Users' Mathboxes Mathbox for Brendan Leahy < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem14 Structured version   Unicode version

Theorem poimirlem14 31874
Description: Lemma for poimir 31893- for at most one simplex associated with a shared face is the opposite vertex last on the walk. (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 ) ) )
Assertion
Ref Expression
poimirlem14  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  N )
Distinct variable groups:    f, j,
t, y, z    ph, j,
y    j, F, y    j, N, y    ph, t    f, K, j, t    f, N, t    ph, z    f, F, t, z    z, K   
z, N    S, j,
t, y, z
Allowed substitution hints:    ph( f)    S( f)    K( y)

Proof of Theorem poimirlem14
Dummy variables  k  m  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimir.0 . . . . . . . . 9  |-  ( ph  ->  N  e.  NN )
21ad2antrr 731 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  N  e.  NN )
3 poimirlem22.s . . . . . . . 8  |-  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 } ) ) ) ) }
4 simplrl 769 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  z  e.  S )
51nngt0d 10655 . . . . . . . . . 10  |-  ( ph  ->  0  <  N )
6 breq2 4425 . . . . . . . . . . 11  |-  ( ( 2nd `  z )  =  N  ->  (
0  <  ( 2nd `  z )  <->  0  <  N ) )
76biimparc 490 . . . . . . . . . 10  |-  ( ( 0  <  N  /\  ( 2nd `  z )  =  N )  -> 
0  <  ( 2nd `  z ) )
85, 7sylan 474 . . . . . . . . 9  |-  ( (
ph  /\  ( 2nd `  z )  =  N )  ->  0  <  ( 2nd `  z ) )
98ad2ant2r 752 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  0  <  ( 2nd `  z
) )
102, 3, 4, 9poimirlem5 31865 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( F `  0 )  =  ( 1st `  ( 1st `  z ) ) )
11 simplrr 770 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  k  e.  S )
12 breq2 4425 . . . . . . . . . . 11  |-  ( ( 2nd `  k )  =  N  ->  (
0  <  ( 2nd `  k )  <->  0  <  N ) )
1312biimparc 490 . . . . . . . . . 10  |-  ( ( 0  <  N  /\  ( 2nd `  k )  =  N )  -> 
0  <  ( 2nd `  k ) )
145, 13sylan 474 . . . . . . . . 9  |-  ( (
ph  /\  ( 2nd `  k )  =  N )  ->  0  <  ( 2nd `  k ) )
1514ad2ant2rl 754 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  0  <  ( 2nd `  k
) )
162, 3, 11, 15poimirlem5 31865 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( F `  0 )  =  ( 1st `  ( 1st `  k ) ) )
1710, 16eqtr3d 2466 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 1st `  ( 1st `  z
) )  =  ( 1st `  ( 1st `  k ) ) )
18 elrabi 3227 . . . . . . . . . . . . 13  |-  ( 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 ) ) )
1918, 3eleq2s 2531 . . . . . . . . . . . 12  |-  ( z  e.  S  ->  z  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
20 xp1st 6835 . . . . . . . . . . . 12  |-  ( 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 ) } ) )
21 xp2nd 6836 . . . . . . . . . . . 12  |-  ( ( 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 ) } )
2219, 20, 213syl 18 . . . . . . . . . . 11  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) )  e.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )
23 fvex 5889 . . . . . . . . . . . 12  |-  ( 2nd `  ( 1st `  z
) )  e.  _V
24 f1oeq1 5820 . . . . . . . . . . . 12  |-  ( f  =  ( 2nd `  ( 1st `  z ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  z ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
2523, 24elab 3219 . . . . . . . . . . 11  |-  ( ( 2nd `  ( 1st `  z ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
2622, 25sylib 200 . . . . . . . . . 10  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
27 f1ofn 5830 . . . . . . . . . 10  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  z
) )  Fn  (
1 ... N ) )
2826, 27syl 17 . . . . . . . . 9  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) )  Fn  (
1 ... N ) )
2928adantr 467 . . . . . . . 8  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( 2nd `  ( 1st `  z ) )  Fn  ( 1 ... N ) )
3029ad2antlr 732 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 2nd `  ( 1st `  z
) )  Fn  (
1 ... N ) )
31 elrabi 3227 . . . . . . . . . . . . 13  |-  ( k  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 } ) ) ) ) }  ->  k  e.  ( ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
