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Theorem eengv 24408
Description: The value of the Euclidean geometry for dimension  N. (Contributed by Thierry Arnoux, 15-Mar-2019.)
Assertion
Ref Expression
eengv  |-  ( 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
>. ) } ) >. } ) )
Distinct variable group:    x, i, y, z, N

Proof of Theorem eengv
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . . 5  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
21opeq2d 4226 . . . 4  |-  ( n  =  N  ->  <. ( Base `  ndx ) ,  ( EE `  n
) >.  =  <. ( Base `  ndx ) ,  ( EE `  N
) >. )
31adantr 465 . . . . . 6  |-  ( ( n  =  N  /\  x  e.  ( EE `  n ) )  -> 
( EE `  n
)  =  ( EE
`  N ) )
4 simpl 457 . . . . . . . 8  |-  ( ( n  =  N  /\  ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n ) ) )  ->  n  =  N )
54oveq2d 6312 . . . . . . 7  |-  ( ( n  =  N  /\  ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n ) ) )  ->  ( 1 ... n )  =  ( 1 ... N ) )
65sumeq1d 13534 . . . . . 6  |-  ( ( n  =  N  /\  ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n ) ) )  ->  sum_ i  e.  ( 1 ... n ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 )  =  sum_ i  e.  ( 1 ... N
) ( ( ( x `  i )  -  ( y `  i ) ) ^
2 ) )
71, 3, 6mpt2eq123dva 6357 . . . . 5  |-  ( n  =  N  ->  (
x  e.  ( EE
`  n ) ,  y  e.  ( EE
`  n )  |->  sum_ i  e.  ( 1 ... n ) ( ( ( x `  i )  -  (
y `  i )
) ^ 2 ) )  =  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N )  |->  sum_ i  e.  ( 1 ... N
) ( ( ( x `  i )  -  ( y `  i ) ) ^
2 ) ) )
87opeq2d 4226 . . . 4  |-  ( n  =  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 ) ) >. )
92, 8preq12d 4119 . . 3  |-  ( n  =  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 ) ) >. }  =  { <. ( Base `  ndx ) ,  ( EE `  N ) >. ,  <. (
dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. } )
101adantr 465 . . . . . . 7  |-  ( ( n  =  N  /\  ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n ) ) )  ->  ( EE `  n )  =  ( EE `  N ) )
11 biidd 237 . . . . . . 7  |-  ( ( ( n  =  N  /\  ( x  e.  ( EE `  n
)  /\  y  e.  ( EE `  n ) ) )  /\  z  e.  ( EE `  n
) )  ->  (
z  Btwn  <. x ,  y >.  <->  z  Btwn  <. x ,  y >. )
)
1210, 11rabeqbidva 3105 . . . . . 6  |-  ( ( n  =  N  /\  ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n ) ) )  ->  { z  e.  ( EE `  n
)  |  z  Btwn  <.
x ,  y >. }  =  { z  e.  ( EE `  N
)  |  z  Btwn  <.
x ,  y >. } )
131, 3, 12mpt2eq123dva 6357 . . . . 5  |-  ( n  =  N  ->  (
x  e.  ( EE
`  n ) ,  y  e.  ( EE
`  n )  |->  { z  e.  ( EE
`  n )  |  z  Btwn  <. x ,  y >. } )  =  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  { z  e.  ( EE `  N )  |  z  Btwn  <. x ,  y >. } ) )
1413opeq2d 4226 . . . 4  |-  ( n  =  N  ->  <. (Itv ` 
ndx ) ,  ( x  e.  ( EE
`  n ) ,  y  e.  ( EE
`  n )  |->  { z  e.  ( EE
`  n )  |  z  Btwn  <. x ,  y >. } ) >.  =  <. (Itv `  ndx ) ,  ( x  e.  ( EE `  N
) ,  y  e.  ( EE `  N
)  |->  { z  e.  ( EE `  N
)  |  z  Btwn  <.
