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Theorem madjusmdetlem2 28728
Description: Lemma for madjusmdet 28731. (Contributed by Thierry Arnoux, 26-Aug-2020.)
Hypotheses
Ref Expression
madjusmdet.b  |-  B  =  ( Base `  A
)
madjusmdet.a  |-  A  =  ( ( 1 ... N ) Mat  R )
madjusmdet.d  |-  D  =  ( ( 1 ... N ) maDet  R )
madjusmdet.k  |-  K  =  ( ( 1 ... N ) maAdju  R )
madjusmdet.t  |-  .x.  =  ( .r `  R )
madjusmdet.z  |-  Z  =  ( ZRHom `  R
)
madjusmdet.e  |-  E  =  ( ( 1 ... ( N  -  1 ) ) maDet  R )
madjusmdet.n  |-  ( ph  ->  N  e.  NN )
madjusmdet.r  |-  ( ph  ->  R  e.  CRing )
madjusmdet.i  |-  ( ph  ->  I  e.  ( 1 ... N ) )
madjusmdet.j  |-  ( ph  ->  J  e.  ( 1 ... N ) )
madjusmdet.m  |-  ( ph  ->  M  e.  B )
madjusmdetlem2.p  |-  P  =  ( i  e.  ( 1 ... N ) 
|->  if ( i  =  1 ,  I ,  if ( i  <_  I ,  ( i  -  1 ) ,  i ) ) )
madjusmdetlem2.s  |-  S  =  ( i  e.  ( 1 ... N ) 
|->  if ( i  =  1 ,  N ,  if ( i  <_  N ,  ( i  - 
1 ) ,  i ) ) )
Assertion
Ref Expression
madjusmdetlem2  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  if ( X  <  I ,  X ,  ( X  +  1 ) )  =  ( ( P  o.  `' S ) `
 X ) )
Distinct variable groups:    B, i    i, I    i, J    i, M    i, N    P, i    R, i    ph, i    S, i
Allowed substitution hints:    A( i)    D( i)    .x. ( i)    E( i)    K( i)    X( i)    Z( i)

Proof of Theorem madjusmdetlem2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 madjusmdet.n . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  NN )
2 nnuz 11218 . . . . . . . . . . . 12  |-  NN  =  ( ZZ>= `  1 )
31, 2syl6eleq 2559 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
4 eluzfz2 11833 . . . . . . . . . . 11  |-  ( N  e.  ( ZZ>= `  1
)  ->  N  e.  ( 1 ... N
) )
53, 4syl 17 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ( 1 ... N ) )
6 eqid 2471 . . . . . . . . . . 11  |-  ( 1 ... N )  =  ( 1 ... N
)
7 madjusmdetlem2.s . . . . . . . . . . 11  |-  S  =  ( i  e.  ( 1 ... N ) 
|->  if ( i  =  1 ,  N ,  if ( i  <_  N ,  ( i  - 
1 ) ,  i ) ) )
8 eqid 2471 . . . . . . . . . . 11  |-  ( SymGrp `  ( 1 ... N
) )  =  (
SymGrp `  ( 1 ... N ) )
9 eqid 2471 . . . . . . . . . . 11  |-  ( Base `  ( SymGrp `  ( 1 ... N ) ) )  =  ( Base `  ( SymGrp `
 ( 1 ... N ) ) )
106, 7, 8, 9fzto1st 28690 . . . . . . . . . 10  |-  ( N  e.  ( 1 ... N )  ->  S  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) ) )
115, 10syl 17 . . . . . . . . 9  |-  ( ph  ->  S  e.  ( Base `  ( SymGrp `  ( 1 ... N ) ) ) )
128, 9symgbasf1o 17102 . . . . . . . . 9  |-  ( S  e.  ( Base `  ( SymGrp `
 ( 1 ... N ) ) )  ->  S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
1311, 12syl 17 . . . . . . . 8  |-  ( ph  ->  S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
1413adantr 472 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  S : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
15 fznatpl1 11876 . . . . . . . 8  |-  ( ( N  e.  NN  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( X  + 
1 )  e.  ( 1 ... N ) )
161, 15sylan 479 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( X  +  1 )  e.  ( 1 ... N ) )
17 eqeq1 2475 . . . . . . . . . . . . 13  |-  ( i  =  x  ->  (
i  =  1  <->  x  =  1 ) )
18 breq1 4398 . . . . . . . . . . . . . 14  |-  ( i  =  x  ->  (
i  <_  N  <->  x  <_  N ) )
