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Theorem poimirlem16 31870
Description: Lemma for poimir 31887 establishing the vertices of the simplex of poimirlem17 31871. (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 )
poimirlem18.3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  E. p  e.  ran  F ( p `
 n )  =/= 
K )
poimirlem18.4  |-  ( ph  ->  ( 2nd `  T
)  =  0 )
Assertion
Ref Expression
poimirlem16  |-  ( ph  ->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  ( ( n  e.  ( 1 ... N )  |->  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  +  if ( n  =  ( ( 2nd `  ( 1st `  T ) ) `
 1 ) ,  1 ,  0 ) ) )  oF  +  ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( 1 ... y ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  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 poimirlem16
StepHypRef Expression
1 poimirlem22.2 . . 3  |-  ( ph  ->  T  e.  S )
2 fveq2 5878 . . . . . . . . . . 11  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
32breq2d 4432 . . . . . . . . . 10  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
43ifbid 3931 . . . . . . . . 9  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
54csbeq1d 3402 . . . . . . . 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 5878 . . . . . . . . . . 11  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
76fveq2d 5882 . . . . . . . . . 10  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
86fveq2d 5882 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
98imaeq1d 5183 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
109xpeq1d 4873 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
118imaeq1d 5183 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
1211xpeq1d 4873 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
1310, 12uneq12d 3621 . . . . . . . . . 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 6320 . . . . . . . . 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 3809 . . . . . . . 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 2463 . . . . . . 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 4508 . . . . . 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 2436 . . . . 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 3231 . . . 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 465 . . 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 3226 . . . . . . . . . . . 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 2530 . . . . . . . . . . 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 6834 . . . . . . . . . 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 6834 . . . . . . . . 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 7499 . . . . . . . 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 466 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( 1st `  ( 1st `  T
) )  Fn  (
1 ... N ) )
33 1ex 9639 . . . . . . . . . 10  |-  1  e.  _V
34 fnconstg 5785 . . . . . . . . . 10  |-  ( 1  e.  _V  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) ) )
3533, 34ax-mp 5 . . . . . . . . 9  |-  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )
36 c0ex 9638 . . . . . . . . . 10  |-  0  e.  _V
37 fnconstg 5785 . . . . . . . . . 10  |-  ( 0  e.  _V  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) ) )
3836, 37ax-mp 5 . . . . . . . . 9  |-  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) )
3935, 38pm3.2i 456 . . . . . . . 8  |-  ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  /\  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( ( y  +  1 )  +  1 ) ... N
) )  X.  {
0 } )  Fn  ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) )
40 xp2nd 6835 . . . . . . . . . . . . 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 5888 . . . . . . . . . . . . 13  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
43 f1oeq1 5819 . . . . . . . . . . . . 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 3218 . . . . . . . . . . . 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 199 . . . . . . . . . . 11  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
46 dff1o3 5834 . . . . . . . . . . . 12  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( ( 2nd `  ( 1st `  T
) ) : ( 1 ... N )
-onto-> ( 1 ... N
)  /\  Fun  `' ( 2nd `  ( 1st `  T ) ) ) )
4746simprbi 465 . . . . . . . . . . 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 5674 . . . . . . . . . 10  |-  ( Fun  `' ( 2nd `  ( 1st `  T ) )  ->  ( ( 2nd `  ( 1st `  T
) ) " (
( 1 ... (
y  +  1 ) )  i^i  ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  i^i  ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) ) )
5048, 49syl 17 . . . . . . . . 9  |-  ( ph  ->  ( ( 2nd `  ( 1st `  T ) )
" ( ( 1 ... ( y  +  1 ) )  i^i  ( ( ( y  +  1 )  +  1 ) ... N
) ) )  =  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  i^i  ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) ) ) )
51 elfznn0 11888 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  y  e.  NN0 )
52 nn0p1nn 10910 . . . . . . . . . . . . . . 15  |-  ( y  e.  NN0  ->  ( y  +  1 )  e.  NN )
5351, 52syl 17 . . . . . . . . . . . . . 14  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
y  +  1 )  e.  NN )
5453nnred 10625 . . . . . . . . . . . . 13  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
y  +  1 )  e.  RR )
5554ltp1d 10538 . . . . . . . . . . . 12  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
y  +  1 )  <  ( ( y  +  1 )  +  1 ) )
56 fzdisj 11827 . . . . . . . . . . . 12  |-  ( ( y  +  1 )  <  ( ( y  +  1 )  +  1 )  ->  (
( 1 ... (
y  +  1 ) )  i^i  ( ( ( y  +  1 )  +  1 ) ... N ) )  =  (/) )
5755, 56syl 17 . . . . . . . . . . 11  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( 1 ... (
y  +  1 ) )  i^i  ( ( ( y  +  1 )  +  1 ) ... N ) )  =  (/) )
5857imaeq2d 5184 . . . . . . . . . 10  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( 1 ... ( y  +  1 ) )  i^i  ( ( ( y  +  1 )  +  1 ) ... N
) ) )  =  ( ( 2nd `  ( 1st `  T ) )
" (/) ) )
59 ima0 5199 . . . . . . . . . 10  |-  ( ( 2nd `  ( 1st `  T ) ) " (/) )  =  (/)
