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Theorem poimirlem21 31973
Description: Lemma for poimir 31985 stating that, given a face not on a back face of the cube and a simplex in which it's opposite the final point of 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 )
poimirlem22.3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  E. p  e.  ran  F ( p `
 n )  =/=  0 )
poimirlem21.4  |-  ( ph  ->  ( 2nd `  T
)  =  N )
Assertion
Ref Expression
poimirlem21  |-  ( 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 poimirlem21
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 poimirlem22.3 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  E. p  e.  ran  F ( p `
 n )  =/=  0 )
6 poimirlem21.4 . . 3  |-  ( ph  ->  ( 2nd `  T
)  =  N )
71, 2, 3, 4, 5, 6poimirlem20 31972 . 2  |-  ( ph  ->  E. z  e.  S  z  =/=  T )
86adantr 467 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  ( 2nd `  T )  =  N )
91nnred 10631 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  RR )
109ltm1d 10546 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N  -  1 )  <  N )
11 nnm1nn0 10918 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
121, 11syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( N  -  1 )  e.  NN0 )
1312nn0red 10933 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( N  -  1 )  e.  RR )
1413, 9ltnled 9787 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( N  - 
1 )  <  N  <->  -.  N  <_  ( N  -  1 ) ) )
1510, 14mpbid 214 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  N  <_  ( N  -  1 ) )
16 elfzle2 11810 . . . . . . . . . . . . . 14  |-  ( N  e.  ( 1 ... ( N  -  1 ) )  ->  N  <_  ( N  -  1 ) )
1715, 16nsyl 125 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  N  e.  ( 1 ... ( N  -  1 ) ) )
18 eleq1 2519 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  z )  =  N  ->  (
( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  <->  N  e.  ( 1 ... ( N  -  1 ) ) ) )
1918notbid 296 . . . . . . . . . . . . 13  |-  ( ( 2nd `  z )  =  N  ->  ( -.  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  <->  -.  N  e.  ( 1 ... ( N  -  1 ) ) ) )
2017, 19syl5ibrcom 226 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 2nd `  z
)  =  N  ->  -.  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) ) )
2120necon2ad 2641 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  -> 
( 2nd `  z
)  =/=  N ) )
2221adantr 467 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  -> 
( 2nd `  z
)  =/=  N ) )
231ad2antrr 733 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  N  e.  NN )
24 fveq2 5870 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  ( 2nd `  t )  =  ( 2nd `  z
) )
2524breq2d 4417 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  z  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  z ) ) )
2625ifbid 3905 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  z  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  z
) ,  y ,  ( y  +  1 ) ) )
2726csbeq1d 3372 . . . . . . . . . . . . . . . . . . . 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 } ) ) ) )
28 fveq2 5870 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  ( 1st `  t )  =  ( 1st `  z
) )
2928fveq2d 5874 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  z  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  z ) ) )
3028fveq2d 5874 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  z  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  z ) ) )
3130imaeq1d 5170 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  z  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... j ) ) )
3231xpeq1d 4860 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  z ) ) "
( 1 ... j
) )  X.  {
1 } ) )
