MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  psmetsym Structured version   Unicode version

Theorem psmetsym 19891
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 988 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  D  e.  (PsMet `  X )
)
2 simp3 990 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
3 simp2 989 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
4 psmettri2 19890 . . . 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 1220 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  <_ 
( ( B D A ) +e
( B D B ) ) )
6 psmet0 19889 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  B  e.  X )  ->  ( B D B )  =  0 )
763adant2 1007 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( B D B )  =  0 )
87oveq2d 6112 . . . 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 19888 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  B  e.  X  /\  A  e.  X )  ->  ( B D A )  e. 
RR* )
10 xaddid1 11214 . . . . . 6  |-  ( ( B D A )  e.  RR*  ->  ( ( B D A ) +e 0 )  =  ( B D A ) )
119, 10syl 16 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  B  e.  X  /\  A  e.  X )  ->  (
( B D A ) +e 0 )  =  ( B D A ) )
12113com23 1193 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  (
( B D A ) +e 0 )  =  ( B D A ) )
138, 12eqtrd 2475 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  (
( B D A ) +e ( B D B ) )  =  ( B D A ) )
145, 13breqtrd 4321 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  <_ 
( B D A ) )
15 psmettri2 19890 . . . 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 1220 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( B D A )  <_ 
( ( A D B ) +e
( A D A ) ) )
17 psmet0 19889 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X )  ->  ( A D A )  =  0 )
18173adant3 1008 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D A )  =  0 )
1918oveq2d 6112 . . . 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 19888 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e. 
RR* )
21 xaddid1 11214 . . . . 5  |-  ( ( A D B )  e.  RR*  ->  ( ( A D B ) +e 0 )  =  ( A D B ) )
2220, 21syl 16 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) +e 0 )  =  ( A D B ) )
2319, 22eqtrd 2475 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) +e ( A D A ) )  =  ( A D B ) )
2416, 23breqtrd 4321 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( B D A )  <_ 
( A D B ) )
2593com23 1193 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( B D A )  e. 
RR* )
26 xrletri3 11134 . . 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 661 . 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 913 1  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  =  ( B D A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4297   ` cfv 5423  (class class class)co 6096   0cc0 9287   RR*cxr 9422    <_ cle 9424   +ecxad 11092  PsMetcpsmet 17805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377  ax-cnex 9343  ax-resscn 9344  ax-1cn 9345  ax-icn 9346  ax-addcl 9347  ax-addrcl 9348  ax-mulcl 9349  ax-mulrcl 9350  ax-mulcom 9351  ax-addass 9352  ax-mulass 9353  ax-distr 9354  ax-i2m1 9355  ax-1ne0 9356  ax-1rid 9357  ax-rnegex 9358  ax-rrecex 9359  ax-cnre 9360  ax-pre-lttri 9361  ax-pre-lttrn 9362  ax-pre-ltadd 9363
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-nel 2614  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-op 3889  df-uni 4097  df-br 4298  df-opab 4356  df-mpt 4357  df-id 4641  df-po 4646  df-so 4647  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-er 7106  df-map 7221  df-en 7316  df-dom 7317  df-sdom 7318  df-pnf 9425  df-mnf 9426  df-xr 9427  df-ltxr 9428  df-le 9429  df-xadd 11095  df-psmet 17814
This theorem is referenced by:  psmettri  19892  elbl3ps  19971  blssps  20004  metustsym  20142  metideq  26325  metider  26326
  Copyright terms: Public domain W3C validator