x ,  y >. } ) >. )
153difeq1d 3617 . . . . . 6  |-  ( ( n  =  N  /\  x  e.  ( EE `  n ) )  -> 
( ( EE `  n )  \  {
x } )  =  ( ( EE `  N )  \  {
x } ) )
16 rabeq 3103 . . . . . . . 8  |-  ( ( EE `  n )  =  ( EE `  N )  ->  { z  e.  ( EE `  n )  |  ( z  Btwn  <. x ,  y >.  \/  x  Btwn  <. z ,  y
>.  \/  y  Btwn  <. x ,  z >. ) }  =  { z  e.  ( EE `  N
)  |  ( z 
Btwn  <. x ,  y
>.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } )
171, 16syl 16 . . . . . . 7  |-  ( n  =  N  ->  { z  e.  ( EE `  n )  |  ( z  Btwn  <. x ,  y >.  \/  x  Btwn  <. z ,  y
>.  \/  y  Btwn  <. x ,  z >. ) }  =  { z  e.  ( EE `  N
)  |  ( z 
Btwn  <. x ,  y
>.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } )
1817adantr 465 . . . . . 6  |-  ( ( n  =  N  /\  ( 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 >. ) }  =  { z  e.  ( EE `  N
)  |  ( z 
Btwn  <. x ,  y
>.  \/  x  Btwn  <. z ,  y >.  \/  y  Btwn  <. x ,  z
>. ) } )
191, 15, 18mpt2eq123dva 6357 . . . . 5  |-  ( n  =  N  ->  (
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
>. ) } )  =  ( 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 >. ) } ) )
2019opeq2d 4226 . . . 4  |-  ( n  =  N  ->  <. (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
>. ) } ) >.  =  <. (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 >. ) } ) >. )
2114, 20preq12d 4119 . . 3  |-  ( n  =  N  ->  { <. (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 >. ) } ) >. }  =  { <. (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
>. ) } ) >. } )
229, 21uneq12d 3655 . 2  |-  ( n  =  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
>. ) } ) >. } )  =  ( { <. ( 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
>. ) } ) >. } ) )
23 df-eeng 24407 . 2  |- EEG  =  ( n  e.  NN  |->  ( { <. ( 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
>. ) } ) >. } ) )
24 prex 4698 . . 3  |-  { <. (
Base `  ndx ) ,  ( EE `  N
) >. ,  <. ( dist `  ndx ) ,  ( x  e.  ( EE `  N ) ,  y  e.  ( EE `  N ) 
|->  sum_ i  e.  ( 1 ... N ) ( ( ( x `
 i )  -  ( y `  i
) ) ^ 2 ) ) >. }  e.  _V
25 prex 4698 . . 3  |-  { <. (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 >. ) } ) >. }  e.  _V
2624, 25unex 6597 . 2  |-  ( {
<. ( 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
>. ) } ) >. } )  e.  _V
2722, 23, 26fvmpt 5956 1  |-  ( 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
>. ) } ) >. } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    \/ w3o 972    = wceq 1395    e. wcel 1819   {crab 2811    \ cdif 3468    u. cun 3469   {csn 4032   {cpr 4034   <.cop 4038   class class class wbr 4456   ` cfv 5594  (class class class)co 6296    |-> cmpt2 6298   1c1 9510    - cmin 9824   NNcn 10556   2c2 10606   ...cfz 11697   ^cexp 12168   sum_csu 13519   ndxcnx 14640   Basecbs 14643   distcds 14720  Itvcitv 23957  LineGclng 23958   EEcee 24317    Btwn cbtwn 24318  EEGceeng 24406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-recs 7060  df-rdg 7094  df-seq 12110  df-sum 13520  df-eeng 24407
This theorem is referenced by:  eengstr  24409  eengbas  24410  ebtwntg  24411  ecgrtg  24412  elntg  24413
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