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Theorem dffprod 25319
Description: Special case of composite over a finite index set. (Contributed by FL, 5-Sep-2010.) (Proof shortened by Mario Carneiro, 26-May-2014.)
Assertion
Ref Expression
dffprod  |-  ( N  e.  ( ZZ>= `  M
)  ->  prod_ k  e.  ( M ... N
) G A  =  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N ) )

Proof of Theorem dffprod
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-prod 25299 . . 3  |-  prod_ k  e.  ( M ... N
) G A  =  if ( ( M ... N )  =  (/) ,  (GId `  G
) ,  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( ( M ... N )  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) } )
2 eluzfz1 10803 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ( M ... N ) )
3 n0i 3460 . . . 4  |-  ( M  e.  ( M ... N )  ->  -.  ( M ... N )  =  (/) )
4 iffalse 3572 . . . 4  |-  ( -.  ( M ... N
)  =  (/)  ->  if ( ( M ... N )  =  (/) ,  (GId `  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n ) ) } )  =  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( ( M ... N )  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) } )
52, 3, 43syl 18 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  if (
( M ... N
)  =  (/) ,  (GId
`  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n ) ) } )  =  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( ( M ... N )  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) } )
61, 5syl5eq 2327 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  prod_ k  e.  ( M ... N
) G A  =  { x  |  E. m E. n  e.  (
ZZ>= `  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n ) ) } )
7 fzopth 10828 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( m ... n
)  <->  ( M  =  m  /\  N  =  n ) ) )
8 simpl 443 . . . . . . . . . . . 12  |-  ( ( M  =  m  /\  N  =  n )  ->  M  =  m )
98seqeq1d 11052 . . . . . . . . . . 11  |-  ( ( M  =  m  /\  N  =  n )  ->  seq  M ( G ,  ( k  e. 
_V  |->  A ) )  =  seq  m ( G ,  ( k  e.  _V  |->  A ) ) )
10 simpr 447 . . . . . . . . . . 11  |-  ( ( M  =  m  /\  N  =  n )  ->  N  =  n )
119, 10fveq12d 5531 . . . . . . . . . 10  |-  ( ( M  =  m  /\  N  =  n )  ->  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N )  =  (  seq  m
( G ,  ( k  e.  _V  |->  A ) ) `  n
) )
1211eleq2d 2350 . . . . . . . . 9  |-  ( ( M  =  m  /\  N  =  n )  ->  ( x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 N )  <->  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) )
1312biimprd 214 . . . . . . . 8  |-  ( ( M  =  m  /\  N  =  n )  ->  ( x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  A ) ) `
 n )  ->  x  e.  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N
) ) )
147, 13syl6bi 219 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( ( M ... N )  =  ( m ... n
)  ->  ( x  e.  (  seq  m ( G ,  ( k  e.  _V  |->  A ) ) `  n )  ->  x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 N ) ) ) )
1514imp3a 420 . . . . . 6  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( (
( M ... N
)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n ) )  ->  x  e.  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N
) ) )
1615rexlimdvw 2670 . . . . 5  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( E. n  e.  ( ZZ>= `  m ) ( ( M ... N )  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e.  _V  |->  A ) ) `  n ) )  ->  x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 N ) ) )
1716exlimdv 1664 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( E. m E. n  e.  (
ZZ>= `  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n ) )  ->  x  e.  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N
) ) )
18 eluzel2 10235 . . . . . . 7  |-  ( N  e.  ( ZZ>= `  M
)  ->  M  e.  ZZ )
1918adantr 451 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  x  e.  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N ) )  ->  M  e.  ZZ )
20 oveq2 5866 . . . . . . . . . 10  |-  ( n  =  N  ->  ( M ... n )  =  ( M ... N
) )
2120eqcomd 2288 . . . . . . . . 9  |-  ( n  =  N  ->  ( M ... N )  =  ( M ... n
) )
2221biantrurd 494 . . . . . . . 8  |-  ( n  =  N  ->  (
x  e.  (  seq 
M ( G , 
( k  e.  _V  |->  A ) ) `  n )  <->  ( ( M ... N )  =  ( M ... n
)  /\  x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) ) )
23 fveq2 5525 . . . . . . . . 9  |-  ( n  =  N  ->  (  seq  M ( G , 
