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Theorem poimirlem8 31912
Description: Lemma for poimir 31937, establishing that away from the opposite vertex the walks in poimirlem9 31913 yield the same vertices. (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 } ) ) ) ) }
poimirlem9.1  |-  ( ph  ->  T  e.  S )
poimirlem9.2  |-  ( ph  ->  ( 2nd `  T
)  e.  ( 1 ... ( N  - 
1 ) ) )
poimirlem9.3  |-  ( ph  ->  U  e.  S )
Assertion
Ref Expression
poimirlem8  |-  ( ph  ->  ( ( 2nd `  ( 1st `  U ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  =  ( ( 2nd `  ( 1st `  T ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) ) )
Distinct variable groups:    f, j,
t, y    ph, j, y   
j, F, y    j, N, y    T, j, y    U, j, y    ph, t    f, K, j, t    f, N, t    T, f    U, f    f, F, t    t, T    t, U    S, j,
t, y
Allowed substitution hints:    ph( f)    S( f)    K( y)

Proof of Theorem poimirlem8
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 poimirlem9.3 . . . . . . . 8  |-  ( ph  ->  U  e.  S )
2 elrabi 3225 . . . . . . . . 9  |-  ( U  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 } ) ) ) ) }  ->  U  e.  ( ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
3 poimirlem22.s . . . . . . . . 9  |-  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 } ) ) ) ) }
42, 3eleq2s 2527 . . . . . . . 8  |-  ( U  e.  S  ->  U  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
51, 4syl 17 . . . . . . 7  |-  ( ph  ->  U  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
6 xp1st 6837 . . . . . . 7  |-  ( U  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  ->  ( 1st `  U )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
75, 6syl 17 . . . . . 6  |-  ( ph  ->  ( 1st `  U
)  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } ) )
8 xp2nd 6838 . . . . . 6  |-  ( ( 1st `  U )  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  -> 
( 2nd `  ( 1st `  U ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
97, 8syl 17 . . . . 5  |-  ( ph  ->  ( 2nd `  ( 1st `  U ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
10 fvex 5891 . . . . . 6  |-  ( 2nd `  ( 1st `  U
) )  e.  _V
11 f1oeq1 5822 . . . . . 6  |-  ( f  =  ( 2nd `  ( 1st `  U ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
1210, 11elab 3217 . . . . 5  |-  ( ( 2nd `  ( 1st `  U ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  U
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
139, 12sylib 199 . . . 4  |-  ( ph  ->  ( 2nd `  ( 1st `  U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
14 f1ofn 5832 . . . 4  |-  ( ( 2nd `  ( 1st `  U ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  U
) )  Fn  (
1 ... N ) )
1513, 14syl 17 . . 3  |-  ( ph  ->  ( 2nd `  ( 1st `  U ) )  Fn  ( 1 ... N ) )
16 difss 3592 . . 3  |-  ( ( 1 ... N ) 
\  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } )  C_  (
1 ... N )
17 fnssres 5707 . . 3  |-  ( ( ( 2nd `  ( 1st `  U ) )  Fn  ( 1 ... N )  /\  (
( 1 ... N
)  \  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } )  C_  (
1 ... N ) )  ->  ( ( 2nd `  ( 1st `  U
) )  |`  (
( 1 ... N
)  \  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } ) )  Fn  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )
1815, 16, 17sylancl 666 . 2  |-  ( ph  ->  ( ( 2nd `  ( 1st `  U ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  Fn  (
( 1 ... N
)  \  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } ) )
19 poimirlem9.1 . . . . . . . 8  |-  ( ph  ->  T  e.  S )
20 elrabi 3225 . . . . . . . . 9  |-  ( T  e.  { t  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  |  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  t
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) }  ->  T  e.  ( ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
2120, 3eleq2s 2527 . . . . . . . 8  |-  ( T  e.  S  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
2219, 21syl 17 . . . . . . 7  |-  ( ph  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
23 xp1st 6837 . . . . . . 7  |-  ( T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  ->  ( 1st `  T )  e.  ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
2422, 23syl 17 . . . . . 6  |-  ( ph  ->  ( 1st `  T
)  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } ) )
25 xp2nd 6838 . . . . . 6  |-  ( ( 1st `  T )  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  -> 
( 2nd `  ( 1st `  T ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
2624, 25syl 17 . . . . 5  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
27 fvex 5891 . . . . . 6  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
28 f1oeq1 5822 . . . . . 6  |-  ( f  =  ( 2nd `  ( 1st `  T ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
2927, 28elab 3217 . . . . 5  |-  ( ( 2nd `  ( 1st `  T ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
3026, 29sylib 199 . . . 4  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
31 f1ofn 5832 . . . 4  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  T
) )  Fn  (
1 ... N ) )
3230, 31syl 17 . . 3  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) )  Fn  ( 1 ... N ) )
33 fnssres 5707 . . 3  |-  ( ( ( 2nd `  ( 1st `  T ) )  Fn  ( 1 ... N )  /\  (
( 1 ... N
)  \  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } )  C_  (
