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Theorem poimirlem18 32022
Description: Lemma for poimir 32037 stating that, given a face not on a front face of the main cube and a simplex in which it's opposite the first vertex on the walk, there exists exactly one other simplex containing it. (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
poimirlem18  |-  ( ph  ->  E! z  e.  S  z  =/=  T )
Distinct variable groups:    f, j, n, p, t, y, z    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    ph, z    f, F, p, t, z    z, K    z, N    t, T, z    S, j, n, p, t, y, z
Allowed substitution hints:    ph( f)    S( f)    K( y)

Proof of Theorem poimirlem18
Dummy variable  s is distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . 3  |-  ( ph  ->  N  e.  NN )
2 poimirlem22.s . . 3  |-  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 } ) ) ) ) }
3 poimirlem22.1 . . 3  |-  ( ph  ->  F : ( 0 ... ( N  - 
1 ) ) --> ( ( 0 ... K
)  ^m  ( 1 ... N ) ) )
4 poimirlem22.2 . . 3  |-  ( ph  ->  T  e.  S )
5 poimirlem18.3 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  E. p  e.  ran  F ( p `
 n )  =/= 
K )
6 poimirlem18.4 . . 3  |-  ( ph  ->  ( 2nd `  T
)  =  0 )
71, 2, 3, 4, 5, 6poimirlem17 32021 . 2  |-  ( ph  ->  E. z  e.  S  z  =/=  T )
86adantr 472 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  ( 2nd `  T )  =  0 )
9 0nnn 10663 . . . . . . . . . . . . 13  |-  -.  0  e.  NN
10 elfznn 11854 . . . . . . . . . . . . 13  |-  ( 0  e.  ( 1 ... ( N  -  1 ) )  ->  0  e.  NN )
119, 10mto 181 . . . . . . . . . . . 12  |-  -.  0  e.  ( 1 ... ( N  -  1 ) )
12 eleq1 2537 . . . . . . . . . . . 12  |-  ( ( 2nd `  z )  =  0  ->  (
( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  <->  0  e.  ( 1 ... ( N  -  1 ) ) ) )
1311, 12mtbiri 310 . . . . . . . . . . 11  |-  ( ( 2nd `  z )  =  0  ->  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )
1413necon2ai 2672 . . . . . . . . . 10  |-  ( ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) )  ->  ( 2nd `  z )  =/=  0 )
151ad2antrr 740 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  N  e.  NN )
16 fveq2 5879 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  ( 2nd `  t )  =  ( 2nd `  z
) )
1716breq2d 4407 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  z  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  z ) ) )
1817ifbid 3894 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  z  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  z
) ,  y ,  ( y  +  1 ) ) )
1918csbeq1d 3356 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  z  ->  [_ 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 `  z ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
20 fveq2 5879 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  ( 1st `  t )  =  ( 1st `  z
) )
2120fveq2d 5883 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  z  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  z ) ) )
2220fveq2d 5883 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  z  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  z ) ) )
2322imaeq1d 5173 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  z  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... j ) ) )
2423xpeq1d 4862 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  z ) ) "
( 1 ... j
) )  X.  {
1 } ) )
2522imaeq1d 5173 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  z  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) ) )
2625xpeq1d 4862 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  z ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
2724, 26uneq12d 3580 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  z  ->  (
( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) )  =  ( ( ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... j ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  z ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) ) )
2821, 27oveq12d 6326 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  z  ->  (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  ( ( 1st `  ( 1st `  z ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
2928csbeq2dv 3785 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  z  ->  [_ if ( y  <  ( 2nd `  z ) ,  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 `  z ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  z ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
3019, 29eqtrd 2505 . . . . . . . . . . . . . . . . . . 19  |-  ( t  =  z  ->  [_ 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 `  z ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  z ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
