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Theorem psmetsym 14314
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 999 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> D e. (PsMet ` X ) )
2 simp3 1001 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> B e. X )
3 simp2 1000 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> A e. X )
4 psmettri2 14313 . . . 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 1251 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( ( B D A ) +e ( B D B ) ) )
6 psmet0 14312 . . . . . 6 |- ( ( D e. (PsMet ` X ) /\ B e. X ) -> ( B D B ) = 0 )
763adant2 1018 . . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D B ) = 0 )
87oveq2d 5916 . . . 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 14311 . . . . . 6 |- ( ( D e. (PsMet ` X ) /\ B e. X /\ A e. X ) -> ( B D A ) e. RR* )
10 xaddid1 9898 . . . . . 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 1211 . . . 4 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e 0 ) = ( B D A ) )
138, 12eqtrd 2222 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( B D A ) +e ( B D B ) ) = ( B D A ) )
145, 13breqtrd 4047 . 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) <_ ( B D A ) )
15 psmettri2 14313 . . . 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 1251 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( ( A D B ) +e ( A D A ) ) )
17 psmet0 14312 . . . . . 6 |- ( ( D e. (PsMet ` X ) /\ A e. X ) -> ( A D A ) = 0 )
18173adant3 1019 . . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D A ) = 0 )
1918oveq2d 5916 . . . 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 14311 . . . . 5 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( A D B ) e. RR* )
21 xaddid1 9898 . . . . 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 2222 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( ( A D B ) +e ( A D A ) ) = ( A D B ) )
2416, 23breqtrd 4047 . 2 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) <_ ( A D B ) )
2593com23 1211 . . 3 |- ( ( D e. (PsMet ` X ) /\ A e. X /\ B e. X ) -> ( B D A ) e. RR* )
26 xrletri3 9840 . . 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. (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 946 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 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2160   class class class wbr 4021   ` cfv 5238  (class class class)co 5900   0cc0 7846   RR*cxr 8026   <_ cle 8028   +ecxad 9806  PsMetcpsmet 13873
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-setind 4557  ax-cnex 7937  ax-resscn 7938  ax-1re 7940  ax-addrcl 7943  ax-0id 7954  ax-rnegex 7955  ax-pre-ltirr 7958  ax-pre-apti 7961
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-nel 2456  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-if 3550  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-mpt 4084  df-id 4314  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-fv 5246  df-ov 5903  df-oprab 5904  df-mpo 5905  df-map 6680  df-pnf 8029  df-mnf 8030  df-xr 8031  df-ltxr 8032  df-le 8033  df-xadd 9809  df-psmet 13881
This theorem is referenced by:  psmettri  14315  distspace  14320  elbl3ps  14379  blssps  14412
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