19 oveq1 6315 . . . . . . . . . . . . . 14  |-  ( i  =  x  ->  (
i  -  1 )  =  ( x  - 
1 ) )
20 id 22 . . . . . . . . . . . . . 14  |-  ( i  =  x  ->  i  =  x )
2118, 19, 20ifbieq12d 3899 . . . . . . . . . . . . 13  |-  ( i  =  x  ->  if ( i  <_  N ,  ( i  - 
1 ) ,  i )  =  if ( x  <_  N , 
( x  -  1 ) ,  x ) )
2217, 21ifbieq2d 3897 . . . . . . . . . . . 12  |-  ( i  =  x  ->  if ( i  =  1 ,  N ,  if ( i  <_  N ,  ( i  - 
1 ) ,  i ) )  =  if ( x  =  1 ,  N ,  if ( x  <_  N , 
( x  -  1 ) ,  x ) ) )
2322cbvmptv 4488 . . . . . . . . . . 11  |-  ( i  e.  ( 1 ... N )  |->  if ( i  =  1 ,  N ,  if ( i  <_  N , 
( i  -  1 ) ,  i ) ) )  =  ( x  e.  ( 1 ... N )  |->  if ( x  =  1 ,  N ,  if ( x  <_  N , 
( x  -  1 ) ,  x ) ) )
247, 23eqtri 2493 . . . . . . . . . 10  |-  S  =  ( x  e.  ( 1 ... N ) 
|->  if ( x  =  1 ,  N ,  if ( x  <_  N ,  ( x  - 
1 ) ,  x
) ) )
2524a1i 11 . . . . . . . . 9  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  S  =  ( x  e.  ( 1 ... N
)  |->  if ( x  =  1 ,  N ,  if ( x  <_  N ,  ( x  -  1 ) ,  x ) ) ) )
26 simpr 468 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  x  =  ( X  +  1 ) )
27 1red 9676 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
1  e.  RR )
28 fz1ssnn 11856 . . . . . . . . . . . . . . . . . . 19  |-  ( 1 ... ( N  - 
1 ) )  C_  NN
29 simpr 468 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  X  e.  ( 1 ... ( N  -  1 ) ) )
3028, 29sseldi 3416 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  X  e.  NN )
3130nnrpd 11362 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  X  e.  RR+ )
3231adantr 472 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  X  e.  RR+ )
3327, 32ltaddrp2d 11395 . . . . . . . . . . . . . . 15  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
1  <  ( X  +  1 ) )
3427, 33ltned 9788 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
1  =/=  ( X  +  1 ) )
3534necomd 2698 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
( X  +  1 )  =/=  1 )
3626, 35eqnetrd 2710 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  x  =/=  1 )
3736neneqd 2648 . . . . . . . . . . 11  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  -.  x  =  1
)
3837iffalsed 3883 . . . . . . . . . 10  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  if ( x  =  1 ,  N ,  if ( x  <_  N , 
( x  -  1 ) ,  x ) )  =  if ( x  <_  N , 
( x  -  1 ) ,  x ) )
391adantr 472 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  N  e.  NN )
4030nnnn0d 10949 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  X  e.  NN0 )
4139nnnn0d 10949 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  N  e.  NN0 )
42 elfzle2 11829 . . . . . . . . . . . . . . . 16  |-  ( X  e.  ( 1 ... ( N  -  1 ) )  ->  X  <_  ( N  -  1 ) )
4329, 42syl 17 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  X  <_  ( N  -  1 ) )
44 nn0ltlem1 11020 . . . . . . . . . . . . . . . 16  |-  ( ( X  e.  NN0  /\  N  e.  NN0 )  -> 
( X  <  N  <->  X  <_  ( N  - 
1 ) ) )
4544biimpar 493 . . . . . . . . . . . . . . 15  |-  ( ( ( X  e.  NN0  /\  N  e.  NN0 )  /\  X  <_  ( N  -  1 ) )  ->  X  <  N
)
4640, 41, 43, 45syl21anc 1291 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  X  <  N )
