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Theorem poimirlem13 32017
Description: Lemma for poimir 32037- 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 740 . . . . . . . 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 740 . . . . . . . 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 778 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  z  e.  S )
7 simprl 772 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  z )  =  0 )
82, 3, 5, 6, 7poimirlem10 32014 . . . . . . 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 779 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  k  e.  S )
10 simprr 774 . . . . . . . 8  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  k )  =  0 )
112, 3, 5, 9, 10poimirlem10 32014 . . . . . . 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 2507 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 1st `  ( 1st `  z
) )  =  ( 1st `  ( 1st `  k ) ) )
13 elrabi 3181 . . . . . . . . . . . . . 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 2567 . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . 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 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 ) } )
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 5818 . . . . . . . . . . . 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 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 ) )
2218, 21sylib 201 . . . . . . . . . 10  |-  ( z  e.  S  ->  ( 2nd `  ( 1st `  z
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
23 f1ofn 5829 . . . . . . . . . 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 472 . . . . . . . 8  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( 2nd `  ( 1st `  z ) )  Fn  ( 1 ... N ) )
2625ad2antlr 741 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  ( 1st `  z
) )  Fn  (
1 ... N ) )
27 elrabi 3181 . . . . . . . . . . . . . 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 2567 . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . 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 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 ) } )
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 5818 . . . . . . . . . . . 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 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 ) )
3632, 35sylib 201 . . . . . . . . . 10  |-  ( k  e.  S  ->  ( 2nd `  ( 1st `  k
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
37 f1ofn 5829 . . . . . . . . . 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 473 . . . . . . . 8  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( 2nd `  ( 1st `  k ) )  Fn  ( 1 ... N ) )
4039ad2antlr 741 . . . . . . 7  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  ( 1st `  k
) )  Fn  (
1 ... N ) )
41 eleq1 2537 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  (
m  e.  ( 1 ... N )  <->  n  e.  ( 1 ... N
) ) )
4241anbi2d 718 . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . 14  |-  ( m  =  n  ->  (
1 ... m )  =  ( 1 ... n
) )
4443imaeq2d 5174 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) ) )
4543imaeq2d 5174 . . . . . . . . . . . . 13  |-  ( m  =  n  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
4644, 45eqeq12d 2486 . . . . . . . . . . . 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 327 . . . . . . . . . . 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 744 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  N  e.  NN )
494ad3antrrr 744 . . . . . . . . . . . . 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 464 . . . . . . . . . . . . . 14  |-  ( ( z  e.  S  /\  k  e.  S )  ->  z  e.  S )
5150ad3antlr 745 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  z  e.  S )
52 simplrl 778 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  ( 2nd `  z )  =  0 )
53 simpr 468 . . . . . . . . . . . . . 14  |-  ( ( z  e.  S  /\  k  e.  S )  ->  k  e.  S )
5453ad3antlr 745 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  k  e.  S )
55 simplrr 779 . . . . . . . . . . . . 13  |-  ( ( ( ( ph  /\  ( z  e.  S  /\  k  e.  S
) )  /\  (
( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 ) )  /\  m  e.  ( 1 ... N
) )  ->  ( 2nd `  k )  =  0 )
56 simpr 468 . . . . . . . . . . . . 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 32015 . . . . . . . . . . . 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 32015 . . . . . . . . . . . 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 3435 . . . . . . . . . . 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 2120 . . . . . . . . . 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 768 . . . . . . . . . . . . . 14  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ph )
62 elfznn 11854 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  ( 1 ... N )  ->  n  e.  NN )
63 nnm1nn0 10935 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  NN0 )
6462, 63syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  (
n  -  1 )  e.  NN0 )
6564adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  ( 1 ... N )  /\  -.  n  =  1
)  ->  ( n  -  1 )  e. 