3330imaeq1d 5170 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  z  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) ) )
3433xpeq1d 4860 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  z ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
3532, 34uneq12d 3591 . . . . . . . . . . . . . . . . . . . . . 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 } ) ) )
3629, 35oveq12d 6313 . . . . . . . . . . . . . . . . . . . . 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 } ) ) ) )
3736csbeq2dv 3783 . . . . . . . . . . . . . . . . . . . 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 } ) ) ) )
3827, 37eqtrd 2487 . . . . . . . . . . . . . . . . . . 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 } ) ) ) )
3938mpteq2dv 4493 . . . . . . . . . . . . . . . . . 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 } ) ) ) ) )
4039eqeq2d 2463 . . . . . . . . . . . . . . . . 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 } ) ) ) ) ) )
4140, 2elrab2 3200 . . . . . . . . . . . . . . . 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 } ) ) ) ) ) )
4241simprbi 466 . . . . . . . . . . . . . . 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 } ) ) ) ) )
4342ad2antlr 734 . . . . . . . . . . . . . 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 } ) ) ) ) )
44 elrabi 3195 . . . . . . . . . . . . . . . . . . . 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 ) ) )
4544, 2eleq2s 2549 . . . . . . . . . . . . . . . . . . 19  |-  ( z  e.  S  ->  z  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
46 xp1st 6828 . . . . . . . . . . . . . . . . . . 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 ) } ) )
4745, 46syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( z  e.  S  ->  ( 1st `  z )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
48 xp1st 6828 . . . . . . . . . . . . . . . . . 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 ) ) )
4947, 48syl 17 . . . . . . . . . . . . . . . . 17  |-  ( z  e.  S  ->  ( 1st `  ( 1st `  z
) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) ) )
50 elmapi 7498 . . . . . . . . . . . . . . . . 17  |-  ( ( 1st `  ( 1st `  z ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  z ) ) : ( 1 ... N ) --> ( 0..^ K ) )
5149, 50syl 17 . . . . . . . . . . . . . . . 16  |-  ( z  e.  S  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ( 0..^ K ) )
52 elfzoelz 11927 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( 0..^ K )  ->  n  e.  ZZ )
5352ssriv 3438 . . . . . . . . . . . . . . . 16  |-  ( 0..^ K )  C_  ZZ
54 fss 5742 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  ( 1st `  z ) ) : ( 1 ... N ) --> ( 0..^ K )  /\  (
0..^ K )  C_  ZZ )  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ZZ )
5551, 53, 54sylancl 669 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ZZ )
5655ad2antlr 734 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ZZ )
57 xp2nd 6829 . . . . . . . . . . . . . . . . 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 ) } )
5847, 57syl 17 . . . . . . . . . . . . . . . 16  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) )  e.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )
59 fvex 5880 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  ( 1st `  z
) )  e.  _V
60 f1oeq1 5810 . . . . . . . . . . . . . . . . 17  |-  ( f  =  ( 2nd `  ( 1st `  z ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  z ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
6159, 60elab 3187 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  ( 1st `  z ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
6258, 61sylib 200 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
6362ad2antlr 734 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
64 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )
6523, 43, 56, 63, 64poimirlem1 31953 . . . . . . . . . . . . 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
) )
661ad2antrr 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  N  e.  NN )
67 fveq2 5870 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
6867breq2d 4417 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
6968ifbid 3905 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
7069csbeq1d 3372 . . . . . . . . . . . . . . . . . . . . . . . 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 } ) ) ) )
71 fveq2 5870 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
7271fveq2d 5874 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
7371fveq2d 5874 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
7473imaeq1d 5170 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
7574xpeq1d 4860 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
7673imaeq1d 5170 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
7776xpeq1d 4860 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
7875, 77uneq12d 3591 . . . . . . . . . . . . . . . . . . . . . . . . . 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 } ) ) )
7972, 78oveq12d 6313 . . . . . . . . . . . . . . . . . . . . . . . . 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 } ) ) ) )
8079csbeq2dv 3783 . . . . . . . . . . . . . . . . . . . . . . . 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 } ) ) ) )
8170, 80eqtrd 2487 . . . . . . . . . . . . . . . . . . . . . . 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 } ) ) ) )
8281mpteq2dv 4493 . . . . . . . . . . . . . . . . . . . . . 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 } ) ) ) ) )
8382eqeq2d 2463 . . . . . . . . . . . . . . . . . . . . 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 } ) ) ) ) ) )
8483, 2elrab2 3200 . . . . . . . . . . . . . . . . . . . 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 } ) ) ) ) ) )
8584simprbi 466 . . . . . . . . . . . . . . . . . . 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 } ) ) ) ) )
864, 85syl 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 } ) ) ) ) )
8786ad2antrr 733 . . . . . . . . . . . . . . . . 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 } ) ) ) ) )
88 elrabi 3195 . . . . . . . . . . . . . . . . . . . . . . . 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 ) ) )
8988, 2eleq2s 2549 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( T  e.  S  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
904, 89syl 17 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ph  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
91 xp1st 6828 . . . . . . . . . . . . . . . . . . . . . 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 ) } ) )
9290, 91syl 17 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ph  ->  ( 1st `  T
)  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } ) )
93 xp1st 6828 . . . . . . . . . . . . . . . . . . . . 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 ) ) )
9492, 93syl 17 . . . . . . . . . . . . . . . . . . . 20  |-  ( ph  ->  ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
95 elmapi 7498 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
9694, 95syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
97 fss 5742 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K )  /\  (
0..^ K )  C_  ZZ )  ->  ( 1st `  ( 1st `  T
) ) : ( 1 ... N ) --> ZZ )
9896, 53, 97sylancl 669 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ZZ )
9998ad2antrr 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 1st `  ( 1st `  T
) ) : ( 1 ... N ) --> ZZ )
100 xp2nd 6829 . . . . . . . . . . . . . . . . . . . 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 ) } )
10192, 100syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
