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Theorem xmstrkgc 24054
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 elex 3102 . 2  |-  ( G  e.  *MetSp  ->  G  e.  _V )
2 eqid 2441 . . . . . 6  |-  ( Base `  G )  =  (
Base `  G )
3 eqid 2441 . . . . . 6  |-  ( dist `  G )  =  (
dist `  G )
42, 3xmssym 20834 . . . . 5  |-  ( ( G  e.  *MetSp  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
x ( dist `  G
) y )  =  ( y ( dist `  G ) x ) )
543expb 1196 . . . 4  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G ) ) )  ->  ( x
( dist `  G )
y )  =  ( y ( dist `  G
) x ) )
65ralrimivva 2862 . . 3  |-  ( G  e.  *MetSp  ->  A. x  e.  ( Base `  G
) A. y  e.  ( Base `  G
) ( x (
dist `  G )
y )  =  ( y ( dist `  G
) x ) )
7 simpl 457 . . . . . . . 8  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  G  e.  *MetSp )
8 simpr3 1003 . . . . . . . 8  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  z  e.  (
Base `  G )
)
9 equid 1775 . . . . . . . . 9  |-  z  =  z
102, 3xmseq0 20833 . . . . . . . . 9  |-  ( ( G  e.  *MetSp  /\  z  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
( z ( dist `  G ) z )  =  0  <->  z  =  z ) )
119, 10mpbiri 233 . . . . . . . 8  |-  ( ( G  e.  *MetSp  /\  z  e.  ( Base `  G )  /\  z  e.  ( Base `  G
) )  ->  (
z ( dist `  G
) z )  =  0 )
127, 8, 8, 11syl3anc 1227 . . . . . . 7  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  ( z (
dist `  G )
z )  =  0 )
1312eqeq2d 2455 . . . . . 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 ) )
142, 3xmseq0 20833 . . . . . . 7  |-  ( ( G  e.  *MetSp  /\  x  e.  ( Base `  G )  /\  y  e.  ( Base `  G
) )  ->  (
( x ( dist `  G ) y )  =  0  <->  x  =  y ) )
15143adant3r3 1206 . . . . . 6  |-  ( ( G  e.  *MetSp  /\  ( x  e.  (
Base `  G )  /\  y  e.  ( Base `  G )  /\  z  e.  ( Base `  G ) ) )  ->  ( ( x ( dist `  G
) y )  =  0  <->  x  =  y
) )
1613, 15bitrd 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 ) )
1716biimpd 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 ) )
1817ralrimivvva 2863 . . 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 ) )
196, 18jca 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 ) ) )
20 eqid 2441 . . 3  |-  (Itv `  G )  =  (Itv
`  G )
212, 3, 20istrkgc 23716 . 2  |-  ( 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 ) ) ) )
221, 19, 21sylanbrc 664 1  |-  ( G  e.  *MetSp  ->  G  e. TarskiGC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   _Vcvv 3093   ` cfv 5574  (class class class)co 6277   0cc0 9490   Basecbs 14504   distcds 14578   *MetSpcxme 20686  TarskiGCcstrkgc 23691  Itvcitv 23697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547  ax-1cn 9548  ax-icn 9549  ax-addcl 9550  ax-addrcl 9551  ax-mulcl 9552  ax-mulrcl 9553  ax-mulcom 9554  ax-addass 9555  ax-mulass 9556  ax-distr 9557  ax-i2m1 9558  ax-1ne0 9559  ax-1rid 9560  ax-rnegex 9561  ax-rrecex 9562  ax-cnre 9563  ax-pre-lttri 9564  ax-pre-lttrn 9565  ax-pre-ltadd 9566  ax-pre-mulgt0 9567  ax-pre-sup 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-1st 6781  df-2nd 6782  df-recs 7040  df-rdg 7074  df-er 7309  df-map 7420  df-en 7515  df-dom 7516  df-sdom 7517  df-sup 7899  df-pnf 9628  df-mnf 9629  df-xr 9630  df-ltxr 9631  df-le 9632  df-sub 9807  df-neg 9808  df-div 10208  df-nn 10538  df-2 10595  df-n0 10797  df-z 10866  df-uz 11086  df-q 11187  df-rp 11225  df-xneg 11322  df-xadd 11323  df-xmul 11324  df-topgen 14713  df-psmet 18279  df-xmet 18280  df-bl 18282  df-mopn 18283  df-top 19266  df-bases 19268  df-topon 19269  df-topsp 19270  df-xms 20689  df-trkgc 23709
This theorem is referenced by: (None)
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