6058, 59syl6eq 2479 . . . . . . . . 9  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( 1 ... ( y  +  1 ) )  i^i  ( ( ( y  +  1 )  +  1 ) ... N
) ) )  =  (/) )
6150, 60sylan9req 2484 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  i^i  ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  (/) )
62 fnun 5697 . . . . . . . 8  |-  ( ( ( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  Fn  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  /\  ( ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } )  Fn  (
( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) )  /\  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... (
y  +  1 ) ) )  i^i  (
( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  (/) )  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) )  Fn  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  u.  ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) ) )
6339, 61, 62sylancr 667 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) )  Fn  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  u.  ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) ) )
64 imaundi 5264 . . . . . . . . 9  |-  ( ( 2nd `  ( 1st `  T ) ) "
( ( 1 ... ( y  +  1 ) )  u.  (
( ( y  +  1 )  +  1 ) ... N ) ) )  =  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  u.  ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) )
65 nnuz 11195 . . . . . . . . . . . . . . 15  |-  NN  =  ( ZZ>= `  1 )
6653, 65syl6eleq 2520 . . . . . . . . . . . . . 14  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
y  +  1 )  e.  ( ZZ>= `  1
) )
67 peano2uz 11213 . . . . . . . . . . . . . 14  |-  ( ( y  +  1 )  e.  ( ZZ>= `  1
)  ->  ( (
y  +  1 )  +  1 )  e.  ( ZZ>= `  1 )
)
6866, 67syl 17 . . . . . . . . . . . . 13  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( y  +  1 )  +  1 )  e.  ( ZZ>= `  1
) )
6968adantl 467 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( y  +  1 )  +  1 )  e.  ( ZZ>= `  1
) )
70 poimir.0 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  NN )
7170nncnd 10626 . . . . . . . . . . . . . . 15  |-  ( ph  ->  N  e.  CC )
72 npcan1 10045 . . . . . . . . . . . . . . 15  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
7371, 72syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
7473adantr 466 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( N  -  1 )  +  1 )  =  N )
75 elfzuz3 11798 . . . . . . . . . . . . . . 15  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  ( N  -  1 )  e.  ( ZZ>= `  y
) )
76 eluzp1p1 11185 . . . . . . . . . . . . . . 15  |-  ( ( N  -  1 )  e.  ( ZZ>= `  y
)  ->  ( ( N  -  1 )  +  1 )  e.  ( ZZ>= `  ( y  +  1 ) ) )
7775, 76syl 17 . . . . . . . . . . . . . 14  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( N  -  1 )  +  1 )  e.  ( ZZ>= `  (
y  +  1 ) ) )
7877adantl 467 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( N  -  1 )  +  1 )  e.  ( ZZ>= `  (
y  +  1 ) ) )
7974, 78eqeltrrd 2511 . . . . . . . . . . . 12  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  N  e.  ( ZZ>= `  ( y  +  1 ) ) )
80 fzsplit2 11825 . . . . . . . . . . . 12  |-  ( ( ( ( y  +  1 )  +  1 )  e.  ( ZZ>= ` 
1 )  /\  N  e.  ( ZZ>= `  ( y  +  1 ) ) )  ->  ( 1 ... N )  =  ( ( 1 ... ( y  +  1 ) )  u.  (
( ( y  +  1 )  +  1 ) ... N ) ) )
8169, 79, 80syl2anc 665 . . . . . . . . . . 11  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
1 ... N )  =  ( ( 1 ... ( y  +  1 ) )  u.  (
( ( y  +  1 )  +  1 ) ... N ) ) )
8281imaeq2d 5184 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( 1 ... ( y  +  1 ) )  u.  ( ( ( y  +  1 )  +  1 ) ... N
) ) ) )
83 f1ofo 5835 . . . . . . . . . . . 12  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N )
-onto-> ( 1 ... N
) )
84 foima 5812 . . . . . . . . . . . 12  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
) -onto-> ( 1 ... N )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
8545, 83, 843syl 18 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
8685adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
8782, 86eqtr3d 2465 . . . . . . . . 9  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( 1 ... ( y  +  1 ) )  u.  ( ( ( y  +  1 )  +  1 ) ... N
) ) )  =  ( 1 ... N
) )
8864, 87syl5eqr 2477 . . . . . . . 8  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  u.  ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  ( 1 ... N ) )
8988fneq2d 5682 . . . . . . 7  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) )  Fn  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  u.  ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) ) )  <->  ( (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( ( y  +  1 )  +  1 ) ... N
) )  X.  {
0 } ) )  Fn  ( 1 ... N ) ) )
9063, 89mpbid 213 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) )  Fn  ( 1 ... N ) )
91 ovex 6330 . . . . . . 7  |-  ( 1 ... N )  e. 
_V
9291a1i 11 . . . . . 6  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
1 ... N )  e. 
_V )
93 inidm 3671 . . . . . 6  |-  ( ( 1 ... N )  i^i  ( 1 ... N ) )  =  ( 1 ... N
)
94 eqidd 2423 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  =  ( ( 1st `  ( 1st `  T ) ) `
 n ) )
95 eqidd 2423 . . . . . 6  |-  ( ( ( ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  n )  =  ( ( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  n ) )
9632, 90, 92, 92, 93, 94, 95offval 6549 . . . . 5  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  ( n  e.  ( 1 ... N )  |->  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  +  ( ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( ( y  +  1 )  +  1 ) ... N
) )  X.  {
0 } ) ) `
 n ) ) ) )
97 oveq1 6309 . . . . . . . . . 10  |-  ( 1  =  if ( n  =  ( ( 2nd `  ( 1st `  T
) ) `  1
) ,  1 ,  0 )  ->  (
1  +  ( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
" ( 1 ... y ) )  X. 
{ 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
) )  =  ( if ( n  =  ( ( 2nd `  ( 1st `  T ) ) `
 1 ) ,  1 ,  0 )  +  ( ( ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( 1 ... y
) )  X.  {
1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
" ( ( y  +  1 ) ... N ) )  X. 
