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Theorem poimirlem19 31879
Description: Lemma for poimir 31893 establishing the vertices of the simplex in poimirlem20 31880. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0  |-  ( ph  ->  N  e.  NN )
poimirlem22.s  |-  S  =  { t  e.  ( ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) )  |  F  =  ( y  e.  ( 0 ... ( N  - 
1 ) )  |->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) }
poimirlem22.1  |-  ( ph  ->  F : ( 0 ... ( N  - 
1 ) ) --> ( ( 0 ... K
)  ^m  ( 1 ... N ) ) )
poimirlem22.2  |-  ( ph  ->  T  e.  S )
poimirlem22.3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  E. p  e.  ran  F ( p `
 n )  =/=  0 )
poimirlem21.4  |-  ( ph  ->  ( 2nd `  T
)  =  N )
Assertion
Ref Expression
poimirlem19  |-  ( ph  ->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  ( ( n  e.  ( 1 ... N )  |->  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  -  if ( n  =  ( ( 2nd `  ( 1st `  T ) ) `
 N ) ,  1 ,  0 ) ) )  oF  +  ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) ) ) )
Distinct variable groups:    f, j, n, p, t, y    ph, j, n, y    j, F, n, y    j, N, n, y    T, j, n, y    ph, p, t    f, K, j, n, p, t   
f, N, p, t    T, f, p    f, F, p, t    t, T    S, j, n, p, t, y
Allowed substitution hints:    ph( f)    S( f)    K( y)

Proof of Theorem poimirlem19
StepHypRef Expression
1 poimirlem22.2 . . 3  |-  ( ph  ->  T  e.  S )
2 fveq2 5879 . . . . . . . . . . 11  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
32breq2d 4433 . . . . . . . . . 10  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
43ifbid 3932 . . . . . . . . 9  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
54csbeq1d 3403 . . . . . . . 8  |-  ( t  =  T  ->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  [_ if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
6 fveq2 5879 . . . . . . . . . . 11  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
76fveq2d 5883 . . . . . . . . . 10  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
86fveq2d 5883 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
98imaeq1d 5184 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
109xpeq1d 4874 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
118imaeq1d 5184 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
1211xpeq1d 4874 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
1310, 12uneq12d 3622 . . . . . . . . . 10  |-  ( t  =  T  ->  (
( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) )  =  ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) ) )
147, 13oveq12d 6321 . . . . . . . . 9  |-  ( t  =  T  ->  (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
1514csbeq2dv 3810 . . . . . . . 8  |-  ( t  =  T  ->  [_ if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  [_ if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
165, 15eqtrd 2464 . . . . . . 7  |-  ( t  =  T  ->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  [_ if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
1716mpteq2dv 4509 . . . . . 6  |-  ( t  =  T  ->  (
y  e.  ( 0 ... ( N  - 
1 ) )  |->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )  =  ( y  e.  ( 0 ... ( N  -  1 ) ) 
|->  [_ if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
1817eqeq2d 2437 . . . . 5  |-  ( t  =  T  ->  ( F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  t
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )  <->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) ) 
|->  [_ if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) ) )
19 poimirlem22.s . . . . 5  |-  S  =  { t  e.  ( ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) )  |  F  =  ( y  e.  ( 0 ... ( N  - 
1 ) )  |->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) }
2018, 19elrab2 3232 . . . 4  |-  ( T  e.  S  <->  ( T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  /\  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) ) )
2120simprbi 466 . . 3  |-  ( T  e.  S  ->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
221, 21syl 17 . 2  |-  ( ph  ->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
23 elrabi 3227 . . . . . . . . . . . 12  |-  ( T  e.  { t  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  |  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  t
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) }  ->  T  e.  ( ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
2423, 19eleq2s 2531 . . . . . . . . . . 11  |-  ( T  e.  S  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
251, 24syl 17 . . . . . . . . . 10  |-  ( ph  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
26 xp1st 6835 . . . . . . . . . 10  |-  ( T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  ->  ( 1st `  T )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
2725, 26syl 17 . . . . . . . . 9  |-  ( ph  ->  ( 1st `  T
)  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } ) )
28 xp1st 6835 . . . . . . . . 9  |-  ( ( 1st `  T )  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  -> 
( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
2927, 28syl 17 . . . . . . . 8  |-  ( ph  ->  ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
30 elmapfn 7500 . . . . . . . 8  |-  ( ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  T ) )  Fn  ( 1 ... N ) )
3129, 30syl 17 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( 1st `  T ) )  Fn  ( 1 ... N ) )
3231adantr 467 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( 1st `  ( 1st `  T
) )  Fn  (
1 ... N ) )
33 1ex 9640 . . . . . . . . . 10  |-  1  e.  _V
34 fnconstg 5786 . . . . . . . . . 10  |-  ( 1  e.  _V  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  X. 
{ 1 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
1 ... y ) ) )
3533, 34ax-mp 5 . . . . . . . . 9  |-  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  X. 
{ 1 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
1 ... y ) )
36 c0ex 9639 . . . . . . . . . 10  |-  0  e.  _V
37 fnconstg 5786 . . . . . . . . . 10  |-  ( 0  e.  _V  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) )  X. 
{ 0 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
( y  +  1 ) ... N ) ) )
3836, 37ax-mp 5 . . . . . . . . 9  |-  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) )  X. 