102 fvex 5880 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
103 f1oeq1 5810 . . . . . . . . . . . . . . . . . . . 20  |-  ( f  =  ( 2nd `  ( 1st `  T ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
104102, 103elab 3187 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  ( 1st `  T ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
105101, 104sylib 200 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
106105ad2antrr 733 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
107 simplr 763 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )
108 xp2nd 6829 . . . . . . . . . . . . . . . . . . . 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
) )
10990, 108syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( 2nd `  T
)  e.  ( 0 ... N ) )
110109adantr 467 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  T )  e.  ( 0 ... N ) )
111 eldifsn 4100 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  T )  e.  ( ( 0 ... N )  \  { ( 2nd `  z
) } )  <->  ( ( 2nd `  T )  e.  ( 0 ... N
)  /\  ( 2nd `  T )  =/=  ( 2nd `  z ) ) )
112111biimpri 210 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  T
)  e.  ( 0 ... N )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  T )  e.  ( ( 0 ... N )  \  {
( 2nd `  z
) } ) )
113110, 112sylan 474 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  T )  e.  ( ( 0 ... N )  \  {
( 2nd `  z
) } ) )
11466, 87, 99, 106, 107, 113poimirlem2 31954 . . . . . . . . . . . . . . . 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
) )
115114ex 436 . . . . . . . . . . . . . . 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 ) ) )
116115necon1bd 2644 . . . . . . . . . . . . . 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 ) ) )
117116adantlr 722 . . . . . . . . . . . . 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 ) ) )
11865, 117mpd 15 . . . . . . . . . . . 12  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  T )  =  ( 2nd `  z
) )
119118neeq1d 2685 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
( 2nd `  T
)  =/=  N  <->  ( 2nd `  z )  =/=  N
) )
120119exbiri 628 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  -> 
( ( 2nd `  z
)  =/=  N  -> 
( 2nd `  T
)  =/=  N ) ) )
12122, 120mpdd 41 . . . . . . . . 9  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  -> 
( 2nd `  T
)  =/=  N ) )
122121necon2bd 2642 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  T
)  =  N  ->  -.  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) ) )
1238, 122mpd 15 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )
124 xp2nd 6829 . . . . . . . . 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
) )
12545, 124syl 17 . . . . . . . 8  |-  ( z  e.  S  ->  ( 2nd `  z )  e.  ( 0 ... N
) )
126 nn0uz 11200 . . . . . . . . . . . . . . . . . 18  |-  NN0  =  ( ZZ>= `  0 )
12712, 126syl6eleq 2541 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( N  -  1 )  e.  ( ZZ>= ` 
0 ) )
128 fzpred 11851 . . . . . . . . . . . . . . . . 17  |-  ( ( N  -  1 )  e.  ( ZZ>= `  0
)  ->  ( 0 ... ( N  - 
1 ) )  =  ( { 0 }  u.  ( ( 0  +  1 ) ... ( N  -  1 ) ) ) )
129127, 128syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 0 ... ( N  -  1 ) )  =  ( { 0 }  u.  (
( 0  +  1 ) ... ( N  -  1 ) ) ) )
130 0p1e1 10728 . . . . . . . . . . . . . . . . . 18  |-  ( 0  +  1 )  =  1
131130oveq1i 6305 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  +  1 ) ... ( N  - 
1 ) )  =  ( 1 ... ( N  -  1 ) )
132131uneq2i 3587 . . . . . . . . . . . . . . . 16  |-  ( { 0 }  u.  (
( 0  +  1 ) ... ( N  -  1 ) ) )  =  ( { 0 }  u.  (
1 ... ( N  - 
1 ) ) )
133129, 132syl6eq 2503 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 0 ... ( N  -  1 ) )  =  ( { 0 }  u.  (
1 ... ( N  - 
1 ) ) ) )
134133eleq2d 2516 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2nd `  z