3130mpteq2dv 4483 . . . . . . . . . . . . . . . . . 18  |-  ( t  =  z  ->  (
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 `  z
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  z ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
3231eqeq2d 2481 . . . . . . . . . . . . . . . . 17  |-  ( t  =  z  ->  ( 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 `  z
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  z ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) ) )
3332, 2elrab2 3186 . . . . . . . . . . . . . . . 16  |-  ( z  e.  S  <->  ( z  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 `  z
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  z ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) ) )
3433simprbi 471 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  z
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  z ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
3534ad2antlr 741 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  z
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  z ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
36 elrabi 3181 . . . . . . . . . . . . . . . . . . . 20  |-  ( z  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 } ) ) ) ) }  ->  z  e.  ( ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
3736, 2eleq2s 2567 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  S  ->  z  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
38 xp1st 6842 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  ->  ( 1st `  z )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
3937, 38syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  S  ->  ( 1st `  z )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
40 xp1st 6842 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  z )  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  -> 
( 1st `  ( 1st `  z ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
4139, 40syl 17 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  S  ->  ( 1st `  ( 1st `  z
) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) ) )
42 elmapi 7511 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  ( 1st `  z ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  z ) ) : ( 1 ... N ) --> ( 0..^ K ) )
4341, 42syl 17 . . . . . . . . . . . . . . . 16  |-  ( z  e.  S  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ( 0..^ K ) )
44 elfzoelz 11947 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( 0..^ K )  ->  n  e.  ZZ )
4544ssriv 3422 . . . . . . . . . . . . . . . 16  |-  ( 0..^ K )  C_  ZZ
46 fss 5749 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  ( 1st `  z ) ) : ( 1 ... N ) --> ( 0..^ K )  /\  (
0..^ K )  C_  ZZ )  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ZZ )
4743, 45, 46sylancl 675 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ZZ )
4847ad2antlr 741 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ZZ )
49 xp2nd 6843 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  z )  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  -> 
( 2nd `  ( 1st `  z ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
5039, 49syl 17 . . . . . . . . . . . . . . . 16  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) )  e.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )
51 fvex 5889 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  ( 1st `  z
) )  e.  _V
52 f1oeq1 5818 . . . . . . . . . . . . . . . . 17  |-  ( f  =  ( 2nd `  ( 1st `  z ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  z ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
5351, 52elab 3173 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  ( 1st `  z ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
5450, 53sylib 201 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
5554ad2antlr 741 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
56 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )
5715, 35, 48, 55, 56poimirlem1 32005 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  -.  E* n  e.  (
1 ... N ) ( ( F `  (
( 2nd `  z
)  -  1 ) ) `  n )  =/=  ( ( F `
 ( 2nd `  z
) ) `  n
) )
581ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  N  e.  NN )
59 fveq2 5879 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
6059breq2d 4407 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
6160ifbid 3894 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
6261csbeq1d 3356 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 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 } ) ) ) )