NN0 )
6662nncnd 10647 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  ( 1 ... N )  ->  n  e.  CC )
67 ax-1cn 9615 . . . . . . . . . . . . . . . . . . . 20  |-  1  e.  CC
68 subeq0 9920 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( n  e.  CC  /\  1  e.  CC )  ->  ( ( n  - 
1 )  =  0  <-> 
n  =  1 ) )
6966, 67, 68sylancl 675 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  ( 1 ... N )  ->  (
( n  -  1 )  =  0  <->  n  =  1 ) )
7069necon3abid 2679 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  (
( n  -  1 )  =/=  0  <->  -.  n  =  1 ) )
7170biimpar 493 . . . . . . . . . . . . . . . . 17  |-  ( ( n  e.  ( 1 ... N )  /\  -.  n  =  1
)  ->  ( n  -  1 )  =/=  0 )
72 elnnne0 10907 . . . . . . . . . . . . . . . . 17  |-  ( ( n  -  1 )  e.  NN  <->  ( (
n  -  1 )  e.  NN0  /\  (
n  -  1 )  =/=  0 ) )
7365, 71, 72sylanbrc 677 . . . . . . . . . . . . . . . 16  |-  ( ( n  e.  ( 1 ... N )  /\  -.  n  =  1
)  ->  ( n  -  1 )  e.  NN )
7473adantl 473 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  -.  n  =  1 ) )  ->  ( n  - 
1 )  e.  NN )
7564nn0red 10950 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  (
n  -  1 )  e.  RR )
7675adantl 473 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
n  -  1 )  e.  RR )
7762nnred 10646 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  n  e.  RR )
7877adantl 473 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  n  e.  RR )
791nnred 10646 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  RR )
8079adantr 472 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  N  e.  RR )
8177lem1d 10562 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  (
n  -  1 )  <_  n )
8281adantl 473 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
n  -  1 )  <_  n )
83 elfzle2 11829 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  ( 1 ... N )  ->  n  <_  N )
8483adantl 473 . . . . . . . . . . . . . . . . 17  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  n  <_  N )
8576, 78, 80, 82, 84letrd 9809 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
n  -  1 )  <_  N )
8685adantrr 731 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  -.  n  =  1 ) )  ->  ( n  - 
1 )  <_  N
)
871nnzd 11062 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  e.  ZZ )
88 fznn 11889 . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  -.  n  =  1 ) )  ->  ( ( n  -  1 )  e.  ( 1 ... N
)  <->  ( ( n  -  1 )  e.  NN  /\  ( n  -  1 )  <_  N ) ) )
9174, 86, 90mpbir2and 936 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( n  e.  ( 1 ... N
)  /\  -.  n  =  1 ) )  ->  ( n  - 
1 )  e.  ( 1 ... N ) )
9261, 91sylan 479 . . . . . . . . . . . . 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 6336 . . . . . . . . . . . . . 14  |-  ( n  -  1 )  e. 
_V
94 eleq1 2537 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( n  - 
1 )  ->  (
m  e.  ( 1 ... N )  <->  ( n  -  1 )  e.  ( 1 ... N
) ) )
9594anbi2d 718 . . . . . . . . . . . . . . 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 6316 . . . . . . . . . . . . . . . . 17  |-  ( m  =  ( n  - 
1 )  ->  (
1 ... m )  =  ( 1 ... (
n  -  1 ) ) )
9796imaeq2d 5174 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( n  - 
1 )  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) ) )
9896imaeq2d 5174 . . . . . . . . . . . . . . . 16  |-  ( m  =  ( n  - 
1 )  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... m ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
9997, 98eqeq12d 2486 . . . . . . . . . . . . . . 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 327 . . . . . . . . . . . . . 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 3086 . . . . . . . . . . . . 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 478 . . . . . . . . . . . 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 626 . . . . . . . . . . 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 5189 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( 1st `  z ) ) " (/) )  =  (/)
105 ima0 5189 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( 1st `  k ) ) " (/) )  =  (/)
106104, 105eqtr4i 2496 . . . . . . . . . . . 12  |-  ( ( 2nd `  ( 1st `  z ) ) " (/) )  =  ( ( 2nd `  ( 1st `  k ) ) " (/) )
