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Theorem poimirlem21 31875
Description: Lemma for poimir 31887 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 31874 . 2  |-  ( ph  ->  E. z  e.  S  z  =/=  T )
86adantr 466 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  ( 2nd `  T )  =  N )
91nnred 10625 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  N  e.  RR )
109ltm1d 10540 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( N  -  1 )  <  N )
11 nnm1nn0 10912 . . . . . . . . . . . . . . . . . 18  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
121, 11syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( N  -  1 )  e.  NN0 )
1312nn0red 10927 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( N  -  1 )  e.  RR )
1413, 9ltnled 9783 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( N  - 
1 )  <  N  <->  -.  N  <_  ( N  -  1 ) ) )
1510, 14mpbid 213 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  N  <_  ( N  -  1 ) )
16 elfzle2 11804 . . . . . . . . . . . . . 14  |-  ( N  e.  ( 1 ... ( N  -  1 ) )  ->  N  <_  ( N  -  1 ) )
1715, 16nsyl 124 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  N  e.  ( 1 ... ( N  -  1 ) ) )
18 eleq1 2494 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  z )  =  N  ->  (
( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  <->  N  e.  ( 1 ... ( N  -  1 ) ) ) )
1918notbid 295 . . . . . . . . . . . . 13  |-  ( ( 2nd `  z )  =  N  ->  ( -.  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  <->  -.  N  e.  ( 1 ... ( N  -  1 ) ) ) )
2017, 19syl5ibrcom 225 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 2nd `  z
)  =  N  ->  -.  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) ) )
2120necon2ad 2637 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  -> 
( 2nd `  z
)  =/=  N ) )
2221adantr 466 . . . . . . . . . 10  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) )  -> 
( 2nd `  z
)  =/=  N ) )
231ad2antrr 730 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  N  e.  NN )
24 fveq2 5878 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  ( 2nd `  t )  =  ( 2nd `  z
) )
2524breq2d 4432 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  z  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  z ) ) )
2625ifbid 3931 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  z  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  z
) ,  y ,  ( y  +  1 ) ) )
2726csbeq1d 3402 . . . . . . . . . . . . . . . . . . . 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 5878 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  ( 1st `  t )  =  ( 1st `  z
) )
2928fveq2d 5882 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  z  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  z ) ) )
3028fveq2d 5882 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  z  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  z ) ) )
3130imaeq1d 5183 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  z  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... j ) ) )
3231xpeq1d 4873 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  z ) ) "
( 1 ... j
) )  X.  {
1 } ) )
3330imaeq1d 5183 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  z  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( ( j  +  1 ) ... N ) ) )
3433xpeq1d 4873 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  z  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  z ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
3532, 34uneq12d 3621 . . . . . . . . . . . . . . . . . . . . . 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 6320 . . . . . . . . . . . . . . . . . . . . 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 3809 . . . . . . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . . . . . . . . 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 4508 . . . . . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . . . 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 3231 . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . 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 731 . . . . . . . . . . . . . 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 3226 . . . . . . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . . . . 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 11921 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( 0..^ K )  ->  n  e.  ZZ )