3231, 3eleq2s 2531 . . . . . . . . . . . 12  |-  ( k  e.  S  ->  k  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
33 xp1st 6835 . . . . . . . . . . . 12  |-  ( k  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  ->  ( 1st `  k )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
34 xp2nd 6836 . . . . . . . . . . . 12  |-  ( ( 1st `  k )  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  -> 
( 2nd `  ( 1st `  k ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
3532, 33, 343syl 18 . . . . . . . . . . 11  |-  ( k  e.  S  ->  ( 2nd `  ( 1st `  k
) )  e.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )
36 fvex 5889 . . . . . . . . . . . 12  |-  ( 2nd `  ( 1st `  k
) )  e.  _V
37 f1oeq1 5820 . . . . . . . . . . . 12  |-  ( f  =  ( 2nd `  ( 1st `  k ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  k ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
3836, 37elab 3219 . . . . . . . . . . 11  |-  ( ( 2nd `  ( 1st `  k ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  k
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
3935, 38sylib 200 . . . . . . . . . 10  |-  ( k  e.  S  ->  ( 2nd `  ( 1st `  k
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
40 f1ofn 5830 . . . . . . . . . 10  |-  ( ( 2nd `  ( 1st `  k ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  k
) )  Fn  (
1 ... N ) )
4139, 40syl 17 . . . . . . . . 9  |-  ( k  e.  S  ->  ( 2nd `  ( 1st `  k
) )  Fn  (
1 ... N ) )
4241adantl 468 . . . . . . . 8  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( 2nd `  ( 1st `  k ) )  Fn  ( 1 ... N ) )
4342ad2antlr 732 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 2nd `  ( 1st `  k
) )  Fn  (
1 ... N ) )
44 simpllr 768 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  ->  ( z  e.  S  /\  k  e.  S ) )
45 oveq2 6311 . . . . . . . . . . . . . . . 16  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
4645imaeq2d 5185 . . . . . . . . . . . . . . 15  |-  ( n  =  N  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... N ) ) )
47 f1ofo 5836 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N )
-onto-> ( 1 ... N
) )
48 foima 5813 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
) -onto-> ( 1 ... N )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
4926, 47, 483syl 18 . . . . . . . . . . . . . . 15  |-  ( z  e.  S  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
5046, 49sylan9eqr 2486 . . . . . . . . . . . . . 14  |-  ( ( z  e.  S  /\  n  =  N )  ->  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( 1 ... N
) )
5150adantlr 720 . . . . . . . . . . . . 13  |-  ( ( ( z  e.  S  /\  k  e.  S
)  /\  n  =  N )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( 1 ... N
) )
5245imaeq2d 5185 . . . . . . . . . . . . . . 15  |-  ( n  =  N  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... N ) ) )
53 f1ofo 5836 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  ( 1st `  k ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  k
) ) : ( 1 ... N )
-onto-> ( 1 ... N
) )
54 foima 5813 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  ( 1st `  k ) ) : ( 1 ... N
) -onto-> ( 1 ... N )  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
5539, 53, 543syl 18 . . . . . . . . . . . . . . 15  |-  ( k  e.  S  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
5652, 55sylan9eqr 2486 . . . . . . . . . . . . . 14  |-  ( ( k  e.  S  /\  n  =  N )  ->  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) )  =  ( 1 ... N
) )
5756adantll 719 . . . . . . . . . . . . 13  |-  ( ( ( z  e.  S  /\  k  e.  S
)  /\  n  =  N )  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) )  =  ( 1 ... N
) )
5851, 57eqtr4d 2467 . . . . . . . . . . . 12  |-  ( ( ( z  e.  S  /\  k  e.  S
)  /\  n  =  N )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
5944, 58sylan 474 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  /\  n  =  N )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
60 simpll 759 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ph )
61 elnnuz 11197 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
621, 61sylib 200 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
