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

Theorem poimirlem14 32018
Description: Lemma for poimir 32037- 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 740 . . . . . . . 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 778 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  z  e.  S )
51nngt0d 10675 . . . . . . . . . 10  |-  ( ph  ->  0  <  N )
6 breq2 4399 . . . . . . . . . . 11  |-  ( ( 2nd `  z )  =  N  ->  (
0  <  ( 2nd `  z )  <->  0  <  N ) )
76biimparc 495 . . . . . . . . . 10  |-  ( ( 0  <  N  /\  ( 2nd `  z )  =  N )  -> 
0  <  ( 2nd `  z ) )
85, 7sylan 479 . . . . . . . . 9  |-  ( (
ph  /\  ( 2nd `  z )  =  N )  ->  0  <  ( 2nd `  z ) )
98ad2ant2r 761 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  0  <  ( 2nd `  z
) )
102, 3, 4, 9poimirlem5 32009 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( F `  0 )  =  ( 1st `  ( 1st `  z ) ) )
11 simplrr 779 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  k  e.  S )
12 breq2 4399 . . . . . . . . . . 11  |-  ( ( 2nd `  k )  =  N  ->  (
0  <  ( 2nd `  k )  <->  0  <  N ) )
1312biimparc 495 . . . . . . . . . 10  |-  ( ( 0  <  N  /\  ( 2nd `  k )  =  N )  -> 
0  <  ( 2nd `  k ) )
145, 13sylan 479 . . . . . . . . 9  |-  ( (
ph  /\  ( 2nd `  k )  =  N )  ->  0  <  ( 2nd `  k ) )
1514ad2ant2rl 763 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  0  <  ( 2nd `  k
) )
162, 3, 11, 15poimirlem5 32009 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( F `  0 )  =  ( 1st `  ( 1st `  k ) ) )
1710, 16eqtr3d 2507 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 1st `  ( 1st `  z
) )  =  ( 1st `  ( 1st `  k ) ) )
18 elrabi 3181 . . . . . . . . . . . . 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 2567 . . . . . . . . . . . 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 6842 . . . . . . . . . . . 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 6843 . . . . . . . . . . . 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 5818 . . . . . . . . . . . 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 3173 . . . . . . . . . . 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 201 . . . . . . . . . 10  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
27 f1ofn 5829 . . . . . . . . . 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 472 . . . . . . . 8  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( 2nd `  ( 1st `  z ) )  Fn  ( 1 ... N ) )
3029ad2antlr 741 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 2nd `  ( 1st `  z
) )  Fn  (
1 ... N ) )
31 elrabi 3181 . . . . . . . . . . . . 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 2567 . . . . . . . . . . . 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 6842 . . . . . . . . . . . 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 6843 . . . . . . . . . . . 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 5818 . . . . . . . . . . . 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 3173 . . . . . . . . . . 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 201 . . . . . . . . . 10  |-  ( k  e.  S  ->  ( 2nd `  ( 1st `  k
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
40 f1ofn 5829 . . . . . . . . . 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 473 . . . . . . . 8  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( 2nd `  ( 1st `  k ) )  Fn  ( 1 ... N ) )
4342ad2antlr 741 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 2nd `  ( 1st `  k
) )  Fn  (
1 ... N ) )
44 simpllr 777 . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . . . 16  |-  ( n  =  N  ->  (
1 ... n )  =  ( 1 ... N
) )
4645imaeq2d 5174 . . . . . . . . . . . . . . 15  |-  ( n  =  N  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... N ) ) )
47 f1ofo 5835 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N )
-onto-> ( 1 ... N
) )
48 foima 5811 . . . . . . . . . . . . . . . 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 2527 . . . . . . . . . . . . . 14  |-  ( ( z  e.  S  /\  n  =  N )  ->  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( 1 ... N
) )
5150adantlr 729 . . . . . . . . . . . . 13  |-  ( ( ( z  e.  S  /\  k  e.  S