5352ssriv 3468 . . . . . . . . . . . . . . . 16  |-  ( 0..^ K )  C_  ZZ
54 fss 5751 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1st `  ( 1st `  z ) ) : ( 1 ... N ) --> ( 0..^ K )  /\  (
0..^ K )  C_  ZZ )  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ZZ )
5551, 53, 54sylancl 666 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ZZ )
5655ad2antlr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 1st `  ( 1st `  z
) ) : ( 1 ... N ) --> ZZ )
57 xp2nd 6835 . . . . . . . . . . . . . . . . 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 5888 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  ( 1st `  z
) )  e.  _V
60 f1oeq1 5819 . . . . . . . . . . . . . . . . 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 3218 . . . . . . . . . . . . . . . 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 199 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
6362ad2antlr 731 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
64 simpr 462 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )
6523, 43, 56, 63, 64poimirlem1 31855 . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  N  e.  NN )
67 fveq2 5878 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
6867breq2d 4432 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
6968ifbid 3931 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
7069csbeq1d 3402 . . . . . . . . . . . . . . . . . . . . . . . 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 5878 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
7271fveq2d 5882 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
7371fveq2d 5882 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
7473imaeq1d 5183 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
7574xpeq1d 4873 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
7673imaeq1d 5183 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
7776xpeq1d 4873 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
7875, 77uneq12d 3621 . . . . . . . . . . . . . . . . . . . . . . . . . 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 6320 . . . . . . . . . . . . . . . . . . . . . . . . 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 3809 . . . . . . . . . . . . . . . . . . . . . . . 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 2463 . . . . . . . . . . . . . . . . . . . . . . 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 4508 . . . . . . . . . . . . . . . . . . . . . 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 2436 . . . . . . . . . . . . . . . . . . . . 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 3231 . . . . . . . . . . . . . . . . . . . 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 465 . . . . . . . . . . . . . . . . . . 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 730 . . . . . . . . . . . . . . . . 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 3226 . . . . . . . . . . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . . . . . . . . . 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 5751 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K )  /\  (
0..^ K )  C_  ZZ )  ->  ( 1st `  ( 1st `  T
) ) : ( 1 ... N ) --> ZZ )
9896, 53, 97sylancl 666 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ZZ )
9998ad2antrr 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 1st `  ( 1st `  T
) ) : ( 1 ... N ) --> ZZ )
100 xp2nd 6835 . . . . . . . . . . . . . . . . . . . 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 5888 . . . . . . . . . . . . . . . . . . . 20  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
103 f1oeq1 5819 . . . . . . . . . . . . . . . . . . . 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 3218 . . . . . . . . . . . . . . . . . . 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 199 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
106105ad2antrr 730 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
107 simplr 760 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )
108 xp2nd 6835 . . . . . . . . . . . . . . . . . . . 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 466 . . . . . . . . . . . . . . . . . 18  |-  ( (