)  e.  ( 0 ... ( N  - 
1 ) )  <->  ( 2nd `  z )  e.  ( { 0 }  u.  ( 1 ... ( N  -  1 ) ) ) ) )
135134notbid 296 . . . . . . . . . . . . 13  |-  ( ph  ->  ( -.  ( 2nd `  z )  e.  ( 0 ... ( N  -  1 ) )  <->  -.  ( 2nd `  z
)  e.  ( { 0 }  u.  (
1 ... ( N  - 
1 ) ) ) ) )
136 ioran 493 . . . . . . . . . . . . . 14  |-  ( -.  ( ( 2nd `  z
)  =  0  \/  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) )  <-> 
( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) )
137 elun 3576 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  z )  e.  ( { 0 }  u.  ( 1 ... ( N  - 
1 ) ) )  <-> 
( ( 2nd `  z
)  e.  { 0 }  \/  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) )
138 fvex 5880 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  z )  e.  _V
139138elsnc 3994 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  z )  e.  { 0 }  <-> 
( 2nd `  z
)  =  0 )
140139orbi1i 523 . . . . . . . . . . . . . . 15  |-  ( ( ( 2nd `  z
)  e.  { 0 }  \/  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  <->  ( ( 2nd `  z )  =  0  \/  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) ) )
141137, 140bitri 253 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  z )  e.  ( { 0 }  u.  ( 1 ... ( N  - 
1 ) ) )  <-> 
( ( 2nd `  z
)  =  0  \/  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) ) )
142136, 141xchnxbir 311 . . . . . . . . . . . . 13  |-  ( -.  ( 2nd `  z
)  e.  ( { 0 }  u.  (
1 ... ( N  - 
1 ) ) )  <-> 
( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) )
143135, 142syl6bb 265 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  ( 2nd `  z )  e.  ( 0 ... ( N  -  1 ) )  <-> 
( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) ) )
144143anbi2d 711 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  -.  ( 2nd `  z )  e.  ( 0 ... ( N  -  1 ) ) )  <->  ( ( 2nd `  z )  e.  ( 0 ... N )  /\  ( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) ) ) )
145 uncom 3580 . . . . . . . . . . . . . . . 16  |-  ( ( 0 ... ( N  -  1 ) )  u.  { N }
)  =  ( { N }  u.  (
0 ... ( N  - 
1 ) ) )
146145difeq1i 3549 . . . . . . . . . . . . . . 15  |-  ( ( ( 0 ... ( N  -  1 ) )  u.  { N } )  \  (
0 ... ( N  - 
1 ) ) )  =  ( ( { N }  u.  (
0 ... ( N  - 
1 ) ) ) 
\  ( 0 ... ( N  -  1 ) ) )
147 difun2 3849 . . . . . . . . . . . . . . 15  |-  ( ( { N }  u.  ( 0 ... ( N  -  1 ) ) )  \  (
0 ... ( N  - 
1 ) ) )  =  ( { N }  \  ( 0 ... ( N  -  1 ) ) )
148146, 147eqtri 2475 . . . . . . . . . . . . . 14  |-  ( ( ( 0 ... ( N  -  1 ) )  u.  { N } )  \  (
0 ... ( N  - 
1 ) ) )  =  ( { N }  \  ( 0 ... ( N  -  1 ) ) )
1491nncnd 10632 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  CC )
150 npcan1 10051 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
151149, 150syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
1521nnnn0d 10932 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  NN0 )
153152, 126syl6eleq 2541 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
154151, 153eqeltrd 2531 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= ` 
0 ) )
15512nn0zd 11045 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
156 uzid 11180 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  -  1 )  e.  ZZ  ->  ( N  -  1 )  e.  ( ZZ>= `  ( N  -  1 ) ) )
157 peano2uz 11219 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  -  1 )  e.  ( ZZ>= `  ( N  -  1 ) )  ->  ( ( N  -  1 )  +  1 )  e.  ( ZZ>= `  ( N  -  1 ) ) )
158155, 156, 1573syl 18 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= `  ( N  -  1
) ) )
159151, 158eqeltrrd 2532 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  ( ZZ>= `  ( N  -  1
) ) )
160 fzsplit2 11831 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= ` 
0 )  /\  N  e.  ( ZZ>= `  ( N  -  1 ) ) )  ->  ( 0 ... N )  =  ( ( 0 ... ( N  -  1 ) )  u.  (
( ( N  - 