)  /\  n  =  N )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( 1 ... N
) )
5245imaeq2d 5174 . . . . . . . . . . . . . . 15  |-  ( n  =  N  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... N ) ) )
53 f1ofo 5835 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  ( 1st `  k ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  k
) ) : ( 1 ... N )
-onto-> ( 1 ... N
) )
54 foima 5811 . . . . . . . . . . . . . . . 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 2527 . . . . . . . . . . . . . 14  |-  ( ( k  e.  S  /\  n  =  N )  ->  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) )  =  ( 1 ... N
) )
5756adantll 728 . . . . . . . . . . . . 13  |-  ( ( ( z  e.  S  /\  k  e.  S
)  /\  n  =  N )  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) )  =  ( 1 ... N
) )
5851, 57eqtr4d 2508 . . . . . . . . . . . 12  |-  ( ( ( z  e.  S  /\  k  e.  S
)  /\  n  =  N )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
5944, 58sylan 479 . . . . . . . . . . 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 768 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ph )
61 elnnuz 11219 . . . . . . . . . . . . . . . . . . . 20  |-  ( N  e.  NN  <->  N  e.  ( ZZ>= `  1 )
)
621, 61sylib 201 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  ( ZZ>= ` 
1 ) )
63 fzm1 11900 . . . . . . . . . . . . . . . . . . 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 719 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( n  e.  ( 1 ... N
)  /\  n  =/=  N )  <->  ( ( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N )  /\  n  =/=  N ) ) )
6665biimpa 492 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  n  =/=  N ) )  ->  (
( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N )  /\  n  =/= 
N ) )
67 df-ne 2643 . . . . . . . . . . . . . . . . . 18  |-  ( n  =/=  N  <->  -.  n  =  N )
6867anbi2i 708 . . . . . . . . . . . . . . . . 17  |-  ( ( ( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N )  /\  n  =/= 
N )  <->  ( (
n  e.  ( 1 ... ( N  - 
1 ) )  \/  n  =  N )  /\  -.  n  =  N ) )
69 pm5.61 727 . . . . . . . . . . . . . . . . 17  |-  ( ( ( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N )  /\  -.  n  =  N )  <->  ( n  e.  ( 1 ... ( N  -  1 ) )  /\  -.  n  =  N ) )
7068, 69bitri 257 . . . . . . . . . . . . . . . 16  |-  ( ( ( n  e.  ( 1 ... ( N  -  1 ) )  \/  n  =  N )  /\  n  =/= 
N )  <->  ( n  e.  ( 1 ... ( N  -  1 ) )  /\  -.  n  =  N ) )
7166, 70sylib 201 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  n  =/=  N ) )  ->  (
n  e.  ( 1 ... ( N  - 
1 ) )  /\  -.  n  =  N
) )
72 1eluzge0 11226 . . . . . . . . . . . . . . . . . 18  |-  1  e.  ( ZZ>= `  0 )
73 fzss1 11863 . . . . . . . . . . . . . . . . . 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 3414 . . . . . . . . . . . . . . . 16  |-  ( n  e.  ( 1 ... ( N  -  1 ) )  ->  n  e.  ( 0 ... ( N  -  1 ) ) )
7675adantr 472 . . . . . . . . . . . . . . 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 479 . . . . . . . . . . . . 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 2537 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
m  e.  ( 0 ... ( N  - 
1 ) )  <->  n  e.  ( 0 ... ( N  -  1 ) ) ) )
8079anbi2d 718 . . . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . . . . 17  |-  ( m  =  n  ->  (
1 ... m )  =  ( 1 ... n
) )
8281imaeq2d 5174 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) ) )
8381imaeq2d 5174 . . . . . . . . . . . . . . . 16  |-  ( m  =  n  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
8482, 83eqeq12d 2486 . . . . . . . . . . . . . . 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 327 . . . . . . . . . . . . . 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 744 . . . . . . . . . . . . . . . 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 744 . . . . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  S  /\  k  e.  S )  ->  z  e.  S )
9089ad3antlr 745 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  z  e.  S )
91 simplrl 778 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( 2nd `  z )  =  N )
92 simpr 468 . . . . . . . . . . . . . . . . 17  |-  ( ( z  e.  S  /\  k  e.  S )  ->  k  e.  S )