1 ... N ) )  ->  ( ( 2nd `  ( 1st `  T
) )  |`  (
( 1 ... N
)  \  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } ) )  Fn  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )
3432, 16, 33sylancl 666 . 2  |-  ( ph  ->  ( ( 2nd `  ( 1st `  T ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  Fn  (
( 1 ... N
)  \  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } ) )
35 poimirlem9.2 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2nd `  T
)  e.  ( 1 ... ( N  - 
1 ) ) )
36 fzp1elp1 11856 . . . . . . . . . . . 12  |-  ( ( 2nd `  T )  e.  ( 1 ... ( N  -  1 ) )  ->  (
( 2nd `  T
)  +  1 )  e.  ( 1 ... ( ( N  - 
1 )  +  1 ) ) )
3735, 36syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2nd `  T
)  +  1 )  e.  ( 1 ... ( ( N  - 
1 )  +  1 ) ) )
38 poimir.0 . . . . . . . . . . . . . 14  |-  ( ph  ->  N  e.  NN )
3938nncnd 10632 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
40 npcan1 10051 . . . . . . . . . . . . 13  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
4139, 40syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
4241oveq2d 6321 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... (
( N  -  1 )  +  1 ) )  =  ( 1 ... N ) )
4337, 42eleqtrd 2509 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd `  T
)  +  1 )  e.  ( 1 ... N ) )
44 fzsplit 11832 . . . . . . . . . 10  |-  ( ( ( 2nd `  T
)  +  1 )  e.  ( 1 ... N )  ->  (
1 ... N )  =  ( ( 1 ... ( ( 2nd `  T
)  +  1 ) )  u.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) ) )
4543, 44syl 17 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... N
)  =  ( ( 1 ... ( ( 2nd `  T )  +  1 ) )  u.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) ) )
4645difeq1d 3582 . . . . . . . 8  |-  ( ph  ->  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( ( ( 1 ... (
( 2nd `  T
)  +  1 ) )  u.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )  \  { ( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )
47 difundir 3726 . . . . . . . . 9  |-  ( ( ( 1 ... (
( 2nd `  T
)  +  1 ) )  u.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )  \  { ( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( ( ( 1 ... (
( 2nd `  T
)  +  1 ) )  \  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } )  u.  ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )
48 elfznn 11835 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  T )  e.  ( 1 ... ( N  -  1 ) )  ->  ( 2nd `  T )  e.  NN )
4935, 48syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 2nd `  T
)  e.  NN )
5049nncnd 10632 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 2nd `  T
)  e.  CC )
51 npcan1 10051 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  T )  e.  CC  ->  (
( ( 2nd `  T
)  -  1 )  +  1 )  =  ( 2nd `  T
) )
5250, 51syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( 2nd `  T )  -  1 )  +  1 )  =  ( 2nd `  T
) )
53 nnuz 11201 . . . . . . . . . . . . . . . 16  |-  NN  =  ( ZZ>= `  1 )
5449, 53syl6eleq 2517 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 2nd `  T
)  e.  ( ZZ>= ` 
1 ) )
5552, 54eqeltrd 2507 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( 2nd `  T )  -  1 )  +  1 )  e.  ( ZZ>= `  1
) )
5649nnzd 11046 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 2nd `  T
)  e.  ZZ )
57 peano2zm 10987 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  T )  e.  ZZ  ->  (
( 2nd `  T
)  -  1 )  e.  ZZ )
5856, 57syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( 2nd `  T
)  -  1 )  e.  ZZ )
59 uzid 11180 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 2nd `  T
)  -  1 )  e.  ZZ  ->  (
( 2nd `  T
)  -  1 )  e.  ( ZZ>= `  (
( 2nd `  T
)  -  1 ) ) )
60 peano2uz 11219 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 2nd `  T
)  -  1 )  e.  ( ZZ>= `  (
( 2nd `  T
)  -  1 ) )  ->  ( (
( 2nd `  T
)  -  1 )  +  1 )  e.  ( ZZ>= `  ( ( 2nd `  T )  - 
1 ) ) )
6158, 59, 603syl 18 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( 2nd `  T )  -  1 )  +  1 )  e.  ( ZZ>= `  (
( 2nd `  T
)  -  1 ) ) )
6252, 61eqeltrrd 2508 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 2nd `  T
)  e.  ( ZZ>= `  ( ( 2nd `  T
)  -  1 ) ) )
63 peano2uz 11219 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  T )  e.  ( ZZ>= `  (
( 2nd `  T
)  -  1 ) )  ->  ( ( 2nd `  T )  +  1 )  e.  (
ZZ>= `  ( ( 2nd `  T )  -  1 ) ) )
6462, 63syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2nd `  T
)  +  1 )  e.  ( ZZ>= `  (
( 2nd `  T
)  -  1 ) ) )
65 fzsplit2 11831 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( 2nd `  T )  -  1 )  +  1 )  e.  ( ZZ>= `  1
)  /\  ( ( 2nd `  T )  +  1 )  e.  (
ZZ>= `  ( ( 2nd `  T )  -  1 ) ) )  -> 
( 1 ... (
( 2nd `  T
)  +  1 ) )  =  ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  ( ( ( ( 2nd `  T
)  -  1 )  +  1 ) ... ( ( 2nd `  T
)  +  1 ) ) ) )
6655, 64, 65syl2anc 665 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1 ... (
( 2nd `  T
)  +  1 ) )  =  ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  ( ( ( ( 2nd `  T
)  -  1 )  +  1 ) ... ( ( 2nd `  T
)  +  1 ) ) ) )
6752oveq1d 6320 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( ( 2nd `  T )  -  1 )  +  1 ) ... (
( 2nd `  T
)  +  1 ) )  =  ( ( 2nd `  T ) ... ( ( 2nd `  T )  +  1 ) ) )