{ 0 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
( y  +  1 ) ... N ) )
3935, 38pm3.2i 457 . . . . . . . 8  |-  ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  X. 
{ 1 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
1 ... y ) )  /\  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( y  +  1 ) ... N
) )  X.  {
0 } )  Fn  ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) ) )
40 xp2nd 6836 . . . . . . . . . . . . 13  |-  ( ( 1st `  T )  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  -> 
( 2nd `  ( 1st `  T ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
4127, 40syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
42 fvex 5889 . . . . . . . . . . . . 13  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
43 f1oeq1 5820 . . . . . . . . . . . . 13  |-  ( f  =  ( 2nd `  ( 1st `  T ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
4442, 43elab 3219 . . . . . . . . . . . 12  |-  ( ( 2nd `  ( 1st `  T ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
4541, 44sylib 200 . . . . . . . . . . 11  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
46 dff1o3 5835 . . . . . . . . . . . 12  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( ( 2nd `  ( 1st `  T
) ) : ( 1 ... N )
-onto-> ( 1 ... N
)  /\  Fun  `' ( 2nd `  ( 1st `  T ) ) ) )
4746simprbi 466 . . . . . . . . . . 11  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  Fun  `' ( 2nd `  ( 1st `  T ) ) )
4845, 47syl 17 . . . . . . . . . 10  |-  ( ph  ->  Fun  `' ( 2nd `  ( 1st `  T
) ) )
49 imain 5675 . . . . . . . . . 10  |-  ( Fun  `' ( 2nd `  ( 1st `  T ) )  ->  ( ( 2nd `  ( 1st `  T
) ) " (
( 1 ... y
)  i^i  ( (
y  +  1 ) ... N ) ) )  =  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  i^i  ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) ) ) )
5048, 49syl 17 . . . . . . . . 9  |-  ( ph  ->  ( ( 2nd `  ( 1st `  T ) )
" ( ( 1 ... y )  i^i  ( ( y  +  1 ) ... N
) ) )  =  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... y ) )  i^i  ( ( 2nd `  ( 1st `  T
) ) " (
( y  +  1 ) ... N ) ) ) )
51 elfznn0 11889 . . . . . . . . . . . . . 14  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  y  e.  NN0 )
5251nn0red 10928 . . . . . . . . . . . . 13  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  y  e.  RR )
5352ltp1d 10539 . . . . . . . . . . . 12  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  y  <  ( y  +  1 ) )
54 fzdisj 11828 . . . . . . . . . . . 12  |-  ( y  <  ( y  +  1 )  ->  (
( 1 ... y
)  i^i  ( (
y  +  1 ) ... N ) )  =  (/) )
5553, 54syl 17 . . . . . . . . . . 11  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( 1 ... y
)  i^i  ( (
y  +  1 ) ... N ) )  =  (/) )
5655imaeq2d 5185 . . . . . . . . . 10  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( 1 ... y )  i^i  ( ( y  +  1 ) ... N
) ) )  =  ( ( 2nd `  ( 1st `  T ) )
" (/) ) )
57 ima0 5200 . . . . . . . . . 10  |-  ( ( 2nd `  ( 1st `  T ) ) " (/) )  =  (/)
5856, 57syl6eq 2480 . . . . . . . . 9  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( 1 ... y )  i^i  ( ( y  +  1 ) ... N
) ) )  =  (/) )
5950, 58sylan9req 2485 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  i^i  ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) ) )  =  (/) )
60 fnun 5698 . . . . . . . 8  |-  ( ( ( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... y
) )  X.  {
1 } )  Fn  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  /\  ( ( ( 2nd `  ( 1st `  T
) ) " (
( y  +  1 ) ... N ) )  X.  { 0 } )  Fn  (
( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) ) )  /\  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... y
) )  i^i  (
( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) ) )  =  (/) )  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... y ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) )  X. 
{ 0 } ) )  Fn  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  u.  ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) ) ) )
6139, 59, 60sylancr 668 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... y ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) )  X. 