47 nnltp1le 11016 . . . . . . . . . . . . . . 15  |-  ( ( X  e.  NN  /\  N  e.  NN )  ->  ( X  <  N  <->  ( X  +  1 )  <_  N ) )
4847biimpa 492 . . . . . . . . . . . . . 14  |-  ( ( ( X  e.  NN  /\  N  e.  NN )  /\  X  <  N
)  ->  ( X  +  1 )  <_  N )
4930, 39, 46, 48syl21anc 1291 . . . . . . . . . . . . 13  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( X  +  1 )  <_  N )
5049adantr 472 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
( X  +  1 )  <_  N )
5126, 50eqbrtrd 4416 . . . . . . . . . . 11  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  x  <_  N )
5251iftrued 3880 . . . . . . . . . 10  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  if ( x  <_  N ,  ( x  - 
1 ) ,  x
)  =  ( x  -  1 ) )
5326oveq1d 6323 . . . . . . . . . . 11  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
( x  -  1 )  =  ( ( X  +  1 )  -  1 ) )
5430nncnd 10647 . . . . . . . . . . . . 13  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  X  e.  CC )
55 1cnd 9677 . . . . . . . . . . . . 13  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  1  e.  CC )
5654, 55pncand 10006 . . . . . . . . . . . 12  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
( X  +  1 )  -  1 )  =  X )
5756adantr 472 . . . . . . . . . . 11  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
( ( X  + 
1 )  -  1 )  =  X )
5853, 57eqtrd 2505 . . . . . . . . . 10  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
( x  -  1 )  =  X )
5938, 52, 583eqtrd 2509 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  if ( x  =  1 ,  N ,  if ( x  <_  N , 
( x  -  1 ) ,  x ) )  =  X )
6025, 59, 16, 29fvmptd 5969 . . . . . . . 8  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( S `  ( X  +  1 ) )  =  X )
6160idi 2 . . . . . . 7  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( S `  ( X  +  1 ) )  =  X )
62 f1ocnvfv 6195 . . . . . . . 8  |-  ( ( S : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  /\  ( X  +  1
)  e.  ( 1 ... N ) )  ->  ( ( S `
 ( X  + 
1 ) )  =  X  ->  ( `' S `  X )  =  ( X  + 
1 ) ) )
6362imp 436 . . . . . . 7  |-  ( ( ( S : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  /\  ( X  + 
1 )  e.  ( 1 ... N ) )  /\  ( S `
 ( X  + 
1 ) )  =  X )  ->  ( `' S `  X )  =  ( X  + 
1 ) )
6414, 16, 61, 63syl21anc 1291 . . . . . 6  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( `' S `  X )  =  ( X  + 
1 ) )
6564fveq2d 5883 . . . . 5  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( P `  ( `' S `  X )
)  =  ( P `
 ( X  + 
1 ) ) )
6665adantr 472 . . . 4  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  X  <  I )  ->  ( P `  ( `' S `  X )
)  =  ( P `
 ( X  + 
1 ) ) )
67 madjusmdetlem2.p . . . . . . 7  |-  P  =  ( i  e.  ( 1 ... N ) 
|->  if ( i  =  1 ,  I ,  if ( i  <_  I ,  ( i  -  1 ) ,  i ) ) )
6820breq1d 4405 . . . . . . . . . 10  |-  ( i  =  x  ->  (
i  <_  I  <->  x  <_  I ) )
6968, 19, 20ifbieq12d 3899 . . . . . . . . 9  |-  ( i  =  x  ->  if ( i  <_  I ,  ( i  - 
1 ) ,  i )  =  if ( x  <_  I , 
( x  -  1 ) ,  x ) )
7017, 69ifbieq2d 3897 . . . . . . . 8  |-  ( i  =  x  ->  if ( i  =  1 ,  I ,  if ( i  <_  I ,  ( i  - 
1 ) ,  i ) )  =  if ( x  =  1 ,  I ,  if ( x  <_  I ,  ( x  -  1 ) ,  x ) ) )
7170cbvmptv 4488 . . . . . . 7  |-  ( i  e.  ( 1 ... N )  |->  if ( i  =  1 ,  I ,  if ( i  <_  I , 