68 fzpr 11858 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  T )  e.  ZZ  ->  (
( 2nd `  T
) ... ( ( 2nd `  T )  +  1 ) )  =  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )
6956, 68syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( 2nd `  T
) ... ( ( 2nd `  T )  +  1 ) )  =  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )
7067, 69eqtrd 2463 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( ( 2nd `  T )  -  1 )  +  1 ) ... (
( 2nd `  T
)  +  1 ) )  =  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } )
7170uneq2d 3620 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 1 ... ( ( 2nd `  T
)  -  1 ) )  u.  ( ( ( ( 2nd `  T
)  -  1 )  +  1 ) ... ( ( 2nd `  T
)  +  1 ) ) )  =  ( ( 1 ... (
( 2nd `  T
)  -  1 ) )  u.  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } ) )
7266, 71eqtrd 2463 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1 ... (
( 2nd `  T
)  +  1 ) )  =  ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } ) )
7372difeq1d 3582 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 1 ... ( ( 2nd `  T
)  +  1 ) )  \  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  \  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } ) )
7449nnred 10631 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 2nd `  T
)  e.  RR )
7574ltm1d 10546 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2nd `  T
)  -  1 )  <  ( 2nd `  T
) )
7658zred 11047 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( 2nd `  T
)  -  1 )  e.  RR )
7776, 74ltnled 9789 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( 2nd `  T )  -  1 )  <  ( 2nd `  T )  <->  -.  ( 2nd `  T )  <_ 
( ( 2nd `  T
)  -  1 ) ) )
7875, 77mpbid 213 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  ( 2nd `  T
)  <_  ( ( 2nd `  T )  - 
1 ) )
79 elfzle2 11810 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  T )  e.  ( 1 ... ( ( 2nd `  T
)  -  1 ) )  ->  ( 2nd `  T )  <_  (
( 2nd `  T
)  -  1 ) )
8078, 79nsyl 124 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  ( 2nd `  T
)  e.  ( 1 ... ( ( 2nd `  T )  -  1 ) ) )
81 difsn 4134 . . . . . . . . . . . . . 14  |-  ( -.  ( 2nd `  T
)  e.  ( 1 ... ( ( 2nd `  T )  -  1 ) )  ->  (
( 1 ... (
( 2nd `  T
)  -  1 ) )  \  { ( 2nd `  T ) } )  =  ( 1 ... ( ( 2nd `  T )  -  1 ) ) )
8280, 81syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 1 ... ( ( 2nd `  T
)  -  1 ) )  \  { ( 2nd `  T ) } )  =  ( 1 ... ( ( 2nd `  T )  -  1 ) ) )
83 peano2re 9813 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  T )  e.  RR  ->  (
( 2nd `  T
)  +  1 )  e.  RR )
8474, 83syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( 2nd `  T
)  +  1 )  e.  RR )
8574ltp1d 10544 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 2nd `  T
)  <  ( ( 2nd `  T )  +  1 ) )
8676, 74, 84, 75, 85lttrd 9803 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2nd `  T
)  -  1 )  <  ( ( 2nd `  T )  +  1 ) )
8776, 84ltnled 9789 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( 2nd `  T )  -  1 )  <  ( ( 2nd `  T )  +  1 )  <->  -.  (
( 2nd `  T
)  +  1 )  <_  ( ( 2nd `  T )  -  1 ) ) )
8886, 87mpbid 213 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  ( ( 2nd `  T )  +  1 )  <_  ( ( 2nd `  T )  - 
1 ) )
89 elfzle2 11810 . . . . . . . . . . . . . . 15  |-  ( ( ( 2nd `  T
)  +  1 )  e.  ( 1 ... ( ( 2nd `  T
)  -  1 ) )  ->  ( ( 2nd `  T )  +  1 )  <_  (
( 2nd `  T
)  -  1 ) )
9088, 89nsyl 124 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  ( ( 2nd `  T )  +  1 )  e.  ( 1 ... ( ( 2nd `  T )  -  1 ) ) )
91 difsn 4134 . . . . . . . . . . . . . 14  |-  ( -.  ( ( 2nd `  T
)  +  1 )  e.  ( 1 ... ( ( 2nd `  T
)  -  1 ) )  ->  ( (
1 ... ( ( 2nd `  T )  -  1 ) )  \  {
( ( 2nd `  T
)  +  1 ) } )  =  ( 1 ... ( ( 2nd `  T )  -  1 ) ) )
9290, 91syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 1 ... ( ( 2nd `  T
)  -  1 ) )  \  { ( ( 2nd `  T
)  +  1 ) } )  =  ( 1 ... ( ( 2nd `  T )  -  1 ) ) )
9382, 92ineq12d 3665 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  \  {
( 2nd `  T
) } )  i^i  ( ( 1 ... ( ( 2nd `  T
)  -  1 ) )  \  { ( ( 2nd `  T
)  +  1 ) } ) )  =  ( ( 1 ... ( ( 2nd `  T
)  -  1 ) )  i^i  ( 1 ... ( ( 2nd `  T )  -  1 ) ) ) )
94 difun2 3877 . . . . . . . . . . . . 13  |-  ( ( ( 1 ... (
( 2nd `  T
)  -  1 ) )  u.  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } )  \  { ( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( ( 1 ... ( ( 2nd `  T )  -  1 ) ) 
\  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } )
95 df-pr 4001 . . . . . . . . . . . . . 14  |-  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) }  =  ( { ( 2nd `  T
) }  u.  {
( ( 2nd `  T
)  +  1 ) } )
9695difeq2i 3580 . . . . . . . . . . . . 13  |-  ( ( 1 ... ( ( 2nd `  T )  -  1 ) ) 
\  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } )  =  ( ( 1 ... (
( 2nd `  T
)  -  1 ) )  \  ( { ( 2nd `  T
) }  u.  {
( ( 2nd `  T
)  +  1 ) } ) )
97 difundi 3725 . . . . . . . . . . . . 13  |-  ( ( 1 ... ( ( 2nd `  T )  -  1 ) ) 
\  ( { ( 2nd `  T ) }  u.  { ( ( 2nd `  T
)  +  1 ) } ) )  =  ( ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  \  {
( 2nd `  T
) } )  i^i  ( ( 1 ... ( ( 2nd `  T
)  -  1 ) )  \  { ( ( 2nd `  T
)  +  1 ) } ) )
9894, 96, 973eqtrri 2456 . . . . . . . . . . . 12  |-  ( ( ( 1 ... (
( 2nd `  T