{ 0 } ) )  Fn  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  u.  ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) ) ) )
62 imaundi 5265 . . . . . . . . 9  |-  ( ( 2nd `  ( 1st `  T ) ) "
( ( 1 ... y )  u.  (
( y  +  1 ) ... N ) ) )  =  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  u.  ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) ) )
63 nn0p1nn 10911 . . . . . . . . . . . . . . 15  |-  ( y  e.  NN0  ->  ( y  +  1 )  e.  NN )
6451, 63syl 17 . . . . . . . . . . . . . 14  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
y  +  1 )  e.  NN )
65 nnuz 11196 . . . . . . . . . . . . . 14  |-  NN  =  ( ZZ>= `  1 )
6664, 65syl6eleq 2521 . . . . . . . . . . . . 13  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
y  +  1 )  e.  ( ZZ>= `  1
) )
6766adantl 468 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
y  +  1 )  e.  ( ZZ>= `  1
) )
68 poimir.0 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN )
6968nncnd 10627 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  CC )
70 npcan1 10046 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
7169, 70syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
7271adantr 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( N  -  1 )  +  1 )  =  N )
73 elfzuz3 11799 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  y
) )
74 peano2uz 11214 . . . . . . . . . . . . . . 15  |-  ( ( N  -  1 )  e.  ( ZZ>= `  y
)  ->  ( ( N  -  1 )  +  1 )  e.  ( ZZ>= `  y )
)
7573, 74syl 17 . . . . . . . . . . . . . 14  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( N  -  1 )  +  1 )  e.  ( ZZ>= `  y
) )
7675adantl 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( N  -  1 )  +  1 )  e.  ( ZZ>= `  y
) )
7772, 76eqeltrrd 2512 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  N  e.  ( ZZ>= `  y )
)
78 fzsplit2 11826 . . . . . . . . . . . 12  |-  ( ( ( y  +  1 )  e.  ( ZZ>= ` 
1 )  /\  N  e.  ( ZZ>= `  y )
)  ->  ( 1 ... N )  =  ( ( 1 ... y )  u.  (
( y  +  1 ) ... N ) ) )
7967, 77, 78syl2anc 666 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
1 ... N )  =  ( ( 1 ... y )  u.  (
( y  +  1 ) ... N ) ) )
8079imaeq2d 5185 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( 1 ... y )  u.  ( ( y  +  1 ) ... N
) ) ) )
81 f1ofo 5836 . . . . . . . . . . . 12  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N )
-onto-> ( 1 ... N
) )
82 foima 5813 . . . . . . . . . . . 12  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
) -onto-> ( 1 ... N )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
8345, 81, 823syl 18 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
8483adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
8580, 84eqtr3d 2466 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( 1 ... y )  u.  ( ( y  +  1 ) ... N
) ) )  =  ( 1 ... N
) )
8662, 85syl5eqr 2478 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  u.  ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) ) )  =  ( 1 ... N ) )
8786fneq2d 5683 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... y
) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
( y  +  1 ) ... N ) )  X.  { 0 } ) )  Fn  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... y ) )  u.  ( ( 2nd `  ( 1st `  T
) ) " (
( y  +  1 ) ... N ) ) )  <->  ( (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( y  +  1 ) ... N
) )  X.  {
0 } ) )  Fn  ( 1 ... N ) ) )
8861, 87mpbid 214 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... y ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) )  X. 
{ 0 } ) )  Fn  ( 1 ... N ) )
89 ovex 6331 . . . . . . 7  |-  ( 1 ... N )  e. 
_V
9089a1i 11 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
1 ... N )  e. 
_V )
91 inidm 3672 . . . . . 6  |-  ( ( 1 ... N )  i^i  ( 1 ... N ) )  =  ( 1 ... N
)
92 eqidd 2424 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  =  ( ( 1st `  ( 1st `  T ) ) `
 n ) )
93 eqidd 2424 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... y
) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
( y  +  1 ) ... N ) )  X.  { 0 } ) ) `  n )  =  ( ( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... y
) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
( y  +  1 ) ... N ) )  X.  { 0 } ) ) `  n ) )
9432, 88, 90, 90, 91, 92, 93offval 6550 . . . . 5  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... y ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( y  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  ( n  e.  ( 1 ... N )  |->  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  +  ( ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( y  +  1 ) ... N
) )  X.  {
0 } ) ) `
 n ) ) ) )
95 elmapi 7499 . . . . . . . . . . . . 13  |-  ( ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
9629, 95syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
9796ffvelrnda 6035 . . . . . . . . . . 11  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  ( 0..^ K ) )
98 elfzonn0 11962 . . . . . . . . . . 11  |-  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  e.  ( 0..^ K )  ->  ( ( 1st `  ( 1st `  T
) ) `  n
)  e.  NN0 )
9997, 98syl 17 . . . . . . . . . 10  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e. 
NN0 )
10099nn0cnd 10929 . . . . . . . . 9  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  CC )
101100adantlr 720 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  CC )
102 ax-1cn 9599 . . . . . . . . . 10  |-  1  e.  CC
103 0cn 9637 . . . . . . . . . 10  |-  0  e.  CC
104102, 103keepel 3977 . . . . . . . . 9  |-  if ( n  =  ( ( 2nd `  ( 1st `  T ) ) `  N ) ,  1 ,  0 )  e.  CC
