Users' Mathboxes Mathbox for Frédéric Liné < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prodeq3ii Unicode version

Theorem prodeq3ii 25308
Description: Equality theorem for a composite. (Contributed by Mario Carneiro, 26-May-2014.)
Assertion
Ref Expression
prodeq3ii  |-  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C
)  ->  prod_ k  e.  A G B  = 
prod_ k  e.  A G C )
Distinct variable group:    A, k
Allowed substitution hints:    B( k)    C( k)    G( k)

Proof of Theorem prodeq3ii
Dummy variables  m  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simplr 731 . . . . . . . . 9  |-  ( ( ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  /\  n  e.  (
ZZ>= `  m ) )  /\  A  =  ( m ... n ) )  ->  n  e.  ( ZZ>= `  m )
)
2 eleq2 2344 . . . . . . . . . . . 12  |-  ( A  =  ( m ... n )  ->  (
x  e.  A  <->  x  e.  ( m ... n
) ) )
32biimprd 214 . . . . . . . . . . 11  |-  ( A  =  ( m ... n )  ->  (
x  e.  ( m ... n )  ->  x  e.  A )
)
4 simpl 443 . . . . . . . . . . . 12  |-  ( ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  /\  n  e.  ( ZZ>= `  m )
)  ->  A. k  e.  A  (  _I  `  B )  =  (  _I  `  C ) )
5 nfmpt1 4109 . . . . . . . . . . . . . . 15  |-  F/_ k
( k  e.  _V  |->  B )
6 nfcv 2419 . . . . . . . . . . . . . . 15  |-  F/_ k
x
75, 6nffv 5532 . . . . . . . . . . . . . 14  |-  F/_ k
( ( k  e. 
_V  |->  B ) `  x )
8 nfmpt1 4109 . . . . . . . . . . . . . . 15  |-  F/_ k
( k  e.  _V  |->  C )
98, 6nffv 5532 . . . . . . . . . . . . . 14  |-  F/_ k
( ( k  e. 
_V  |->  C ) `  x )
107, 9nfeq 2426 . . . . . . . . . . . . 13  |-  F/ k ( ( k  e. 
_V  |->  B ) `  x )  =  ( ( k  e.  _V  |->  C ) `  x
)
11 vex 2791 . . . . . . . . . . . . . . 15  |-  k  e. 
_V
12 eqid 2283 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  _V  |->  B )  =  ( k  e. 
_V  |->  B )
1312fvmpt2i 5607 . . . . . . . . . . . . . . . 16  |-  ( k  e.  _V  ->  (
( k  e.  _V  |->  B ) `  k
)  =  (  _I 
`  B ) )
14 eqid 2283 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  _V  |->  C )  =  ( k  e. 
_V  |->  C )
1514fvmpt2i 5607 . . . . . . . . . . . . . . . 16  |-  ( k  e.  _V  ->  (
( k  e.  _V  |->  C ) `  k
)  =  (  _I 
`  C ) )
1613, 15eqeq12d 2297 . . . . . . . . . . . . . . 15  |-  ( k  e.  _V  ->  (
( ( k  e. 
_V  |->  B ) `  k )  =  ( ( k  e.  _V  |->  C ) `  k
)  <->  (  _I  `  B )  =  (  _I  `  C ) ) )
1711, 16ax-mp 8 . . . . . . . . . . . . . 14  |-  ( ( ( k  e.  _V  |->  B ) `  k
)  =  ( ( k  e.  _V  |->  C ) `  k )  <-> 
(  _I  `  B
)  =  (  _I 
`  C ) )
18 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( k  =  x  ->  (
( k  e.  _V  |->  B ) `  k
)  =  ( ( k  e.  _V  |->  B ) `  x ) )
19 fveq2 5525 . . . . . . . . . . . . . . 15  |-  ( k  =  x  ->  (
( k  e.  _V  |->  C ) `  k
)  =  ( ( k  e.  _V  |->  C ) `  x ) )
2018, 19eqeq12d 2297 . . . . . . . . . . . . . 14  |-  ( k  =  x  ->  (
( ( k  e. 
_V  |->  B ) `  k )  =  ( ( k  e.  _V  |->  C ) `  k
)  <->  ( ( k  e.  _V  |->  B ) `
 x )  =  ( ( k  e. 
