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Theorem ecgrtg 24859
Description: The congruence relation used in the Tarski structure for the Euclidean geometry is the same as Cgr. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Hypotheses
Ref Expression
ecgrtg.1  |-  ( ph  ->  N  e.  NN )
ecgrtg.2  |-  P  =  ( Base `  (EEG `  N ) )
ecgrtg.3  |-  .-  =  ( dist `  (EEG `  N
) )
ecgrtg.a  |-  ( ph  ->  A  e.  P )
ecgrtg.b  |-  ( ph  ->  B  e.  P )
ecgrtg.c  |-  ( ph  ->  C  e.  P )
ecgrtg.d  |-  ( ph  ->  D  e.  P )
Assertion
Ref Expression
ecgrtg  |-  ( ph  ->  ( <. A ,  B >.Cgr
<. C ,  D >.  <->  ( A  .-  B )  =  ( C  .-  D
) ) )

Proof of Theorem ecgrtg
Dummy variables  x  i  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ecgrtg.a . . . 4  |-  ( ph  ->  A  e.  P )
2 ecgrtg.1 . . . . . 6  |-  ( ph  ->  N  e.  NN )
3 eengbas 24857 . . . . . 6  |-  ( N  e.  NN  ->  ( EE `  N )  =  ( Base `  (EEG `  N ) ) )
42, 3syl 17 . . . . 5  |-  ( ph  ->  ( EE `  N
)  =  ( Base `  (EEG `  N )
) )
5 ecgrtg.2 . . . . 5  |-  P  =  ( Base `  (EEG `  N ) )
64, 5syl6eqr 2488 . . . 4  |-  ( ph  ->  ( EE `  N
)  =  P )
71, 6eleqtrrd 2520 . . 3  |-  ( ph  ->  A  e.  ( EE
`  N ) )
8 ecgrtg.b . . . 4  |-  ( ph  ->  B  e.  P )
98, 6eleqtrrd 2520 . . 3  |-  ( ph  ->  B  e.  ( EE
`  N ) )
10 ecgrtg.c . . . 4  |-  ( ph  ->  C  e.  P )
1110, 6eleqtrrd 2520 . . 3  |-  ( ph  ->  C  e.  ( EE
`  N ) )
12 ecgrtg.d . . . 4  |-  ( ph  ->  D  e.  P )
1312, 6eleqtrrd 2520 . . 3  |-  ( ph  ->  D  e.  ( EE
`  N ) )
14 brcgr 24776 . . 3  |-  ( ( ( A  e.  ( EE `  N )  /\  B  e.  ( EE `  N ) )  /\  ( C  e.  ( EE `  N )  /\  D  e.  ( EE `  N
) ) )  -> 
( <. A ,  B >.Cgr
<. C ,  D >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) ) )
157, 9, 11, 13, 14syl22anc 1265 . 2  |-  ( ph  ->  ( <. A ,  B >.Cgr
<. C ,  D >.  <->  sum_ i  e.  ( 1 ... N ) ( ( ( A `  i )  -  ( B `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) ) )
16 dsid 15260 . . . . . . 7  |-  dist  = Slot  ( dist `  ndx )
17 fvex 5891 . . . . . . . 8  |-  (EEG `  N )  e.  _V
1817a1i 11 . . . . . . 7  |-  ( ph  ->  (EEG `  N )  e.  _V )
19 eengstr 24856 . . . . . . . . . 10  |-  ( N  e.  NN  ->  (EEG `  N ) Struct  <. 1 , ; 1 7 >. )
202, 19syl 17 . . . . . . . . 9  |-  ( ph  ->  (EEG `  N ) Struct  <.
