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Theorem prodeq3ii 25411
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 2357 . . . . . . . . . . . 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 4125 . . . . . . . . . . . . . . 15  |-  F/_ k
( k  e.  _V  |->  B )
6 nfcv 2432 . . . . . . . . . . . . . . 15  |-  F/_ k
x
75, 6nffv 5548 . . . . . . . . . . . . . 14  |-  F/_ k
( ( k  e. 
_V  |->  B ) `  x )
8 nfmpt1 4125 . . . . . . . . . . . . . . 15  |-  F/_ k
( k  e.  _V  |->  C )
98, 6nffv 5548 . . . . . . . . . . . . . 14  |-  F/_ k
( ( k  e. 
_V  |->  C ) `  x )
107, 9nfeq 2439 . . . . . . . . . . . . 13  |-  F/ k ( ( k  e. 
_V  |->  B ) `  x )  =  ( ( k  e.  _V  |->  C ) `  x
)
11 vex 2804 . . . . . . . . . . . . . . 15  |-  k  e. 
_V
12 eqid 2296 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  _V  |->  B )  =  ( k  e. 
_V  |->  B )
1312fvmpt2i 5623 . . . . . . . . . . . . . . . 16  |-  ( k  e.  _V  ->  (
( k  e.  _V  |->  B ) `  k
)  =  (  _I 
`  B ) )
14 eqid 2296 . . . . . . . . . . . . . . . . 17  |-  ( k  e.  _V  |->  C )  =  ( k  e. 
_V  |->  C )
1514fvmpt2i 5623 . . . . . . . . . . . . . . . 16  |-  ( k  e.  _V  ->  (
( k  e.  _V  |->  C ) `  k
)  =  (  _I 
`  C ) )
1613, 15eqeq12d 2310 . . . . . . . . . . . . . . 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 5541 . . . . . . . . . . . . . . 15  |-  ( k  =  x  ->  (
( k  e.  _V  |->  B ) `  k
)  =  ( ( k  e.  _V  |->  B ) `  x ) )
19 fveq2 5541 . . . . . . . . . . . . . . 15  |-  ( k  =  x  ->  (
( k  e.  _V  |->  C ) `  k
)  =  ( ( k  e.  _V  |->  C ) `  x ) )
2018, 19eqeq12d 2310 . . . . . . . . . . . . . 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 2891 . . . . . . . . . . . 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 11086 . . . . . . . 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 2363 . . . . . . 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 2573 . . . . 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 1616 . . . 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 2410 . . 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 3593 . 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 25402 . 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 25402 . 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 2353 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801   (/)c0 3468   ifcif 3578    e. cmpt 4093    _I cid 4320   ` cfv 5271  (class class class)co 5874   ZZ>=cuz 10246   ...cfz 10798    seq cseq 11062  GIdcgi 20870   prod_cprd 25401
This theorem is referenced by:  prodeq3  25412  prodeqfv  25421
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-prod 25402
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