105104a1i 11 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  /\  n  e.  ( 1 ... N
) )  ->  if ( n  =  (
( 2nd `  ( 1st `  T ) ) `
 N ) ,  1 ,  0 )  e.  CC )
106 snssi 4142 . . . . . . . . . . 11  |-  ( 1  e.  CC  ->  { 1 }  C_  CC )
107102, 106ax-mp 5 . . . . . . . . . 10  |-  { 1 }  C_  CC
108 snssi 4142 . . . . . . . . . . 11  |-  ( 0  e.  CC  ->  { 0 }  C_  CC )
109103, 108ax-mp 5 . . . . . . . . . 10  |-  { 0 }  C_  CC
110107, 109unssi 3642 . . . . . . . . 9  |-  ( { 1 }  u.  {
0 } )  C_  CC
11133fconst 5784 . . . . . . . . . . . . 13  |-  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  X.  { 1 } ) : ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) ) --> { 1 }
11236fconst 5784 . . . . . . . . . . . . 13  |-  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) : ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) ) --> { 0 }
113111, 112pm3.2i 457 . . . . . . . . . . . 12  |-  ( ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } ) : ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) ) --> { 1 }  /\  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) : ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) ) --> { 0 } )
114 simpr 463 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  n  e.  ( ( 1  +  1 ) ... N
) )  ->  n  e.  ( ( 1  +  1 ) ... N
) )
11568nnzd 11041 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  N  e.  ZZ )
116 1z 10969 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  1  e.  ZZ
117 peano2z 10980 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1  e.  ZZ  ->  (
1  +  1 )  e.  ZZ )
118116, 117ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 1  +  1 )  e.  ZZ
119115, 118jctil 540 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( ( 1  +  1 )  e.  ZZ  /\  N  e.  ZZ ) )
120 elfzelz 11802 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  e.  ( ( 1  +  1 ) ... N )  ->  n  e.  ZZ )
121120, 116jctir 541 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  ( ( 1  +  1 ) ... N )  ->  (
n  e.  ZZ  /\  1  e.  ZZ )
)
122 fzsubel 11836 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( 1  +  1 )  e.  ZZ  /\  N  e.  ZZ )  /\  ( n  e.  ZZ  /\  1  e.  ZZ ) )  -> 
( n  e.  ( ( 1  +  1 ) ... N )  <-> 
( n  -  1 )  e.  ( ( ( 1  +  1 )  -  1 ) ... ( N  - 
1 ) ) ) )
123119, 121, 122syl2an 480 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  n  e.  ( ( 1  +  1 ) ... N
) )  ->  (
n  e.  ( ( 1  +  1 ) ... N )  <->  ( n  -  1 )  e.  ( ( ( 1  +  1 )  - 
1 ) ... ( N  -  1 ) ) ) )
124114, 123mpbid 214 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  n  e.  ( ( 1  +  1 ) ... N
) )  ->  (
n  -  1 )  e.  ( ( ( 1  +  1 )  -  1 ) ... ( N  -  1 ) ) )
125102, 102pncan3oi 9893 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 1  +  1 )  -  1 )  =  1
126125oveq1i 6313 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 1  +  1 )  -  1 ) ... ( N  - 
1 ) )  =  ( 1 ... ( N  -  1 ) )
127124, 126syl6eleq 2521 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  n  e.  ( ( 1  +  1 ) ... N
) )  ->  (
n  -  1 )  e.  ( 1 ... ( N  -  1 ) ) )
128127ralrimiva 2840 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A. n  e.  ( ( 1  +  1 ) ... N ) ( n  -  1 )  e.  ( 1 ... ( N  - 
1 ) ) )
129 simpr 463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  y  e.  ( 1 ... ( N  -  1 ) ) )  ->  y  e.  ( 1 ... ( N  -  1 ) ) )
130 peano2zm 10982 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
131115, 130syl 17 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
132131, 116jctil 540 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( 1  e.  ZZ  /\  ( N  -  1 )  e.  ZZ ) )
133 elfzelz 11802 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  e.  ( 1 ... ( N  -  1 ) )  ->  y  e.  ZZ )
134133, 116jctir 541 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  ( 1 ... ( N  -  1 ) )  ->  (
y  e.  ZZ  /\  1  e.  ZZ )
)
135 fzaddel 11835 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( 1  e.  ZZ  /\  ( N  -  1 )  e.  ZZ )  /\  ( y  e.  ZZ  /\  1  e.  ZZ ) )  -> 
( y  e.  ( 1 ... ( N  -  1 ) )  <-> 
( y  +  1 )  e.  ( ( 1  +  1 ) ... ( ( N  -  1 )  +  1 ) ) ) )
136132, 134, 135syl2an 480 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( (
ph  /\  y  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
y  e.  ( 1 ... ( N  - 
1 ) )  <->  ( y  +  1 )  e.  ( ( 1  +  1 ) ... (
( N  -  1 )  +  1 ) ) ) )
137129, 136mpbid 214 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  y  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
y  +  1 )  e.  ( ( 1  +  1 ) ... ( ( N  - 
1 )  +  1 ) ) )
13871oveq2d 6319 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( ( 1  +  1 ) ... (
( N  -  1 )  +  1 ) )  =  ( ( 1  +  1 ) ... N ) )
139138adantr 467 . . . . . . . . . . . . . . . . . . . . 21  |-  ( (
ph  /\  y  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
( 1  +  1 ) ... ( ( N  -  1 )  +  1 ) )  =  ( ( 1  +  1 ) ... N ) )
140137, 139eleqtrd 2513 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  y  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
y  +  1 )  e.  ( ( 1  +  1 ) ... N ) )
141120zcnd 11043 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  e.  ( ( 1  +  1 ) ... N )  ->  n  e.  CC )
142133zcnd 11043 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( y  e.  ( 1 ... ( N  -  1 ) )  ->  y  e.  CC )
143 subadd2 9881 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( n  e.  CC  /\  1  e.  CC  /\  y  e.  CC )  ->  (
( n  -  1 )  =  y  <->  ( y  +  1 )  =  n ) )
144102, 143mp3an2 1349 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( n  e.  CC  /\  y  e.  CC )  ->  ( ( n  - 
1 )  =  y  <-> 