1 , ; 1 7 >. )
21 isstruct 15094 . . . . . . . . . 10  |-  ( (EEG
`  N ) Struct  <. 1 , ; 1 7 >.  <->  ( (
1  e.  NN  /\ ; 1 7  e.  NN  /\  1  <_ ; 1
7 )  /\  Fun  ( (EEG `  N )  \  { (/) } )  /\  dom  (EEG `  N )  C_  ( 1 ...; 1 7 ) ) )
2221simp2bi 1021 . . . . . . . . 9  |-  ( (EEG
`  N ) Struct  <. 1 , ; 1 7 >.  ->  Fun  ( (EEG `  N )  \  { (/) } ) )
2320, 22syl 17 . . . . . . . 8  |-  ( ph  ->  Fun  ( (EEG `  N )  \  { (/)
} ) )
24 structcnvcnv 15095 . . . . . . . . . 10  |-  ( (EEG
`  N ) Struct  <. 1 , ; 1 7 >.  ->  `' `' (EEG `  N )  =  ( (EEG `  N )  \  { (/)
} ) )
2520, 24syl 17 . . . . . . . . 9  |-  ( ph  ->  `' `' (EEG `  N )  =  ( (EEG `  N )  \  { (/)
} ) )
2625funeqd 5622 . . . . . . . 8  |-  ( ph  ->  ( Fun  `' `' (EEG `  N )  <->  Fun  ( (EEG
`  N )  \  { (/) } ) ) )
2723, 26mpbird 235 . . . . . . 7  |-  ( ph  ->  Fun  `' `' (EEG
`  N ) )
28 opex 4686 . . . . . . . . . 10  |-  <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  _V
2928prid2 4112 . . . . . . . . 9  |-  <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }
30 elun1 3639 . . . . . . . . 9  |-  ( <.
( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  ->  <.
( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  ( { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  u.  {
<. (Itv `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  { z  e.  ( EE `  N )  |  z  Btwn  <. x ,  y >. } )
>. ,  <. (LineG `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( ( EE
`  N )  \  { x } ) 
|->  { z  e.  ( EE `  N )  |  ( z  Btwn  <.
x ,  y >.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } ) >. } ) )
3129, 30ax-mp 5 . . . . . . . 8  |-  <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  ( { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  u.  {
<. (Itv `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  { z  e.  ( EE `  N )  |  z  Btwn  <. x ,  y >. } )
>. ,  <. (LineG `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( ( EE
`  N )  \  { x } ) 
|->  { z  e.  ( EE `  N )  |  ( z  Btwn  <.
x ,  y >.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } ) >. } )
32 eengv 24855 . . . . . . . . 9  |-  ( N  e.  NN  ->  (EEG `  N )  =  ( { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  u.  {
<. (Itv `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  { z  e.  ( EE `  N )  |  z  Btwn  <. x ,  y >. } )
>. ,  <. (LineG `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( ( EE
`  N )  \  { x } ) 
|->  { z  e.  ( EE `  N )  |  ( z  Btwn  <.
x ,  y >.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } ) >. } ) )
332, 32syl 17 . . . . . . . 8  |-  ( ph  ->  (EEG `  N )  =  ( { <. (
Base `  ndx ) ,  ( EE `  N
) >. ,  <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  u.  {
<. (Itv `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  { z  e.  ( EE `  N )  |  z  Btwn  <. x ,  y >. } )
>. ,  <. (LineG `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( ( EE
`  N )  \  { x } ) 
|->  { z  e.  ( EE `  N )  |  ( z  Btwn  <.
x ,  y >.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } ) >. } ) )
3431, 33syl5eleqr 2524 . . . . . . 7  |-  ( ph  -> 
<. ( dist `  ndx ) ,  ( x  e.  ( EE `  N
) ,  y  e.  ( EE `  N
)  |->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >.  e.  (EEG
`  N ) )
35 fvex 5891 . . . . . . . . 9  |-  ( EE
`  N )  e. 
_V
3635, 35mpt2ex 6884 . . . . . . . 8  |-  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N )  |->  sum_ i  e.  ( 1 ... N
) ( ( ( x `  i )  -  ( y `  i ) ) ^
2 ) )  e. 