9392ad3antlr 745 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  k  e.  S )
94 simplrr 779 . . . . . . . . . . . . . . . 16  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N ) )  /\  m  e.  ( 0 ... ( N  -  1 ) ) )  ->  ( 2nd `  k )  =  N )
95 simpr 468 . . . . . . . . . . . . . . . 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 32016 . . . . . . . . . . . . . . 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 32016 . . . . . . . . . . . . . . 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 3435 . . . . . . . . . . . . . 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 2120 . . . . . . . . . . . . 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 478 . . . . . . . . . . . 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 660 . . . . . . . . . . 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 2730 . . . . . . . . . 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 468 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  n  e.  ( 1 ... N
) )
104 elfzelz 11826 . . . . . . . . . . . . . 14  |-  ( n  e.  ( 1 ... N )  ->  n  e.  ZZ )
1051nnzd 11062 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  ZZ )
106 elfzm1b 11898 . . . . . . . . . . . . . 14  |-  ( ( n  e.  ZZ  /\  N  e.  ZZ )  ->  ( n  e.  ( 1 ... N )  <-> 
( n  -  1 )  e.  ( 0 ... ( N  - 
1 ) ) ) )
107104, 105, 106syl2anr 486 . . . . . . . . . . . . 13  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
n  e.  ( 1 ... N )  <->  ( n  -  1 )  e.  ( 0 ... ( N  -  1 ) ) ) )
108103, 107mpbid 215 . . . . . . . . . . . 12  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
n  -  1 )  e.  ( 0 ... ( N  -  1 ) ) )
10960, 108sylan 479 . . . . . . . . . . 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 6336 . . . . . . . . . . . 12  |-  ( n  -  1 )  e. 
_V
111 eleq1 2537 . . . . . . . . . . . . . 14  |-  ( m  =  ( n  - 
1 )  ->  (
m  e.  ( 0 ... ( N  - 
1 ) )  <->  ( n  -  1 )  e.  ( 0 ... ( N  -  1 ) ) ) )
112111anbi2d 718 . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . . 15  |-  ( m  =  ( n  - 
1 )  ->  (
1 ... m )  =  ( 1 ... (
n  -  1 ) ) )
114113imaeq2d 5174 . . . . . . . . . . . . . 14  |-  ( m  =  ( n  - 
1 )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) )
115113imaeq2d 5174 . . . . . . . . . . . . . 14  |-  ( m  =  ( n  - 
1 )  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
116114, 115eqeq12d 2486 . . . . . . . . . . . . 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 327 . . . . . . . . . . . 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 3086 . . . . . . . . . . 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 478 . . . . . . . . . 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 3541 . . . . . . . . 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 5940 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  ( 1st `  z ) )  Fn  ( 1 ... N )  /\  n  e.  ( 1 ... N
) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  ( ( 2nd `  ( 1st `  z
) ) " {
n } ) )
12228, 121sylan 479 . . . . . . . . . . 11  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `  n ) }  =  ( ( 2nd `  ( 1st `  z ) )
" { n }
) )
123 elfznn 11854 . . . . . . . . . . . . . 14  |-  ( n  e.  ( 1 ... N )  ->  n  e.  NN )
124 uncom 3569 . . . . . . . . . . . . . . . . 17  |-  ( ( 1 ... ( n  -  1 ) )  u.  { n }
)  =  ( { n }  u.  (
1 ... ( n  - 
1 ) ) )
125124difeq1i 3536 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1 ... (
n  -  1 ) )  u.  { n } )  \  (
1 ... ( n  - 
1 ) ) )  =  ( ( { n }  u.  (
1 ... ( n  - 
1 ) ) ) 
\  ( 1 ... ( n  -  1 ) ) )
126 difun2 3838 . . . . . . . . . . . . . . . 16  |-  ( ( { n }  u.  ( 1 ... (
n  -  1 ) ) )  \  (
1 ... ( n  - 
1 ) ) )  =  ( { n }  \  ( 1 ... ( n  -  1 ) ) )
127125, 126eqtri 2493 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... (
n  -  1 ) )  u.  { n } )  \  (
1 ... ( n  - 
1 ) ) )  =  ( { n }  \  ( 1 ... ( n  -  1 ) ) )
128 nncn 10639 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  ->  n  e.  CC )
129 npcan1 10065 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  CC  ->  (
( n  -  1 )  +  1 )  =  n )
130128, 129syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  (
( n  -  1 )  +  1 )  =  n )
131 elnnuz 11219 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
132131biimpi 199 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
133130, 132eqeltrd 2549 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
( n  -  1 )  +  1 )  e.  ( ZZ>= `  1
) )
134 nnm1nn0 10935 . . . . . . . . . . . . . . . . . . . . 21  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
135134nn0zd 11061 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  ZZ )
136 uzid 11197 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  -  1 )  e.  ZZ  ->  (
n  -  1 )  e.  ( ZZ>= `  (
n  -  1 ) ) )
137 peano2uz 11235 . . . . . . . . . . . . . . . . . . . 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 2550 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  ( n  -  1 ) ) )
140 fzsplit2 11850 . . . . . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  (
1 ... n )  =  ( ( 1 ... ( n  -  1 ) )  u.  (
( ( n  - 
1 )  +  1 ) ... n ) ) )
142130oveq1d 6323 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  (
( ( n  - 
1 )  +  1 ) ... n )  =  ( n ... n ) )
143 nnz 10983 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  ->  n  e.  ZZ )
144 fzsn 11866 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  ZZ  ->  (
n ... n )  =  { n } )
145143, 144syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  (
n ... n )  =  { n } )
146142, 145eqtrd 2505 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
( ( n  - 
1 )  +  1 ) ... n )  =  { n }
)
147146uneq2d 3579 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  (
( 1 ... (
n  -  1 ) )  u.  ( ( ( n  -  1 )  +  1 ) ... n ) )  =  ( ( 1 ... ( n  - 
1 ) )  u. 
{ n } ) )
148141, 147eqtrd 2505 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (
1 ... n )  =  ( ( 1 ... ( n  -  1 ) )  u.  {
n } ) )
149148difeq1d 3539 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) )  =  ( ( ( 1 ... ( n  -  1 ) )  u.  { n }
)  \  ( 1 ... ( n  - 
1 ) ) ) )
150 nnre 10638 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  n  e.  RR )
151 ltm1 10467 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  RR  ->  (
n  -  1 )  <  n )
152 peano2rem 9961 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  RR  ->  (
n  -  1 )  e.  RR )
153 ltnle 9731 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( n  -  1 )  e.  RR  /\  n  e.  RR )  ->  ( ( n  - 
1 )  <  n  <->  -.  n  <_  ( n  -  1 ) ) )
154152, 153mpancom 682 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  RR  ->  (
( n  -  1 )  <  n  <->  -.  n  <_  ( n  -  1 ) ) )
155151, 154mpbid 215 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  RR  ->  -.  n  <_  ( n  - 
1 ) )
156 elfzle2 11829 . . . . . . . . . . . . . . . . . 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 3616 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1 ... ( n  -  1 ) )  i^i  { n }
)  =  ( { n }  i^i  (
1 ... ( n  - 
1 ) ) )
160159eqeq1i 2476 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1 ... (
n  -  1 ) )  i^i  { n } )  =  (/)  <->  ( { n }  i^i  ( 1 ... (
n  -  1 ) ) )  =  (/) )
161 disjsn 4023 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1 ... (
n  -  1 ) )  i^i  { n } )  =  (/)  <->  -.  n  e.  ( 1 ... ( n  - 
1 ) ) )
162 disj3 3813 . . . . . . . . . . . . . . . . 17  |-  ( ( { n }  i^i  ( 1 ... (
n  -  1 ) ) )  =  (/)  <->  {
n }  =  ( { n }  \ 
( 1 ... (
n  -  1 ) ) ) )
163160, 161, 1623bitr3i 283 . . . . . . . . . . . . . . . 16  |-  ( -.  n  e.  ( 1 ... ( n  - 
1 ) )  <->  { n }  =  ( {
n }  \  (
1 ... ( n  - 
1 ) ) ) )
164158, 163sylib 201 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  { n }  =  ( {
n }  \  (
1 ... ( n  - 
1 ) ) ) )
165127, 149, 1643eqtr4a 2531 . . . . . . . . . . . . . 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 5174 . . . . . . . . . . . 12  |-  ( n  e.  ( 1 ... N )  ->  (
( 2nd `  ( 1st `  z ) )
" ( ( 1 ... n )  \ 
( 1 ... (
n  -  1 ) ) ) )  =  ( ( 2nd `  ( 1st `  z ) )
" { n }
) )
168167adantl 473 . . . . . . . . . . 11  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) ) )  =  ( ( 2nd `  ( 1st `  z ) ) " { n } ) )