63 fveq2 5879 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
6463fveq2d 5883 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
6563fveq2d 5883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
6665imaeq1d 5173 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
6766xpeq1d 4862 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
6865imaeq1d 5173 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
6968xpeq1d 4862 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
7067, 69uneq12d 3580 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( 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 } ) ) )
7164, 70oveq12d 6326 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( 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 } ) ) ) )
7271csbeq2dv 3785 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 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 } ) ) ) )
7362, 72eqtrd 2505 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( 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 } ) ) ) )
7473mpteq2dv 4483 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 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 } ) ) ) ) )
7574eqeq2d 2481 . . . . . . . . . . . . . . . . . . . . 21  |-  ( 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 } ) ) ) ) ) )
7675, 2elrab2 3186 . . . . . . . . . . . . . . . . . . . 20  |-  ( 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 } ) ) ) ) ) )
7776simprbi 471 . . . . . . . . . . . . . . . . . . 19  |-  ( 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 } ) ) ) ) )
784, 77syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( 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 } ) ) ) ) )
7978ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  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 } ) ) ) ) )
80 elrabi 3181 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( 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 ) ) )
8180, 2eleq2s 2567 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( T  e.  S  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
824, 81syl 17 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
83 xp1st 6842 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( 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 ) } ) )
8482, 83syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( 1st `  T
)  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } ) )
85 xp1st 6842 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( 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 ) ) )
8684, 85syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
87 elmapi 7511 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
8886, 87syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
89 fss 5749 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K )  /\  (
0..^ K )  C_  ZZ )  ->  ( 1st `  ( 1st `  T
) ) : ( 1 ... N ) --> ZZ )
9088, 45, 89sylancl 675 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ZZ )
9190ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 1st `  ( 1st `  T
) ) : ( 1 ... N ) --> ZZ )
92 xp2nd 6843 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 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 ) } )
9384, 92syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
94 fvex 5889 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
95 f1oeq1 5818 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  =  ( 2nd `  ( 1st `  T ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
9694, 95elab 3173 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  ( 1st `  T ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
9793, 96sylib 201 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
9897ad2antrr 740 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
99 simplr 770 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )
100 xp2nd 6843 . . . . . . . . . . . . . . . . . . . 20  |-  ( T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  ->  ( 2nd `  T )  e.  ( 0 ... N
) )
10182, 100syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( 2nd `  T
)  e.  ( 0 ... N ) )
102101adantr 472 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  T )  e.  ( 0 ... N ) )
103 eldifsn 4088 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  T )  e.  ( ( 0 ... N )  \  { ( 2nd `  z
) } )  <->  ( ( 2nd `  T )  e.  ( 0 ... N
)  /\  ( 2nd `  T )  =/=  ( 2nd `  z ) ) )
104103biimpri 211 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  T
)  e.  ( 0 ... N )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  T )  e.  ( ( 0 ... N )  \  {
