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

Theorem xmstrkgc 23144
Description: Any metric space fulfills Tarski's geometry axioms of congruence. (Contributed by Thierry Arnoux, 13-Mar-2019.)
Assertion
Ref Expression
xmstrkgc  |-  ( G  e.  *MetSp  ->  G  e. TarskiGC )

Proof of Theorem xmstrkgc
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2443 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
2 eqid 2443 . . . . . 6  |-  ( dist `  G )  =  (
dist `  G )
31, 2xmssym 20052 . . . . 5  |-  ( ( G  e.  *MetSp  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( dist `  G
) y )  =  ( y ( dist `  G ) x ) )
433expb 1188 . . . 4  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G ) ) )  ->  ( x
( dist `  G )
y )  =  ( y ( dist `  G
) x ) )
54ralrimivva 2820 . . 3  |-  ( G  e.  *MetSp  ->  A. x  e.  ( Base `  G
) A. y  e.  ( Base `  G
) ( x (
dist `  G )
y )  =  ( y ( dist `  G
) x ) )
6 simpl 457 . . . . . . . 8  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  G  e.  *MetSp )
7 simpr3 996 . . . . . . . 8  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  z  e.  (
Base `  G )
)
8 equid 1729 . . . . . . . . 9  |-  z  =  z
91, 2xmseq0 20051 . . . . . . . . 9  |-  ( ( G  e.  *MetSp  /\  z  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
( z ( dist `  G ) z )  =  0  <->  z  =  z ) )
108, 9mpbiri 233 . . . . . . . 8  |-  ( ( G  e.  *MetSp  /\  z  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
z ( dist `  G
) z )  =  0 )
116, 7, 7, 10syl3anc 1218 . . . . . . 7  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  ( z (
dist `  G )
z )  =  0 )
1211eqeq2d 2454 . . . . . 6  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  ( ( x ( dist `  G
) y )  =  ( z ( dist `  G ) z )  <-> 
( x ( dist `  G ) y )  =  0 ) )
13 simpr1 994 . . . . . . 7  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  x  e.  (
Base `  G )
)
14 simpr2 995 . . . . . . 7  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  y  e.  (
Base `  G )
)
151, 2xmseq0 20051 . . . . . . 7  |-  ( ( G  e.  *MetSp  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
( x ( dist `  G ) y )  =  0  <->  x  =  y ) )
166, 13, 14, 15syl3anc 1218 . . . . . 6  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  ( ( x ( dist `  G
) y )  =  0  <->  x  =  y
) )
1712, 16bitrd 253 . . . . 5  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  ( ( x ( dist `  G
) y )  =  ( z ( dist `  G ) z )  <-> 
x  =  y ) )
1817biimpd 207 . . . 4  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  ( ( x ( dist `  G
) y )  =  ( z ( dist `  G ) z )  ->  x  =  y ) )
1918ralrimivvva 2821 . . 3  |-  ( G  e.  *MetSp  ->  A. x  e.  ( Base `  G
) A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( x ( dist `  G
) y )  =  ( z ( dist `  G ) z )  ->  x  =  y ) )
205, 19jca 532 . 2  |-  ( G  e.  *MetSp  ->  ( A. x  e.  ( Base `  G ) A. y  e.  ( Base `  G ) ( x ( dist `  G
) y )  =  ( y ( dist `  G ) x )  /\  A. x  e.  ( Base `  G
) A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( x ( dist `  G
) y )  =  ( z ( dist `  G ) z )  ->  x  =  y ) ) )
21 elex 2993 . . . 4  |-  ( G  e.  *MetSp  ->  G  e.  _V )
2221anim1i 568 . . 3  |-  ( ( G  e.  *MetSp  /\  ( A. x  e.  ( Base `  G
) A. y  e.  ( Base `  G
) ( x (
dist `  G )
y )  =  ( y ( dist `  G
) x )  /\  A. x  e.  ( Base `  G ) A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( x ( dist `  G
) y )  =  ( z ( dist `  G ) z )  ->  x  =  y ) ) )  -> 
( G  e.  _V  /\  ( A. x  e.  ( Base `  G
) A. y  e.  ( Base `  G
) ( x (
dist `  G )
y )  =  ( y ( dist `  G
) x )  /\  A. x  e.  ( Base `  G ) A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( x ( dist `  G
) y )  =  ( z ( dist `  G ) z )  ->  x  =  y ) ) ) )
23 eqid 2443 . . . 4  |-  (Itv `  G )  =  (Itv
`  G )
241, 2, 23istrkgc 22929 . . 3  |-  ( G  e. TarskiGC  <->  ( G  e.  _V  /\  ( A. x  e.  (
Base `  G ) A. y  e.  ( Base `  G ) ( x ( dist `  G
) y )  =  ( y ( dist `  G ) x )  /\  A. x  e.  ( Base `  G
) A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( x ( dist `  G
) y )  =  ( z ( dist `  G ) z )  ->  x  =  y ) ) ) )
2522, 24sylibr 212 . 2  |-  ( ( G  e.  *MetSp  /\  ( A. x  e.  ( Base `  G
) A. y  e.  ( Base `  G
) ( x (
dist `  G )
y )  =  ( y ( dist `  G
) x )  /\  A. x  e.  ( Base `  G ) A. y  e.  ( Base `  G
) A. z  e.  ( Base `  G
) ( ( x ( dist `  G
) y )  =  ( z ( dist `  G ) z )  ->  x  =  y ) ) )  ->  G  e. TarskiGC )
2620, 25mpdan 668 1  |-  ( G  e.  *MetSp  ->  G  e. TarskiGC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2727   _Vcvv 2984   ` cfv 5430  (class class class)co 6103   0cc0 9294   Basecbs 14186   distcds 14259   *MetSpcxme 19904  TarskiGCcstrkgc 22902  Itvcitv 22909
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 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384  ax-cnex 9350  ax-resscn 9351  ax-1cn 9352  ax-icn 9353  ax-addcl 9354  ax-addrcl 9355  ax-mulcl 9356  ax-mulrcl 9357  ax-mulcom 9358  ax-addass 9359  ax-mulass 9360  ax-distr 9361  ax-i2m1 9362  ax-1ne0 9363  ax-1rid 9364  ax-rnegex 9365  ax-rrecex 9366  ax-cnre 9367  ax-pre-lttri 9368  ax-pre-lttrn 9369  ax-pre-ltadd 9370  ax-pre-mulgt0 9371  ax-pre-sup 9372
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 2577  df-ne 2620  df-nel 2621  df-ral 2732  df-rex 2733  df-reu 2734  df-rmo 2735  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-riota 6064  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-1st 6589  df-2nd 6590  df-recs 6844  df-rdg 6878  df-er 7113  df-map 7228  df-en 7323  df-dom 7324  df-sdom 7325  df-sup 7703  df-pnf 9432  df-mnf 9433  df-xr 9434  df-ltxr 9435  df-le 9436  df-sub 9609  df-neg 9610  df-div 10006  df-nn 10335  df-2 10392  df-n0 10592  df-z 10659  df-uz 10874  df-q 10966  df-rp 11004  df-xneg 11101  df-xadd 11102  df-xmul 11103  df-topgen 14394  df-psmet 17821  df-xmet 17822  df-bl 17824  df-mopn 17825  df-top 18515  df-bases 18517  df-topon 18518  df-topsp 18519  df-xms 19907  df-trkgc 22921
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator