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Theorem poimirlem13 31873
Description: Lemma for poimir 31893- for at most one simplex associated with a shared face is the opposite vertex first 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
poimirlem13  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  0 )
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 poimirlem13
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 )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  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 poimirlem22.1 . . . . . . . . 9  |-  ( ph  ->  F : ( 0 ... ( N  - 
1 ) ) --> ( ( 0 ... K
)  ^m  ( 1 ... N ) ) )
54ad2antrr 731 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  F : ( 0 ... ( N  -  1 ) ) --> ( ( 0 ... K )  ^m  ( 1 ... N ) ) )
6 simplrl 769 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  z  e.  S )
7 simprl 763 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  z )  =  0 )
82, 3, 5, 6, 7poimirlem10 31870 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  (
( F `  ( N  -  1 ) )  oF  -  ( ( 1 ... N )  X.  {
1 } ) )  =  ( 1st `  ( 1st `  z ) ) )
9 simplrr 770 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  k  e.  S )
10 simprr 765 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  k )  =  0 )
112, 3, 5, 9, 10poimirlem10 31870 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  (
( F `  ( N  -  1 ) )  oF  -  ( ( 1 ... N )  X.  {
1 } ) )  =  ( 1st `  ( 1st `  k ) ) )
128, 11eqtr3d 2466 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 1st `  ( 1st `  z
) )  =  ( 1st `  ( 1st `  k ) ) )
13 elrabi 3227 . . . . . . . . . . . . . 14  |-  ( 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 ) ) )
1413, 3eleq2s 2531 . . . . . . . . . . . . 13  |-  ( z  e.  S  ->  z  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
15 xp1st 6835 . . . . . . . . . . . . 13  |-  ( 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 ) } ) )
1614, 15syl 17 . . . . . . . . . . . 12  |-  ( z  e.  S  ->  ( 1st `  z )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
17 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 ) } )
1816, 17syl 17 . . . . . . . . . . 11  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) )  e.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )
19 fvex 5889 . . . . . . . . . . . 12  |-  ( 2nd `  ( 1st `  z
) )  e.  _V
20 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
) ) )
2119, 20elab 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 ) )
2218, 21sylib 200 . . . . . . . . . 10  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
23 f1ofn 5830 . . . . . . . . . 10  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  z
) )  Fn  (
1 ... N ) )
2422, 23syl 17 . . . . . . . . 9  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) )  Fn  (
1 ... N ) )
2524adantr 467 . . . . . . . 8  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( 2nd `  ( 1st `  z ) )  Fn  ( 1 ... N ) )
2625ad2antlr 732 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  ( 1st `  z
) )  Fn  (
1 ... N ) )
27 elrabi 3227 . . . . . . . . . . . . . 14  |-  ( 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 ) ) )
2827, 3eleq2s 2531 . . . . . . . . . . . . 13  |-  ( k  e.  S  ->  k  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
29 xp1st 6835 . . . . . . . . . . . . 13  |-  ( 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 ) } ) )
3028, 29syl 17 . . . . . . . . . . . 12  |-  ( k  e.  S  ->  ( 1st `  k )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
31 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 ) } )
3230, 31syl 17 . . . . . . . . . . 11  |-  ( k  e.  S  ->  ( 2nd `  ( 1st `  k
) )  e.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )
33 fvex 5889 . . . . . . . . . . . 12  |-  ( 2nd `  ( 1st `  k
) )  e.  _V
34 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
) ) )
3533, 34elab 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 ) )
3632, 35sylib 200 . . . . . . . . . 10  |-  ( k  e.  S  ->  ( 2nd `  ( 1st `  k
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
37 f1ofn 5830 . . . . . . . . . 10  |-  ( ( 2nd `  ( 1st `  k ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  k
) )  Fn  (
1 ... N ) )
3836, 37syl 17 . . . . . . . . 9  |-  ( k  e.  S  ->  ( 2nd `  ( 1st `  k
) )  Fn  (