{ 0 } ) ) `  n ) ) )
9897eqeq2d 2436 . . . . . . . . 9  |-  ( 1  =  if ( n  =  ( ( 2nd `  ( 1st `  T
) ) `  1
) ,  1 ,  0 )  ->  (
( ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( ( y  +  1 )  +  1 ) ... N
) )  X.  {
0 } ) ) `
 n )  =  ( 1  +  ( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( 1 ... y ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
) )  <->  ( (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) ) `  n )  =  ( if ( n  =  ( ( 2nd `  ( 1st `  T ) ) ` 
1 ) ,  1 ,  0 )  +  ( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( 1 ... y ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
) ) ) )
99 oveq1 6309 . . . . . . . . . 10  |-  ( 0  =  if ( n  =  ( ( 2nd `  ( 1st `  T
) ) `  1
) ,  1 ,  0 )  ->  (
0  +  ( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
" ( 1 ... y ) )  X. 
{ 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
) )  =  ( if ( n  =  ( ( 2nd `  ( 1st `  T ) ) `
 1 ) ,  1 ,  0 )  +  ( ( ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( 1 ... y
) )  X.  {
1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
" ( ( y  +  1 ) ... N ) )  X. 
{ 0 } ) ) `  n ) ) )
10099eqeq2d 2436 . . . . . . . . 9  |-  ( 0  =  if ( n  =  ( ( 2nd `  ( 1st `  T
) ) `  1
) ,  1 ,  0 )  ->  (
( ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( ( y  +  1 )  +  1 ) ... N
) )  X.  {
0 } ) ) `
 n )  =  ( 0  +  ( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( 1 ... y ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
) )  <->  ( (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) ) `  n )  =  ( if ( n  =  ( ( 2nd `  ( 1st `  T ) ) ` 
1 ) ,  1 ,  0 )  +  ( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( 1 ... y ) )  X.  { 1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) )  X.  { 0 } ) ) `  n
) ) ) )
101 1p0e1 10723 . . . . . . . . . . . . . 14  |-  ( 1  +  0 )  =  1
102101eqcomi 2435 . . . . . . . . . . . . 13  |-  1  =  ( 1  +  0 )
103 f1ofn 5829 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  T
) )  Fn  (
1 ... N ) )
10445, 103syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) )  Fn  ( 1 ... N ) )
105104adantr 466 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( 2nd `  ( 1st `  T
) )  Fn  (
1 ... N ) )
106 fzss2 11839 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ( ZZ>= `  (
y  +  1 ) )  ->  ( 1 ... ( y  +  1 ) )  C_  ( 1 ... N
) )
10779, 106syl 17 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
1 ... ( y  +  1 ) )  C_  ( 1 ... N
) )
108 eluzfz1 11807 . . . . . . . . . . . . . . . . . 18  |-  ( ( y  +  1 )  e.  ( ZZ>= `  1
)  ->  1  e.  ( 1 ... (
y  +  1 ) ) )
10966, 108syl 17 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  1  e.  ( 1 ... (
y  +  1 ) ) )
110109adantl 467 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  1  e.  ( 1 ... (
y  +  1 ) ) )
111 fnfvima 6155 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  ( 1st `  T ) )  Fn  ( 1 ... N )  /\  (
1 ... ( y  +  1 ) )  C_  ( 1 ... N
)  /\  1  e.  ( 1 ... (
y  +  1 ) ) )  ->  (
( 2nd `  ( 1st `  T ) ) `
 1 )  e.  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) ) )
112105, 107, 110, 111syl3anc 1264 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  T ) ) `
 1 )  e.  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) ) )
113 fvun1 5949 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  X.  { 1 } )  Fn  ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... (
y  +  1 ) ) )  /\  (
( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } )  Fn  ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) )  /\  ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  i^i  ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) ) )  =  (/)  /\  (
( 2nd `  ( 1st `  T ) ) `
 1 )  e.  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) ) ) )  ->  ( (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( ( y  +  1 )  +  1 ) ... N ) )  X. 
{ 0 } ) ) `  ( ( 2nd `  ( 1st `  T ) ) ` 
1 ) )  =  ( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } ) `  ( ( 2nd `  ( 1st `  T ) ) `
 1 ) ) )
11435, 38, 113mp3an12 1350 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  i^i  ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) ) )  =  (/)  /\  ( ( 2nd `  ( 1st `  T ) ) `
 1 )  e.  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) ) )  ->  ( ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( ( y  +  1 )  +  1 ) ... N
) )  X.  {
0 } ) ) `
 ( ( 2nd `  ( 1st `  T
) ) `  1
) )  =  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  X.  { 1 } ) `  ( ( 2nd `  ( 1st `  T ) ) ` 
1 ) ) )
11561, 112, 114syl2anc 665 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  ( ( 2nd `  ( 1st `  T ) ) `
 1 ) )  =  ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  X. 