107 oveq1 6315 . . . . . . . . . . . . . . . 16  |-  ( n  =  1  ->  (
n  -  1 )  =  ( 1  -  1 ) )
108 1m1e0 10700 . . . . . . . . . . . . . . . 16  |-  ( 1  -  1 )  =  0
109107, 108syl6eq 2521 . . . . . . . . . . . . . . 15  |-  ( n  =  1  ->  (
n  -  1 )  =  0 )
110109oveq2d 6324 . . . . . . . . . . . . . 14  |-  ( n  =  1  ->  (
1 ... ( n  - 
1 ) )  =  ( 1 ... 0
) )
111 fz10 11846 . . . . . . . . . . . . . 14  |-  ( 1 ... 0 )  =  (/)
112110, 111syl6eq 2521 . . . . . . . . . . . . 13  |-  ( n  =  1  ->  (
1 ... ( n  - 
1 ) )  =  (/) )
113112imaeq2d 5174 . . . . . . . . . . . 12  |-  ( n  =  1  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  z ) )
" (/) ) )
114112imaeq2d 5174 . . . . . . . . . . . 12  |-  ( n  =  1  ->  (
( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" (/) ) )
115106, 113, 1143eqtr4a 2531 . . . . . . . . . . 11  |-  ( n  =  1  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
116103, 115pm2.61d2 165 . . . . . . . . . 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 3541 . . . . . . . . 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 5940 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  ( 1st `  z ) )  Fn  ( 1 ... N )  /\  n  e.  ( 1 ... N
) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  ( ( 2nd `  ( 1st `  z
) ) " {
n } ) )
11924, 118sylan 479 . . . . . . . . . . 11  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  { ( ( 2nd `  ( 1st `  z ) ) `  n ) }  =  ( ( 2nd `  ( 1st `  z ) )
" { n }
) )
12062adantl 473 . . . . . . . . . . . . 13  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  n  e.  NN )
121 uncom 3569 . . . . . . . . . . . . . . . 16  |-  ( ( 1 ... ( n  -  1 ) )  u.  { n }
)  =  ( { n }  u.  (
1 ... ( n  - 
1 ) ) )
122121difeq1i 3536 . . . . . . . . . . . . . . 15  |-  ( ( ( 1 ... (
n  -  1 ) )  u.  { n } )  \  (
1 ... ( n  - 
1 ) ) )  =  ( ( { n }  u.  (
1 ... ( n  - 
1 ) ) ) 
\  ( 1 ... ( n  -  1 ) ) )
123 difun2 3838 . . . . . . . . . . . . . . 15  |-  ( ( { n }  u.  ( 1 ... (
n  -  1 ) ) )  \  (
1 ... ( n  - 
1 ) ) )  =  ( { n }  \  ( 1 ... ( n  -  1 ) ) )
124122, 123eqtri 2493 . . . . . . . . . . . . . 14  |-  ( ( ( 1 ... (
n  -  1 ) )  u.  { n } )  \  (
1 ... ( n  - 
1 ) ) )  =  ( { n }  \  ( 1 ... ( n  -  1 ) ) )
125 nncn 10639 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  n  e.  CC )
126 npcan1 10065 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  CC  ->  (
( n  -  1 )  +  1 )  =  n )
127125, 126syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
( n  -  1 )  +  1 )  =  n )
128 elnnuz 11219 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  <->  n  e.  ( ZZ>= `  1 )
)
129128biimpi 199 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  1 )
)
130127, 129eqeltrd 2549 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  (
( n  -  1 )  +  1 )  e.  ( ZZ>= `  1
) )
13163nn0zd 11061 . . . . . . . . . . . . . . . . . . . 20  |-  ( n  e.  NN  ->  (
n  -  1 )  e.  ZZ )
132 uzid 11197 . . . . . . . . . . . . . . . . . . . 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 11235 . . . . . . . . . . . . . . . . . . 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 2550 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  n  e.  ( ZZ>= `  ( n  -  1 ) ) )
137 fzsplit2 11850 . . . . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (
1 ... n )  =  ( ( 1 ... ( n  -  1 ) )  u.  (
( ( n  - 
1 )  +  1 ) ... n ) ) )
139127oveq1d 6323 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
( ( n  - 
1 )  +  1 ) ... n )  =  ( n ... n ) )
140 nnz 10983 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  NN  ->  n  e.  ZZ )
141 fzsn 11866 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  ZZ  ->  (
n ... n )  =  { n } )
142140, 141syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  NN  ->  (
n ... n )  =  { n } )
143139, 142eqtrd 2505 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  NN  ->  (
( ( n  - 
1 )  +  1 ) ... n )  =  { n }
)
144143uneq2d 3579 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  (
( 1 ... (
n  -  1 ) )  u.  ( ( ( n  -  1 )  +  1 ) ... n ) )  =  ( ( 1 ... ( n  - 
1 ) )  u. 