)  -  1 ) )  \  { ( 2nd `  T ) } )  i^i  (
( 1 ... (
( 2nd `  T
)  -  1 ) )  \  { ( ( 2nd `  T
)  +  1 ) } ) )  =  ( ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  \  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } )
99 inidm 3671 . . . . . . . . . . . 12  |-  ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  i^i  ( 1 ... ( ( 2nd `  T
)  -  1 ) ) )  =  ( 1 ... ( ( 2nd `  T )  -  1 ) )
10093, 98, 993eqtr3g 2486 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  \  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )
10173, 100eqtrd 2463 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1 ... ( ( 2nd `  T
)  +  1 ) )  \  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )
102 peano2re 9813 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 2nd `  T
)  +  1 )  e.  RR  ->  (
( ( 2nd `  T
)  +  1 )  +  1 )  e.  RR )
10384, 102syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( 2nd `  T )  +  1 )  +  1 )  e.  RR )
10484ltp1d 10544 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2nd `  T
)  +  1 )  <  ( ( ( 2nd `  T )  +  1 )  +  1 ) )
10574, 84, 103, 85, 104lttrd 9803 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 2nd `  T
)  <  ( (
( 2nd `  T
)  +  1 )  +  1 ) )
10674, 103ltnled 9789 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( 2nd `  T
)  <  ( (
( 2nd `  T
)  +  1 )  +  1 )  <->  -.  (
( ( 2nd `  T
)  +  1 )  +  1 )  <_ 
( 2nd `  T
) ) )
107105, 106mpbid 213 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  ( ( ( 2nd `  T )  +  1 )  +  1 )  <_  ( 2nd `  T ) )
108 elfzle1 11809 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  T )  e.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  ->  (
( ( 2nd `  T
)  +  1 )  +  1 )  <_ 
( 2nd `  T
) )
109107, 108nsyl 124 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  ( 2nd `  T
)  e.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )
110 difsn 4134 . . . . . . . . . . . . 13  |-  ( -.  ( 2nd `  T
)  e.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  ->  (
( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
)  \  { ( 2nd `  T ) } )  =  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )
111109, 110syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  \  {
( 2nd `  T
) } )  =  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )
11284, 103ltnled 9789 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( 2nd `  T )  +  1 )  <  ( ( ( 2nd `  T
)  +  1 )  +  1 )  <->  -.  (
( ( 2nd `  T
)  +  1 )  +  1 )  <_ 
( ( 2nd `  T
)  +  1 ) ) )
113104, 112mpbid 213 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  ( ( ( 2nd `  T )  +  1 )  +  1 )  <_  (
( 2nd `  T
)  +  1 ) )
114 elfzle1 11809 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  T
)  +  1 )  e.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  ->  (
( ( 2nd `  T
)  +  1 )  +  1 )  <_ 
( ( 2nd `  T
)  +  1 ) )
115113, 114nsyl 124 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  ( ( 2nd `  T )  +  1 )  e.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )
116 difsn 4134 . . . . . . . . . . . . 13  |-  ( -.  ( ( 2nd `  T
)  +  1 )  e.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  ->  (
( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
)  \  { (
( 2nd `  T
)  +  1 ) } )  =  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N ) )
117115, 116syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  \  {
( ( 2nd `  T
)  +  1 ) } )  =  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N ) )
118111, 117ineq12d 3665 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  \  {
( 2nd `  T
) } )  i^i  ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  \  {
( ( 2nd `  T
)  +  1 ) } ) )  =  ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  i^i  (
( ( ( 2nd `  T )  +  1 )  +  1 ) ... N ) ) )
11995difeq2i 3580 . . . . . . . . . . . 12  |-  ( ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )  \  { ( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )  \ 
( { ( 2nd `  T ) }  u.  { ( ( 2nd `  T
)  +  1 ) } ) )
120 difundi 3725 . . . . . . . . . . . 12  |-  ( ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )  \ 
( { ( 2nd `  T ) }  u.  { ( ( 2nd `  T
)  +  1 ) } ) )  =  ( ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  \  {
( 2nd `  T
) } )  i^i  ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  \  {
( ( 2nd `  T
)  +  1 ) } ) )
121119, 120eqtr2i 2452 . . . . . . . . . . 11  |-  ( ( ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
)  \  { ( 2nd `  T ) } )  i^i  ( ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )  \  { ( ( 2nd `  T )  +  1 ) } ) )  =  ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )
122 inidm 3671 . . . . . . . . . . 11  |-  ( ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )  i^i  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  =  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )
123118, 121, 1223eqtr3g 2486 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )
124101, 123uneq12d 3621 . . . . . . . . 9  |-  ( ph  ->  ( ( ( 1 ... ( ( 2nd `  T )  +  1 ) )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  u.  ( ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )  \  { ( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  =  ( ( 1 ... (
( 2nd `  T
