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Theorem etransclem5 38216
Description: A change of bound variable, often used in proofs for etransc 38261. (Contributed by Glauco Siliprandi, 5-Apr-2020.)
Assertion
Ref Expression
etransclem5  |-  ( j  e.  ( 0 ... M )  |->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) ) )  =  ( k  e.  ( 0 ... M )  |->  ( y  e.  X  |->  ( ( y  -  k
) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
Distinct variable groups:    j, M, k    P, j, k, x, y    j, X, k, x, y
Allowed substitution hints:    M( x, y)

Proof of Theorem etransclem5
StepHypRef Expression
1 oveq1 6315 . . . . 5  |-  ( x  =  y  ->  (
x  -  j )  =  ( y  -  j ) )
21oveq1d 6323 . . . 4  |-  ( x  =  y  ->  (
( x  -  j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) )  =  ( ( y  -  j ) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) )
32cbvmptv 4488 . . 3  |-  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) )  =  ( y  e.  X  |->  ( ( y  -  j ) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) )
4 oveq2 6316 . . . . 5  |-  ( j  =  k  ->  (
y  -  j )  =  ( y  -  k ) )
5 eqeq1 2475 . . . . . 6  |-  ( j  =  k  ->  (
j  =  0  <->  k  =  0 ) )
65ifbid 3894 . . . . 5  |-  ( j  =  k  ->  if ( j  =  0 ,  ( P  - 
1 ) ,  P
)  =  if ( k  =  0 ,  ( P  -  1 ) ,  P ) )
74, 6oveq12d 6326 . . . 4  |-  ( j  =  k  ->  (
( y  -  j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) )  =  ( ( y  -  k ) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) )
87mpteq2dv 4483 . . 3  |-  ( j  =  k  ->  (
y  e.  X  |->  ( ( y  -  j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) )  =  ( y  e.  X  |->  ( ( y  -  k
) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
93, 8syl5eq 2517 . 2  |-  ( j  =  k  ->  (
x  e.  X  |->  ( ( x  -  j
) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) )  =  ( y  e.  X  |->  ( ( y  -  k
) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
109cbvmptv 4488 1  |-  ( j  e.  ( 0 ... M )  |->  ( x  e.  X  |->  ( ( x  -  j ) ^ if ( j  =  0 ,  ( P  -  1 ) ,  P ) ) ) )  =  ( k  e.  ( 0 ... M )  |->  ( y  e.  X  |->  ( ( y  -  k
) ^ if ( k  =  0 ,  ( P  -  1 ) ,  P ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    = wceq 1452   ifcif 3872    |-> cmpt 4454  (class class class)co 6308   0cc0 9557   1c1 9558    - cmin 9880   ...cfz 11810   ^cexp 12310
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-iota 5553  df-fv 5597  df-ov 6311
This theorem is referenced by:  etransclem27  38238  etransclem29  38240  etransclem31  38242  etransclem32  38243  etransclem33  38244  etransclem34  38245  etransclem35  38246  etransclem38  38249  etransclem40  38251  etransclem42  38253  etransclem44  38255  etransclem45  38256  etransclem46  38257
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