( 2nd `  z
) } ) )
105102, 104sylan 479 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  T )  e.  ( ( 0 ... N )  \  {
( 2nd `  z
) } ) )
10658, 79, 91, 98, 99, 105poimirlem2 32006 . . . . . . . . . . . . . . . 16  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  E* n  e.  ( 1 ... N ) ( ( F `  (
( 2nd `  z
)  -  1 ) ) `  n )  =/=  ( ( F `
 ( 2nd `  z
) ) `  n
) )
107106ex 441 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( ( 2nd `  T )  =/=  ( 2nd `  z
)  ->  E* n  e.  ( 1 ... N
) ( ( F `
 ( ( 2nd `  z )  -  1 ) ) `  n
)  =/=  ( ( F `  ( 2nd `  z ) ) `  n ) ) )
108107necon1bd 2661 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( -.  E* n  e.  (
1 ... N ) ( ( F `  (
( 2nd `  z
)  -  1 ) ) `  n )  =/=  ( ( F `
 ( 2nd `  z
) ) `  n
)  ->  ( 2nd `  T )  =  ( 2nd `  z ) ) )
109108adantlr 729 . . . . . . . . . . . . 13  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( -.  E* n  e.  ( 1 ... N ) ( ( F `  ( ( 2nd `  z
)  -  1 ) ) `  n )  =/=  ( ( F `
 ( 2nd `  z
) ) `  n
)  ->  ( 2nd `  T )  =  ( 2nd `  z ) ) )
11057, 109mpd 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  T )  =  ( 2nd `  z
) )
111110neeq1d 2702 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
( 2nd `  T
)  =/=  0  <->  ( 2nd `  z )  =/=  0 ) )
112111exbiri 634 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  -> 
( ( 2nd `  z
)  =/=  0  -> 
( 2nd `  T
)  =/=  0 ) ) )
11314, 112mpdi 42 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  -> 
( 2nd `  T
)  =/=  0 ) )
114113necon2bd 2659 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  T
)  =  0  ->  -.  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) ) )
1158, 114mpd 15 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )
116 xp2nd 6843 . . . . . . . . 9  |-  ( z  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  ->  ( 2nd `  z )  e.  ( 0 ... N
) )
11737, 116syl 17 . . . . . . . 8  |-  ( z  e.  S  ->  ( 2nd `  z )  e.  ( 0 ... N
) )
1181nncnd 10647 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  CC )
119 npcan1 10065 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
120118, 119syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
121 nnuz 11218 . . . . . . . . . . . . . . . . . . 19  |-  NN  =  ( ZZ>= `  1 )
1221, 121syl6eleq 2559 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
123120, 122eqeltrd 2549 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= ` 
1 ) )
1241nnzd 11062 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  N  e.  ZZ )
125 peano2zm 11004 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  ZZ  ->  ( N  -  1 )  e.  ZZ )
126124, 125syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
127 uzid 11197 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  -  1 )  e.  ZZ  ->  ( N  -  1 )  e.  ( ZZ>= `  ( N  -  1 ) ) )
128 peano2uz 11235 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  -  1 )  e.  ( ZZ>= `  ( N  -  1 ) )  ->  ( ( N  -  1 )  +  1 )  e.  ( ZZ>= `  ( N  -  1 ) ) )
129126, 127, 1283syl 18 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= `  ( N  -  1
) ) )
130120, 129eqeltrrd 2550 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  ( ZZ>= `  ( N  -  1
) ) )
131 fzsplit2 11850 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= ` 
1 )  /\  N  e.  ( ZZ>= `  ( N  -  1 ) ) )  ->  ( 1 ... N )  =  ( ( 1 ... ( N  -  1 ) )  u.  (
( ( N  - 
1 )  +  1 ) ... N ) ) )
132123, 130, 131syl2anc 673 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 1 ... N
)  =  ( ( 1 ... ( N  -  1 ) )  u.  ( ( ( N  -  1 )  +  1 ) ... N ) ) )
133120oveq1d 6323 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( ( N  -  1 )  +  1 ) ... N
)  =  ( N ... N ) )
134 fzsn 11866 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
135124, 134syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( N ... N
)  =  { N } )
136133, 135eqtrd 2505 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( N  -  1 )  +  1 ) ... N
)  =  { N } )
137136uneq2d 3579 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 1 ... ( N  -  1 ) )  u.  (
( ( N  - 
1 )  +  1 ) ... N ) )  =  ( ( 1 ... ( N  -  1 ) )  u.  { N }
) )
138132, 137eqtrd 2505 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1 ... N
)  =  ( ( 1 ... ( N  -  1 ) )  u.  { N }
) )
139138eleq2d 2534 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2nd `  z
)  e.  ( 1 ... N )  <->  ( 2nd `  z )  e.  ( ( 1 ... ( N  -  1 ) )  u.  { N } ) ) )
140139notbid 301 . . . . . . . . . . . . 13  |-  ( ph  ->  ( -.  ( 2nd `  z )  e.  ( 1 ... N )  <->  -.  ( 2nd `  z
)  e.  ( ( 1 ... ( N  -  1 ) )  u.  { N }
) ) )