{ 1 } ) `
 ( ( 2nd `  ( 1st `  T
) ) `  1
) ) )
11633fvconst2 6132 . . . . . . . . . . . . . . 15  |-  ( ( ( 2nd `  ( 1st `  T ) ) `
 1 )  e.  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... ( y  +  1 ) ) )  -> 
( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } ) `  ( ( 2nd `  ( 1st `  T ) ) `
 1 ) )  =  1 )
117112, 116syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... ( y  +  1 ) ) )  X.  { 1 } ) `  ( ( 2nd `  ( 1st `  T ) ) ` 
1 ) )  =  1 )
118115, 117eqtrd 2463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... (
y  +  1 ) ) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
( ( y  +  1 )  +  1 ) ... N ) )  X.  { 0 } ) ) `  ( ( 2nd `  ( 1st `  T ) ) `
 1 ) )  =  1 )
119 simpr 462 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
ph  /\  n  e.  ( 1 ... ( N  -  1 ) ) )  ->  n  e.  ( 1 ... ( N  -  1 ) ) )
12070nnzd 11040 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ph  ->  N  e.  ZZ )
121 peano2zm 10981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
122120, 121syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
123 1z 10968 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  1  e.  ZZ
124122, 123jctil 539 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( 1  e.  ZZ  /\  ( N  -  1 )  e.  ZZ ) )
125 elfzelz 11801 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( n  e.  ( 1 ... ( N  -  1 ) )  ->  n  e.  ZZ )
126125, 123jctir 540 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( n  e.  ( 1 ... ( N  -  1 ) )  ->  (
n  e.  ZZ  /\  1  e.  ZZ )
)
127 fzaddel 11834 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( ( 1  e.  ZZ  /\  ( N  -  1 )  e.  ZZ )  /\  ( n  e.  ZZ  /\  1  e.  ZZ ) )  -> 
( n  e.  ( 1 ... ( N  -  1 ) )  <-> 
( n  +  1 )  e.  ( ( 1  +  1 ) ... ( ( N  -  1 )  +  1 ) ) ) )
128124, 126, 127syl2an 479 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
ph  /\  n  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
n  e.  ( 1 ... ( N  - 
1 ) )  <->  ( n  +  1 )  e.  ( ( 1  +  1 ) ... (
( N  -  1 )  +  1 ) ) ) )
129119, 128mpbid 213 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
ph  /\  n  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
n  +  1 )  e.  ( ( 1  +  1 ) ... ( ( N  - 
1 )  +  1 ) ) )
13073oveq2d 6318 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( ( 1  +  1 ) ... (
( N  -  1 )  +  1 ) )  =  ( ( 1  +  1 ) ... N ) )
131130adantr 466 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
ph  /\  n  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
( 1  +  1 ) ... ( ( N  -  1 )  +  1 ) )  =  ( ( 1  +  1 ) ... N ) )
132129, 131eleqtrd 2512 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  n  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
n  +  1 )  e.  ( ( 1  +  1 ) ... N ) )
133132ralrimiva 2839 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  A. n  e.  ( 1 ... ( N  -  1 ) ) ( n  +  1 )  e.  ( ( 1  +  1 ) ... N ) )
134 simpr 462 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
ph  /\  y  e.  ( ( 1  +  1 ) ... N
) )  ->  y  e.  ( ( 1  +  1 ) ... N
) )
135 peano2z 10979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( 1  e.  ZZ  ->  (
1  +  1 )  e.  ZZ )
136123, 135ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( 1  +  1 )  e.  ZZ
137120, 136jctil 539 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  ( ( 1  +  1 )  e.  ZZ  /\  N  e.  ZZ ) )
138 elfzelz 11801 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( y  e.  ( ( 1  +  1 ) ... N )  ->  y  e.  ZZ )
139138, 123jctir 540 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( y  e.  ( ( 1  +  1 ) ... N )  ->  (
y  e.  ZZ  /\  1  e.  ZZ )
)
140 fzsubel 11835 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( ( ( 1  +  1 )  e.  ZZ  /\  N  e.  ZZ )  /\  ( y  e.  ZZ  /\  1  e.  ZZ ) )  -> 
( y  e.  ( ( 1  +  1 ) ... N )  <-> 
( y  -  1 )  e.  ( ( ( 1  +  1 )  -  1 ) ... ( N  - 
1 ) ) ) )
141137, 139, 140syl2an 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( (
ph  /\  y  e.  ( ( 1  +  1 ) ... N
) )  ->  (
y  e.  ( ( 1  +  1 ) ... N )  <->  ( y  -  1 )  e.  ( ( ( 1  +  1 )  - 
1 ) ... ( N  -  1 ) ) ) )
142134, 141mpbid 213 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
ph  /\  y  e.  ( ( 1  +  1 ) ... N
) )  ->  (
y  -  1 )  e.  ( ( ( 1  +  1 )  -  1 ) ... ( N  -  1 ) ) )
143 ax-1cn 9598 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  1  e.  CC
144143, 143pncan3oi 9892 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( 1  +  1 )  -  1 )  =  1
145144oveq1i 6312 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( 1  +  1 )  -  1 ) ... ( N  - 
1 ) )  =  ( 1 ... ( N  -  1 ) )
146142, 145syl6eleq 2520 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
ph  /\  y  e.  ( ( 1  +  1 ) ... N
) )  ->  (
y  -  1 )  e.  ( 1 ... ( N  -  1 ) ) )
147138zcnd 11042 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( y  e.  ( ( 1  +  1 ) ... N )  ->  y  e.  CC )
148 elfznn 11829 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( n  e.  ( 1 ... ( N  -  1 ) )  ->  n  e.  NN )
149148nncnd 10626 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( n  e.  ( 1 ... ( N  -  1 ) )  ->  n  e.  CC )