63 fzm1 11876 . . . . . . . . . . . . . . . . . . 19  |-  ( N  e.  ( ZZ>= `  1
)  ->  ( n  e.  ( 1 ... N
)  <->  ( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N ) ) )
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( n  e.  ( 1 ... N )  <-> 
( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N ) ) )
6564anbi1d 710 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( n  e.  ( 1 ... N
)  /\  n  =/=  N )  <->  ( ( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N )  /\  n  =/=  N ) ) )
6665biimpa 487 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  n  =/=  N ) )  ->  (
( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N )  /\  n  =/= 
N ) )
67 df-ne 2621 . . . . . . . . . . . . . . . . . 18  |-  ( n  =/=  N  <->  -.  n  =  N )
6867anbi2i 699 . . . . . . . . . . . . . . . . 17  |-  ( ( ( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N )  /\  n  =/= 
N )  <->  ( (
n  e.  ( 1 ... ( N  - 
1 ) )  \/  n  =  N )  /\  -.  n  =  N ) )
69 pm5.61 718 . . . . . . . . . . . . . . . . 17  |-  ( ( ( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N )  /\  -.  n  =  N )  <->  ( n  e.  ( 1 ... ( N  -  1 ) )  /\  -.  n  =  N ) )
7068, 69bitri 253 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N )  /\  n  =/= 
N )  <->  ( n  e.  ( 1 ... ( N  -  1 ) )  /\  -.  n  =  N ) )
7166, 70sylib 200 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  n  =/=  N ) )  ->  (
n  e.  ( 1 ... ( N  - 
1 ) )  /\  -.  n  =  N
) )
72 1eluzge0 11204 . . . . . . . . . . . . . . . . . 18  |-  1  e.  ( ZZ>= `  0 )
73 fzss1 11839 . . . . . . . . . . . . . . . . . 18  |-  ( 1  e.  ( ZZ>= `  0
)  ->  ( 1 ... ( N  - 
1 ) )  C_  ( 0 ... ( N  -  1 ) ) )
7472, 73ax-mp 5 . . . . . . . . . . . . . . . . 17  |-  ( 1 ... ( N  - 
1 ) )  C_  ( 0 ... ( N  -  1 ) )
7574sseli 3461 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( 1 ... ( N  -  1 ) )  ->  n  e.  ( 0 ... ( N  -  1 ) ) )
7675adantr 467 . . . . . . . . . . . . . . 15  |-  ( ( n  e.  ( 1 ... ( N  - 
1 ) )  /\  -.  n  =  N
)  ->  n  e.  ( 0 ... ( N  -  1 ) ) )
7771, 76syl 17 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  n  =/=  N ) )  ->  n  e.  ( 0 ... ( N  -  1 ) ) )
7860, 77sylan 474 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  ( n  e.  ( 1 ... N
)  /\  n  =/=  N ) )  ->  n  e.  ( 0 ... ( N  -  1 ) ) )
79 eleq1 2495 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
m  e.  ( 0 ... ( N  - 
1 ) )  <->  n  e.  ( 0 ... ( N  -  1 ) ) ) )
8079anbi2d 709 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  (
( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  <->  ( ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  /\  ( ( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 0 ... ( N  -  1 ) ) ) ) )
81 oveq2 6311 . . . . . . . . . . . . . . . . 17  |-  ( m  =  n  ->  (
1 ... m )  =  ( 1 ... n
) )
8281imaeq2d 5185 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) ) )
8381imaeq2d 5185 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
8482, 83eqeq12d 2445 . . . . . . . . . . . . . . 15  |-  ( m  =  n  ->  (
( ( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  <->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... n ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... n ) ) ) )
8580, 84imbi12d 322 . . . . . . . . . . . . . 14  |-  ( m  =  n  ->  (
( ( ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  /\  ( ( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... m ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... m ) ) )  <->  ( ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 0 ... ( N  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) ) ) )
861ad3antrrr 735 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  N  e.  NN )
87 poimirlem22.1 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  F : ( 0 ... ( N  - 
1 ) ) --> ( ( 0 ... K
)  ^m  ( 1 ... N ) ) )
8887ad3antrrr 735 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  F :
( 0 ... ( N  -  1 ) ) --> ( ( 0 ... K )  ^m  ( 1 ... N
) ) )
89 simpl 459 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  S  /\  k  e.  S )  ->  z  e.  S )
9089ad3antlr 736 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  z  e.  S )
91 simplrl 769 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( 2nd `  z )  =  N )
92 simpr 463 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  S  /\  k  e.  S )  ->  k  e.  S )
9392ad3antlr 736 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  S )
94 simplrr 770 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( 2nd `  k )  =  N )
95 simpr 463 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  m  e.  ( 0 ... ( N  -  1 ) ) )
9686, 3, 88, 90, 91, 93, 94, 95poimirlem12 31872 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... m ) ) 
C_  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... m ) ) )
9786, 3, 88, 93, 94, 90, 91, 95poimirlem12 31872 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... m ) ) 
C_  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... m ) ) )
9896, 97eqssd 3482 . . . . . . . . . . . . . 14  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... m ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... m ) ) )
9985, 98chvarv 2069 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... n ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... n ) ) )
10078, 99syldan 473 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  ( n  e.  ( 1 ... N
)  /\  n  =/=  N ) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
101100anassrs 653 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  /\  n  =/= 
N )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
10259, 101pm2.61dane 2743 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... n ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... n ) ) )
103 simpr 463 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  n  e.  ( 1 ... N
) )
104 elfzelz 11802 . . . . . . . . . . . . . 14  |-  ( n  e.  ( 1 ... N )  ->  n  e.  ZZ )
1051nnzd 11041 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  ZZ )
106 elfzm1b 11874 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ZZ  /\  N  e.  ZZ )  ->  ( n  e.  ( 1 ... N )  <-> 
( n  -  1 )  e.  ( 0 ... ( N  - 
1 ) ) ) )
107104, 105, 106syl2anr 481 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
n  e.  ( 1 ... N )  <->  ( n  -  1 )  e.  ( 0 ... ( N  -  1 ) ) ) )
108103, 107mpbid 214 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
n  -  1 )  e.  ( 0 ... ( N  -  1 ) ) )
10960, 108sylan 474 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  ->  ( n  -  1 )  e.  ( 0 ... ( N  -  1 ) ) )
110 ovex 6331 . . . . . . . . . . . 12  |-  ( n  -  1 )  e. 
_V
111 eleq1 2495 . . . . . . . . . . . . . 14  |-  ( m  =  ( n  - 
1 )  ->  (
m  e.  ( 0 ... ( N  - 
1 ) )  <->  ( n  -  1 )  e.  ( 0 ... ( N  -  1 ) ) ) )
112111anbi2d 709 . . . . . . . . . . . . 13  |-  ( m  =  ( n  - 
1 )  ->  (
( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  <->  ( ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  /\  ( ( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  ( n  - 
1 )  e.  ( 0 ... ( N  -  1 ) ) ) ) )
113 oveq2 6311 . . . . . . . . . . . . . . 15  |-  ( m  =  ( n  - 
1 )  ->  (
1 ... m )  =  ( 1 ... (
n  -  1 ) ) )
114113imaeq2d 5185 . . . . . . . . . . . . . 14  |-  ( m  =  ( n  - 
1 )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) )
115113imaeq2d 5185 . . . . . . . . . . . . . 14  |-  ( m  =  ( n  - 
1 )  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
116114, 115eqeq12d 2445 . . . . . . . . . . . . 13  |-  ( m  =  ( n  - 
1 )  ->  (
( ( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  <->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... ( n  - 
1 ) ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... ( n  - 
1 ) ) ) ) )
117112, 116imbi12d 322 . . . . . . . . . . . 12  |-  ( m  =  ( n  - 
1 )  ->  (
( ( ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  /\  ( ( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... m ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... m ) ) )  <->  ( ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  /\  (
n  -  1 )  e.  ( 0 ... ( N  -  1 ) ) )  -> 
( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) ) ) )
118110, 117, 98vtocl 3134 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  ( n  - 
1 )  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... ( n  - 
1 ) ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... ( n  - 
1 ) ) ) )
119109, 118syldan 473 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... ( n  - 
1 ) ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... ( n  - 
1 ) ) ) )
120102, 119difeq12d 3585 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  ->  ( (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  \ 
( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) )  =  ( ( ( 2nd `  ( 1st `  k ) ) "
( 1 ... n
) )  \  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) ) )
121 fnsnfv 5939 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  ( 1st `  z ) )  Fn  ( 1 ... N )  /\  n  e.  ( 1 ... N
) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  ( ( 2nd `  ( 1st `  z
) ) " {
n } ) )
12228, 121sylan 474 . . . . . . . . . . 11  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `  n ) }  =  ( ( 2nd `  ( 1st `  z ) )
" { n }
) )
123 elfznn 11830 . . . . . . . . . . . . . 14  |-  ( n  e.  ( 1 ... N )  ->  n  e.  NN )
124 uncom 3611 . . . . . . . . . . . . . . . . 17  |-  ( ( 1 ... ( n  -  1 ) )  u.  { n }
)  =  ( { n }  u.  (
1 ... ( n  - 
1 ) ) )
125124difeq1i 3580 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1 ... (
n  -  1 ) )  u.  { n } )  \  (
1 ... ( n  - 
1 ) ) )  =  ( ( { n }  u.  (
1 ... ( n  - 
1 ) ) ) 
\  ( 1 ... ( n  -  1 ) ) )
126 difun2 3876 . . . . . . . . . . . . . . . 16  |-  ( ( { n }  u.  ( 1 ... (
n  -  1 ) ) )  \  (
1 ... ( n  - 
1 ) ) )  =  ( { n }  \  ( 1 ... ( n  -  1 ) ) )
127125, 126eqtri 2452 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... (
n  -  1 ) )  u.  { n } )  \  (
1 ... ( n  - 
1 ) ) )  =  ( { n }  \  ( 1 ... ( n  -  1 ) ) )
128 nncn 10619 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  ->  n  e.  CC )
129 npcan1 10046 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  CC  ->  (
( n  -  1 )  +  1 )  =  n )
130128, 129syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  (
( n  -  1 )  +  1 )  =  n )
131 elnnuz 11197 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
132131biimpi 198 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
133130, 132eqeltrd 2511 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
( n  -  1 )  +  1 )  e.  ( ZZ>= `  1
) )
134 nnm1nn0 10913 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
135134nn0zd 11040 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  ZZ )
136 uzid 11175 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  -  1 )  e.  ZZ  ->  (
n  -  1 )  e.  ( ZZ>= `  (
n  -  1 ) ) )
137 peano2uz 11214 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  -  1 )  e.  ( ZZ>= `  (
n  -  1 ) )  ->  ( (
n  -  1 )  +  1 )  e.  ( ZZ>= `  ( n  -  1 ) ) )
138135, 136, 1373syl 18 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  (
( n  -  1 )  +  1 )  e.  ( ZZ>= `  (
n  -  1 ) ) )
139130, 138eqeltrrd 2512 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  ( n  -  1 ) ) )
140 fzsplit2 11826 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( n  - 
1 )  +  1 )  e.  ( ZZ>= ` 
1 )  /\  n  e.  ( ZZ>= `  ( n  -  1 ) ) )  ->  ( 1 ... n )  =  ( ( 1 ... ( n  -  1 ) )  u.  (
( ( n  - 
1 )  +  1 ) ... n ) ) )
141133, 139, 140syl2anc 666 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  (
1 ... n )  =  ( ( 1 ... ( n  -  1 ) )  u.  (
( ( n  - 
1 )  +  1 ) ... n ) ) )
142130oveq1d 6318 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  (
( ( n  - 
1 )  +  1 ) ... n )  =  ( n ... n ) )
143 nnz 10961 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  ->  n  e.  ZZ )
144 fzsn 11842 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  ZZ  ->  (
n ... n )  =  { n } )
145143, 144syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  (
n ... n )  =  { n } )
146142, 145eqtrd 2464 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
( ( n  - 
1 )  +  1 ) ... n )  =  { n }
)
147146uneq2d 3621 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  (
( 1 ... (
n  -  1 ) )  u.  ( ( ( n  -  1 )  +  1 ) ... n ) )  =  ( ( 1 ... ( n  - 
1 ) )  u. 
{ n } ) )
148141, 147eqtrd 2464 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (
1 ... n )  =  ( ( 1 ... ( n  -  1 ) )  u.  {
n } ) )
149148difeq1d 3583 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) )  =  ( ( ( 1 ... ( n  -  1 ) )  u.  { n }
)  \  ( 1 ... ( n  - 
1 ) ) ) )
150 nnre 10618 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  n  e.  RR )
151 ltm1 10447 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  RR  ->  (
n  -  1 )  <  n )
152 peano2rem 9943 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  RR  ->  (
n  -  1 )  e.  RR )
153 ltnle 9715 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( n  -  1 )  e.  RR  /\  n  e.  RR )  ->  ( ( n  - 
1 )  <  n  <->  -.  n  <_  ( n  -  1 ) ) )
154152, 153mpancom 674 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  RR  ->  (
( n  -  1 )  <  n  <->  -.  n  <_  ( n  -  1 ) ) )
155151, 154mpbid 214 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  RR  ->  -.  n  <_  ( n  - 
1 ) )
156 elfzle2 11805 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... ( n  -  1 ) )  ->  n  <_  ( n  -  1 ) )
157155, 156nsyl 125 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR  ->  -.  n  e.  ( 1 ... ( n  - 
1 ) ) )
158150, 157syl 17 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  -.  n  e.  ( 1 ... ( n  - 
1 ) ) )
159 incom 3656 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1 ... ( n  -  1 ) )  i^i  { n }
)  =  ( { n }  i^i  (
1 ... ( n  - 
1 ) ) )
160159eqeq1i 2430 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1 ... (
n  -  1 ) )  i^i  { n } )  =  (/)  <->  ( { n }  i^i  ( 1 ... (
n  -  1 ) ) )  =  (/) )
161 disjsn 4058 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1 ... (
n  -  1 ) )  i^i  { n } )  =  (/)  <->  -.  n  e.  ( 1 ... ( n  - 
1 ) ) )
162 disj3 3838 . . . . . . . . . . . . . . . . 17  |-  ( ( { n }  i^i  ( 1 ... (
n  -  1 ) ) )  =  (/)  <->  {
n }  =  ( { n }  \ 
( 1 ... (
n  -  1 ) ) ) )
163160, 161, 1623bitr3i 279 . . . . . . . . . . . . . . . 16  |-  ( -.  n  e.  ( 1 ... ( n  - 
1 ) )  <->  { n }  =  ( {
n }  \  (
1 ... ( n  - 
1 ) ) ) )
164158, 163sylib 200 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  { n }  =  ( {
n }  \  (
1 ... ( n  - 
1 ) ) ) )
165127, 149, 1643eqtr4a 2490 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) )  =  { n }
)
166123, 165syl 17 . . . . . . . . . . . . 13  |-  ( n  e.  ( 1 ... N )  ->  (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) )  =  { n }
)
167166imaeq2d 5185 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... N )  ->  (
( 2nd `  ( 1st `  z ) )
" ( ( 1 ... n )  \ 
( 1 ... (
n  -  1 ) ) ) )  =  ( ( 2nd `  ( 1st `  z ) )
" { n }
) )
168167adantl 468 . . . . . . . . . . 11  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) ) )  =  ( ( 2nd `  ( 1st `  z ) ) " { n } ) )
169 dff1o3 5835 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( ( 2nd `  ( 1st `  z
) ) : ( 1 ... N )
-onto-> ( 1 ... N
)  /\  Fun  `' ( 2nd `  ( 1st `  z ) ) ) )
170169simprbi 466 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  Fun  `' ( 2nd `  ( 1st `  z ) ) )
171 imadif 5674 . . . . . . . . . . . . 13  |-  ( Fun  `' ( 2nd `  ( 1st `  z ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) ) )  =  ( ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  \ 
( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) ) )
17226, 170, 1713syl 18 . . . . . . . . . . . 12  |-  ( z  e.  S  ->  (
( 2nd `  ( 1st `  z ) )
" ( ( 1 ... n )  \ 
( 1 ... (
n  -  1 ) ) ) )  =  ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... n ) ) 
\  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... ( n  - 
1 ) ) ) ) )
173172adantr 467 . . . . . . . . . . 11  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) ) )  =  ( ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  \ 
( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) ) )
