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Theorem psmetsym 20577
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 996 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  D  e.  (PsMet `  X )
)
2 simp3 998 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  B  e.  X )
3 simp2 997 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  A  e.  X )
4 psmettri2 20576 . . . 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 1230 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  <_ 
( ( B D A ) +e
( B D B ) ) )
6 psmet0 20575 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  B  e.  X )  ->  ( B D B )  =  0 )
763adant2 1015 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( B D B )  =  0 )
87oveq2d 6300 . . . 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 20574 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  B  e.  X  /\  A  e.  X )  ->  ( B D A )  e. 
RR* )
10 xaddid1 11438 . . . . . 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 1202 . . . 4  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  (
( B D A ) +e 0 )  =  ( B D A ) )
138, 12eqtrd 2508 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  (
( B D A ) +e ( B D B ) )  =  ( B D A ) )
145, 13breqtrd 4471 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  <_ 
( B D A ) )
15 psmettri2 20576 . . . 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 1230 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( B D A )  <_ 
( ( A D B ) +e
( A D A ) ) )
17 psmet0 20575 . . . . . 6  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X )  ->  ( A D A )  =  0 )
18173adant3 1016 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D A )  =  0 )
1918oveq2d 6300 . . . 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 20574 . . . . 5  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( A D B )  e. 
RR* )
21 xaddid1 11438 . . . . 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 2508 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  (
( A D B ) +e ( A D A ) )  =  ( A D B ) )
2416, 23breqtrd 4471 . 2  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( B D A )  <_ 
( A D B ) )
2593com23 1202 . . 3  |-  ( ( D  e.  (PsMet `  X )  /\  A  e.  X  /\  B  e.  X )  ->  ( B D A )  e. 
RR* )
26 xrletri3 11358 . . 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 920 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5588  (class class class)co 6284   0cc0 9492   RR*cxr 9627    <_ cle 9629   +ecxad 11316  PsMetcpsmet 18201
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6576  ax-cnex 9548  ax-resscn 9549  ax-1cn 9550  ax-icn 9551  ax-addcl 9552  ax-addrcl 9553  ax-mulcl 9554  ax-mulrcl 9555  ax-mulcom 9556  ax-addass 9557  ax-mulass 9558  ax-distr 9559  ax-i2m1 9560  ax-1ne0 9561  ax-1rid 9562  ax-rnegex 9563  ax-rrecex 9564  ax-cnre 9565  ax-pre-lttri 9566  ax-pre-lttrn 9567  ax-pre-ltadd 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-er 7311  df-map 7422  df-en 7517  df-dom 7518  df-sdom 7519  df-pnf 9630  df-mnf 9631  df-xr 9632  df-ltxr 9633  df-le 9634  df-xadd 11319  df-psmet 18210
This theorem is referenced by:  psmettri  20578  elbl3ps  20657  blssps  20690  metustsym  20828  metideq  27536  metider  27537
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