150 subadd2 9880 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31  |-  ( ( y  e.  CC  /\  1  e.  CC  /\  n  e.  CC )  ->  (
( y  -  1 )  =  n  <->  ( n  +  1 )  =  y ) )
151143, 150mp3an2 1348 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( ( y  e.  CC  /\  n  e.  CC )  ->  ( ( y  - 
1 )  =  n  <-> 
( n  +  1 )  =  y ) )
152151bicomd 204 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( y  e.  CC  /\  n  e.  CC )  ->  ( ( n  + 
1 )  =  y  <-> 
( y  -  1 )  =  n ) )
153 eqcom 2431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( n  +  1 )  =  y  <->  y  =  ( n  +  1
) )
154 eqcom 2431 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( y  -  1 )  =  n  <->  n  =  ( y  -  1 ) )
155152, 153, 1543bitr3g 290 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( y  e.  CC  /\  n  e.  CC )  ->  ( y  =  ( n  +  1 )  <-> 
n  =  ( y  -  1 ) ) )
156147, 149, 155syl2an 479 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ( y  e.  ( ( 1  +  1 ) ... N )  /\  n  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( y  =  ( n  +  1 )  <->  n  =  (
y  -  1 ) ) )
157156ralrimiva 2839 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( y  e.  ( ( 1  +  1 ) ... N )  ->  A. n  e.  ( 1 ... ( N  -  1 ) ) ( y  =  ( n  +  1 )  <->  n  =  (
y  -  1 ) ) )
158157adantl 467 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
ph  /\  y  e.  ( ( 1  +  1 ) ... N
) )  ->  A. n  e.  ( 1 ... ( N  -  1 ) ) ( y  =  ( n  +  1 )  <->  n  =  (
y  -  1 ) ) )
159 reu6i 3262 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ( ( y  -  1 )  e.  ( 1 ... ( N  - 
1 ) )  /\  A. n  e.  ( 1 ... ( N  - 
1 ) ) ( y  =  ( n  +  1 )  <->  n  =  ( y  -  1 ) ) )  ->  E! n  e.  (
1 ... ( N  - 
1 ) ) y  =  ( n  + 
1 ) )
160146, 158, 159syl2anc 665 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  y  e.  ( ( 1  +  1 ) ... N
) )  ->  E! n  e.  ( 1 ... ( N  - 
1 ) ) y  =  ( n  + 
1 ) )
161160ralrimiva 2839 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  A. y  e.  ( ( 1  +  1 ) ... N ) E! n  e.  ( 1 ... ( N  -  1 ) ) y  =  ( n  +  1 ) )
162 eqid 2422 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( n  e.  ( 1 ... ( N  -  1 ) )  |->  ( n  +  1 ) )  =  ( n  e.  ( 1 ... ( N  -  1 ) )  |->  ( n  + 
1 ) )
163162f1ompt 6056 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( n  e.  ( 1 ... ( N  - 
1 ) )  |->  ( n  +  1 ) ) : ( 1 ... ( N  - 
1 ) ) -1-1-onto-> ( ( 1  +  1 ) ... N )  <->  ( A. n  e.  ( 1 ... ( N  - 
1 ) ) ( n  +  1 )  e.  ( ( 1  +  1 ) ... N )  /\  A. y  e.  ( (
1  +  1 ) ... N ) E! n  e.  ( 1 ... ( N  - 
1 ) ) y  =  ( n  + 
1 ) ) )
164133, 161, 163sylanbrc 668 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( n  e.  ( 1 ... ( N  -  1 ) ) 
|->  ( n  +  1 ) ) : ( 1 ... ( N  -  1 ) ) -1-1-onto-> ( ( 1  +  1 ) ... N ) )
165 f1osng 5866 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( N  e.  NN  /\  1  e.  _V )  ->  { <. N ,  1
>. } : { N }
-1-1-onto-> { 1 } )
16670, 33, 165sylancl 666 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  { <. N ,  1
>. } : { N }
-1-1-onto-> { 1 } )
16770nnred 10625 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  N  e.  RR )
168167ltm1d 10540 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  ( N  -  1 )  <  N )
169122zred 11041 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( N  -  1 )  e.  RR )
170169, 167ltnled 9783 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  ( ( N  - 
1 )  <  N  <->  -.  N  <_  ( N  -  1 ) ) )
171168, 170mpbid 213 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  -.  N  <_  ( N  -  1 ) )
172 elfzle2 11804 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( N  e.  ( 1 ... ( N  -  1 ) )  ->  N  <_  ( N  -  1 ) )
173171, 172nsyl 124 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  -.  N  e.  ( 1 ... ( N  -  1 ) ) )
174 disjsn 4057 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( 1 ... ( N  -  1 ) )  i^i  { N } )  =  (/)  <->  -.  N  e.  ( 1 ... ( N  - 
1 ) ) )
175173, 174sylibr 215 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( ( 1 ... ( N  -  1 ) )  i^i  { N } )  =  (/) )
176 1re 9643 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  1  e.  RR
177176ltp1i 10511 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  1  <  ( 1  +  1 )
178176, 176readdcli 9657 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( 1  +  1 )  e.  RR
179176, 178ltnlei 9756 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1  <  ( 1  +  1 )  <->  -.  (
1  +  1 )  <_  1 )
180177, 179mpbi 211 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  -.  (
1  +  1 )  <_  1
181 elfzle1 11803 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 1  e.  ( ( 1  +  1 ) ... N )  ->  (
1  +  1 )  <_  1 )
182180, 181mto 179 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  -.  1  e.  ( ( 1  +  1 ) ... N
)
183 disjsn 4057 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( ( 1  +  1 ) ... N
)  i^i  { 1 } )  =  (/)  <->  -.  1  e.  ( (
1  +  1 ) ... N ) )
184182, 183mpbir 212 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( 1  +  1 ) ... N )  i^i  { 1 } )  =  (/)
185 f1oun 5847 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( ( ( n  e.  ( 1 ... ( N  -  1 ) )  |->  ( n  + 
