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Theorem eengv 23176
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 5686 . . . . 5  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
21opeq2d 4061 . . . 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 6102 . . . . . . 7  |-  ( ( n  =  N  /\  ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n ) ) )  ->  ( 1 ... n )  =  ( 1 ... N ) )
65sumeq1d 13170 . . . . . 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 6142 . . . . 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 4061 . . . 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 3957 . . 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 ) ) >. } )
104, 1syl 16 . . . . . . 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 2963 . . . . . 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 6142 . . . . 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 4061 . . . 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 3468 . . . . . 6  |-  ( ( n  =  N  /\  x  e.  ( EE `  n ) )  -> 
( ( EE `  n )  \  {
x } )  =  ( ( EE `  N )  \  {
x } ) )
16 rabeq 2961 . . . . . . . 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 6142 . . . . 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 4061 . . . 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 3957 . . 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 3506 . 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 23175 . 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 4529 . . 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 4529 . . 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 6373 . 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 5769 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 964    = wceq 1369    e. wcel 1756   {crab 2714    \ cdif 3320    u. cun 3321   {csn 3872   {cpr 3874   <.cop 3878   class class class wbr 4287   ` cfv 5413  (class class class)co 6086    e. cmpt2 6088   1c1 9275    - cmin 9587   NNcn 10314   2c2 10363   ...cfz 11429   ^cexp 11857   sum_csu 13155   ndxcnx 14163   Basecbs 14166   distcds 14239  Itvcitv 22872  LineGclng 22873   EEcee 23085    Btwn cbtwn 23086  EEGceeng 23174
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 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-recs 6824  df-rdg 6858  df-seq 11799  df-sum 13156  df-eeng 23175
This theorem is referenced by:  eengstr  23177  eengbas  23178  ebtwntg  23179  ecgrtg  23180  elntg  23181
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