)  -  1 ) )  u.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) ) )
12547, 124syl5eq 2475 . . . . . . . 8  |-  ( ph  ->  ( ( ( 1 ... ( ( 2nd `  T )  +  1 ) )  u.  (
( ( ( 2nd `  T )  +  1 )  +  1 ) ... N ) ) 
\  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } )  =  ( ( 1 ... (
( 2nd `  T
)  -  1 ) )  u.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) ) )
12646, 125eqtrd 2463 . . . . . . 7  |-  ( ph  ->  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) ) )
127126eleq2d 2492 . . . . . 6  |-  ( ph  ->  ( k  e.  ( ( 1 ... N
)  \  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } )  <->  k  e.  ( ( 1 ... ( ( 2nd `  T
)  -  1 ) )  u.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) ) ) )
128 elun 3606 . . . . . 6  |-  ( k  e.  ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  (
( ( ( 2nd `  T )  +  1 )  +  1 ) ... N ) )  <-> 
( k  e.  ( 1 ... ( ( 2nd `  T )  -  1 ) )  \/  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N ) ) )
129127, 128syl6bb 264 . . . . 5  |-  ( ph  ->  ( k  e.  ( ( 1 ... N
)  \  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } )  <->  ( k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) )  \/  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) ) ) )
130129biimpa 486 . . . 4  |-  ( (
ph  /\  k  e.  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  ->  (
k  e.  ( 1 ... ( ( 2nd `  T )  -  1 ) )  \/  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) ) )
131 fveq2 5881 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
132131breq2d 4435 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
133132ifbid 3933 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
134133csbeq1d 3402 . . . . . . . . . . . . . . . . . . 19  |-  ( t  =  T  ->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  [_ if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
135 fveq2 5881 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
136135fveq2d 5885 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
137135fveq2d 5885 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
138137imaeq1d 5186 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
139138xpeq1d 4876 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
140137imaeq1d 5186 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
141140xpeq1d 4876 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
142139, 141uneq12d 3621 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  T  ->  (
( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) )  =  ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) ) )
143136, 142oveq12d 6323 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  T  ->  (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
144143csbeq2dv 3811 . . . . . . . . . . . . . . . . . . 19  |-  ( t  =  T  ->  [_ if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  [_ if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
145134, 144eqtrd 2463 . . . . . . . . . . . . . . . . . 18  |-  ( t  =  T  ->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  [_ if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
146145mpteq2dv 4511 . . . . . . . . . . . . . . . . 17  |-  ( t  =  T  ->  (
y  e.  ( 0 ... ( N  - 
1 ) )  |->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )  =  ( y  e.  ( 0 ... ( N  -  1 ) ) 
|->  [_ if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
147146eqeq2d 2436 . . . . . . . . . . . . . . . 16  |-  ( t  =  T  ->  ( F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  t
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )  <->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) ) 
|->  [_ if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) ) )
148147, 3elrab2 3230 . . . . . . . . . . . . . . 15  |-  ( T  e.  S  <->  ( T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  /\  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) ) )
149148simprbi 465 . . . . . . . . . . . . . 14  |-  ( T  e.  S  ->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
15019, 149syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
151 xp1st 6837 . . . . . . . . . . . . . . . 16  |-  ( ( 1st `  T )  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  -> 
( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
15224, 151syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
153 elmapi 7504 . . . . . . . . . . . . . . 15  |-  ( ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
154152, 153syl 17 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
155 elfzoelz 11927 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( 0..^ K )  ->  n  e.  ZZ )
156155ssriv 3468 . . . . . . . . . . . . . 14  |-  ( 0..^ K )  C_  ZZ
157 fss 5754 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K )  /\  (
0..^ K )  C_  ZZ )  ->  ( 1st `  ( 1st `  T
) ) : ( 1 ... N ) --> ZZ )
158154, 156, 157sylancl 666 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ZZ )
15938, 150, 158, 30, 35poimirlem1 31905 . . . . . . . . . . . 12  |-  ( ph  ->  -.  E* n  e.  ( 1 ... N
) ( ( F `
 ( ( 2nd `  T )  -  1 ) ) `  n
)  =/=  ( ( F `  ( 2nd `  T ) ) `  n ) )
16038adantr 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  N  e.  NN )
161 fveq2 5881 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  U  ->  ( 2nd `  t )  =  ( 2nd `  U