{ n } ) )
145138, 144eqtrd 2505 . . . . . . . . . . . . . . 15  |-  ( n  e.  NN  ->  (
1 ... n )  =  ( ( 1 ... ( n  -  1 ) )  u.  {
n } ) )
146145difeq1d 3539 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) )  =  ( ( ( 1 ... ( n  -  1 ) )  u.  { n }
)  \  ( 1 ... ( n  - 
1 ) ) ) )
147 nnre 10638 . . . . . . . . . . . . . . . 16  |-  ( n  e.  NN  ->  n  e.  RR )
148 ltm1 10467 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  RR  ->  (
n  -  1 )  <  n )
149 peano2rem 9961 . . . . . . . . . . . . . . . . . . 19  |-  ( n  e.  RR  ->  (
n  -  1 )  e.  RR )
150 ltnle 9731 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( n  -  1 )  e.  RR  /\  n  e.  RR )  ->  ( ( n  - 
1 )  <  n  <->  -.  n  <_  ( n  -  1 ) ) )
151149, 150mpancom 682 . . . . . . . . . . . . . . . . . 18  |-  ( n  e.  RR  ->  (
( n  -  1 )  <  n  <->  -.  n  <_  ( n  -  1 ) ) )
152148, 151mpbid 215 . . . . . . . . . . . . . . . . 17  |-  ( n  e.  RR  ->  -.  n  <_  ( n  - 
1 ) )
153 elfzle2 11829 . . . . . . . . . . . . . . . . 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 3616 . . . . . . . . . . . . . . . . 17  |-  ( ( 1 ... ( n  -  1 ) )  i^i  { n }
)  =  ( { n }  i^i  (
1 ... ( n  - 
1 ) ) )
157156eqeq1i 2476 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1 ... (
n  -  1 ) )  i^i  { n } )  =  (/)  <->  ( { n }  i^i  ( 1 ... (
n  -  1 ) ) )  =  (/) )
158 disjsn 4023 . . . . . . . . . . . . . . . 16  |-  ( ( ( 1 ... (
n  -  1 ) )  i^i  { n } )  =  (/)  <->  -.  n  e.  ( 1 ... ( n  - 
1 ) ) )
159 disj3 3813 . . . . . . . . . . . . . . . 16  |-  ( ( { n }  i^i  ( 1 ... (
n  -  1 ) ) )  =  (/)  <->  {
n }  =  ( { n }  \ 
( 1 ... (
n  -  1 ) ) ) )
160157, 158, 1593bitr3i 283 . . . . . . . . . . . . . . 15  |-  ( -.  n  e.  ( 1 ... ( n  - 
1 ) )  <->  { n }  =  ( {
n }  \  (
1 ... ( n  - 
1 ) ) ) )
161155, 160sylib 201 . . . . . . . . . . . . . 14  |-  ( n  e.  NN  ->  { n }  =  ( {
n }  \  (
1 ... ( n  - 
1 ) ) ) )
162124, 146, 1613eqtr4a 2531 . . . . . . . . . . . . 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 5174 . . . . . . . . . . 11  |-  ( ( z  e.  S  /\  n  e.  ( 1 ... N ) )  ->  ( ( 2nd `  ( 1st `  z
) ) " (
( 1 ... n
)  \  ( 1 ... ( n  - 
1 ) ) ) )  =  ( ( 2nd `  ( 1st `  z ) ) " { n } ) )
165 dff1o3 5834 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( 1st `  z ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( ( 2nd `  ( 1st `  z
) ) : ( 1 ... N )
-onto-> ( 1 ... N
)  /\  Fun  `' ( 2nd `  ( 1st `  z ) ) ) )
166165simprbi 471 . . . . . . . . . . . . . 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 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 ) ) ) ) )
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 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 ) ) ) ) )
171119, 164, 1703eqtr2d 2511 . . . . . . . . . 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 479 . . . . . . . . 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 2537 . . . . . . . . . . . . 13  |-  ( z  =  k  ->  (
z  e.  S  <->  k  e.  S ) )