( i  -  1 ) ,  i ) ) )  =  ( x  e.  ( 1 ... N )  |->  if ( x  =  1 ,  I ,  if ( x  <_  I ,  ( x  -  1 ) ,  x ) ) )
7267, 71eqtri 2493 . . . . . 6  |-  P  =  ( x  e.  ( 1 ... N ) 
|->  if ( x  =  1 ,  I ,  if ( x  <_  I ,  ( x  -  1 ) ,  x ) ) )
7372a1i 11 . . . . 5  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  X  <  I )  ->  P  =  ( x  e.  ( 1 ... N
)  |->  if ( x  =  1 ,  I ,  if ( x  <_  I ,  ( x  -  1 ) ,  x ) ) ) )
7433, 26breqtrrd 4422 . . . . . . . . . . 11  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
1  <  x )
7527, 74ltned 9788 . . . . . . . . . 10  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
1  =/=  x )
7675necomd 2698 . . . . . . . . 9  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  x  =/=  1 )
7776neneqd 2648 . . . . . . . 8  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  -.  x  =  1
)
7877iffalsed 3883 . . . . . . 7  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  ->  if ( x  =  1 ,  I ,  if ( x  <_  I ,  ( x  -  1 ) ,  x ) )  =  if ( x  <_  I , 
( x  -  1 ) ,  x ) )
7978adantlr 729 . . . . . 6  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  X  <  I
)  /\  x  =  ( X  +  1
) )  ->  if ( x  =  1 ,  I ,  if ( x  <_  I , 
( x  -  1 ) ,  x ) )  =  if ( x  <_  I , 
( x  -  1 ) ,  x ) )
80 simpr 468 . . . . . . . 8  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  X  <  I
)  /\  x  =  ( X  +  1
) )  ->  x  =  ( X  + 
1 ) )
8130ad2antrr 740 . . . . . . . . 9  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  X  <  I
)  /\  x  =  ( X  +  1
) )  ->  X  e.  NN )
82 fz1ssnn 11856 . . . . . . . . . . 11  |-  ( 1 ... N )  C_  NN
83 madjusmdet.i . . . . . . . . . . 11  |-  ( ph  ->  I  e.  ( 1 ... N ) )
8482, 83sseldi 3416 . . . . . . . . . 10  |-  ( ph  ->  I  e.  NN )
8584ad3antrrr 744 . . . . . . . . 9  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  X  <  I
)  /\  x  =  ( X  +  1
) )  ->  I  e.  NN )
86 simplr 770 . . . . . . . . 9  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  X  <  I
)  /\  x  =  ( X  +  1
) )  ->  X  <  I )
87 nnltp1le 11016 . . . . . . . . . 10  |-  ( ( X  e.  NN  /\  I  e.  NN )  ->  ( X  <  I  <->  ( X  +  1 )  <_  I ) )
8887biimpa 492 . . . . . . . . 9  |-  ( ( ( X  e.  NN  /\  I  e.  NN )  /\  X  <  I
)  ->  ( X  +  1 )  <_  I )
8981, 85, 86, 88syl21anc 1291 . . . . . . . 8  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  X  <  I
)  /\  x  =  ( X  +  1
) )  ->  ( X  +  1 )  <_  I )
9080, 89eqbrtrd 4416 . . . . . . 7  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  X  <  I
)  /\  x  =  ( X  +  1
) )  ->  x  <_  I )
9190iftrued 3880 . . . . . 6  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  X  <  I
)  /\  x  =  ( X  +  1
) )  ->  if ( x  <_  I ,  ( x  -  1 ) ,  x )  =  ( x  - 
1 ) )
9258adantlr 729 . . . . . 6  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  X  <  I
)  /\  x  =  ( X  +  1
) )  ->  (
x  -  1 )  =  X )
9379, 91, 923eqtrd 2509 . . . . 5  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  X  <  I
)  /\  x  =  ( X  +  1
) )  ->  if ( x  =  1 ,  I ,  if ( x  <_  I , 
( x  -  1 ) ,  x ) )  =  X )
9416adantr 472 . . . . 5  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  X  <  I )  ->  ( X  +  1 )  e.  ( 1 ... N ) )