1 ... N ) )
3938adantl 468 . . . . . . . 8  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( 2nd `  ( 1st `  k ) )  Fn  ( 1 ... N ) )
4039ad2antlr 732 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  ( 1st `  k
) )  Fn  (
1 ... N ) )
41 eleq1 2495 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  (
m  e.  ( 1 ... N )  <->  n  e.  ( 1 ... N
) ) )
4241anbi2d 709 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  <->  ( (
( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  /\  n  e.  ( 1 ... N
) ) ) )
43 oveq2 6311 . . . . . . . . . . . . . 14  |-  ( m  =  n  ->  (
1 ... m )  =  ( 1 ... n
) )
4443imaeq2d 5185 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) ) )
4543imaeq2d 5185 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
4644, 45eqeq12d 2445 . . . . . . . . . . . 12  |-  ( m  =  n  ->  (
( ( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  <->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... n ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... n ) ) ) )
4742, 46imbi12d 322 . . . . . . . . . . 11  |-  ( m  =  n  ->  (
( ( ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  /\  ( ( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) ) )  <-> 
( ( ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  /\  ( ( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) ) ) )
481ad3antrrr 735 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  N  e.  NN )
494ad3antrrr 735 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  F : ( 0 ... ( N  -  1 ) ) --> ( ( 0 ... K )  ^m  ( 1 ... N ) ) )
50 simpl 459 . . . . . . . . . . . . . 14  |-  ( ( z  e.  S  /\  k  e.  S )  ->  z  e.  S )
5150ad3antlr 736 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  z  e.  S )
52 simplrl 769 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  ( 2nd `  z )  =  0 )
53 simpr 463 . . . . . . . . . . . . . 14  |-  ( ( z  e.  S  /\  k  e.  S )  ->  k  e.  S )
5453ad3antlr 736 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  k  e.  S )
55 simplrr 770 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  ( 2nd `  k )  =  0 )
56 simpr 463 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  m  e.  ( 1 ... N
) )
5748, 3, 49, 51, 52, 54, 55, 56poimirlem11 31871 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  C_  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) ) )
5848, 3, 49, 54, 55, 51, 52, 56poimirlem11 31871 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  C_  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) ) )
5957, 58eqssd 3482 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) ) )
6047, 59chvarv 2069 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
61 simpll 759 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ph )
62 elfznn 11830 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  ( 1 ... N )  ->  n  e.  NN )
63 nnm1nn0 10913 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  (
n  -  1 )  e.  NN0 )
6564adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  ( 1 ... N )  /\  -.  n  =  1
)  ->  ( n  -  1 )  e. 
NN0 )
6662nncnd 10627 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  ( 1 ... N )  ->  n  e.  CC )
67 ax-1cn 9599 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
68 subeq0 9902 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  - 
1 )  =  0  <-> 
n  =  1 ) )
6966, 67, 68sylancl 667 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  ( 1 ... N )  ->  (
( n  -  1 )  =  0  <->  n  =  1 ) )
7069necon3abid 2671 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  (
( n  -  1 )  =/=  0  <->  -.  n  =  1 ) )
7170biimpar 488 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  ( 1 ... N )  /\  -.  n  =  1
)  ->  ( n  -  1 )  =/=  0 )
72 elnnne0 10885 . . . . . . . . . . . . . . . . 17  |-  ( ( n  -  1 )  e.  NN  <->  ( (
n  -  1 )  e.  NN0  /\  (
n  -  1 )  =/=  0 ) )
7365, 71, 72sylanbrc 669 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( 1 ... N )  /\  -.  n  =  1
)  ->  ( n  -  1 )  e.  NN )
7473adantl 468 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  -.  n  =  1 ) )  ->  ( n  - 
1 )  e.  NN )
7564nn0red 10928 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  (
n  -  1 )  e.  RR )
7675adantl 468 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
n  -  1 )  e.  RR )
