MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eengv Structured version   Unicode version

Theorem eengv 23974
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 5865 . . . . 5  |-  ( n  =  N  ->  ( EE `  n )  =  ( EE `  N
) )
21opeq2d 4220 . . . 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 6299 . . . . . . 7  |-  ( ( n  =  N  /\  ( x  e.  ( EE `  n )  /\  y  e.  ( EE `  n ) ) )  ->  ( 1 ... n )  =  ( 1 ... N ) )
65sumeq1d 13485 . . . . . 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 6341 . . . . 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 4220 . . . 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 4114 . . 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 3109 . . . . . 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 6341 . . . . 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 4220 . . . 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 3621 . . . . . 6  |-  ( ( n  =  N  /\  x  e.  ( EE `  n ) )  -> 
( ( EE `  n )  \  {
x } )  =  ( ( EE `  N )  \  {
x } ) )
16 rabeq 3107 . . . . . . . 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 6341 . . . . 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 4220 . . . 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 4114 . . 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 3659 . 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 23973 . 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 4689 . . 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 4689 . . 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 6581 . 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 5949 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 1379    e. wcel 1767   {crab 2818    \ cdif 3473    u. cun 3474   {csn 4027   {cpr 4029   <.cop 4033   class class class wbr 4447   ` cfv 5587  (class class class)co 6283    |-> cmpt2 6285   1c1 9492    - cmin 9804   NNcn 10535   2c2 10584   ...cfz 11671   ^cexp 12133   sum_csu 13470   ndxcnx 14486   Basecbs 14489   distcds 14563  Itvcitv 23576  LineGclng 23577   EEcee 23883    Btwn cbtwn 23884  EEGceeng 23972
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-recs 7042  df-rdg 7076  df-seq 12075  df-sum 13471  df-eeng 23973
This theorem is referenced by:  eengstr  23975  eengbas  23976  ebtwntg  23977  ecgrtg  23978  elntg  23979
  Copyright terms: Public domain W3C validator