174173anbi1d 719 . . . . . . . . . . . 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 3971 . . . . . . . . . . . . 13  |-  ( z  =  k  ->  { ( ( 2nd `  ( 1st `  z ) ) `
 n ) }  =  { ( ( 2nd `  ( 1st `  k ) ) `  n ) } )
179176imaeq1d 5173 . . . . . . . . . . . . . 14  |-  ( z  =  k  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... n ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... n ) ) )
180176imaeq1d 5173 . . . . . . . . . . . . . 14  |-  ( z  =  k  ->  (
( 2nd `  ( 1st `  z ) )
" ( 1 ... ( n  -  1 ) ) )  =  ( ( 2nd `  ( 1st `  k ) )
" ( 1 ... ( n  -  1 ) ) ) )
181179, 180difeq12d 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 ) ) ) ) )
182178, 181eqeq12d 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 ) ) ) ) ) )
183174, 182imbi12d 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 ) ) ) ) ) ) )
184183, 171chvarv 2120 . . . . . . . . . 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 479 . . . . . . . . 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 2515 . . . . . . . 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 4131 . . . . . . . 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 5994 . . . . . 6  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  ( 1st `  z
) )  =  ( 2nd `  ( 1st `  k ) ) )
191 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 ) ) )
19216, 30, 191syl2an 485 . . . . . . 7  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( ( ( 1st `  ( 1st `  z
) )  =  ( 1st `  ( 1st `  k ) )  /\  ( 2nd `  ( 1st `  z ) )  =  ( 2nd `  ( 1st `  k ) ) )  <->  ( 1st `  z
)  =  ( 1st `  k ) ) )
193192ad2antlr 741 . . . . . 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 935 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 1st `  z )  =  ( 1st `  k
) )
195 eqtr3 2492 . . . . . 6  |-  ( ( ( 2nd `  z
)  =  0  /\  ( 2nd `  k
)  =  0 )  ->  ( 2nd `  z
)  =  ( 2nd `  k ) )
196195adantl 473 . . . . 5  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  ( 2nd `  z )  =  ( 2nd `  k
) )
197 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 ) )
19814, 28, 197syl2an 485 . . . . . 6  |-  ( ( z  e.  S  /\  k  e.  S )  ->  ( ( ( 1st `  z )  =  ( 1st `  k )  /\  ( 2nd `  z
)  =  ( 2nd `  k ) )  <->  z  =  k ) )
199198ad2antlr 741 . . . . 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 935 . . . 4  |-  ( ( ( ph  /\  (
z  e.  S  /\  k  e.  S )
)  /\  ( ( 2nd `  z )  =  0  /\  ( 2nd `  k )  =  0 ) )  ->  z  =  k )
201200ex 441 . . 3  |-  ( (
ph  /\  ( z  e.  S  /\  k  e.  S ) )  -> 
( ( ( 2nd `  z )  =  0  /\  ( 2nd `  k
)  =  0 )  ->  z  =  k ) )
202201ralrimivva 2814 . 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 2473 . . 3  |-  ( z  =  k  ->  (
( 2nd `  z
)  =  0  <->  ( 2nd `  k )  =  0 ) )
205204rmo4 3219 . 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 217 1  |-  ( ph  ->  E* z  e.  S  ( 2nd `  z )  =  0 )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ 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   (/)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   NN0cn0 10893   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
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