1 ) ) : ( 1 ... ( N  -  1 ) ) -1-1-onto-> ( ( 1  +  1 ) ... N
)  /\  { <. N , 
1 >. } : { N } -1-1-onto-> { 1 } )  /\  ( ( ( 1 ... ( N  -  1 ) )  i^i  { N }
)  =  (/)  /\  (
( ( 1  +  1 ) ... N
)  i^i  { 1 } )  =  (/) ) )  ->  (
( n  e.  ( 1 ... ( N  -  1 ) ) 
|->  ( n  +  1 ) )  u.  { <. N ,  1 >. } ) : ( ( 1 ... ( N  -  1 ) )  u.  { N } ) -1-1-onto-> ( ( ( 1  +  1 ) ... N )  u.  {
1 } ) )
186184, 185mpanr2 688 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( ( ( n  e.  ( 1 ... ( N  -  1 ) )  |->  ( n  + 
1 ) ) : ( 1 ... ( N  -  1 ) ) -1-1-onto-> ( ( 1  +  1 ) ... N
)  /\  { <. N , 
1 >. } : { N } -1-1-onto-> { 1 } )  /\  ( ( 1 ... ( N  - 
1 ) )  i^i 
{ N } )  =  (/) )  ->  (
( n  e.  ( 1 ... ( N  -  1 ) ) 
|->  ( n  +  1 ) )  u.  { <. N ,  1 >. } ) : ( ( 1 ... ( N  -  1 ) )  u.  { N } ) -1-1-onto-> ( ( ( 1  +  1 ) ... N )  u.  {
1 } ) )
187164, 166, 175, 186syl21anc 1263 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( N  -  1 ) )  |->  ( n  + 
1 ) )  u. 
{ <. N ,  1
>. } ) : ( ( 1 ... ( N  -  1 ) )  u.  { N } ) -1-1-onto-> ( ( ( 1  +  1 ) ... N )  u.  {
1 } ) )
188 ssv 3484 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  NN  C_  _V
189188, 70sseldi 3462 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  N  e.  _V )
19033a1i 11 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  1  e.  _V )
19170, 65syl6eleq 2520 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
19273, 191eqeltrd 2510 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= ` 
1 ) )
193 uzid 11174 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( N  -  1 )  e.  ZZ  ->  ( N  -  1 )  e.  ( ZZ>= `  ( N  -  1 ) ) )
194 peano2uz 11213 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( N  -  1 )  e.  ( ZZ>= `  ( N  -  1 ) )  ->  ( ( N  -  1 )  +  1 )  e.  ( ZZ>= `  ( N  -  1 ) ) )
195122, 193, 1943syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= `  ( N  -  1
) ) )
19673, 195eqeltrrd 2511 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  N  e.  ( ZZ>= `  ( N  -  1
) ) )
197 fzsplit2 11825 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ( ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= ` 
1 )  /\  N  e.  ( ZZ>= `  ( N  -  1 ) ) )  ->  ( 1 ... N )  =  ( ( 1 ... ( N  -  1 ) )  u.  (
( ( N  - 
1 )  +  1 ) ... N ) ) )
198192, 196, 197syl2anc 665 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  ( 1 ... N
)  =  ( ( 1 ... ( N  -  1 ) )  u.  ( ( ( N  -  1 )  +  1 ) ... N ) ) )
19973oveq1d 6317 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( ( ( N  -  1 )  +  1 ) ... N
)  =  ( N ... N ) )
200 fzsn 11841 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
201120, 200syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( N ... N
)  =  { N } )
202199, 201eqtrd 2463 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( ph  ->  ( ( ( N  -  1 )  +  1 ) ... N
)  =  { N } )
203202uneq2d 3620 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( ph  ->  ( ( 1 ... ( N  -  1 ) )  u.  (
( ( N  - 
1 )  +  1 ) ... N ) )  =  ( ( 1 ... ( N  -  1 ) )  u.  { N }
) )
204198, 203eqtr2d 2464 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  ( ( 1 ... ( N  -  1 ) )  u.  { N } )  =  ( 1 ... N ) )
205 iftrue 3915 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( n  =  N  ->  if ( n  =  N ,  1 ,  ( n  +  1 ) )  =  1 )
206205adantl 467 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( (
ph  /\  n  =  N )  ->  if ( n  =  N ,  1 ,  ( n  +  1 ) )  =  1 )
207189, 190, 204, 206fmptapd 6100 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( N  -  1 ) )  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) )  u.  { <. N , 
1 >. } )  =  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
208 eleq1 2494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30  |-  ( n  =  N  ->  (
n  e.  ( 1 ... ( N  - 
1 ) )  <->  N  e.  ( 1 ... ( N  -  1 ) ) ) )
209208notbid 295 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( n  =  N  ->  ( -.  n  e.  (
1 ... ( N  - 
1 ) )  <->  -.  N  e.  ( 1 ... ( N  -  1 ) ) ) )
210173, 209syl5ibrcom 225 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ph  ->  ( n  =  N  ->  -.  n  e.  ( 1 ... ( N  -  1 ) ) ) )
211210necon2ad 2637 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( ph  ->  ( n  e.  ( 1 ... ( N  -  1 ) )  ->  n  =/=  N
) )
212211imp 430 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( (
ph  /\  n  e.  ( 1 ... ( N  -  1 ) ) )  ->  n  =/=  N )
213 ifnefalse 3921 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( n  =/=  N  ->  if ( n  =  N ,  1 ,  ( n  +  1 ) )  =  ( n  +  1 ) )
214212, 213syl 17 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( (
ph  /\  n  e.  ( 1 ... ( N  -  1 ) ) )  ->  if ( n  =  N ,  1 ,  ( n  +  1 ) )  =  ( n  +  1 ) )
215214mpteq2dva 4507 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ph  ->  ( n  e.  ( 1 ... ( N  -  1 ) ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) )  =  ( n  e.  ( 1 ... ( N  -  1 ) ) 
|->  ( n  +  1 ) ) )
216215uneq1d 3619 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( ( n  e.  ( 1 ... ( N  -  1 ) )  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) )  u.  { <. N , 
1 >. } )  =  ( ( n  e.  ( 1 ... ( N  -  1 ) )  |->  ( n  + 
1 ) )  u. 