169 dff1o3 5834 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( ( 2nd `  ( 1st `  z
) ) : ( 1 ... N )
-onto-> ( 1 ... N
)  /\  Fun  `' ( 2nd `  ( 1st `  z ) ) ) )
170169simprbi 471 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  Fun  `' ( 2nd `  ( 1st `  z ) ) )
171 imadif 5668 . . . . . . . . . . . . 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 472 . . . . . . . . . . 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 2511 . . . . . . . . . 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 479 . . . . . . . . 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 2537 . . . . . . . . . . . . 13  |-  ( z  =  k  ->  (
z  e.  S  <->  k  e.  S ) )
177176anbi1d 719 . . . . . . . . . . . 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 3971 . . . . . . . . . . . . 13  |-  ( z  =  k  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  { ( ( 2nd `  ( 1st `  k ) ) `  n ) } )
182179imaeq1d 5173 . . . . . . . . . . . . . 14  |-  ( z  =  k  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
183179imaeq1d 5173 . . . . . . . . . . . . . 14  |-  ( z  =  k  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
184182, 183difeq12d 3541 . . . . . . . . . . . . 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 2486 . . . . . . . . . . . 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 327 . . . . . . . . . . 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 2120 . . . . . . . . . 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 479 . . . . . . . . 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 2515 . . . . . . . 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 4131 . . . . . . . 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 5994 . . . . . 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 6851 . . . . . . . 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 485 . . . . . . 7  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( ( ( 1st `  ( 1st `  z
) )  =  ( 1st `  ( 1st `  k ) )  /\  ( 2nd `  ( 1st `  z ) )  =  ( 2nd `  ( 1st `  k ) ) )  <->  ( 1st `  z
)  =  ( 1st `  k ) ) )
198197ad2antlr 741 . . . . . 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 935 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 1st `  z )  =  ( 1st `  k
) )
200 eqtr3 2492 . . . . . 6  |-  ( ( ( 2nd `  z
)  =  N  /\  ( 2nd `  k )  =  N )  -> 
( 2nd `  z
)  =  ( 2nd `  k ) )
201200adantl 473 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  ( 2nd `  z )  =  ( 2nd `  k
) )
202 xpopth 6851 . . . . . . 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 485 . . . . . 6  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( ( ( 1st `  z )  =  ( 1st `  k )  /\  ( 2nd `  z
)  =  ( 2nd `  k ) )  <->  z  =  k ) )
204203ad2antlr 741 . . . . 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 935 . . . 4  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  N  /\  ( 2nd `  k )  =  N ) )  ->  z  =  k )
206205ex 441 . . 3  |-  ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  -> 
( ( ( 2nd `  z )  =  N  /\  ( 2nd `  k
)  =  N )  ->  z  =  k ) )
207206ralrimivva 2814 . 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 2473 . . 3  |-  ( z  =  k  ->  (
( 2nd `  z
)  =  N  <->  ( 2nd `  k )  =  N ) )
210209rmo4 3219 . 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 217 1  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  N )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457    =/= wne 2641   A.wral 2756   E*wrmo 2759   {crab 2760   [_csb 3349    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   (/)c0 3722   ifcif 3872   {csn 3959   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   `'ccnv 4838   "cima 4842   Fun wfun 5583    Fn wfn 5584   -->wf 5585   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308    oFcof 6548   1stc1st 6810   2ndc2nd 6811    ^m cmap 7490   CCcc 9555   RRcr 9556   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   ZZcz 10961   ZZ>=cuz 11182   ...cfz 11810  ..^cfzo 11942
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943
This theorem is referenced by:  poimirlem18  32022  poimirlem21  32025
  Copyright terms: Public domain W3C validator