( y  +  1 )  =  n ) )
145 eqcom 2432 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( y  =  ( n  - 
1 )  <->  ( n  -  1 )  =  y )
146 eqcom 2432 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( n  =  ( y  +  1 )  <->  ( y  +  1 )  =  n )
147144, 145, 1463bitr4g 292 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( n  e.  CC  /\  y  e.  CC )  ->  ( y  =  ( n  -  1 )  <-> 
n  =  ( y  +  1 ) ) )
148141, 142, 147syl2anr 481 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  e.  ( 1 ... ( N  - 
1 ) )  /\  n  e.  ( (
1  +  1 ) ... N ) )  ->  ( y  =  ( n  -  1 )  <->  n  =  (
y  +  1 ) ) )
149148ralrimiva 2840 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  ( 1 ... ( N  -  1 ) )  ->  A. n  e.  ( ( 1  +  1 ) ... N
) ( y  =  ( n  -  1 )  <->  n  =  (
y  +  1 ) ) )
150149adantl 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  y  e.  ( 1 ... ( N  -  1 ) ) )  ->  A. n  e.  ( ( 1  +  1 ) ... N
) ( y  =  ( n  -  1 )  <->  n  =  (
y  +  1 ) ) )
151 reu6i 3263 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( y  +  1 )  e.  ( ( 1  +  1 ) ... N )  /\  A. n  e.  ( ( 1  +  1 ) ... N ) ( y  =  ( n  -  1 )  <->  n  =  ( y  +  1 ) ) )  ->  E! n  e.  (
( 1  +  1 ) ... N ) y  =  ( n  -  1 ) )
152140, 150, 151syl2anc 666 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  y  e.  ( 1 ... ( N  -  1 ) ) )  ->  E! n  e.  ( (
1  +  1 ) ... N ) y  =  ( n  - 
1 ) )
153152ralrimiva 2840 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  A. y  e.  ( 1 ... ( N  -  1 ) ) E! n  e.  ( ( 1  +  1 ) ... N ) y  =  ( n  -  1 ) )
154 eqid 2423 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  ( ( 1  +  1 ) ... N )  |->  ( n  -  1 ) )  =  ( n  e.  ( ( 1  +  1 ) ... N
)  |->  ( n  - 
1 ) )
155154f1ompt 6057 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  ( ( 1  +  1 ) ... N )  |->  ( n  -  1 ) ) : ( ( 1  +  1 ) ... N ) -1-1-onto-> ( 1 ... ( N  - 
1 ) )  <->  ( A. n  e.  ( (
1  +  1 ) ... N ) ( n  -  1 )  e.  ( 1 ... ( N  -  1 ) )  /\  A. y  e.  ( 1 ... ( N  - 
1 ) ) E! n  e.  ( ( 1  +  1 ) ... N ) y  =  ( n  - 
1 ) ) )
156128, 153, 155sylanbrc 669 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( n  e.  ( ( 1  +  1 ) ... N ) 
|->  ( n  -  1 ) ) : ( ( 1  +  1 ) ... N ) -1-1-onto-> ( 1 ... ( N  -  1 ) ) )
157 f1osng 5867 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1  e.  _V  /\  N  e.  NN )  ->  { <. 1 ,  N >. } : { 1 } -1-1-onto-> { N } )
15833, 68, 157sylancr 668 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  { <. 1 ,  N >. } : { 1 } -1-1-onto-> { N } )
15968nnred 10626 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  N  e.  RR )
160159ltm1d 10541 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( N  -  1 )  <  N )
161131zred 11042 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( N  -  1 )  e.  RR )
162161, 159ltnled 9784 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( N  - 
1 )  <  N  <->  -.  N  <_  ( N  -  1 ) ) )
163160, 162mpbid 214 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  -.  N  <_  ( N  -  1 ) )
164 elfzle2 11805 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  ( 1 ... ( N  -  1 ) )  ->  N  <_  ( N  -  1 ) )
165163, 164nsyl 125 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  -.  N  e.  ( 1 ... ( N  -  1 ) ) )
166 disjsn 4058 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1 ... ( N  -  1 ) )  i^i  { N } )  =  (/)  <->  -.  N  e.  ( 1 ... ( N  - 
1 ) ) )
167165, 166sylibr 216 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( 1 ... ( N  -  1 ) )  i^i  { N } )  =  (/) )
168 1re 9644 . . . . . . . . . . . . . . . . . . . . . 22  |-  1  e.  RR
169168ltp1i 10512 . . . . . . . . . . . . . . . . . . . . 21  |-  1  <  ( 1  +  1 )
170118zrei 10945 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 1  +  1 )  e.  RR
171168, 170ltnlei 9757 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 1  <  ( 1  +  1 )  <->  -.  (
1  +  1 )  <_  1 )
172169, 171mpbi 212 . . . . . . . . . . . . . . . . . . . 20  |-  -.  (
1  +  1 )  <_  1
173 elfzle1 11804 . . . . . . . . . . . . . . . . . . . 20  |-  ( 1  e.  ( ( 1  +  1 ) ... N )  ->  (
1  +  1 )  <_  1 )
174172, 173mto 180 . . . . . . . . . . . . . . . . . . 19  |-  -.  1  e.  ( ( 1  +  1 ) ... N
)
175 disjsn 4058 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( 1  +  1 ) ... N
)  i^i  { 1 } )  =  (/)  <->  -.  1  e.  ( (
1  +  1 ) ... N ) )
176174, 175mpbir 213 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1  +  1 ) ... N )  i^i  { 1 } )  =  (/)
177 f1oun 5848 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  e.  ( ( 1  +  1 ) ... N
)  |->  ( n  - 
1 ) ) : ( ( 1  +  1 ) ... N
)
-1-1-onto-> ( 1 ... ( N  -  1 ) )  /\  { <. 1 ,  N >. } : { 1 } -1-1-onto-> { N } )  /\  ( ( ( ( 1  +  1 ) ... N )  i^i 
{ 1 } )  =  (/)  /\  (
( 1 ... ( N  -  1 ) )  i^i  { N } )  =  (/) ) )  ->  (
( n  e.  ( ( 1  +  1 ) ... N ) 
|->  ( n  -  1 ) )  u.  { <. 1 ,  N >. } ) : ( ( ( 1  +  1 ) ... N )  u.  { 1 } ) -1-1-onto-> ( ( 1 ... ( N  -  1 ) )  u.  { N } ) )
178176, 177mpanr1 688 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( n  e.  ( ( 1  +  1 ) ... N
)  |->  ( n  - 
1 ) ) : ( ( 1  +  1 ) ... N
)
-1-1-onto-> ( 1 ... ( N  -  1 ) )  /\  { <. 1 ,  N >. } : { 1 } -1-1-onto-> { N } )  /\  ( ( 1 ... ( N  -  1 ) )  i^i  { N } )  =  (/) )  ->  ( ( n  e.  ( ( 1  +  1 ) ... N )  |->  ( n  -  1 ) )  u.  { <. 1 ,  N >. } ) : ( ( ( 1  +  1 ) ... N )  u.  {
1 } ) -1-1-onto-> ( ( 1 ... ( N  -  1 ) )  u.  { N }
) )
179156, 158, 167, 178syl21anc 1264 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( n  e.  ( ( 1  +  1 ) ... N
)  |->  ( n  - 
1 ) )  u. 