141 ioran 498 . . . . . . . . . . . . . 14  |-  ( -.  ( ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  \/  ( 2nd `  z
)  =  N )  <-> 
( -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) )  /\  -.  ( 2nd `  z )  =  N ) )
142 elun 3565 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  z )  e.  ( ( 1 ... ( N  - 
1 ) )  u. 
{ N } )  <-> 
( ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  \/  ( 2nd `  z
)  e.  { N } ) )
143 fvex 5889 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  z )  e.  _V
144143elsnc 3984 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  z )  e.  { N }  <->  ( 2nd `  z )  =  N )
145144orbi2i 528 . . . . . . . . . . . . . . 15  |-  ( ( ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  \/  ( 2nd `  z
)  e.  { N } )  <->  ( ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) )  \/  ( 2nd `  z )  =  N ) )
146142, 145bitri 257 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  z )  e.  ( ( 1 ... ( N  - 
1 ) )  u. 
{ N } )  <-> 
( ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  \/  ( 2nd `  z
)  =  N ) )
147141, 146xchnxbir 316 . . . . . . . . . . . . 13  |-  ( -.  ( 2nd `  z
)  e.  ( ( 1 ... ( N  -  1 ) )  u.  { N }
)  <->  ( -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) )  /\  -.  ( 2nd `  z )  =  N ) )
148140, 147syl6bb 269 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  ( 2nd `  z )  e.  ( 1 ... N )  <-> 
( -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) )  /\  -.  ( 2nd `  z )  =  N ) ) )
149148anbi2d 718 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  -.  ( 2nd `  z )  e.  ( 1 ... N ) )  <->  ( ( 2nd `  z )  e.  ( 0 ... N )  /\  ( -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) )  /\  -.  ( 2nd `  z )  =  N ) ) ) )
1501nnnn0d 10949 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  NN0 )
151 nn0uz 11217 . . . . . . . . . . . . . . . . 17  |-  NN0  =  ( ZZ>= `  0 )
152150, 151syl6eleq 2559 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
153 fzpred 11870 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( ZZ>= `  0
)  ->  ( 0 ... N )  =  ( { 0 }  u.  ( ( 0  +  1 ) ... N ) ) )
154152, 153syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 0 ... N
)  =  ( { 0 }  u.  (
( 0  +  1 ) ... N ) ) )
155154difeq1d 3539 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 0 ... N )  \  (
1 ... N ) )  =  ( ( { 0 }  u.  (
( 0  +  1 ) ... N ) )  \  ( 1 ... N ) ) )
156 difun2 3838 . . . . . . . . . . . . . . 15  |-  ( ( { 0 }  u.  ( 1 ... N
) )  \  (
1 ... N ) )  =  ( { 0 }  \  ( 1 ... N ) )
157 0p1e1 10743 . . . . . . . . . . . . . . . . . 18  |-  ( 0  +  1 )  =  1
158157oveq1i 6318 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  +  1 ) ... N )  =  ( 1 ... N
)
159158uneq2i 3576 . . . . . . . . . . . . . . . 16  |-  ( { 0 }  u.  (
( 0  +  1 ) ... N ) )  =  ( { 0 }  u.  (
1 ... N ) )
160159difeq1i 3536 . . . . . . . . . . . . . . 15  |-  ( ( { 0 }  u.  ( ( 0  +  1 ) ... N
) )  \  (
1 ... N ) )  =  ( ( { 0 }  u.  (
1 ... N ) ) 
\  ( 1 ... N ) )
161 incom 3616 . . . . . . . . . . . . . . . . 17  |-  ( { 0 }  i^i  (
1 ... N ) )  =  ( ( 1 ... N )  i^i 
{ 0 } )
162 elfznn 11854 . . . . . . . . . . . . . . . . . . 19  |-  ( 0  e.  ( 1 ... N )  ->  0  e.  NN )
1639, 162mto 181 . . . . . . . . . . . . . . . . . 18  |-  -.  0  e.  ( 1 ... N
)
164 disjsn 4023 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 1 ... N
)  i^i  { 0 } )  =  (/)  <->  -.  0  e.  ( 1 ... N ) )
165163, 164mpbir 214 . . . . . . . . . . . . . . . . 17  |-  ( ( 1 ... N )  i^i  { 0 } )  =  (/)
166161, 165eqtri 2493 . . . . . . . . . . . . . . . 16  |-  ( { 0 }  i^i  (
1 ... N ) )  =  (/)
167 disj3 3813 . . . . . . . . . . . . . . . 16  |-  ( ( { 0 }  i^i  ( 1 ... N
) )  =  (/)  <->  {
0 }  =  ( { 0 }  \ 
( 1 ... N
) ) )
168166, 167mpbi 213 . . . . . . . . . . . . . . 15  |-  { 0 }  =  ( { 0 }  \  (
1 ... N ) )
169156, 160, 1683eqtr4i 2503 . . . . . . . . . . . . . 14  |-  ( ( { 0 }  u.  ( ( 0  +  1 ) ... N
) )  \  (
1 ... N ) )  =  { 0 }
170155, 169syl6eq 2521 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 0 ... N )  \  (
1 ... N ) )  =  { 0 } )
171170eleq2d 2534 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 2nd `  z
)  e.  ( ( 0 ... N ) 
\  ( 1 ... N ) )  <->  ( 2nd `  z )  e.  {
0 } ) )
172 eldif 3400 . . . . . . . . . . . 12  |-  ( ( 2nd `  z )  e.  ( ( 0 ... N )  \ 
( 1 ... N
) )  <->  ( ( 2nd `  z )  e.  ( 0 ... N
)  /\  -.  ( 2nd `  z )  e.  ( 1 ... N
) ) )
173143elsnc 3984 . . . . . . . . . . . 12  |-  ( ( 2nd `  z )  e.  { 0 }  <-> 
( 2nd `  z
)  =  0 )