7762nnred 10626 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  n  e.  RR )
7877adantl 468 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  n  e.  RR )
791nnred 10626 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  RR )
8079adantr 467 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  N  e.  RR )
8177lem1d 10542 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  (
n  -  1 )  <_  n )
8281adantl 468 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
n  -  1 )  <_  n )
83 elfzle2 11805 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  n  <_  N )
8483adantl 468 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  n  <_  N )
8576, 78, 80, 82, 84letrd 9794 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
n  -  1 )  <_  N )
8685adantrr 722 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  -.  n  =  1 ) )  ->  ( n  - 
1 )  <_  N
)
871nnzd 11041 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  ZZ )
88 fznn 11865 . . . . . . . . . . . . . . . . 17  |-  ( N  e.  ZZ  ->  (
( n  -  1 )  e.  ( 1 ... N )  <->  ( (
n  -  1 )  e.  NN  /\  (
n  -  1 )  <_  N ) ) )
8987, 88syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( n  - 
1 )  e.  ( 1 ... N )  <-> 
( ( n  - 
1 )  e.  NN  /\  ( n  -  1 )  <_  N )
) )
9089adantr 467 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  -.  n  =  1 ) )  ->  ( ( n  -  1 )  e.  ( 1 ... N
)  <->  ( ( n  -  1 )  e.  NN  /\  ( n  -  1 )  <_  N ) ) )
9174, 86, 90mpbir2and 931 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  -.  n  =  1 ) )  ->  ( n  - 
1 )  e.  ( 1 ... N ) )
9261, 91sylan 474 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  ( n  e.  ( 1 ... N )  /\  -.  n  =  1 ) )  ->  ( n  -  1 )  e.  ( 1 ... N
) )
93 ovex 6331 . . . . . . . . . . . . . 14  |-  ( n  -  1 )  e. 
_V
94 eleq1 2495 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( n  - 
1 )  ->  (
m  e.  ( 1 ... N )  <->  ( n  -  1 )  e.  ( 1 ... N
) ) )
9594anbi2d 709 . . . . . . . . . . . . . . 15  |-  ( m  =  ( n  - 
1 )  ->  (
( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  <->  ( (
( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  /\  (
n  -  1 )  e.  ( 1 ... N ) ) ) )
96 oveq2 6311 . . . . . . . . . . . . . . . . 17  |-  ( m  =  ( n  - 
1 )  ->  (
1 ... m )  =  ( 1 ... (
n  -  1 ) ) )
9796imaeq2d 5185 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( n  - 
1 )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) )
9896imaeq2d 5185 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( n  - 
1 )  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
9997, 98eqeq12d 2445 . . . . . . . . . . . . . . 15  |-  ( 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 ) ) ) ) )
10095, 99imbi12d 322 . . . . . . . . . . . . . 14  |-  ( m  =  ( n  - 
1 )  ->  (
( ( ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  /\  ( ( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) ) )  <-> 
( ( ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  /\  ( ( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  ( n  -  1 )  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) ) ) )
10193, 100, 59vtocl 3134 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  ( n  -  1 )  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
10292, 101syldan 473 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  ( n  e.  ( 1 ... N )  /\  -.  n  =  1 ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... ( n  - 
1 ) ) )  =  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... ( n  - 
1 ) ) ) )
103102expr 619 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  n  e.  ( 1 ... N
) )  ->  ( -.  n  =  1  ->  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) ) )
104 ima0 5200 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( 1st `  z ) ) " (/) )  =  (/)
105 ima0 5200 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( 1st `  k ) ) " (/) )  =  (/)
106104, 105eqtr4i 2455 . . . . . . . . . . . 12  |-  ( ( 2nd `  ( 1st `  z ) ) " (/) )  =  ( ( 2nd `  ( 1st `  k ) ) " (/) )
107 oveq1 6310 . . . . . . . . . . . . . . . 16  |-  ( n  =  1  ->  (
n  -  1 )  =  ( 1  -  1 ) )
108 1m1e0 10680 . . . . . . . . . . . . . . . 16  |-  ( 1  -  1 )  =  0
109107, 108syl6eq 2480 . . . . . . . . . . . . . . 15  |-  ( n  =  1  ->  (
n  -  1 )  =  0 )
110109oveq2d 6319 . . . . . . . . . . . . . 14  |-  ( n  =  1  ->  (
1 ... ( n  - 
1 ) )  =  ( 1 ... 0
) )
111 fz10 11822 . . . . . . . . . . . . . 14  |-  ( 1 ... 0 )  =  (/)
112110, 111syl6eq 2480 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  (
1 ... ( n  - 
1 ) )  =  (/) )
113112imaeq2d 5185 . . . . . . . . . . . 12  |-  ( n  =  1  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  z ) )
" (/) ) )
114112imaeq2d 5185 . . . . . . . . . . . 12  |-  ( n  =  1  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" (/) ) )
115106, 113, 1143eqtr4a 2490 . . . . . . . . . . 11  |-  ( n  =  1  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
116103, 115pm2.61d2 164 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
11760, 116difeq12d 3585 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  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 ) ) ) ) )
118 fnsnfv 5939 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  ( 1st `  z ) )  Fn  ( 1 ... N )  /\  n  e.  ( 1 ... N
) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  ( ( 2nd `  ( 1st `  z
) ) " {
n } ) )
11924, 118sylan 474 . . . . . . . . . . 11  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `  n ) }  =  ( ( 2nd `  ( 1st `  z ) )
" { n }
) )
12062adantl 468 . . . . . . . . . . . . 13  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  n  e.  NN )
121 uncom 3611 . . . . . . . . . . . . . . . 16  |-  ( ( 1 ... ( n  -  1 ) )  u.  { n }
)  =  ( { n }  u.  (
1 ... ( n  - 
1 ) ) )
122121difeq1i 3580 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... (
n  -  1 ) )  u.  { n } )  \  (
1 ... ( n  - 
1 ) ) )  =  ( ( { n }  u.  (
1 ... ( n  - 
1 ) ) ) 
\  ( 1 ... ( n  -  1 ) ) )
123 difun2 3876 . . . . . . . . . . . . . . 15  |-  ( ( { n }  u.  ( 1 ... (
n  -  1 ) ) )  \  (
1 ... ( n  - 
1 ) ) )  =  ( { n }  \  ( 1 ... ( n  -  1 ) ) )
124122, 123eqtri 2452 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... (
n  -  1 ) )  u.  { n } )  \  (
1 ... ( n  - 
1 ) ) )  =  ( { n }  \  ( 1 ... ( n  -  1 ) ) )
125 nncn 10619 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  n  e.  CC )
126 npcan1 10046 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  CC  ->  (
( n  -  1 )  +  1 )  =  n )
127125, 126syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
( n  -  1 )  +  1 )  =  n )
128 elnnuz 11197 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
129128biimpi 198 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
130127, 129eqeltrd 2511 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  (
( n  -  1 )  +  1 )  e.  ( ZZ>= `  1
) )
13163nn0zd 11040 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  ZZ )
132 uzid 11175 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  -  1 )  e.  ZZ  ->  (
n  -  1 )  e.  ( ZZ>= `  (
n  -  1 ) ) )
133131, 132syl 17 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  ( ZZ>= `  (
n  -  1 ) ) )
134 peano2uz 11214 . . . . . . . . . . . . . . . . . . 19  |-  ( ( n  -  1 )  e.  ( ZZ>= `  (
n  -  1 ) )  ->  ( (
n  -  1 )  +  1 )  e.  ( ZZ>= `  ( n  -  1 ) ) )
135133, 134syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
( n  -  1 )  +  1 )  e.  ( ZZ>= `  (
n  -  1 ) ) )
136127, 135eqeltrrd 2512 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  ( n  -  1 ) ) )
137 fzsplit2 11826 . . . . . . . . . . . . . . . . 17  |-  ( ( ( ( n  - 
1 )  +  1 )  e.  ( ZZ>= ` 
1 )  /\  n  e.  ( ZZ>= `  ( n  -  1 ) ) )  ->  ( 1 ... n )  =  ( ( 1 ... ( n  -  1 ) )  u.  (
( ( n  - 
1 )  +  1 ) ... n ) ) )
138130, 136, 137syl2anc 666 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (
1 ... n )  =  ( ( 1 ... ( n  -  1 ) )  u.  (
( ( n  - 
1 )  +  1 ) ... n ) ) )
139127oveq1d 6318 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
( ( n  - 
1 )  +  1 ) ... n )  =  ( n ... n ) )
140 nnz 10961 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  n  e.  ZZ )
141 fzsn 11842 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  ZZ  ->  (
n ... n )  =  { n } )
142140, 141syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
n ... n )  =  { n } )
143139, 142eqtrd 2464 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  (
( ( n  - 
1 )  +  1 ) ... n )  =  { n }
)
144143uneq2d 3621 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (
( 1 ... (
n  -  1 ) )  u.  ( ( ( n  -  1 )  +  1 ) ... n ) )  =  ( ( 1 ... ( n  - 
1 ) )  u. 
{ n } ) )
145138, 144eqtrd 2464 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
1 ... n )  =  ( ( 1 ... ( n  -  1 ) )  u.  {
n } ) )
146145difeq1d 3583 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) )  =  ( ( ( 1 ... ( n  -  1 ) )  u.  { n }
)  \  ( 1 ... ( n  - 
1 ) ) ) )
147 nnre 10618 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  n  e.  RR )
148 ltm1 10447 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  RR  ->  (
n  -  1 )  <  n )
149 peano2rem 9943 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  RR  ->  (
n  -  1 )  e.  RR )
150 ltnle 9715 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( n  -  1 )  e.  RR  /\  n  e.  RR )  ->  ( ( n  - 
1 )  <  n  <->  -.  n  <_  ( n  -  1 ) ) )
151149, 150mpancom 674 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  RR  ->  (
( n  -  1 )  <  n  <->  -.  n  <_  ( n  -  1 ) ) )
152148, 151mpbid 214 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR  ->  -.  n  <_  ( n  - 
1 ) )
153 elfzle2 11805 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  ( 1 ... ( n  -  1 ) )  ->  n  <_  ( n  -  1 ) )
154152, 153nsyl 125 . . . . . . . . . . . . . . . 16  |-  ( n  e.  RR  ->  -.  n  e.  ( 1 ... ( n  - 
1 ) ) )
155147, 154syl 17 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  -.  n  e.  ( 1 ... ( n  - 
1 ) ) )
156 incom 3656 . . . . . . . . . . . . . . . . 17  |-  ( ( 1 ... ( n  -  1 ) )  i^i  { n }
)  =  ( { n }  i^i  (
1 ... ( n  - 
1 ) ) )
157156eqeq1i 2430 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1 ... (
n  -  1 ) )  i^i  { n } )  =  (/)  <->  ( { n }  i^i  ( 1 ... (
n  -  1 ) ) )  =  (/) )
158 disjsn 4058 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1 ... (
n  -  1 ) )  i^i  { n } )  =  (/)  <->  -.  n  e.  ( 1 ... ( n  - 
1 ) ) )
159 disj3 3838 . . . . . . . . . . . . . . . 16  |-  ( ( { n }  i^i  ( 1 ... (
n  -  1 ) ) )  =  (/)  <->  {
n }  =  ( { n }  \ 
( 1 ... (
n  -  1 ) ) ) )
160157, 158, 1593bitr3i 279 . . . . . . . . . . . . . . 15  |-  ( -.  n  e.  ( 1 ... ( n  - 
1 ) )  <->  { n }  =  ( {
n }  \  (
1 ... ( n  - 
1 ) ) ) )
161155, 160sylib 200 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  { n }  =  ( {
n }  \  (
1 ... ( n  - 
1 ) ) ) )
162124, 146, 1613eqtr4a 2490 . . . . . . . . . . . . 13  |-  ( n  e.  NN  ->  (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) )  =  { n }
)
163120, 162syl 17 . . . . . . . . . . . 12  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  ( ( 1 ... n )  \ 
( 1 ... (
n  -  1 ) ) )  =  {
n } )
164163imaeq2d 5185 . . . . . . . . . . 11  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) ) )  =  ( ( 2nd `  ( 1st `  z ) ) " { n } ) )
165 dff1o3 5835 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( ( 2nd `  ( 1st `  z
) ) : ( 1 ... N )
-onto-> ( 1 ... N
)  /\  Fun  `' ( 2nd `  ( 1st `  z ) ) ) )
166165simprbi 466 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  Fun  `' ( 2nd `  ( 1st `  z ) ) )
16722, 166syl 17 . . . . . . . . . . . . 13  |-  ( z  e.  S  ->  Fun  `' ( 2nd `  ( 1st `  z ) ) )
168 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 ) ) ) ) )
169167, 168syl 17 . . . . . . . . . . . 12  |-  ( z  e.  S  ->  (
( 2nd `  ( 1st `  z ) )
" ( ( 1 ... n )  \ 
( 1 ... (
n  -  1 ) ) ) )  =  ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... n ) ) 
\  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... ( n  - 
1 ) ) ) ) )
170169adantr 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 ) ) ) ) )
171119, 164, 1703eqtr2d 2470 . . . . . . . . . 10  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `  n ) }  =  ( ( ( 2nd `  ( 1st `  z
) ) " (
1 ... n ) ) 
\  ( ( 2nd `  ( 1st `  z
) ) " (
1 ... ( n  - 
1 ) ) ) ) )
1726, 171sylan 474 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  n  e.  ( 1 ... N
) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  ( ( ( 2nd `  ( 1st `  z ) ) "
( 1 ... n
) )  \  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) ) )
173 eleq1 2495 . . . . . . . . . . . . 13  |-  ( z  =  k  ->  (
z  e.  S  <->  k  e.  S ) )
174173anbi1d 710 . . . . . . . . . . . 12  |-  ( z  =  k  ->  (
( z  e.  S  /\  n  e.  (
1 ... N ) )  <-> 
( k  e.  S  /\  n  e.  (
1 ... N ) ) ) )
175 fveq2 5879 . . . . . . . . . . . . . . . 16  |-  ( z  =  k  ->  ( 1st `  z )  =  ( 1st `  k
) )
176175fveq2d 5883 . . . . . . . . . . . . . . 15  |-  ( z  =  k  ->  ( 2nd `  ( 1st `  z
) )  =  ( 2nd `  ( 1st `  k ) ) )
177176fveq1d 5881 . . . . . . . . . . . . . 14  |-  ( z  =  k  ->  (
( 2nd `  ( 1st `  z ) ) `
 n )  =  ( ( 2nd `  ( 1st `  k ) ) `
 n ) )
178177sneqd 4009 . . . . . . . . . . . . 13  |-  ( z  =  k  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  { ( ( 2nd `  ( 1st `  k ) ) `  n ) } )
179176imaeq1d 5184 . . . . . . . . . . . . . 14  |-  ( z  =  k  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
180176imaeq1d 5184 . . . . . . . . . . . . . 14  |-  ( z  =  k  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
181179, 180difeq12d 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 ) ) ) ) )
182178, 181eqeq12d 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 ) ) ) ) ) )
183174, 182imbi12d 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 ) ) ) ) ) ) )
184183, 171chvarv 2069 . . . . . . . . . 10  |-  ( ( k  e.  S  /\  n  e.  ( 1 ... N ) )  ->  { ( ( 2nd `  ( 1st `  k ) ) `  n ) }  =  ( ( ( 2nd `  ( 1st `  k
) ) " (
1 ... n ) ) 
\  ( ( 2nd `  ( 1st `  k
) ) " (
1 ... ( n  - 
1 ) ) ) ) )
1859, 184sylan 474 . . . . . . . . 9  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  n  e.  ( 1 ... N
) )  ->  { ( ( 2nd `  ( 1st `  k ) ) `
 n ) }  =  ( ( ( 2nd `  ( 1st `  k ) ) "
( 1 ... n
) )  \  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) ) )
186117, 172, 1853eqtr4d 2474 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  n  e.  ( 1 ... N
) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  { ( ( 2nd `  ( 1st `  k ) ) `  n ) } )