{ <. 1 ,  N >. } ) : ( ( ( 1  +  1 ) ... N
)  u.  { 1 } ) -1-1-onto-> ( ( 1 ... ( N  -  1 ) )  u.  { N } ) )
180 eleq1 2495 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( n  =  1  ->  (
n  e.  ( ( 1  +  1 ) ... N )  <->  1  e.  ( ( 1  +  1 ) ... N
) ) )
181174, 180mtbiri 305 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  =  1  ->  -.  n  e.  ( (
1  +  1 ) ... N ) )
182181necon2ai 2660 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  ( ( 1  +  1 ) ... N )  ->  n  =/=  1 )
183 ifnefalse 3922 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  =/=  1  ->  if ( n  =  1 ,  N ,  ( n  -  1 ) )  =  ( n  - 
1 ) )
184182, 183syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  ( ( 1  +  1 ) ... N )  ->  if ( n  =  1 ,  N ,  ( n  -  1 ) )  =  ( n  - 
1 ) )
185184mpteq2ia 4504 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  ( ( 1  +  1 ) ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) )  =  ( n  e.  ( ( 1  +  1 ) ... N )  |->  ( n  -  1 ) )
186185uneq1i 3617 . . . . . . . . . . . . . . . . . 18  |-  ( ( n  e.  ( ( 1  +  1 ) ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) )  u.  { <. 1 ,  N >. } )  =  ( ( n  e.  ( ( 1  +  1 ) ... N )  |->  ( n  -  1 ) )  u.  { <. 1 ,  N >. } )
18733a1i 11 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  1  e.  _V )
188 ssv 3485 . . . . . . . . . . . . . . . . . . . 20  |-  NN  C_  _V
189188, 68sseldi 3463 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  _V )
19068, 65syl6eleq 2521 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
191 fzpred 11846 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  ( ZZ>= `  1
)  ->  ( 1 ... N )  =  ( { 1 }  u.  ( ( 1  +  1 ) ... N ) ) )
192190, 191syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( 1 ... N
)  =  ( { 1 }  u.  (
( 1  +  1 ) ... N ) ) )
193 uncom 3611 . . . . . . . . . . . . . . . . . . . 20  |-  ( { 1 }  u.  (
( 1  +  1 ) ... N ) )  =  ( ( ( 1  +  1 ) ... N )  u.  { 1 } )
194192, 193syl6req 2481 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( ( 1  +  1 ) ... N )  u.  {
1 } )  =  ( 1 ... N
) )
195 iftrue 3916 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  =  1  ->  if ( n  =  1 ,  N ,  ( n  -  1 ) )  =  N )
196195adantl 468 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  n  = 
1 )  ->  if ( n  =  1 ,  N ,  ( n  -  1 ) )  =  N )
197187, 189, 194, 196fmptapd 6101 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( n  e.  ( ( 1  +  1 ) ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) )  u.  { <. 1 ,  N >. } )  =  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
198186, 197syl5eqr 2478 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( n  e.  ( ( 1  +  1 ) ... N
)  |->  ( n  - 
1 ) )  u. 