174171, 172, 1733bitr3g 295 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  -.  ( 2nd `  z )  e.  ( 1 ... N ) )  <->  ( 2nd `  z
)  =  0 ) )
175149, 174bitr3d 263 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  ( -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) )  /\  -.  ( 2nd `  z )  =  N ) )  <->  ( 2nd `  z )  =  0 ) )
176175biimpd 212 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  ( -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) )  /\  -.  ( 2nd `  z )  =  N ) )  -> 
( 2nd `  z
)  =  0 ) )
177176expdimp 444 . . . . . . . 8  |-  ( (
ph  /\  ( 2nd `  z )  e.  ( 0 ... N ) )  ->  ( ( -.  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  /\  -.  ( 2nd `  z
)  =  N )  ->  ( 2nd `  z
)  =  0 ) )
178117, 177sylan2 482 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  (
( -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) )  /\  -.  ( 2nd `  z )  =  N )  ->  ( 2nd `  z )  =  0 ) )
179115, 178mpand 689 . . . . . 6  |-  ( (
ph  /\  z  e.  S )  ->  ( -.  ( 2nd `  z
)  =  N  -> 
( 2nd `  z
)  =  0 ) )
1801, 2, 3poimirlem13 32017 . . . . . . . . . 10  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  0 )
181 fveq2 5879 . . . . . . . . . . . 12  |-  ( z  =  s  ->  ( 2nd `  z )  =  ( 2nd `  s
) )
182181eqeq1d 2473 . . . . . . . . . . 11  |-  ( z  =  s  ->  (
( 2nd `  z
)  =  0  <->  ( 2nd `  s )  =  0 ) )
183182rmo4 3219 . . . . . . . . . 10  |-  ( E* z  e.  S  ( 2nd `  z )  =  0  <->  A. z  e.  S  A. s  e.  S  ( (
( 2nd `  z
)  =  0  /\  ( 2nd `  s
)  =  0 )  ->  z  =  s ) )
184180, 183sylib 201 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  S  A. s  e.  S  ( ( ( 2nd `  z )  =  0  /\  ( 2nd `  s
)  =  0 )  ->  z  =  s ) )
185184r19.21bi 2776 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  A. s  e.  S  ( (
( 2nd `  z
)  =  0  /\  ( 2nd `  s
)  =  0 )  ->  z  =  s ) )
1864adantr 472 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  T  e.  S )
187 fveq2 5879 . . . . . . . . . . . 12  |-  ( s  =  T  ->  ( 2nd `  s )  =  ( 2nd `  T
) )
188187eqeq1d 2473 . . . . . . . . . . 11  |-  ( s  =  T  ->  (
( 2nd `  s
)  =  0  <->  ( 2nd `  T )  =  0 ) )
189188anbi2d 718 . . . . . . . . . 10  |-  ( s  =  T  ->  (
( ( 2nd `  z
)  =  0  /\  ( 2nd `  s
)  =  0 )  <-> 
( ( 2nd `  z
)  =  0  /\  ( 2nd `  T
)  =  0 ) ) )
190 eqeq2 2482 . . . . . . . . . 10  |-  ( s  =  T  ->  (
z  =  s  <->  z  =  T ) )
191189, 190imbi12d 327 . . . . . . . . 9  |-  ( s  =  T  ->  (
( ( ( 2nd `  z )  =  0  /\  ( 2nd `  s
)  =  0 )  ->  z  =  s )  <->  ( ( ( 2nd `  z )  =  0  /\  ( 2nd `  T )  =  0 )  ->  z  =  T ) ) )
192191rspccv 3133 . . . . . . . 8  |-  ( A. s  e.  S  (
( ( 2nd `  z
)  =  0  /\  ( 2nd `  s
)  =  0 )  ->  z  =  s )  ->  ( T  e.  S  ->  ( ( ( 2nd `  z
)  =  0  /\  ( 2nd `  T
)  =  0 )  ->  z  =  T ) ) )
193185, 186, 192sylc 61 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  (
( ( 2nd `  z
)  =  0  /\  ( 2nd `  T
)  =  0 )  ->  z  =  T ) )
1948, 193mpan2d 688 . . . . . 6  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  z
)  =  0  -> 
z  =  T ) )
195179, 194syld 44 . . . . 5  |-  ( (
ph  /\  z  e.  S )  ->  ( -.  ( 2nd `  z
)  =  N  -> 
z  =  T ) )
196195necon1ad 2660 . . . 4  |-  ( (
ph  /\  z  e.  S )  ->  (
z  =/=  T  -> 
( 2nd `  z
)  =  N ) )
197196ralrimiva 2809 . . 3  |-  ( ph  ->  A. z  e.  S  ( z  =/=  T  ->  ( 2nd `  z
)  =  N ) )
1981, 2, 3poimirlem14 32018 . . 3  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  N )
199 rmoim 3227 . . 3  |-  ( A. z  e.  S  (
z  =/=  T  -> 
( 2nd `  z
)  =  N )  ->  ( E* z  e.  S  ( 2nd `  z )  =  N  ->  E* z  e.  S  z  =/=  T
) )
200197, 198, 199sylc 61 . 2  |-  ( ph  ->  E* z  e.  S  z  =/=  T )
201 reu5 2994 . 2  |-  ( E! z  e.  S  z  =/=  T  <->  ( E. z  e.  S  z  =/=  T  /\  E* z  e.  S  z  =/=  T ) )
2027, 200, 201sylanbrc 677 1  |-  ( ph  ->  E! z  e.  S  z  =/=  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756   E.wrex 2757   E!wreu 2758   E*wrmo 2759   {crab 2760   [_csb 3349    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   ifcif 3872   {csn 3959   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   ran crn 4840   "cima 4842   -->wf 5585   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308    oFcof 6548   1stc1st 6810   2ndc2nd 6811    ^m cmap 7490   CCcc 9555   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    - cmin 9880   NNcn 10631   NN0cn0 10893   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810  ..^cfzo 11942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943
This theorem is referenced by:  poimirlem22  32026
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