Theorem etransclem5 40456

Description: A change of bound variable, often used in proofs for etransc 40500. (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:   x, k   j, M, k   P, j, k, x, y   j, X, k, x, y

Allowed substitution hints:   M( x, y)

Proof of Theorem etransclem5

Step	Hyp	Ref	Expression
1		oveq1 6657	. . . . 5 |- ( x = y -> ( x - j ) = ( y - j ) )
2	1	oveq1d 6665	. . . 4 |- ( x = y -> ( ( x - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) )
3	2	cbvmptv 4750	. . 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 6658	. . . . 5 |- ( j = k -> ( y - j ) = ( y - k ) )
5		eqeq1 2626	. . . . . 6 |- ( j = k -> ( j = 0 <-> k = 0 ) )
6	5	ifbid 4108	. . . . 5 |- ( j = k -> if ( j = 0 , ( P - 1 ) , P ) = if ( k = 0 , ( P - 1 ) , P ) )
7	4, 6	oveq12d 6668	. . . 4 |- ( j = k -> ( ( y - j ) ^ if ( j = 0 , ( P - 1 ) , P ) ) = ( ( y - k ) ^ if ( k = 0 , ( P - 1 ) , P ) ) )
8	7	mpteq2dv 4745	. . 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 ) ) ) )
9	3, 8	syl5eq 2668	. 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 ) ) ) )
10	9	cbvmptv 4750	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 ) ) ) )

Syntax hints:   = wceq 1483   ifcif 4086   |-> cmpt 4729  (class class class)co 6650   0cc0 9936   1c1 9937   - cmin 10266   ...cfz 12326   ^cexp 12860

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  etransclem27  40478  etransclem29  40480  etransclem31  40482  etransclem32  40483  etransclem33  40484  etransclem34  40485  etransclem35  40486  etransclem38  40489  etransclem40  40491  etransclem42  40493  etransclem44  40495  etransclem45  40496  etransclem46  40497