{ <. 1 ,  N >. } )  =  ( n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) )
19971, 190eqeltrd 2511 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= ` 
1 ) )
200 uzid 11175 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  -  1 )  e.  ZZ  ->  ( N  -  1 )  e.  ( ZZ>= `  ( N  -  1 ) ) )
201 peano2uz 11214 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( N  -  1 )  e.  ( ZZ>= `  ( N  -  1 ) )  ->  ( ( N  -  1 )  +  1 )  e.  ( ZZ>= `  ( N  -  1 ) ) )
202131, 200, 2013syl 18 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= `  ( N  -  1
) ) )
20371, 202eqeltrrd 2512 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  ( ZZ>= `  ( N  -  1
) ) )
204 fzsplit2 11826 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= ` 
1 )  /\  N  e.  ( ZZ>= `  ( N  -  1 ) ) )  ->  ( 1 ... N )  =  ( ( 1 ... ( N  -  1 ) )  u.  (
( ( N  - 
1 )  +  1 ) ... N ) ) )
205199, 203, 204syl2anc 666 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 1 ... N
)  =  ( ( 1 ... ( N  -  1 ) )  u.  ( ( ( N  -  1 )  +  1 ) ... N ) ) )
20671oveq1d 6318 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( ( N  -  1 )  +  1 ) ... N
)  =  ( N ... N ) )
207 fzsn 11842 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
208115, 207syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( N ... N
)  =  { N } )
209206, 208eqtrd 2464 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( ( N  -  1 )  +  1 ) ... N
)  =  { N } )
210209uneq2d 3621 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( 1 ... ( N  -  1 ) )  u.  (
( ( N  - 
1 )  +  1 ) ... N ) )  =  ( ( 1 ... ( N  -  1 ) )  u.  { N }
) )
211205, 210eqtr2d 2465 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( 1 ... ( N  -  1 ) )  u.  { N } )  =  ( 1 ... N ) )
212198, 194, 211f1oeq123d 5826 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( n  e.  ( ( 1  +  1 ) ... N )  |->  ( n  -  1 ) )  u.  { <. 1 ,  N >. } ) : ( ( ( 1  +  1 ) ... N )  u.  {
1 } ) -1-1-onto-> ( ( 1 ... ( N  -  1 ) )  u.  { N }
)  <->  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
213179, 212mpbid 214 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) )
214 f1oco 5851 . . . . . . . . . . . . . . 15  |-  ( ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  /\  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
21545, 213, 214syl2anc 666 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
216 dff1o3 5835 . . . . . . . . . . . . . . 15  |-  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  <->  ( (
( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) : ( 1 ... N ) -onto-> ( 1 ... N )  /\  Fun  `' ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) ) )
217216simprbi 466 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  ->  Fun  `' ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) )
218 imain 5675 . . . . . . . . . . . . . 14  |-  ( Fun  `' ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) )  -> 
( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( ( 1 ... ( y  +  1 ) )  i^i  (
( ( y  +  1 )  +  1 ) ... N ) ) )  =  ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  i^i  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) ) ) )
219215, 217, 2183syl 18 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( ( 1 ... ( y  +  1 ) )  i^i  (
( ( y  +  1 )  +  1 ) ... N ) ) )  =  ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  i^i  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) ) ) )
22064nnred 10626 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
y  +  1 )  e.  RR )
221220ltp1d 10539 . . . . . . . . . . . . . . . 16  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
y  +  1 )  <  ( ( y  +  1 )  +  1 ) )
222 fzdisj 11828 . . . . . . . . . . . . . . . 16  |-  ( ( y  +  1 )  <  ( ( y  +  1 )  +  1 )  ->  (
( 1 ... (
y  +  1 ) )  i^i  ( ( ( y  +  1 )  +  1 ) ... N ) )  =  (/) )
223221, 222syl 17 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( 1 ... (
y  +  1 ) )  i^i  ( ( ( y  +  1 )  +  1 ) ... N ) )  =  (/) )
224223imaeq2d 5185 . . . . . . . . . . . . . 14  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( 1 ... ( y  +  1 ) )  i^i  ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" (/) ) )
225 ima0 5200 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " (/) )  =  (/)
226224, 225syl6eq 2480 . . . . . . . . . . . . 13  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( 1 ... ( y  +  1 ) )  i^i  ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  (/) )
227219, 226sylan9req 2485 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  i^i  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  (/) )
228 fun 5761 . . . . . . . . . . . 12  |-  ( ( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  X.  { 1 } ) : ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) ) --> { 1 }  /\  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) : ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) --> { 0 } )  /\  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( 1 ... ( y  +  1 ) ) )  i^i  ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( ( ( y  +  1 )  +  1 ) ... N
) ) )  =  (/) )  ->  ( ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) ) : ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  u.  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) ) --> ( { 1 }  u.  { 0 } ) )
229113, 227, 228sylancr 668 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) : ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  u.  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) ) ) --> ( { 1 }  u.  { 0 } ) )
230 imaundi 5265 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( 1 ... ( y  +  1 ) )  u.  ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  u.  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) )
23164peano2nnd 10628 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( y  +  1 )  +  1 )  e.  NN )
232231, 65syl6eleq 2521 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( y  +  1 )  +  1 )  e.  ( ZZ>= `  1
) )
233232adantl 468 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( y  +  1 )  +  1 )  e.  ( ZZ>= `  1
) )
234 eluzp1p1 11186 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  -  1 )  e.  ( ZZ>= `  y