{ <. N ,  1
>. } ) )
217207, 216eqtr3d 2465 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) )  =  ( ( n  e.  ( 1 ... ( N  -  1 ) )  |->  ( n  + 
1 ) )  u. 
{ <. N ,  1
>. } ) )
218198, 203eqtrd 2463 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( 1 ... N
)  =  ( ( 1 ... ( N  -  1 ) )  u.  { N }
) )
219 uzid 11174 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 1  e.  ZZ  ->  1  e.  ( ZZ>= `  1 )
)
220 peano2uz 11213 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 1  e.  ( ZZ>= `  1
)  ->  ( 1  +  1 )  e.  ( ZZ>= `  1 )
)
221123, 219, 220mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 1  +  1 )  e.  ( ZZ>= `  1 )
222 fzsplit2 11825 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( ( 1  +  1 )  e.  ( ZZ>= ` 
1 )  /\  N  e.  ( ZZ>= `  1 )
)  ->  ( 1 ... N )  =  ( ( 1 ... 1 )  u.  (
( 1  +  1 ) ... N ) ) )
223221, 191, 222sylancr 667 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ph  ->  ( 1 ... N
)  =  ( ( 1 ... 1 )  u.  ( ( 1  +  1 ) ... N ) ) )
224 fzsn 11841 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 1  e.  ZZ  ->  (
1 ... 1 )  =  { 1 } )
225123, 224ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 1 ... 1 )  =  { 1 }
226225uneq1i 3616 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( ( 1 ... 1 )  u.  ( ( 1  +  1 ) ... N ) )  =  ( { 1 }  u.  ( ( 1  +  1 ) ... N ) )
227226equncomi 3612 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( ( 1 ... 1 )  u.  ( ( 1  +  1 ) ... N ) )  =  ( ( ( 1  +  1 ) ... N )  u.  {
1 } )
228223, 227syl6eq 2479 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  ( 1 ... N
)  =  ( ( ( 1  +  1 ) ... N )  u.  { 1 } ) )
229217, 218, 228f1oeq123d 5825 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  <->  ( ( n  e.  ( 1 ... ( N  -  1 ) )  |->  ( n  +  1 ) )  u.  { <. N , 
1 >. } ) : ( ( 1 ... ( N  -  1 ) )  u.  { N } ) -1-1-onto-> ( ( ( 1  +  1 ) ... N )  u.  {
1 } ) ) )
230187, 229mpbird 235 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) )
231 f1oco 5850 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
)  /\  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
23245, 230, 231syl2anc 665 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
233 dff1o3 5834 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  <->  ( (
( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) : ( 1 ... N ) -onto-> ( 1 ... N )  /\  Fun  `' ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) ) )
234233simprbi 465 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  ->  Fun  `' ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) )
235232, 234syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  Fun  `' ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) )
236 imain 5674 . . . . . . . . . . . . . . . . . 18  |-  ( Fun  `' ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )  -> 
( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( ( 1 ... y )  i^i  (
( y  +  1 ) ... N ) ) )  =  ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( 1 ... y
) )  i^i  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) ) ) )
237235, 236syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( ( 1 ... y )  i^i  (
( y  +  1 ) ... N ) ) )  =  ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( 1 ... y
) )  i^i  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) ) ) )
23851nn0red 10927 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  y  e.  RR )
239238ltp1d 10538 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  y  <  ( y  +  1 ) )
240 fzdisj 11827 . . . . . . . . . . . . . . . . . . . 20  |-  ( y  <  ( y  +  1 )  ->  (
( 1 ... y
)  i^i  ( (
y  +  1 ) ... N ) )  =  (/) )
241239, 240syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( 1 ... y
)  i^i  ( (
y  +  1 ) ... N ) )  =  (/) )
242241imaeq2d 5184 . . . . . . . . . . . . . . . . . 18  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( 1 ... y )  i^i  ( ( y  +  1 ) ... N ) ) )  =  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
" (/) ) )
243 ima0 5199 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " (/) )  =  (/)
244242, 243syl6eq 2479 . . . . . . . . . . . . . . . . 17  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( 1 ... y )  i^i  ( ( y  +  1 ) ... N ) ) )  =  (/) )
245237, 244sylan9req 2484 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( 1 ... y
) )  i^i  (
( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) ) )  =  (/) )
246 imassrn 5195 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) " (
( y  +  1 ) ... N ) )  C_  ran  ( n  e.  ( 1 ... N )  |->  if ( n  =  N , 
1 ,  ( n  +  1 ) ) )
247 f1of 5828 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  ->  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) : ( 1 ... N ) --> ( 1 ... N ) )
248 frn 5749 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) : ( 1 ... N ) --> ( 1 ... N
)  ->  ran  ( n  e.  ( 1 ... N )  |->  if ( n  =  N , 
1 ,  ( n  +  1 ) ) )  C_  ( 1 ... N ) )
249230, 247, 2483syl 18 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ran  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) 
C_  ( 1 ... N ) )
250246, 249syl5ss 3475 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) )
" ( ( y  +  1 ) ... N ) )  C_  ( 1 ... N
) )
251250adantr 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) "
( ( y  +  1 ) ... N
) )  C_  (
1 ... N ) )
