Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prodex Unicode version

Theorem prodex 25304
Description: A finite composite is a set. (Contributed by FL, 19-Sep-2011.) (Revised by Mario Carneiro, 26-May-2014.)
Assertion
Ref Expression
prodex  |-  prod_ k  e.  A G B  e. 
_V

Proof of Theorem prodex
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prod 25299 . 2  |-  prod_ k  e.  A G B  =  if ( A  =  (/) ,  (GId `  G
) ,  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) ) } )
2 fvex 5539 . . 3  |-  (GId `  G )  e.  _V
3 2rexuz 10271 . . . . 5  |-  ( E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) )  <->  E. m  e.  ZZ  E. n  e.  ZZ  (
m  <_  n  /\  ( A  =  (
m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) ) ) )
43abbii 2395 . . . 4  |-  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) ) }  =  { x  |  E. m  e.  ZZ  E. n  e.  ZZ  (
m  <_  n  /\  ( A  =  (
m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) ) ) }
5 zex 10033 . . . . 5  |-  ZZ  e.  _V
6 fvex 5539 . . . . . 6  |-  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n )  e.  _V
7 simprr 733 . . . . . . 7  |-  ( ( m  <_  n  /\  ( A  =  (
m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) ) )  ->  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) )
87abssi 3248 . . . . . 6  |-  { x  |  ( m  <_  n  /\  ( A  =  ( m ... n
)  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) ) ) }  C_  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n )
96, 8ssexi 4159 . . . . 5  |-  { x  |  ( m  <_  n  /\  ( A  =  ( m ... n
)  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) ) ) }  e.  _V
105, 5, 9ab2rexex2 6001 . . . 4  |-  { x  |  E. m  e.  ZZ  E. n  e.  ZZ  (
m  <_  n  /\  ( A  =  (
m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) ) ) }  e.  _V
114, 10eqeltri 2353 . . 3  |-  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  B ) ) `
 n ) ) }  e.  _V
122, 11ifex 3623 . 2  |-  if ( A  =  (/) ,  (GId
`  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) ) } )  e.  _V
131, 12eqeltri 2353 1  |-  prod_ k  e.  A G B  e. 
_V
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   (/)c0 3455   ifcif 3565   class class class wbr 4023    e. cmpt 4077   ` cfv 5255  (class class class)co 5858    <_ cle 8868   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046  GIdcgi 20854   prod_cprd 25298
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-neg 9040  df-z 10025  df-uz 10231  df-prod 25299
  Copyright terms: Public domain W3C validator