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

Theorem eqeefv 24071
Description: Two points are equal iff they agree in all dimensions. (Contributed by Scott Fenton, 10-Jun-2013.)
Assertion
Ref Expression
eqeefv  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
Distinct variable groups:    A, i    B, i    i, N

Proof of Theorem eqeefv
StepHypRef Expression
1 eleei 24065 . . 3  |-  ( A  e.  ( EE `  N )  ->  A : ( 1 ... N ) --> RR )
2 ffn 5717 . . 3  |-  ( A : ( 1 ... N ) --> RR  ->  A  Fn  ( 1 ... N ) )
31, 2syl 16 . 2  |-  ( A  e.  ( EE `  N )  ->  A  Fn  ( 1 ... N
) )
4 eleei 24065 . . 3  |-  ( B  e.  ( EE `  N )  ->  B : ( 1 ... N ) --> RR )
5 ffn 5717 . . 3  |-  ( B : ( 1 ... N ) --> RR  ->  B  Fn  ( 1 ... N ) )
64, 5syl 16 . 2  |-  ( B  e.  ( EE `  N )  ->  B  Fn  ( 1 ... N
) )
7 eqfnfv 5962 . 2  |-  ( ( A  Fn  ( 1 ... N )  /\  B  Fn  ( 1 ... N ) )  ->  ( A  =  B  <->  A. i  e.  ( 1 ... N ) ( A `  i
)  =  ( B `
 i ) ) )
83, 6, 7syl2an 477 1  |-  ( ( A  e.  ( EE
`  N )  /\  B  e.  ( EE `  N ) )  -> 
( A  =  B  <->  A. i  e.  (
1 ... N ) ( A `  i )  =  ( B `  i ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   A.wral 2791    Fn wfn 5569   -->wf 5570   ` cfv 5574  (class class class)co 6277   RRcr 9489   1c1 9491   ...cfz 11676   EEcee 24056
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573  ax-cnex 9546  ax-resscn 9547
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-map 7420  df-ee 24059
This theorem is referenced by:  eqeelen  24072  brbtwn2  24073  colinearalg  24078  axcgrid  24084  ax5seglem4  24100  ax5seglem5  24101  axbtwnid  24107  axeuclid  24131  axcontlem2  24133  axcontlem4  24135  axcontlem7  24138
  Copyright terms: Public domain W3C validator