252 elfz1end 11830 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( N  e.  NN  <->  N  e.  ( 1 ... N
) )
25370, 252sylib 199 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  N  e.  ( 1 ... N ) )
254 eqid 2422 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( n  e.  ( 1 ... N )  |->  if ( n  =  N , 
1 ,  ( n  +  1 ) ) )  =  ( n  e.  ( 1 ... N )  |->  if ( n  =  N , 
1 ,  ( n  +  1 ) ) )
255205, 254, 33fvmpt 5961 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  ( 1 ... N )  ->  (
( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) `  N )  =  1 )
256253, 255syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) `
 N )  =  1 )
257256adantr 466 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) `  N )  =  1 )
258 f1ofn 5829 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N )  ->  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) )  Fn  ( 1 ... N ) )
259230, 258syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) )  Fn  ( 1 ... N
) )
260259adantr 466 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) )  Fn  (
1 ... N ) )
261 fzss1 11838 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( y  +  1 )  e.  ( ZZ>= `  1
)  ->  ( (
y  +  1 ) ... N )  C_  ( 1 ... N
) )
26266, 261syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( y  e.  ( 0 ... ( N  -  1 ) )  ->  (
( y  +  1 ) ... N ) 
C_  ( 1 ... N ) )
263262adantl 467 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( y  +  1 ) ... N ) 
C_  ( 1 ... N ) )
264 eluzfz2 11808 . . . . . . . . . . . . . . . . . . . . 21  |-  ( N  e.  ( ZZ>= `  (
y  +  1 ) )  ->  N  e.  ( ( y  +  1 ) ... N
) )
26579, 264syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  N  e.  ( ( y  +  1 ) ... N
) )
266 fnfvima 6155 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) )  Fn  ( 1 ... N
)  /\  ( (
y  +  1 ) ... N )  C_  ( 1 ... N
)  /\  N  e.  ( ( y  +  1 ) ... N
) )  ->  (
( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) `  N )  e.  ( ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) "
( ( y  +  1 ) ... N
) ) )
267260, 263, 265, 266syl3anc 1264 . . . . . . . . . . . . . . . . . . 19  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) `  N )  e.  ( ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) "
( ( y  +  1 ) ... N
) ) )
268257, 267eqeltrrd 2511 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  1  e.  ( ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) )
" ( ( y  +  1 ) ... N ) ) )
269 fnfvima 6155 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  ( 1st `  T ) )  Fn  ( 1 ... N )  /\  (
( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) "
( ( y  +  1 ) ... N
) )  C_  (
1 ... N )  /\  1  e.  ( (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) " (
( y  +  1 ) ... N ) ) )  ->  (
( 2nd `  ( 1st `  T ) ) `
 1 )  e.  ( ( 2nd `  ( 1st `  T ) )
" ( ( n  e.  ( 1 ... N )  |->  if ( n  =  N , 
1 ,  ( n  +  1 ) ) ) " ( ( y  +  1 ) ... N ) ) ) )
270105, 251, 268, 269syl3anc 1264 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  T ) ) `
 1 )  e.  ( ( 2nd `  ( 1st `  T ) )
" ( ( n  e.  ( 1 ... N )  |->  if ( n  =  N , 
1 ,  ( n  +  1 ) ) ) " ( ( y  +  1 ) ... N ) ) ) )
271 imaco 5356 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T
) ) " (
( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) "
( ( y  +  1 ) ... N
) ) )
272270, 271syl6eleqr 2521 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  y  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  T ) ) `
 1 )  e.  ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( ( y  +  1 ) ... N
) ) )
273 fnconstg 5785 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  _V  ->  (
( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( 1 ... y
) )  X.  {
1 } )  Fn  ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( 1 ... y
) ) )
27433, 273ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( 1 ... y ) )  X.  { 1 } )  Fn  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( 1 ... y ) )
275 fnconstg 5785 . . . . . . . . . . . . . . . . . 18  |-  ( 0  e.  _V  ->  (
( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( ( y  +  1 ) ... N
) )  X.  {
0 } )  Fn  ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( ( y  +  1 ) ... N
) ) )
27636, 275ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) )  X.  { 0 } )  Fn  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( ( y  +  1 ) ... N ) )
277 fvun2 5950 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
" ( 1 ... y ) )  X. 
{ 1 } )  Fn  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
" ( 1 ... y ) )  /\  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
" ( ( y  +  1 ) ... N ) )  X. 
{ 0 } )  Fn  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
" ( ( y  +  1 ) ... N ) )  /\  ( ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N
)  |->  if ( n  =  N ,  1 ,  ( n  + 
1 ) ) ) ) " ( 1 ... y ) )  i^i  ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  ( 1 ... N ) 
|->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) )
" ( ( y  +  1 ) ... N ) ) )  =  (/)  /\  (
( 2nd `  ( 1st `  T ) ) `
 1 )  e.  ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( ( y  +  1 ) ... N
) ) ) )  ->  ( ( ( ( ( ( 2nd `  ( 1st `  T
) )  o.  (
n  e.  ( 1 ... N )  |->  if ( n  =  N ,  1 ,  ( n  +  1 ) ) ) ) "
( 1 ... y
) )  X.  {
1 } )  u.  ( ( ( ( 2nd `  ( 1st `  T ) )  o.  ( n  e.  (