ph  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  T )  e.  ( 0 ... N ) )
111 eldifsn 4122 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 2nd `  T )  e.  ( ( 0 ... N )  \  { ( 2nd `  z
) } )  <->  ( ( 2nd `  T )  e.  ( 0 ... N
)  /\  ( 2nd `  T )  =/=  ( 2nd `  z ) ) )
112111biimpri 209 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( 2nd `  T
)  e.  ( 0 ... N )  /\  ( 2nd `  T )  =/=  ( 2nd `  z
) )  ->  ( 2nd `  T )  e.  ( ( 0 ... N )  \  {
( 2nd `  z
) } ) )
113110, 112sylan 473 . . . . . . . . . . . . . . . . 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 31856 . . . . . . . . . . . . . . . 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 435 . . . . . . . . . . . . . . 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 2642 . . . . . . . . . . . . . 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 719 . . . . . . . . . . . . 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 2701 . . . . . . . . . . 11  |-  ( ( ( ph  /\  z  e.  S )  /\  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  (
( 2nd `  T
)  =/=  N  <->  ( 2nd `  z )  =/=  N
) )
120119exbiri 626 . . . . . . . . . 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 2639 . . . . . . . 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 6835 . . . . . . . . 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 11194 . . . . . . . . . . . . . . . . . 18  |-  NN0  =  ( ZZ>= `  0 )
12712, 126syl6eleq 2520 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( N  -  1 )  e.  ( ZZ>= ` 
0 ) )
128 fzpred 11845 . . . . . . . . . . . . . . . . 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 10722 . . . . . . . . . . . . . . . . . 18  |-  ( 0  +  1 )  =  1
131130oveq1i 6312 . . . . . . . . . . . . . . . . 17  |-  ( ( 0  +  1 ) ... ( N  - 
1 ) )  =  ( 1 ... ( N  -  1 ) )
132131uneq2i 3617 . . . . . . . . . . . . . . . 16  |-  ( { 0 }  u.  (
( 0  +  1 ) ... ( N  -  1 ) ) )  =  ( { 0 }  u.  (
1 ... ( N  - 
1 ) ) )
133129, 132syl6eq 2479 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 0 ... ( N  -  1 ) )  =  ( { 0 }  u.  (
1 ... ( N  - 
1 ) ) ) )
134133eleq2d 2492 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2nd `  z
)  e.  ( 0 ... ( N  - 
1 ) )  <->  ( 2nd `  z )  e.  ( { 0 }  u.  ( 1 ... ( N  -  1 ) ) ) ) )
135134notbid 295 . . . . . . . . . . . . 13  |-  ( ph  ->  ( -.  ( 2nd `  z )  e.  ( 0 ... ( N  -  1 ) )  <->  -.  ( 2nd `  z
)  e.  ( { 0 }  u.  (
1 ... ( N  - 
1 ) ) ) ) )
136 ioran 492 . . . . . . . . . . . . . 14  |-  ( -.  ( ( 2nd `  z
)  =  0  \/  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) )  <-> 
( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) )
137 elun 3606 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  z )  e.  ( { 0 }  u.  ( 1 ... ( N  - 
1 ) ) )  <-> 
( ( 2nd `  z
)  e.  { 0 }  \/  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) )
138 fvex 5888 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  z )  e.  _V
139138elsnc 4020 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  z )  e.  { 0 }  <-> 
( 2nd `  z
)  =  0 )
140139orbi1i 522 . . . . . . . . . . . . . . 15  |-  ( ( ( 2nd `  z
)  e.  { 0 }  \/  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  <->  ( ( 2nd `  z )  =  0  \/  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) ) )
141137, 140bitri 252 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  z )  e.  ( { 0 }  u.  ( 1 ... ( N  - 
1 ) ) )  <-> 
( ( 2nd `  z
)  =  0  \/  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) ) )
142136, 141xchnxbir 310 . . . . . . . . . . . . 13  |-  ( -.  ( 2nd `  z
)  e.  ( { 0 }  u.  (
1 ... ( N  - 
1 ) ) )  <-> 
( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) )
143135, 142syl6bb 264 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  ( 2nd `  z )  e.  ( 0 ... ( N  -  1 ) )  <-> 
( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) ) )
144143anbi2d 708 . . . . . . . . . . 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 3610 . . . . . . . . . . . . . . . 16  |-  ( ( 0 ... ( N  -  1 ) )  u.  { N }
)  =  ( { N }  u.  (
0 ... ( N  - 
1 ) ) )
146145difeq1i 3579 . . . . . . . . . . . . . . 15  |-  ( ( ( 0 ... ( N  -  1 ) )  u.  { N } )  \  (
0 ... ( N  - 
