, 9breqtrd 3899 1  |-  ( ( D  e.  (PsMet `  X )  /\  ( A  e.  X  /\  B  e.  X  /\  C  e.  X )
)  ->  ( A D B )  <_  (
( A D C ) +e ( C D B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    /\ w3a 930    = wceq 1299    e. wcel 1448   class class class wbr 3875   ` cfv 5059  (class class class)co 5706    <_ 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: (None)
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