95 simplr 770 . . . . 5  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  X  <  I )  ->  X  e.  ( 1 ... ( N  -  1 ) ) )
9673, 93, 94, 95fvmptd 5969 . . . 4  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  X  <  I )  ->  ( P `  ( X  +  1 ) )  =  X )
9766, 96eqtr2d 2506 . . 3  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  X  <  I )  ->  X  =  ( P `  ( `' S `  X ) ) )
9865adantr 472 . . . 4  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  -.  X  <  I )  -> 
( P `  ( `' S `  X ) )  =  ( P `
 ( X  + 
1 ) ) )
9972a1i 11 . . . . 5  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  -.  X  <  I )  ->  P  =  ( x  e.  ( 1 ... N
)  |->  if ( x  =  1 ,  I ,  if ( x  <_  I ,  ( x  -  1 ) ,  x ) ) ) )
10078adantlr 729 . . . . . 6  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  -.  X  < 
I )  /\  x  =  ( X  + 
1 ) )  ->  if ( x  =  1 ,  I ,  if ( x  <_  I ,  ( x  -  1 ) ,  x ) )  =  if ( x  <_  I , 
( x  -  1 ) ,  x ) )
10130ad2antrr 740 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  x  =  ( X  +  1 ) )  /\  x  <_  I )  ->  X  e.  NN )
10284ad3antrrr 744 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  x  =  ( X  +  1 ) )  /\  x  <_  I )  ->  I  e.  NN )
10326adantr 472 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  x  =  ( X  +  1 ) )  /\  x  <_  I )  ->  x  =  ( X  + 
1 ) )
104 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  x  =  ( X  +  1 ) )  /\  x  <_  I )  ->  x  <_  I )
105103, 104eqbrtrrd 4418 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  x  =  ( X  +  1 ) )  /\  x  <_  I )  ->  ( X  +  1 )  <_  I )
10687biimpar 493 . . . . . . . . . . . . 13  |-  ( ( ( X  e.  NN  /\  I  e.  NN )  /\  ( X  + 
1 )  <_  I
)  ->  X  <  I )
107101, 102, 105, 106syl21anc 1291 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  x  =  ( X  +  1 ) )  /\  x  <_  I )  ->  X  <  I )
108107ex 441 . . . . . . . . . . 11  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
( x  <_  I  ->  X  <  I ) )
109108con3d 140 . . . . . . . . . 10  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  x  =  ( X  + 
1 ) )  -> 
( -.  X  < 
I  ->  -.  x  <_  I ) )
110109imp 436 . . . . . . . . 9  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  x  =  ( X  +  1 ) )  /\  -.  X  <  I )  ->  -.  x  <_  I )
111110an32s 821 . . . . . . . 8  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  -.  X  < 
I )  /\  x  =  ( X  + 
1 ) )  ->  -.  x  <_  I )
112111iffalsed 3883 . . . . . . 7  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  -.  X  < 
I )  /\  x  =  ( X  + 
1 ) )  ->  if ( x  <_  I ,  ( x  - 
1 ) ,  x
)  =  x )
113 simpr 468 . . . . . . 7  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  -.  X  < 
I )  /\  x  =  ( X  + 
1 ) )  ->  x  =  ( X  +  1 ) )
114112, 113eqtrd 2505 . . . . . 6  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  -.  X  < 
I )  /\  x  =  ( X  + 
1 ) )  ->  if ( x  <_  I ,  ( x  - 
1 ) ,  x
)  =  ( X  +  1 ) )
115100, 114eqtrd 2505 . . . . 5  |-  ( ( ( ( ph  /\  X  e.  ( 1 ... ( N  - 
1 ) ) )  /\  -.  X  < 
I )  /\  x  =  ( X  + 
1 ) )  ->  if ( x  =  1 ,  I ,  if ( x  <_  I ,  ( x  -  1 ) ,  x ) )  =  ( X  +  1 ) )
11616adantr 472 . . . . 5  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  -.  X  <  I )  -> 
( X  +  1 )  e.  ( 1 ... N ) )