) )
162161breq2d 4435 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  U  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  U ) ) )
163162ifbid 3933 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  U  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  U
) ,  y ,  ( y  +  1 ) ) )
164163csbeq1d 3402 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  U  ->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  [_ if ( y  <  ( 2nd `  U ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
165 fveq2 5881 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  U  ->  ( 1st `  t )  =  ( 1st `  U
) )
166165fveq2d 5885 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  U  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  U ) ) )
167165fveq2d 5885 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  U  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  U ) ) )
168167imaeq1d 5186 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  U  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  U ) )
" ( 1 ... j ) ) )
169168xpeq1d 4876 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  U  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  U ) ) "
( 1 ... j
) )  X.  {
1 } ) )
170167imaeq1d 5186 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  U  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) ) )
171170xpeq1d 4876 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  U  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  U ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
172169, 171uneq12d 3621 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  U  ->  (
( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) )  =  ( ( ( ( 2nd `  ( 1st `  U ) )
" ( 1 ... j ) )  X. 
{ 1 } )  u.  ( ( ( 2nd `  ( 1st `  U ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) ) )
173166, 172oveq12d 6323 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  U  ->  (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  ( ( 1st `  ( 1st `  U ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  U
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
174173csbeq2dv 3811 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  U  ->  [_ if ( y  <  ( 2nd `  U ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  [_ if ( y  <  ( 2nd `  U ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  U ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  U
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
175164, 174eqtrd 2463 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  U  ->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  [_ if ( y  <  ( 2nd `  U ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  U ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  U
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
176175mpteq2dv 4511 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  U  ->  (
y  e.  ( 0 ... ( N  - 
1 ) )  |->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )  =  ( y  e.  ( 0 ... ( N  -  1 ) ) 
|->  [_ if ( y  <  ( 2nd `  U
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  U ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  U
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
177176eqeq2d 2436 . . . . . . . . . . . . . . . . . . 19  |-  ( t  =  U  ->  ( F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  t
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )  <->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) ) 
|->  [_ if ( y  <  ( 2nd `  U
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  U ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  U
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) ) )
178177, 3elrab2 3230 . . . . . . . . . . . . . . . . . 18  |-  ( U  e.  S  <->  ( U  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 `  U
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  U ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  U
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) ) )
179178simprbi 465 . . . . . . . . . . . . . . . . 17  |-  ( U  e.  S  ->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  U
) ,  y ,  ( y  +  1 ) )  /  j ]_ ( ( 1st `  ( 1st `  U ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  U
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
1801, 179syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  F  =  ( y  e.  ( 0 ... ( N  -  1 ) )  |->  [_ if ( y  <  ( 2nd `  U ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  U ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  U
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
181180adantr 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  F  =  ( y  e.  ( 0 ... ( N  - 
1 ) )  |->  [_ if ( y  <  ( 2nd `  U ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  U ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  U
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) )
182 xp1st 6837 . . . . . . . . . . . . . . . . . . 19  |-  ( ( 1st `  U )  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  -> 
( 1st `  ( 1st `  U ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
1837, 182syl 17 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 1st `  ( 1st `  U ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