( k  e.  _V  |->  A ) ) `  n )  =  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 N ) )
2423eleq2d 2350 . . . . . . . 8  |-  ( n  =  N  ->  (
x  e.  (  seq 
M ( G , 
( k  e.  _V  |->  A ) ) `  n )  <->  x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 N ) ) )
2522, 24bitr3d 246 . . . . . . 7  |-  ( n  =  N  ->  (
( ( M ... N )  =  ( M ... n )  /\  x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) )  <-> 
x  e.  (  seq 
M ( G , 
( k  e.  _V  |->  A ) ) `  N ) ) )
2625rspcev 2884 . . . . . 6  |-  ( ( N  e.  ( ZZ>= `  M )  /\  x  e.  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N ) )  ->  E. n  e.  ( ZZ>= `  M )
( ( M ... N )  =  ( M ... n )  /\  x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) )
27 fveq2 5525 . . . . . . . 8  |-  ( m  =  M  ->  ( ZZ>=
`  m )  =  ( ZZ>= `  M )
)
28 oveq1 5865 . . . . . . . . . 10  |-  ( m  =  M  ->  (
m ... n )  =  ( M ... n
) )
2928eqeq2d 2294 . . . . . . . . 9  |-  ( m  =  M  ->  (
( M ... N
)  =  ( m ... n )  <->  ( M ... N )  =  ( M ... n ) ) )
30 seqeq1 11049 . . . . . . . . . . 11  |-  ( m  =  M  ->  seq  m ( G , 
( k  e.  _V  |->  A ) )  =  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) )
3130fveq1d 5527 . . . . . . . . . 10  |-  ( m  =  M  ->  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n )  =  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) )
3231eleq2d 2350 . . . . . . . . 9  |-  ( m  =  M  ->  (
x  e.  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n )  <->  x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) )
3329, 32anbi12d 691 . . . . . . . 8  |-  ( m  =  M  ->  (
( ( M ... N )  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) )  <-> 
( ( M ... N )  =  ( M ... n )  /\  x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) ) )
3427, 33rexeqbidv 2749 . . . . . . 7  |-  ( m  =  M  ->  ( E. n  e.  ( ZZ>=
`  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n ) )  <->  E. n  e.  ( ZZ>= `  M )
( ( M ... N )  =  ( M ... n )  /\  x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) ) )
3534spcegv 2869 . . . . . 6  |-  ( M  e.  ZZ  ->  ( E. n  e.  ( ZZ>=
`  M ) ( ( M ... N
)  =  ( M ... n )  /\  x  e.  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  n
) )  ->  E. m E. n  e.  ( ZZ>=
`  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n ) ) ) )
3619, 26, 35sylc 56 . . . . 5  |-  ( ( N  e.  ( ZZ>= `  M )  /\  x  e.  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N ) )  ->  E. m E. n  e.  ( ZZ>=
`  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n ) ) )
3736ex 423 . . . 4  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( x  e.  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N )  ->  E. m E. n  e.  ( ZZ>= `  m )
( ( M ... N )  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) ) )
3817, 37impbid 183 . . 3  |-  ( N  e.  ( ZZ>= `  M
)  ->  ( E. m E. n  e.  (
ZZ>= `  m ) ( ( M ... N
)  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  A ) ) `  n ) )  <->  x  e.  (  seq  M ( G ,  ( k  e. 
_V  |->  A ) ) `
 N ) ) )
3938abbi1dv 2399 . 2  |-  ( N  e.  ( ZZ>= `  M
)  ->  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( ( M ... N )  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  A ) ) `
 n ) ) }  =  (  seq 
M ( G , 
( k  e.  _V  |->  A ) ) `  N ) )
406, 39eqtrd 2315 1  |-  ( N  e.  ( ZZ>= `  M
)  ->  prod_ k  e.  ( M ... N
) G A  =  (  seq  M ( G ,  ( k  e.  _V  |->  A ) ) `  N ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   E.wrex 2544   _Vcvv 2788   (/)c0 3455   ifcif 3565    e. cmpt 4077   ` cfv 5255  (class class class)co 5858   ZZcz 10024   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046  GIdcgi 20854   prod_cprd 25298
This theorem is referenced by:  fprodser  25320  fprod1i  25322  fprodp1  25323
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-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-pre-lttri 8811  ax-pre-lttrn 8812
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-nel 2449  df-ral 2548  df-rex 2549  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-po 4314  df-so 4315  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-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-neg 9040  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-prod 25299
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