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Theorem eengv 23397
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 5802 . . . . 5  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
21opeq2d 4177 . . . 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 6219 . . . . . . 7  |-  ( ( n  =  N  /\  ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n ) ) )  ->  ( 1 ... n )  =  ( 1 ... N ) )
65sumeq1d 13299 . . . . . 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 6259 . . . . 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 4177 . . . 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 4073 . . 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 3074 . . . . . 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 6259 . . . . 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 4177 . . . 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 3584 . . . . . 6  |-  ( ( n  =  N  /\  x  e.  ( EE `  n ) )  -> 
( ( EE `  n )  \  {
x } )  =  ( ( EE `  N )  \  {
x } ) )
16 rabeq 3072 . . . . . . . 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 6259 . . . . 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 4177 . . . 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 4073 . . 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 3622 . 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 23396 . 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 4645 . . 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 4645 . . 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 6491 . 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 5886 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 1370    e. wcel 1758   {crab 2803    \ cdif 3436    u. cun 3437   {csn 3988   {cpr 3990   <.cop 3994   class class class wbr 4403   ` cfv 5529  (class class class)co 6203    |-> cmpt2 6205   1c1 9397    - cmin 9709   NNcn 10436   2c2 10485   ...cfz 11557   ^cexp 11985   sum_csu 13284   ndxcnx 14292   Basecbs 14295   distcds 14369  Itvcitv 23032  LineGclng 23033   EEcee 23306    Btwn cbtwn 23307  EEGceeng 23395
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-recs 6945  df-rdg 6979  df-seq 11927  df-sum 13285  df-eeng 23396
This theorem is referenced by:  eengstr  23398  eengbas  23399  ebtwntg  23400  ecgrtg  23401  elntg  23402
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