1 )  +  1 ) ... N ) ) )
161154, 159, 160syl2anc 667 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 0 ... N
)  =  ( ( 0 ... ( N  -  1 ) )  u.  ( ( ( N  -  1 )  +  1 ) ... N ) ) )
162151oveq1d 6310 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( ( N  -  1 )  +  1 ) ... N
)  =  ( N ... N ) )
1631nnzd 11046 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  ZZ )
164 fzsn 11847 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
165163, 164syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( N ... N
)  =  { N } )
166162, 165eqtrd 2487 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( N  -  1 )  +  1 ) ... N
)  =  { N } )
167166uneq2d 3590 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 0 ... ( N  -  1 ) )  u.  (
( ( N  - 
1 )  +  1 ) ... N ) )  =  ( ( 0 ... ( N  -  1 ) )  u.  { N }
) )
168161, 167eqtrd 2487 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 0 ... N
)  =  ( ( 0 ... ( N  -  1 ) )  u.  { N }
) )
169168difeq1d 3552 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 0 ... N )  \  (
0 ... ( N  - 
1 ) ) )  =  ( ( ( 0 ... ( N  -  1 ) )  u.  { N }
)  \  ( 0 ... ( N  - 
1 ) ) ) )
170 elfzle2 11810 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( 0 ... ( N  -  1 ) )  ->  N  <_  ( N  -  1 ) )
17115, 170nsyl 125 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  N  e.  ( 0 ... ( N  -  1 ) ) )
172 incom 3627 . . . . . . . . . . . . . . . . 17  |-  ( ( 0 ... ( N  -  1 ) )  i^i  { N }
)  =  ( { N }  i^i  (
0 ... ( N  - 
1 ) ) )
173172eqeq1i 2458 . . . . . . . . . . . . . . . 16  |-  ( ( ( 0 ... ( N  -  1 ) )  i^i  { N } )  =  (/)  <->  ( { N }  i^i  (
0 ... ( N  - 
1 ) ) )  =  (/) )
174 disjsn 4034 . . . . . . . . . . . . . . . 16  |-  ( ( ( 0 ... ( N  -  1 ) )  i^i  { N } )  =  (/)  <->  -.  N  e.  ( 0 ... ( N  - 
1 ) ) )
175 disj3 3811 . . . . . . . . . . . . . . . 16  |-  ( ( { N }  i^i  ( 0 ... ( N  -  1 ) ) )  =  (/)  <->  { N }  =  ( { N }  \  (
0 ... ( N  - 
1 ) ) ) )
176173, 174, 1753bitr3i 279 . . . . . . . . . . . . . . 15  |-  ( -.  N  e.  ( 0 ... ( N  - 
1 ) )  <->  { N }  =  ( { N }  \  (
0 ... ( N  - 
1 ) ) ) )
177171, 176sylib 200 . . . . . . . . . . . . . 14  |-  ( ph  ->  { N }  =  ( { N }  \ 
( 0 ... ( N  -  1 ) ) ) )
178148, 169, 1773eqtr4a 2513 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 0 ... N )  \  (
0 ... ( N  - 
1 ) ) )  =  { N }
)
179178eleq2d 2516 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 2nd `  z
)  e.  ( ( 0 ... N ) 
\  ( 0 ... ( N  -  1 ) ) )  <->  ( 2nd `  z )  e.  { N } ) )
180 eldif 3416 . . . . . . . . . . . 12  |-  ( ( 2nd `  z )  e.  ( ( 0 ... N )  \ 
( 0 ... ( N  -  1 ) ) )  <->  ( ( 2nd `  z )  e.  ( 0 ... N
)  /\  -.  ( 2nd `  z )  e.  ( 0 ... ( N  -  1 ) ) ) )
181138elsnc 3994 . . . . . . . . . . . 12  |-  ( ( 2nd `  z )  e.  { N }  <->  ( 2nd `  z )  =  N )
182179, 180, 1813bitr3g 291 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  -.  ( 2nd `  z )  e.  ( 0 ... ( N  -  1 ) ) )  <->  ( 2nd `  z
)  =  N ) )
183144, 182bitr3d 259 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  ( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) )  <->  ( 2nd `  z )  =  N ) )
184183biimpd 211 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  ( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( 2nd `  z
)  =  N ) )
185184expdimp 439 . . . . . . . 8  |-  ( (
ph  /\  ( 2nd `  z )  e.  ( 0 ... N ) )  ->  ( ( -.  ( 2nd `  z
)  =  0  /\ 
-.  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( 2nd `  z
)  =  N ) )
186125, 185sylan2 477 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  (
( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  z )  =  N ) )
187123, 186mpan2d 681 . . . . . 6  |-  ( (
ph  /\  z  e.  S )  ->  ( -.  ( 2nd `  z
)  =  0  -> 
( 2nd `  z
)  =  N ) )