1 ) ) )  =  ( ( { N }  u.  (
0 ... ( N  - 
1 ) ) ) 
\  ( 0 ... ( N  -  1 ) ) )
147 difun2 3875 . . . . . . . . . . . . . . 15  |-  ( ( { N }  u.  ( 0 ... ( N  -  1 ) ) )  \  (
0 ... ( N  - 
1 ) ) )  =  ( { N }  \  ( 0 ... ( N  -  1 ) ) )
148146, 147eqtri 2451 . . . . . . . . . . . . . 14  |-  ( ( ( 0 ... ( N  -  1 ) )  u.  { N } )  \  (
0 ... ( N  - 
1 ) ) )  =  ( { N }  \  ( 0 ... ( N  -  1 ) ) )
1491nncnd 10626 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  CC )
150 npcan1 10045 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
151149, 150syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
1521nnnn0d 10926 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  NN0 )
153152, 126syl6eleq 2520 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  ( ZZ>= ` 
0 ) )
154151, 153eqeltrd 2510 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  e.  ( ZZ>= ` 
0 ) )
15512nn0zd 11039 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  ( N  -  1 )  e.  ZZ )
156 uzid 11174 . . . . . . . . . . . . . . . . . . 19  |-  ( ( N  -  1 )  e.  ZZ  ->  ( N  -  1 )  e.  ( ZZ>= `  ( N  -  1 ) ) )
157 peano2uz 11213 . . . . . . . . . . . . . . . . . . 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 2511 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  ( ZZ>= `  ( N  -  1
) ) )
160 fzsplit2 11825 . . . . . . . . . . . . . . . . 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 665 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 0 ... N
)  =  ( ( 0 ... ( N  -  1 ) )  u.  ( ( ( N  -  1 )  +  1 ) ... N ) ) )
162151oveq1d 6317 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( ( ( N  -  1 )  +  1 ) ... N
)  =  ( N ... N ) )
1631nnzd 11040 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  ZZ )
164 fzsn 11841 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  ZZ  ->  ( N ... N )  =  { N } )
165163, 164syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( N ... N
)  =  { N } )
166162, 165eqtrd 2463 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( ( N  -  1 )  +  1 ) ... N
)  =  { N } )
167166uneq2d 3620 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 0 ... ( N  -  1 ) )  u.  (
( ( N  - 
1 )  +  1 ) ... N ) )  =  ( ( 0 ... ( N  -  1 ) )  u.  { N }
) )
168161, 167eqtrd 2463 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 0 ... N
)  =  ( ( 0 ... ( N  -  1 ) )  u.  { N }
) )
169168difeq1d 3582 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 0 ... N )  \  (
0 ... ( N  - 
1 ) ) )  =  ( ( ( 0 ... ( N  -  1 ) )  u.  { N }
)  \  ( 0 ... ( N  - 
1 ) ) ) )
170 elfzle2 11804 . . . . . . . . . . . . . . . 16  |-  ( N  e.  ( 0 ... ( N  -  1 ) )  ->  N  <_  ( N  -  1 ) )
17115, 170nsyl 124 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  N  e.  ( 0 ... ( N  -  1 ) ) )
172 incom 3655 . . . . . . . . . . . . . . . . 17  |-  ( ( 0 ... ( N  -  1 ) )  i^i  { N }
)  =  ( { N }  i^i  (
0 ... ( N  - 
1 ) ) )
173172eqeq1i 2429 . . . . . . . . . . . . . . . 16  |-  ( ( ( 0 ... ( N  -  1 ) )  i^i  { N } )  =  (/)  <->  ( { N }  i^i  (
0 ... ( N  - 
1 ) ) )  =  (/) )
174 disjsn 4057 . . . . . . . . . . . . . . . 16  |-  ( ( ( 0 ... ( N  -  1 ) )  i^i  { N } )  =  (/)  <->  -.  N  e.  ( 0 ... ( N  - 
1 ) ) )
175 disj3 3837 . . . . . . . . . . . . . . . 16  |-  ( ( { N }  i^i  ( 0 ... ( N  -  1 ) ) )  =  (/)  <->  { N }  =  ( { N }  \  (
0 ... ( N  - 
1 ) ) ) )
176173, 174, 1753bitr3i 278 . . . . . . . . . . . . . . 15  |-  ( -.  N  e.  ( 0 ... ( N  - 
1 ) )  <->  { N }  =  ( { N }  \  (
0 ... ( N  - 
1 ) ) ) )
177171, 176sylib 199 . . . . . . . . . . . . . 14  |-  ( ph  ->  { N }  =  ( { N }  \ 
( 0 ... ( N  -  1 ) ) ) )
178148, 169, 1773eqtr4a 2489 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 0 ... N )  \  (
0 ... ( N  - 
1 ) ) )  =  { N }
)
179178eleq2d 2492 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( 2nd `  z
)  e.  ( ( 0 ... N ) 
\  ( 0 ... ( N  -  1 ) ) )  <->  ( 2nd `  z )  e.  { N } ) )
180 eldif 3446 . . . . . . . . . . . 12  |-  ( ( 2nd `  z )  e.  ( ( 0 ... N )  \ 
( 0 ... ( N  -  1 ) ) )  <->  ( ( 2nd `  z )  e.  ( 0 ... N
)  /\  -.  ( 2nd `  z )  e.  ( 0 ... ( N  -  1 ) ) ) )
181138elsnc 4020 . . . . . . . . . . . 12  |-  ( ( 2nd `  z )  e.  { N }  <->  ( 2nd `  z )  =  N )