)  ->  ( ( N  -  1 )  +  1 )  e.  ( ZZ>= `  ( y  +  1 ) ) )
23573, 234syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( N  -  1 )  +  1 )  e.  ( ZZ>= `  (
y  +  1 ) ) )
236235adantl 468 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( N  -  1 )  +  1 )  e.  ( ZZ>= `  (
y  +  1 ) ) )
23772, 236eqeltrrd 2512 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  N  e.  ( ZZ>= `  ( y  +  1 ) ) )
238 fzsplit2 11826 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( y  +  1 )  +  1 )  e.  ( ZZ>= ` 
1 )  /\  N  e.  ( ZZ>= `  ( y  +  1 ) ) )  ->  ( 1 ... N )  =  ( ( 1 ... ( y  +  1 ) )  u.  (
( ( y  +  1 )  +  1 ) ... N ) ) )
239233, 237, 238syl2anc 666 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
1 ... N )  =  ( ( 1 ... ( y  +  1 ) )  u.  (
( ( y  +  1 )  +  1 ) ... N ) ) )
240239imaeq2d 5185 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... N ) )  =  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( 1 ... ( y  +  1 ) )  u.  ( ( ( y  +  1 )  +  1 ) ... N
) ) ) )
241 f1ofo 5836 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  -> 
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) )
242 foima 5813 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) : ( 1 ... N ) -onto-> ( 1 ... N )  ->  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
243215, 241, 2423syl 18 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... N
) )  =  ( 1 ... N ) )
244243adantr 467 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... N ) )  =  ( 1 ... N ) )
245240, 244eqtr3d 2466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( 1 ... ( y  +  1 ) )  u.  ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  ( 1 ... N ) )
246230, 245syl5eqr 2478 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  u.  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  ( 1 ... N ) )
247246feq2d 5731 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) : ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  u.  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) ) ) --> ( { 1 }  u.  { 0 } )  <->  ( (
( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) ) : ( 1 ... N ) --> ( { 1 }  u.  { 0 } ) ) )
248229, 247mpbid 214 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) : ( 1 ... N ) --> ( { 1 }  u.  { 0 } ) )
249248ffvelrnda 6035 . . . . . . . . 9  |-  ( ( ( ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
)  e.  ( { 1 }  u.  {
0 } ) )
250110, 249sseldi 3463 . . . . . . . 8  |-  ( ( ( ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
)  e.  CC )
251101, 105, 250subadd23d 10010 . . . . . . 7  |-  ( ( ( ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( ( 1st `  ( 1st `  T
) ) `  n
)  -  if ( n  =  ( ( 2nd `  ( 1st `  T ) ) `  N ) ,  1 ,  0 ) )  +  ( ( ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) ) `  n ) )  =  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  +  ( ( ( ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) ) `  n )  -  if ( n  =  ( ( 2nd `  ( 1st `  T
) ) `  N
) ,  1 ,  0 ) ) ) )
252 oveq2 6311 . . . . . . . . . 10  |-  ( 1  =  if ( n  =  ( ( 2nd `  ( 1st `  T
) ) `  N
) ,  1 ,  0 )  ->  (
( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
)  -  1 )  =  ( ( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
)  -  if ( n  =  ( ( 2nd `  ( 1st `  T ) ) `  N ) ,  1 ,  0 ) ) )
253252eqeq1d 2425 . . . . . . . . 9  |-  ( 1  =  if ( n  =  ( ( 2nd `  ( 1st `  T
) ) `  N
) ,  1 ,  0 )  ->  (
( ( ( ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) ) `  n )  -  1 )  =  ( ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( y  +  1 ) ... N
) )  X.  {
0 } ) ) `
 n )  <->  ( (
( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
)  -  if ( n  =  ( ( 2nd `  ( 1st `  T ) ) `  N ) ,  1 ,  0 ) )  =  ( ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( y  +  1 ) ... N
) )  X.  {
0 } ) ) `
 n ) ) )
254 oveq2 6311 . . . . . . . . . 10  |-  ( 0  =  if ( n  =  ( ( 2nd `  ( 1st `  T
) ) `  N
) ,  1 ,  0 )  ->  (
( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
)  -  0 )  =  ( ( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
)  -  if ( n  =  ( ( 2nd `  ( 1st `  T ) ) `  N ) ,  1 ,  0 ) ) )
255254eqeq1d 2425 . . . . . . . . 9  |-  ( 0  =  if ( n  =  ( ( 2nd `  ( 1st `  T
) ) `  N
) ,  1 ,  0 )  ->  (
( ( ( ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) ) `  n )  -  0 )  =  ( ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( y  +  1 ) ... N
) )  X.  {
0 } ) ) `
 n )  <->  ( (
( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( 1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) ) " ( ( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
)  -  if ( n  =  ( ( 2nd `  ( 1st `  T ) ) `  N ) ,  1 ,  0 ) )  =  ( ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... y ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( y  +  1 ) ... N
) )  X.  {
0 } ) ) `
 n ) ) )
256 1m1e0 10680 . . . . . . . . . . . . 13  |-  ( 1  -  1 )  =  0
257 f1ofn 5830 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  T
) )  Fn  (
1 ... N ) )
25845, 257syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) )  Fn  ( 1 ... N ) )
259258adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( 2nd `  ( 1st `  T
) )  Fn  (
1 ... N ) )
260 imassrn 5196 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) " (
1 ... ( y  +  1 ) ) ) 
C_  ran  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) )
261 f1of 5829 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  ->  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) : ( 1 ... N ) --> ( 1 ... N ) )
262213, 261syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  1 ,  N , 
( n  -  1 ) ) ) : ( 1 ... N
) --> ( 1 ... N ) )
263 frn 5750 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) ) : ( 1 ... N ) --> ( 1 ... N
)  ->  ran  ( n  e.  ( 1 ... N )  |->  if ( n  =  1 ,  N ,  ( n  -  1 ) ) )  C_  ( 1 ... N ) )
264262, 263syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ran  ( n  e.  ( 1 ... N
)  |->  if ( n  =  1 ,  N ,  ( n  - 
1 ) ) ) 
C_  ( 1 ... N ) )
265260, 264syl5ss 3476 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( n  e.  ( 1 ... N
)  |->  if (