11799, 115, 116, 116fvmptd 5969 . . . 4  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  -.  X  <  I )  -> 
( P `  ( X  +  1 ) )  =  ( X  +  1 ) )
11898, 117eqtr2d 2506 . . 3  |-  ( ( ( ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  /\  -.  X  <  I )  -> 
( X  +  1 )  =  ( P `
 ( `' S `  X ) ) )
11997, 118ifeqda 3905 . 2  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  if ( X  <  I ,  X ,  ( X  +  1 ) )  =  ( P `  ( `' S `  X ) ) )
120 f1ocnv 5840 . . . . . 6  |-  ( S : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  ->  `' S : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
12111, 12, 1203syl 18 . . . . 5  |-  ( ph  ->  `' S : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
122 f1ofun 5830 . . . . 5  |-  ( `' S : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  ->  Fun  `' S )
123121, 122syl 17 . . . 4  |-  ( ph  ->  Fun  `' S )
124123adantr 472 . . 3  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  Fun  `' S )
125 fzdif2 28444 . . . . . . . 8  |-  ( N  e.  ( ZZ>= `  1
)  ->  ( (
1 ... N )  \  { N } )  =  ( 1 ... ( N  -  1 ) ) )
1263, 125syl 17 . . . . . . 7  |-  ( ph  ->  ( ( 1 ... N )  \  { N } )  =  ( 1 ... ( N  -  1 ) ) )
127 difss 3549 . . . . . . 7  |-  ( ( 1 ... N ) 
\  { N }
)  C_  ( 1 ... N )
128126, 127syl6eqssr 3469 . . . . . 6  |-  ( ph  ->  ( 1 ... ( N  -  1 ) )  C_  ( 1 ... N ) )
129 f1odm 5832 . . . . . . 7  |-  ( `' S : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  ->  dom  `' S  =  (
1 ... N ) )
130121, 129syl 17 . . . . . 6  |-  ( ph  ->  dom  `' S  =  ( 1 ... N
) )
131128, 130sseqtr4d 3455 . . . . 5  |-  ( ph  ->  ( 1 ... ( N  -  1 ) )  C_  dom  `' S
)
132131adantr 472 . . . 4  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
1 ... ( N  - 
1 ) )  C_  dom  `' S )
133132, 29sseldd 3419 . . 3  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  X  e.  dom  `' S )
134 fvco 5956 . . 3  |-  ( ( Fun  `' S  /\  X  e.  dom  `' S
)  ->  ( ( P  o.  `' S
) `  X )  =  ( P `  ( `' S `  X ) ) )
135124, 133, 134syl2anc 673 . 2  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
( P  o.  `' S ) `  X
)  =  ( P `
 ( `' S `  X ) ) )
136119, 135eqtr4d 2508 1  |-  ( (
ph  /\  X  e.  ( 1 ... ( N  -  1 ) ) )  ->  if ( X  <  I ,  X ,  ( X  +  1 ) )  =  ( ( P  o.  `' S ) `
 X ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904    \ cdif 3387    C_ wss 3390   ifcif 3872   {csn 3959   class class class wbr 4395    |-> cmpt 4454   `'ccnv 4838   dom cdm 4839    o. ccom 4843   Fun wfun 5583   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   NN0cn0 10893   ZZ>=cuz 11182   RR+crp 11325   ...cfz 11810   Basecbs 15199   .rcmulr 15269   SymGrpcsymg 17096   CRingccrg 17859   ZRHomczrh 19148   Mat cmat 19509   maDet cmdat 19686   maAdju cmadu 19734
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-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-2o 7201  df-oadd 7204  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-n0 10894  df-z 10962  df-uz 11183  df-rp 11326  df-fz 11811  df-struct 15201  df-ndx 15202  df-slot 15203  df-base 15204  df-plusg 15281  df-tset 15287  df-symg 17097  df-pmtr 17161
This theorem is referenced by:  madjusmdetlem3  28729
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