187 fvex 5889 . . . . . . . . 9  |-  ( ( 2nd `  ( 1st `  z ) ) `  n )  e.  _V
188187sneqr 4165 . . . . . . . 8  |-  ( { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  { ( ( 2nd `  ( 1st `  k ) ) `  n ) }  ->  ( ( 2nd `  ( 1st `  z ) ) `
 n )  =  ( ( 2nd `  ( 1st `  k ) ) `
 n ) )
189186, 188syl 17 . . . . . . 7  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  n  e.  ( 1 ... N
) )  ->  (
( 2nd `  ( 1st `  z ) ) `
 n )  =  ( ( 2nd `  ( 1st `  k ) ) `
 n ) )
19026, 40, 189eqfnfvd 5992 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  ( 1st `  z
) )  =  ( 2nd `  ( 1st `  k ) ) )
191 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 ) ) )
19216, 30, 191syl2an 480 . . . . . . 7  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( ( ( 1st `  ( 1st `  z
) )  =  ( 1st `  ( 1st `  k ) )  /\  ( 2nd `  ( 1st `  z ) )  =  ( 2nd `  ( 1st `  k ) ) )  <->  ( 1st `  z
)  =  ( 1st `  k ) ) )
193192ad2antlr 732 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  (
( ( 1st `  ( 1st `  z ) )  =  ( 1st `  ( 1st `  k ) )  /\  ( 2nd `  ( 1st `  z ) )  =  ( 2nd `  ( 1st `  k ) ) )  <->  ( 1st `  z
)  =  ( 1st `  k ) ) )
19412, 190, 193mpbi2and 930 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 1st `  z )  =  ( 1st `  k
) )
195 eqtr3 2451 . . . . . 6  |-  ( ( ( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 )  ->  ( 2nd `  z
)  =  ( 2nd `  k ) )
196195adantl 468 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  z )  =  ( 2nd `  k
) )
197 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 ) )
19814, 28, 197syl2an 480 . . . . . 6  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( ( ( 1st `  z )  =  ( 1st `  k )  /\  ( 2nd `  z
)  =  ( 2nd `  k ) )  <->  z  =  k ) )
199198ad2antlr 732 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  (
( ( 1st `  z
)  =  ( 1st `  k )  /\  ( 2nd `  z )  =  ( 2nd `  k
) )  <->  z  =  k ) )
200194, 196, 199mpbi2and 930 . . . 4  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  z  =  k )
201200ex 436 . . 3  |-  ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  -> 
( ( ( 2nd `  z )  =  0  /\  ( 2nd `  k
)  =  0 )  ->  z  =  k ) )
202201ralrimivva 2847 . 2  |-  ( ph  ->  A. z  e.  S  A. k  e.  S  ( ( ( 2nd `  z )  =  0  /\  ( 2nd `  k
)  =  0 )  ->  z  =  k ) )
203 fveq2 5879 . . . 4  |-  ( z  =  k  ->  ( 2nd `  z )  =  ( 2nd `  k
) )
204203eqeq1d 2425 . . 3  |-  ( z  =  k  ->  (
( 2nd `  z
)  =  0  <->  ( 2nd `  k )  =  0 ) )
205204rmo4 3265 . 2  |-  ( E* z  e.  S  ( 2nd `  z )  =  0  <->  A. z  e.  S  A. k  e.  S  ( (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 )  ->  z  =  k ) )
206202, 205sylibr 216 1  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ 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   (/)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   NN0cn0 10871   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
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