182179, 180, 1813bitr3g 290 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  -.  ( 2nd `  z )  e.  ( 0 ... ( N  -  1 ) ) )  <->  ( 2nd `  z
)  =  N ) )
183144, 182bitr3d 258 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  ( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) )  <->  ( 2nd `  z )  =  N ) )
184183biimpd 210 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 2nd `  z )  e.  ( 0 ... N )  /\  ( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) ) )  -> 
( 2nd `  z
)  =  N ) )
185184expdimp 438 . . . . . . . 8  |-  ( (
ph  /\  ( 2nd `  z )  e.  ( 0 ... N ) )  ->  ( ( -.  ( 2nd `  z
)  =  0  /\ 
-.  ( 2nd `  z
)  e.  ( 1 ... ( N  - 
1 ) ) )  ->  ( 2nd `  z
)  =  N ) )
186125, 185sylan2 476 . . . . . . 7  |-  ( (
ph  /\  z  e.  S )  ->  (
( -.  ( 2nd `  z )  =  0  /\  -.  ( 2nd `  z )  e.  ( 1 ... ( N  -  1 ) ) )  ->  ( 2nd `  z )  =  N ) )
187123, 186mpan2d 678 . . . . . 6  |-  ( (
ph  /\  z  e.  S )  ->  ( -.  ( 2nd `  z
)  =  0  -> 
( 2nd `  z
)  =  N ) )
1881, 2, 3poimirlem14 31868 . . . . . . . . . 10  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  N )
189 fveq2 5878 . . . . . . . . . . . 12  |-  ( z  =  s  ->  ( 2nd `  z )  =  ( 2nd `  s
) )
190189eqeq1d 2424 . . . . . . . . . . 11  |-  ( z  =  s  ->  (
( 2nd `  z
)  =  N  <->  ( 2nd `  s )  =  N ) )
191190rmo4 3264 . . . . . . . . . 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 199 . . . . . . . . 9  |-  ( ph  ->  A. z  e.  S  A. s  e.  S  ( ( ( 2nd `  z )  =  N  /\  ( 2nd `  s
)  =  N )  ->  z  =  s ) )
193192r19.21bi 2794 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  A. s  e.  S  ( (
( 2nd `  z
)  =  N  /\  ( 2nd `  s )  =  N )  -> 
z  =  s ) )
1944adantr 466 . . . . . . . 8  |-  ( (
ph  /\  z  e.  S )  ->  T  e.  S )
195 fveq2 5878 . . . . . . . . . . . 12  |-  ( s  =  T  ->  ( 2nd `  s )  =  ( 2nd `  T
) )
196195eqeq1d 2424 . . . . . . . . . . 11  |-  ( s  =  T  ->  (
( 2nd `  s
)  =  N  <->  ( 2nd `  T )  =  N ) )
197196anbi2d 708 . . . . . . . . . 10  |-  ( s  =  T  ->  (
( ( 2nd `  z
)  =  N  /\  ( 2nd `  s )  =  N )  <->  ( ( 2nd `  z )  =  N  /\  ( 2nd `  T )  =  N ) ) )
198 eqeq2 2437 . . . . . . . . . 10  |-  ( s  =  T  ->  (
z  =  s  <->  z  =  T ) )
199197, 198imbi12d 321 . . . . . . . . 9  |-  ( s  =  T  ->  (
( ( ( 2nd `  z )  =  N  /\  ( 2nd `  s
)  =  N )  ->  z  =  s )  <->  ( ( ( 2nd `  z )  =  N  /\  ( 2nd `  T )  =  N )  ->  z  =  T ) ) )
200199rspccv 3179 . . . . . . . 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 678 . . . . . 6  |-  ( (
ph  /\  z  e.  S )  ->  (
( 2nd `  z
)  =  N  -> 
z  =  T ) )
203187, 202syld 45 . . . . 5  |-  ( (
ph  /\  z  e.  S )  ->  ( -.  ( 2nd `  z
)  =  0  -> 
z  =  T ) )
204203necon1ad 2640 . . . 4  |-  ( (
ph  /\  z  e.  S )  ->  (
z  =/=  T  -> 
( 2nd `  z
)  =  0 ) )
205204ralrimiva 2839 . . 3  |-  ( ph  ->  A. z  e.  S  ( z  =/=  T  ->  ( 2nd `  z
)  =  0 ) )
2061, 2, 3poimirlem13 31867 . . 3  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  0 )
207 rmoim 3271 . . 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 3044 . 2  |-  ( E! z  e.  S  z  =/=  T  <->  ( E. z  e.  S  z  =/=  T  /\  E* z  e.  S  z  =/=  T ) )
2107, 208, 209sylanbrc 668 1  |-  ( ph  ->  E! z  e.  S  z  =/=  T )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1868   {cab 2407    =/= wne 2618   A.wral 2775   E.wrex 2776   E!wreu 2777   E*wrmo 2778   {crab 2779   [_csb 3395    \ cdif 3433    u. cun 3434    i^i cin 3435    C_ wss 3436   (/)c0 3761   ifcif 3909   {csn 3996   class class class wbr 4420    |-> cmpt 4479    X. cxp 4848   ran crn 4851   "cima 4853   -->wf 5594   -1-1-onto->wf1o 5597   ` cfv 5598  (class class class)co 6302    oFcof 6540   1stc1st 6802   2ndc2nd 6803    ^m cmap 7477   CCcc 9538   0cc0 9540   1c1 9541    + caddc 9543    < clt 9676    <_ cle 9677    - cmin 9861   NNcn 10610   NN0cn0 10870   ZZcz 10938   ZZ>=cuz 11160   ...cfz 11785  ..^cfzo 11916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rmo 2783  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  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 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-fzo 11917
This theorem is referenced by:  poimirlem22  31876
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