1881, 2, 3poimirlem14 31966 . . . . . . . . . 10  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  N )
189 fveq2 5870 . . . . . . . . . . . 12  |-  ( z  =  s  ->  ( 2nd `  z )  =  ( 2nd `  s
) )
190189eqeq1d 2455 . . . . . . . . . . 11  |-  ( z  =  s  ->  (
( 2nd `  z
)  =  N  <->  ( 2nd `  s )  =  N ) )
191190rmo4 3233 . . . . . . . . . 10  |-  ( E* z  e.  S  ( 2nd `  z )  =  N  <->  A. z  e.  S  A. s  e.  S  ( (
( 2nd `  z
)  =  N  /\  ( 2nd `  s )  =  N )  -> 
z  =  s ) )
192188, 191sylib 200 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  S  A. s  e.  S  ( ( ( 2nd `  z )  =  N  /\  ( 2nd `  s
)  =  N )  ->  z  =  s ) )
193192r19.21bi 2759 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  A. s  e.  S  ( (
( 2nd `  z
)  =  N  /\  ( 2nd `  s )  =  N )  -> 
z  =  s ) )
1944adantr 467 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  T  e.  S )
195 fveq2 5870 . . . . . . . . . . . 12  |-  ( s  =  T  ->  ( 2nd `  s )  =  ( 2nd `  T
) )
196195eqeq1d 2455 . . . . . . . . . . 11  |-  ( s  =  T  ->  (
( 2nd `  s
)  =  N  <->  ( 2nd `  T )  =  N ) )
197196anbi2d 711 . . . . . . . . . 10  |-  ( s  =  T  ->  (
( ( 2nd `  z
)  =  N  /\  ( 2nd `  s )  =  N )  <->  ( ( 2nd `  z )  =  N  /\  ( 2nd `  T )  =  N ) ) )
198 eqeq2 2464 . . . . . . . . . 10  |-  ( s  =  T  ->  (
z  =  s  <->  z  =  T ) )
199197, 198imbi12d 322 . . . . . . . . 9  |-  ( s  =  T  ->  (
( ( ( 2nd `  z )  =  N  /\  ( 2nd `  s
)  =  N )  ->  z  =  s )  <->  ( ( ( 2nd `  z )  =  N  /\  ( 2nd `  T )  =  N )  ->  z  =  T ) ) )
200199rspccv 3149 . . . . . . . 8  |-  ( A. s  e.  S  (
( ( 2nd `  z
)  =  N  /\  ( 2nd `  s )  =  N )  -> 
z  =  s )  ->  ( T  e.  S  ->  ( (
( 2nd `  z
)  =  N  /\  ( 2nd `  T )  =  N )  -> 
z  =  T ) ) )
201193, 194, 200sylc 62 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  (
( ( 2nd `  z
)  =  N  /\  ( 2nd `  T )  =  N )  -> 
z  =  T ) )
2028, 201mpan2d 681 . . . . . 6  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  z
)  =  N  -> 
z  =  T ) )
203187, 202syld 45 . . . . 5  |-  ( (
ph  /\  z  e.  S )  ->  ( -.  ( 2nd `  z
)  =  0  -> 
z  =  T ) )
204203necon1ad 2643 . . . 4  |-  ( (
ph  /\  z  e.  S )  ->  (
z  =/=  T  -> 
( 2nd `  z
)  =  0 ) )
205204ralrimiva 2804 . . 3  |-  ( ph  ->  A. z  e.  S  ( z  =/=  T  ->  ( 2nd `  z
)  =  0 ) )
2061, 2, 3poimirlem13 31965 . . 3  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  0 )
207 rmoim 3241 . . 3  |-  ( A. z  e.  S  (
z  =/=  T  -> 
( 2nd `  z
)  =  0 )  ->  ( E* z  e.  S  ( 2nd `  z )  =  0  ->  E* z  e.  S  z  =/=  T
) )
208205, 206, 207sylc 62 . 2  |-  ( ph  ->  E* z  e.  S  z  =/=  T )
209 reu5 3010 . 2  |-  ( E! z  e.  S  z  =/=  T  <->  ( E. z  e.  S  z  =/=  T  /\  E* z  e.  S  z  =/=  T ) )
2107, 208, 209sylanbrc 671 1  |-  ( ph  ->  E! z  e.  S  z  =/=  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 370    /\ wa 371    = wceq 1446    e. wcel 1889   {cab 2439    =/= wne 2624   A.wral 2739   E.wrex 2740   E!wreu 2741   E*wrmo 2742   {crab 2743   [_csb 3365    \ cdif 3403    u. cun 3404    i^i cin 3405    C_ wss 3406   (/)c0 3733   ifcif 3883   {csn 3970   class class class wbr 4405    |-> cmpt 4464    X. cxp 4835   ran crn 4838   "cima 4840   -->wf 5581   -1-1-onto->wf1o 5584   ` cfv 5585  (class class class)co 6295    oFcof 6534   1stc1st 6796   2ndc2nd 6797    ^m cmap 7477   CCcc 9542   0cc0 9544   1c1 9545    + caddc 9547    < clt 9680    <_ cle 9681    - cmin 9865   NNcn 10616   NN0cn0 10876   ZZcz 10944   ZZ>=cuz 11166   ...cfz 11791  ..^cfzo 11922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-rep 4518  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-of 6536  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923
This theorem is referenced by:  poimirlem22  31974
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