184 elmapi 7504 . . . . . . . . . . . . . . . . . 18  |-  ( ( 1st `  ( 1st `  U ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  U ) ) : ( 1 ... N ) --> ( 0..^ K ) )
185183, 184syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 1st `  ( 1st `  U ) ) : ( 1 ... N ) --> ( 0..^ K ) )
186 fss 5754 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1st `  ( 1st `  U ) ) : ( 1 ... N ) --> ( 0..^ K )  /\  (
0..^ K )  C_  ZZ )  ->  ( 1st `  ( 1st `  U
) ) : ( 1 ... N ) --> ZZ )
187185, 156, 186sylancl 666 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 1st `  ( 1st `  U ) ) : ( 1 ... N ) --> ZZ )
188187adantr 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  ( 1st `  ( 1st `  U ) ) : ( 1 ... N ) --> ZZ )
18913adantr 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  ( 2nd `  ( 1st `  U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
19035adantr 466 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  ( 2nd `  T
)  e.  ( 1 ... ( N  - 
1 ) ) )
191 xp2nd 6838 . . . . . . . . . . . . . . . . 17  |-  ( U  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) )  ->  ( 2nd `  U )  e.  ( 0 ... N
) )
1925, 191syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 2nd `  U
)  e.  ( 0 ... N ) )
193 eldifsn 4125 . . . . . . . . . . . . . . . . 17  |-  ( ( 2nd `  U )  e.  ( ( 0 ... N )  \  { ( 2nd `  T
) } )  <->  ( ( 2nd `  U )  e.  ( 0 ... N
)  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) ) )
194193biimpri 209 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  U
)  e.  ( 0 ... N )  /\  ( 2nd `  U )  =/=  ( 2nd `  T
) )  ->  ( 2nd `  U )  e.  ( ( 0 ... N )  \  {
( 2nd `  T
) } ) )
195192, 194sylan 473 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  ( 2nd `  U
)  e.  ( ( 0 ... N ) 
\  { ( 2nd `  T ) } ) )
196160, 181, 188, 189, 190, 195poimirlem2 31906 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  E* n  e.  ( 1 ... N
) ( ( F `
 ( ( 2nd `  T )  -  1 ) ) `  n
)  =/=  ( ( F `  ( 2nd `  T ) ) `  n ) )
197196ex 435 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 2nd `  U
)  =/=  ( 2nd `  T )  ->  E* n  e.  ( 1 ... N ) ( ( F `  (
( 2nd `  T
)  -  1 ) ) `  n )  =/=  ( ( F `
 ( 2nd `  T
) ) `  n
) ) )
198197necon1bd 2638 . . . . . . . . . . . 12  |-  ( ph  ->  ( -.  E* n  e.  ( 1 ... N
) ( ( F `
 ( ( 2nd `  T )  -  1 ) ) `  n
)  =/=  ( ( F `  ( 2nd `  T ) ) `  n )  ->  ( 2nd `  U )  =  ( 2nd `  T
) ) )
199159, 198mpd 15 . . . . . . . . . . 11  |-  ( ph  ->  ( 2nd `  U
)  =  ( 2nd `  T ) )
200199oveq1d 6320 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd `  U
)  -  1 )  =  ( ( 2nd `  T )  -  1 ) )
201200oveq2d 6321 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... (
( 2nd `  U
)  -  1 ) )  =  ( 1 ... ( ( 2nd `  T )  -  1 ) ) )
202201eleq2d 2492 . . . . . . . 8  |-  ( ph  ->  ( k  e.  ( 1 ... ( ( 2nd `  U )  -  1 ) )  <-> 
k  e.  ( 1 ... ( ( 2nd `  T )  -  1 ) ) ) )
203202biimpar 487 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )
20438adantr 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )  ->  N  e.  NN )
2051adantr 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )  ->  U  e.  S )
206199, 35eqeltrd 2507 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  U
)  e.  ( 1 ... ( N  - 
1 ) ) )
207206adantr 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )  ->  ( 2nd `  U )  e.  ( 1 ... ( N  -  1 ) ) )
208 simpr 462 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )  ->  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )
209204, 3, 205, 207, 208poimirlem6 31910 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )  ->  ( iota_ n  e.  ( 1 ... N ) ( ( F `  (
k  -  1 ) ) `  n )  =/=  ( ( F `
 k ) `  n ) )  =  ( ( 2nd `  ( 1st `  U ) ) `
 k ) )
210203, 209syldan 472 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  ( iota_ n  e.  ( 1 ... N ) ( ( F `  (
k  -  1 ) ) `  n )  =/=  ( ( F `
 k ) `  n ) )  =  ( ( 2nd `  ( 1st `  U ) ) `
 k ) )
21138adantr 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  N  e.  NN )
21219adantr 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  T  e.  S )
21335adantr 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  ( 2nd `  T )  e.  ( 1 ... ( N  -  1 ) ) )
214 simpr 462 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )
215211, 3, 212, 213, 214poimirlem6 31910 . . . . . 6  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  ( iota_ n  e.  ( 1 ... N ) ( ( F `  (
k  -  1 ) ) `  n )  =/=  ( ( F `
 k ) `  n ) )  =  ( ( 2nd `  ( 1st `  T ) ) `
 k ) )
216210, 215eqtr3d 2465 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  U ) ) `
 k )  =  ( ( 2nd `  ( 1st `  T ) ) `
 k ) )
217199oveq1d 6320 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2nd `  U
)  +  1 )  =  ( ( 2nd `  T )  +  1 ) )
218217oveq1d 6320 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 2nd `  U )  +  1 )  +  1 )  =  ( ( ( 2nd `  T )  +  1 )  +  1 ) )
219218oveq1d 6320 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
)  =  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )
220219eleq2d 2492 . . . . . . . 8  |-  ( ph  ->  ( k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N )  <->  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) ) )