174122, 168, 1733eqtr2d 2470 . . . . . . . . . 10  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `  n ) }  =  ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... n ) ) 
\  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... ( n  - 
1 ) ) ) ) )
1754, 174sylan 474 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  ->  { (
( 2nd `  ( 1st `  z ) ) `
 n ) }  =  ( ( ( 2nd `  ( 1st `  z ) ) "
( 1 ... n
) )  \  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) ) )
176 eleq1 2495 . . . . . . . . . . . . 13  |-  ( z  =  k  ->  (
z  e.  S  <->  k  e.  S ) )
177176anbi1d 710 . . . . . . . . . . . 12  |-  ( z  =  k  ->  (
( z  e.  S  /\  n  e.  (
1 ... N ) )  <-> 
( k  e.  S  /\  n  e.  (
1 ... N ) ) ) )
178 fveq2 5879 . . . . . . . . . . . . . . . 16  |-  ( z  =  k  ->  ( 1st `  z )  =  ( 1st `  k
) )
179178fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( z  =  k  ->  ( 2nd `  ( 1st `  z
) )  =  ( 2nd `  ( 1st `  k ) ) )
180179fveq1d 5881 . . . . . . . . . . . . . 14  |-  ( z  =  k  ->  (
( 2nd `  ( 1st `  z ) ) `
 n )  =  ( ( 2nd `  ( 1st `  k ) ) `
 n ) )
181180sneqd 4009 . . . . . . . . . . . . 13  |-  ( z  =  k  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  { ( ( 2nd `  ( 1st `  k ) ) `  n ) } )
182179imaeq1d 5184 . . . . . . . . . . . . . 14  |-  ( z  =  k  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
183179imaeq1d 5184 . . . . . . . . . . . . . 14  |-  ( z  =  k  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
184182, 183difeq12d 3585 . . . . . . . . . . . . 13  |-  ( z  =  k  ->  (
( ( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  \ 
( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) )  =  ( ( ( 2nd `  ( 1st `  k ) ) "
( 1 ... n
) )  \  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) ) )
185181, 184eqeq12d 2445 . . . . . . . . . . . 12  |-  ( z  =  k  ->  ( { ( ( 2nd `  ( 1st `  z
) ) `  n
) }  =  ( ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  \ 
( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) )  <->  { ( ( 2nd `  ( 1st `  k
) ) `  n
) }  =  ( ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) )  \ 
( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) ) ) )
186177, 185imbi12d 322 . . . . . . . . . . 11  |-  ( z  =  k  ->  (
( ( z  e.  S  /\  n  e.  ( 1 ... N
) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  ( ( ( 2nd `  ( 1st `  z ) ) "
( 1 ... n
) )  \  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) ) )  <->  ( ( k  e.  S  /\  n  e.  ( 1 ... N
) )  ->  { ( ( 2nd `  ( 1st `  k ) ) `
 n ) }  =  ( ( ( 2nd `  ( 1st `  k ) ) "
( 1 ... n
) )  \  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) ) ) ) )
187186, 174chvarv 2069 . . . . . . . . . 10  |-  ( ( k  e.  S  /\  n  e.  ( 1 ... N ) )  ->  { ( ( 2nd `  ( 1st `  k ) ) `  n ) }  =  ( ( ( 2nd `  ( 1st `  k
) ) " (
1 ... n ) ) 
\  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... ( n  - 
1 ) ) ) ) )
18811, 187sylan 474 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  ->  { (
( 2nd `  ( 1st `  k ) ) `
 n ) }  =  ( ( ( 2nd `  ( 1st `  k ) ) "
( 1 ... n
) )  \  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) ) )
189120, 175, 1883eqtr4d 2474 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  ->  { (
( 2nd `  ( 1st `  z ) ) `
 n ) }  =  { ( ( 2nd `  ( 1st `  k ) ) `  n ) } )
190 fvex 5889 . . . . . . . . 9  |-  ( ( 2nd `  ( 1st `  z ) ) `  n )  e.  _V
191190sneqr 4165 . . . . . . . 8  |-  ( { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  { ( ( 2nd `  ( 1st `  k ) ) `  n ) }  ->  ( ( 2nd `  ( 1st `  z ) ) `
 n )  =  ( ( 2nd `  ( 1st `  k ) ) `
 n ) )
192189, 191syl 17 . . . . . . 7  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  n  e.  ( 1 ... N ) )  ->  ( ( 2nd `  ( 1st `  z
) ) `  n
)  =  ( ( 2nd `  ( 1st `  k ) ) `  n ) )