_V
3736a1i 11 . . . . . . 7  |-  ( ph  ->  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) )  e.  _V )
3816, 18, 27, 34, 37strfv2d 15118 . . . . . 6  |-  ( ph  ->  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) )  =  (
dist `  (EEG `  N
) ) )
39 ecgrtg.3 . . . . . 6  |-  .-  =  ( dist `  (EEG `  N
) )
4038, 39syl6reqr 2489 . . . . 5  |-  ( ph  ->  .-  =  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N )  |->  sum_ i  e.  ( 1 ... N
) ( ( ( x `  i )  -  ( y `  i ) ) ^
2 ) ) )
41 simplrl 768 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  x  =  A )
4241fveq1d 5883 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
x `  i )  =  ( A `  i ) )
43 simplrr 769 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  y  =  B )
4443fveq1d 5883 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
y `  i )  =  ( B `  i ) )
4542, 44oveq12d 6323 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( x `  i
)  -  ( y `
 i ) )  =  ( ( A `
 i )  -  ( B `  i ) ) )
4645oveq1d 6320 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  A  /\  y  =  B )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( ( x `  i )  -  (
y `  i )
) ^ 2 )  =  ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )
4746sumeq2dv 13747 . . . . 5  |-  ( (
ph  /\  ( x  =  A  /\  y  =  B ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( x `  i )  -  (
y `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 ) )
48 sumex 13732 . . . . . 6  |-  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 )  e.  _V
4948a1i 11 . . . . 5  |-  ( ph  -> 
sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  e.  _V )
5040, 47, 7, 9, 49ovmpt2d 6438 . . . 4  |-  ( ph  ->  ( A  .-  B
)  =  sum_ i  e.  ( 1 ... N
) ( ( ( A `  i )  -  ( B `  i ) ) ^
2 ) )
5150eqcomd 2437 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  ( A  .-  B ) )
52 simplrl 768 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  x  =  C )
5352fveq1d 5883 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
x `  i )  =  ( C `  i ) )
54 simplrr 769 . . . . . . . . 9  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  y  =  D )
5554fveq1d 5883 . . . . . . . 8  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
y `  i )  =  ( D `  i ) )
5653, 55oveq12d 6323 . . . . . . 7  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( x `  i
)  -  ( y `
 i ) )  =  ( ( C `
 i )  -  ( D `  i ) ) )
5756oveq1d 6320 . . . . . 6  |-  ( ( ( ph  /\  (
x  =  C  /\  y  =  D )
)  /\  i  e.  ( 1 ... N
) )  ->  (
( ( x `  i )  -  (
y `  i )
) ^ 2 )  =  ( ( ( C `  i )  -  ( D `  i ) ) ^
2 ) )
5857sumeq2dv 13747 . . . . 5  |-  ( (
ph  /\  ( x  =  C  /\  y  =  D ) )  ->  sum_ i  e.  ( 1 ... N ) ( ( ( x `  i )  -  (
y `  i )
) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 ) )
59 sumex 13732 . . . . . 6  |-  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( D `  i ) ) ^
2 )  e.  _V
6059a1i 11 . . . . 5  |-  ( ph  -> 
sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  e.  _V )
6140, 58, 11, 13, 60ovmpt2d 6438 . . . 4  |-  ( ph  ->  ( C  .-  D
)  =  sum_ i  e.  ( 1 ... N
) ( ( ( C `  i )  -  ( D `  i ) ) ^
2 ) )
6261eqcomd 2437 . . 3  |-  ( ph  -> 
sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  =  ( C  .-  D ) )
6351, 62eqeq12d 2451 . 2  |-  ( ph  ->  ( sum_ i  e.  ( 1 ... N ) ( ( ( A `
 i )  -  ( B `  i ) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N ) ( ( ( C `
 i )  -  ( D `  i ) ) ^ 2 )  <-> 
( A  .-  B
)  =  ( C 
.-  D ) ) )
6415, 63bitrd 256 1  |-  ( ph  ->  ( <. A ,  B >.Cgr
<. C ,  D >.  <->  ( A  .-  B )  =  ( C  .-  D
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    \/ w3o 981    /\ w3a 982    = wceq 1437    e. wcel 1870   {crab 2786   _Vcvv 3087    \ cdif 3439    u. cun 3440    C_ wss 3442   (/)c0 3767   {csn 4002   {cpr 4004   <.cop 4008   class class class wbr 4426   `'ccnv 4853   dom cdm 4854   Fun wfun 5595   ` cfv 5601  (class class class)co 6305    |-> cmpt2 6307   1c1 9539    <_ cle 9675    - cmin 9859   NNcn 10609   2c2 10659   7c7 10664  ;cdc 11051   ...cfz 11782   ^cexp 12269   sum_csu 13730   Struct cstr 15080   ndxcnx 15081   Basecbs 15084   distcds 15161  Itvcitv 24347  LineGclng 24348   EEcee 24764    Btwn cbtwn 24765  Cgrccgr 24766  EEGceeng 24853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-rep 4538  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-fal 1443  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-oadd 7194  df-er 7371  df-map 7482  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-nn 10610  df-2 10668  df-3 10669  df-4 10670  df-5 10671  df-6 10672  df-7 10673  df-8 10674  df-9 10675  df-10 10676  df-n0 10870  df-z 10938  df-dec 11052  df-uz 11160  df-fz 11783  df-seq 12211  df-sum 13731  df-struct 15086  df-ndx 15087  df-slot 15088  df-base 15089  df-ds 15174  df-itv 24349  df-lng 24350  df-ee 24767  df-cgr 24769  df-eeng 24854
This theorem is referenced by:  eengtrkg  24861
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