221220biimpar 487 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )
22238adantr 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )  ->  N  e.  NN )
2231adantr 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )  ->  U  e.  S )
224206adantr 466 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )  ->  ( 2nd `  U )  e.  ( 1 ... ( N  -  1 ) ) )
225 simpr 462 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )  ->  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )
226222, 3, 223, 224, 225poimirlem7 31911 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )  ->  ( iota_ n  e.  ( 1 ... N ) ( ( F `  (
k  -  2 ) ) `  n )  =/=  ( ( F `
 ( k  - 
1 ) ) `  n ) )  =  ( ( 2nd `  ( 1st `  U ) ) `
 k ) )
227221, 226syldan 472 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  ( iota_ n  e.  ( 1 ... N ) ( ( F `  (
k  -  2 ) ) `  n )  =/=  ( ( F `
 ( k  - 
1 ) ) `  n ) )  =  ( ( 2nd `  ( 1st `  U ) ) `
 k ) )
22838adantr 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  N  e.  NN )
22919adantr 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  T  e.  S )
23035adantr 466 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  ( 2nd `  T )  e.  ( 1 ... ( N  -  1 ) ) )
231 simpr 462 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )
232228, 3, 229, 230, 231poimirlem7 31911 . . . . . 6  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  ( iota_ n  e.  ( 1 ... N ) ( ( F `  (
k  -  2 ) ) `  n )  =/=  ( ( F `
 ( k  - 
1 ) ) `  n ) )  =  ( ( 2nd `  ( 1st `  T ) ) `
 k ) )
233227, 232eqtr3d 2465 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  (
( 2nd `  ( 1st `  U ) ) `
 k )  =  ( ( 2nd `  ( 1st `  T ) ) `
 k ) )
234216, 233jaodan 792 . . . 4  |-  ( (
ph  /\  ( k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) )  \/  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) ) )  -> 
( ( 2nd `  ( 1st `  U ) ) `
 k )  =  ( ( 2nd `  ( 1st `  T ) ) `
 k ) )
235130, 234syldan 472 . . 3  |-  ( (
ph  /\  k  e.  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  ->  (
( 2nd `  ( 1st `  U ) ) `
 k )  =  ( ( 2nd `  ( 1st `  T ) ) `
 k ) )
236 fvres 5895 . . . 4  |-  ( k  e.  ( ( 1 ... N )  \  { ( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  ->  ( (
( 2nd `  ( 1st `  U ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) ) `  k
)  =  ( ( 2nd `  ( 1st `  U ) ) `  k ) )
237236adantl 467 . . 3  |-  ( (
ph  /\  k  e.  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  ->  (
( ( 2nd `  ( 1st `  U ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) ) `  k
)  =  ( ( 2nd `  ( 1st `  U ) ) `  k ) )
238 fvres 5895 . . . 4  |-  ( k  e.  ( ( 1 ... N )  \  { ( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  ->  ( (
( 2nd `  ( 1st `  T ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) ) `  k
)  =  ( ( 2nd `  ( 1st `  T ) ) `  k ) )
239238adantl 467 . . 3  |-  ( (
ph  /\  k  e.  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  ->  (
( ( 2nd `  ( 1st `  T ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) ) `  k
)  =  ( ( 2nd `  ( 1st `  T ) ) `  k ) )
240235, 237, 2393eqtr4d 2473 . 2  |-  ( (
ph  /\  k  e.  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  ->  (
( ( 2nd `  ( 1st `  U ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) ) `  k
)  =  ( ( ( 2nd `  ( 1st `  T ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) ) `  k
) )
24118, 34, 240eqfnfvd 5994 1  |-  ( ph  ->  ( ( 2nd `  ( 1st `  U ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  =  ( ( 2nd `  ( 1st `  T ) )  |`  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 369    /\ wa 370    = wceq 1437    e. wcel 1872   {cab 2407    =/= wne 2614   E*wrmo 2774   {crab 2775   [_csb 3395    \ cdif 3433    u. cun 3434    i^i cin 3435    C_ wss 3436   ifcif 3911   {csn 3998   {cpr 4000   class class class wbr 4423    |-> cmpt 4482    X. cxp 4851    |` cres 4855   "cima 4856    Fn wfn 5596   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601   iota_crio 6266  (class class class)co 6305    oFcof 6543   1stc1st 6805   2ndc2nd 6806    ^m cmap 7483   CCcc 9544   RRcr 9545   0cc0 9546   1c1 9547    + caddc 9549    < clt 9682    <_ cle 9683    - cmin 9867   NNcn 10616   2c2 10666   ZZcz 10944   ZZ>=cuz 11166   ...cfz 11791  ..^cfzo 11922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-rep 4536  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6597  ax-cnex 9602  ax-resscn 9603  ax-1cn 9604  ax-icn 9605  ax-addcl 9606  ax-addrcl 9607  ax-mulcl 9608  ax-mulrcl 9609  ax-mulcom 9610  ax-addass 9611  ax-mulass 9612  ax-distr 9613  ax-i2m1 9614  ax-1ne0 9615  ax-1rid 9616  ax-rnegex 9617  ax-rrecex 9618  ax-cnre 9619  ax-pre-lttri 9620  ax-pre-lttrn 9621  ax-pre-ltadd 9622  ax-pre-mulgt0 9623
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rmo 2779  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-of 6545  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7039  df-recs 7101  df-rdg 7139  df-1o 7193  df-er 7374  df-map 7485  df-en 7581  df-dom 7582  df-sdom 7583  df-fin 7584  df-pnf 9684  df-mnf 9685  df-xr 9686  df-ltxr 9687  df-le 9688  df-sub 9869  df-neg 9870  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-fzo 11923
This theorem is referenced by:  poimirlem9  31913
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