19330, 43, 192eqfnfvd 5992 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 2nd `  ( 1st `  z
) )  =  ( 2nd `  ( 1st `  k ) ) )
19419, 20syl 17 . . . . . . . 8  |-  ( z  e.  S  ->  ( 1st `  z )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
19532, 33syl 17 . . . . . . . 8  |-  ( k  e.  S  ->  ( 1st `  k )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
196 xpopth 6844 . . . . . . . 8  |-  ( ( ( 1st `  z
)  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } )  /\  ( 1st `  k )  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } ) )  ->  ( ( ( 1st `  ( 1st `  z ) )  =  ( 1st `  ( 1st `  k ) )  /\  ( 2nd `  ( 1st `  z ) )  =  ( 2nd `  ( 1st `  k ) ) )  <->  ( 1st `  z
)  =  ( 1st `  k ) ) )
197194, 195, 196syl2an 480 . . . . . . 7  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( ( ( 1st `  ( 1st `  z
) )  =  ( 1st `  ( 1st `  k ) )  /\  ( 2nd `  ( 1st `  z ) )  =  ( 2nd `  ( 1st `  k ) ) )  <->  ( 1st `  z
)  =  ( 1st `  k ) ) )
198197ad2antlr 732 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  (
( ( 1st `  ( 1st `  z ) )  =  ( 1st `  ( 1st `  k ) )  /\  ( 2nd `  ( 1st `  z ) )  =  ( 2nd `  ( 1st `  k ) ) )  <->  ( 1st `  z
)  =  ( 1st `  k ) ) )
19917, 193, 198mpbi2and 930 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 1st `  z )  =  ( 1st `  k
) )
200 eqtr3 2451 . . . . . 6  |-  ( ( ( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N )  -> 
( 2nd `  z
)  =  ( 2nd `  k ) )
201200adantl 468 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 2nd `  z )  =  ( 2nd `  k
) )
202 xpopth 6844 . . . . . . 7  |-  ( ( z  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) )  /\  k  e.  ( ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )  ->  ( (
( 1st `  z
)  =  ( 1st `  k )  /\  ( 2nd `  z )  =  ( 2nd `  k
) )  <->  z  =  k ) )
20319, 32, 202syl2an 480 . . . . . 6  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( ( ( 1st `  z )  =  ( 1st `  k )  /\  ( 2nd `  z
)  =  ( 2nd `  k ) )  <->  z  =  k ) )
204203ad2antlr 732 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  (
( ( 1st `  z
)  =  ( 1st `  k )  /\  ( 2nd `  z )  =  ( 2nd `  k
) )  <->  z  =  k ) )
205199, 201, 204mpbi2and 930 . . . 4  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  z  =  k )
206205ex 436 . . 3  |-  ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  -> 
( ( ( 2nd `  z )  =  N  /\  ( 2nd `  k
)  =  N )  ->  z  =  k ) )
207206ralrimivva 2847 . 2  |-  ( ph  ->  A. z  e.  S  A. k  e.  S  ( ( ( 2nd `  z )  =  N  /\  ( 2nd `  k
)  =  N )  ->  z  =  k ) )
208 fveq2 5879 . . . 4  |-  ( z  =  k  ->  ( 2nd `  z )  =  ( 2nd `  k
) )
209208eqeq1d 2425 . . 3  |-  ( z  =  k  ->  (
( 2nd `  z
)  =  N  <->  ( 2nd `  k )  =  N ) )
210209rmo4 3265 . 2  |-  ( E* z  e.  S  ( 2nd `  z )  =  N  <->  A. z  e.  S  A. k  e.  S  ( (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N )  -> 
z  =  k ) )
211207, 210sylibr 216 1  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    \/ wo 370    /\ wa 371    = wceq 1438    e. wcel 1869   {cab 2408    =/= wne 2619   A.wral 2776   E*wrmo 2779   {crab 2780   [_csb 3396    \ cdif 3434    u. cun 3435    i^i cin 3436    C_ wss 3437   (/)c0 3762   ifcif 3910   {csn 3997   class class class wbr 4421    |-> cmpt 4480    X. cxp 4849   `'ccnv 4850   "cima 4854   Fun wfun 5593    Fn wfn 5594   -->wf 5595   -onto->wfo 5597   -1-1-onto->wf1o 5598   ` cfv 5599  (class class class)co 6303    oFcof 6541   1stc1st 6803   2ndc2nd 6804    ^m cmap 7478   CCcc 9539   RRcr 9540   0cc0 9541   1c1 9542    + caddc 9544    < clt 9677    <_ cle 9678    - cmin 9862   NNcn 10611   ZZcz 10939   ZZ>=cuz 11161   ...cfz 11786  ..^cfzo 11917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rmo 2784  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-fzo 11918
This theorem is referenced by:  poimirlem18  31878  poimirlem21  31881
  Copyright terms: Public domain W3C validator