_V  |->  C ) `  x ) ) )
2117, 20syl5bbr 250 . . . . . . . . . . . . 13  |-  ( k  =  x  ->  (
(  _I  `  B
)  =  (  _I 
`  C )  <->  ( (
k  e.  _V  |->  B ) `  x )  =  ( ( k  e.  _V  |->  C ) `
 x ) ) )
2210, 21rspc 2878 . . . . . . . . . . . 12  |-  ( x  e.  A  ->  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  ->  (
( k  e.  _V  |->  B ) `  x
)  =  ( ( k  e.  _V  |->  C ) `  x ) ) )
234, 22syl5com 26 . . . . . . . . . . 11  |-  ( ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  /\  n  e.  ( ZZ>= `  m )
)  ->  ( x  e.  A  ->  ( ( k  e.  _V  |->  B ) `  x )  =  ( ( k  e.  _V  |->  C ) `
 x ) ) )
243, 23sylan9r 639 . . . . . . . . . 10  |-  ( ( ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  /\  n  e.  (
ZZ>= `  m ) )  /\  A  =  ( m ... n ) )  ->  ( x  e.  ( m ... n
)  ->  ( (
k  e.  _V  |->  B ) `  x )  =  ( ( k  e.  _V  |->  C ) `
 x ) ) )
2524imp 418 . . . . . . . . 9  |-  ( ( ( ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  /\  n  e.  (
ZZ>= `  m ) )  /\  A  =  ( m ... n ) )  /\  x  e.  ( m ... n
) )  ->  (
( k  e.  _V  |->  B ) `  x
)  =  ( ( k  e.  _V  |->  C ) `  x ) )
261, 25seqfveq 11070 . . . . . . . 8  |-  ( ( ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  /\  n  e.  (
ZZ>= `  m ) )  /\  A  =  ( m ... n ) )  ->  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n )  =  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) )
2726eleq2d 2350 . . . . . . 7  |-  ( ( ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  /\  n  e.  (
ZZ>= `  m ) )  /\  A  =  ( m ... n ) )  ->  ( x  e.  (  seq  m ( G ,  ( k  e.  _V  |->  B ) ) `  n )  <-> 
x  e.  (  seq  m ( G , 
( k  e.  _V  |->  C ) ) `  n ) ) )
2827pm5.32da 622 . . . . . 6  |-  ( ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C )  /\  n  e.  ( ZZ>= `  m )
)  ->  ( ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e.  _V  |->  B ) ) `  n ) )  <->  ( A  =  ( m ... n
)  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) ) )
2928rexbidva 2560 . . . . 5  |-  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C
)  ->  ( E. n  e.  ( ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e.  _V  |->  B ) ) `  n ) )  <->  E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  C ) ) `  n ) ) ) )
3029exbidv 1612 . . . 4  |-  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C
)  ->  ( E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  B ) ) `  n ) )  <->  E. m E. n  e.  ( ZZ>=
`  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  C ) ) `  n ) ) ) )
3130abbidv 2397 . . 3  |-  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C
)  ->  { 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. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) } )
3231ifeq2d 3580 . 2  |-  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C
)  ->  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 ) ) } )  =  if ( A  =  (/) ,  (GId
`  G ) ,  { x  |  E. m E. n  e.  (
ZZ>= `  m ) ( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G , 
( k  e.  _V  |->  C ) ) `  n ) ) } ) )
33 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 ) ) } )
34 df-prod 25299 . 2  |-  prod_ k  e.  A G C  =  if ( A  =  (/) ,  (GId `  G
) ,  { x  |  E. m E. n  e.  ( ZZ>= `  m )
( A  =  ( m ... n )  /\  x  e.  (  seq  m ( G ,  ( k  e. 
_V  |->  C ) ) `
 n ) ) } )
3532, 33, 343eqtr4g 2340 1  |-  ( A. k  e.  A  (  _I  `  B )  =  (  _I  `  C
)  ->  prod_ k  e.  A G B  = 
prod_ k  e.  A G C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788   (/)c0 3455   ifcif 3565    e. cmpt 4077    _I cid 4304   ` cfv 5255  (class class class)co 5858   ZZ>=cuz 10230   ...cfz 10782    seq cseq 11046  GIdcgi 20854   prod_cprd 25298
This theorem is referenced by:  prodeq3  25309  prodeqfv  25318
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-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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-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-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  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-riota 6304  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-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-prod 25299
  Copyright terms: Public domain W3C validator