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Theorem poimirlem8 32012
Description: Lemma for poimir 32037, establishing that away from the opposite vertex the walks in poimirlem9 32013 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 3181 . . . . . . . . 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 2567 . . . . . . . 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 6842 . . . . . . 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 6843 . . . . . 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 5889 . . . . . 6  |-  ( 2nd `  ( 1st `  U
) )  e.  _V
11 f1oeq1 5818 . . . . . 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 3173 . . . . 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 201 . . . 4  |-  ( ph  ->  ( 2nd `  ( 1st `  U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
14 f1ofn 5829 . . . 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 3549 . . 3  |-  ( ( 1 ... N ) 
\  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } )  C_  (
1 ... N )
17 fnssres 5699 . . 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 675 . 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 3181 . . . . . . . . 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 2567 . . . . . . . 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 6842 . . . . . . 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 6843 . . . . . 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 5889 . . . . . 6  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
28 f1oeq1 5818 . . . . . 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 3173 . . . . 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 201 . . . 4  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
31 f1ofn 5829 . . . 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 5699 . . 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 675 . 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 11875 . . . . . . . . . . . 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 10647 . . . . . . . . . . . . 13  |-  ( ph  ->  N  e.  CC )
40 npcan1 10065 . . . . . . . . . . . . 13  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
4139, 40syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
4241oveq2d 6324 . . . . . . . . . . 11  |-  ( ph  ->  ( 1 ... (
( N  -  1 )  +  1 ) )  =  ( 1 ... N ) )
4337, 42eleqtrd 2551 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd `  T
)  +  1 )  e.  ( 1 ... N ) )
44 fzsplit 11851 . . . . . . . . . 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 3539 . . . . . . . 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 3687 . . . . . . . . 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 11854 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  T )  e.  ( 1 ... ( N  -  1 ) )  ->  ( 2nd `  T )  e.  NN )
4935, 48syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 2nd `  T
)  e.  NN )
5049nncnd 10647 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 2nd `  T
)  e.  CC )
51 npcan1 10065 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  T )  e.  CC  ->  (
( ( 2nd `  T
)  -  1 )  +  1 )  =  ( 2nd `  T
) )
5250, 51syl 17 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( 2nd `  T )  -  1 )  +  1 )  =  ( 2nd `  T
) )
53 nnuz 11218 . . . . . . . . . . . . . . . 16  |-  NN  =  ( ZZ>= `  1 )
5449, 53syl6eleq 2559 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 2nd `  T
)  e.  ( ZZ>= ` 
1 ) )
5552, 54eqeltrd 2549 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( 2nd `  T )  -  1 )  +  1 )  e.  ( ZZ>= `  1
) )
5649nnzd 11062 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( 2nd `  T
)  e.  ZZ )
57 peano2zm 11004 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  T )  e.  ZZ  ->  (
( 2nd `  T
)  -  1 )  e.  ZZ )
5856, 57syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( 2nd `  T
)  -  1 )  e.  ZZ )
59 uzid 11197 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 2nd `  T
)  -  1 )  e.  ZZ  ->  (
( 2nd `  T
)  -  1 )  e.  ( ZZ>= `  (
( 2nd `  T
)  -  1 ) ) )
60 peano2uz 11235 . . . . . . . . . . . . . . . . 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 2550 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 2nd `  T
)  e.  ( ZZ>= `  ( ( 2nd `  T
)  -  1 ) ) )
63 peano2uz 11235 . . . . . . . . . . . . . . 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 11850 . . . . . . . . . . . . . 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 673 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1 ... (
( 2nd `  T
)  +  1 ) )  =  ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  ( ( ( ( 2nd `  T
)  -  1 )  +  1 ) ... ( ( 2nd `  T
)  +  1 ) ) ) )
6752oveq1d 6323 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( ( 2nd `  T )  -  1 )  +  1 ) ... (
( 2nd `  T
)  +  1 ) )  =  ( ( 2nd `  T ) ... ( ( 2nd `  T )  +  1 ) ) )
68 fzpr 11877 . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( ( ( 2nd `  T )  -  1 )  +  1 ) ... (
( 2nd `  T
)  +  1 ) )  =  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } )
7170uneq2d 3579 . . . . . . . . . . . . 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 2505 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1 ... (
( 2nd `  T
)  +  1 ) )  =  ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } ) )
7372difeq1d 3539 . . . . . . . . . . 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 10646 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 2nd `  T
)  e.  RR )
7574ltm1d 10561 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2nd `  T
)  -  1 )  <  ( 2nd `  T
) )
7658zred 11063 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( 2nd `  T
)  -  1 )  e.  RR )
7776, 74ltnled 9799 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( 2nd `  T )  -  1 )  <  ( 2nd `  T )  <->  -.  ( 2nd `  T )  <_ 
( ( 2nd `  T
)  -  1 ) ) )
7875, 77mpbid 215 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  ( 2nd `  T
)  <_  ( ( 2nd `  T )  - 
1 ) )
79 elfzle2 11829 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  T )  e.  ( 1 ... ( ( 2nd `  T
)  -  1 ) )  ->  ( 2nd `  T )  <_  (
( 2nd `  T
)  -  1 ) )
8078, 79nsyl 125 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  ( 2nd `  T
)  e.  ( 1 ... ( ( 2nd `  T )  -  1 ) ) )
81 difsn 4097 . . . . . . . . . . . . . 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 9824 . . . . . . . . . . . . . . . . . 18  |-  ( ( 2nd `  T )  e.  RR  ->  (
( 2nd `  T
)  +  1 )  e.  RR )
8474, 83syl 17 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( ( 2nd `  T
)  +  1 )  e.  RR )
8574ltp1d 10559 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( 2nd `  T
)  <  ( ( 2nd `  T )  +  1 ) )
8676, 74, 84, 75, 85lttrd 9813 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2nd `  T
)  -  1 )  <  ( ( 2nd `  T )  +  1 ) )
8776, 84ltnled 9799 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( 2nd `  T )  -  1 )  <  ( ( 2nd `  T )  +  1 )  <->  -.  (
( 2nd `  T
)  +  1 )  <_  ( ( 2nd `  T )  -  1 ) ) )
8886, 87mpbid 215 . . . . . . . . . . . . . . 15  |-  ( ph  ->  -.  ( ( 2nd `  T )  +  1 )  <_  ( ( 2nd `  T )  - 
1 ) )
89 elfzle2 11829 . . . . . . . . . . . . . . 15  |-  ( ( ( 2nd `  T
)  +  1 )  e.  ( 1 ... ( ( 2nd `  T
)  -  1 ) )  ->  ( ( 2nd `  T )  +  1 )  <_  (
( 2nd `  T
)  -  1 ) )
9088, 89nsyl 125 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  ( ( 2nd `  T )  +  1 )  e.  ( 1 ... ( ( 2nd `  T )  -  1 ) ) )
91 difsn 4097 . . . . . . . . . . . . . 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 3626 . . . . . . . . . . . 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 3838 . . . . . . . . . . . . 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 3962 . . . . . . . . . . . . . 14  |-  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) }  =  ( { ( 2nd `  T
) }  u.  {
( ( 2nd `  T
)  +  1 ) } )
9695difeq2i 3537 . . . . . . . . . . . . 13  |-  ( ( 1 ... ( ( 2nd `  T )  -  1 ) ) 
\  { ( 2nd `  T ) ,  ( ( 2nd `  T
)  +  1 ) } )  =  ( ( 1 ... (
( 2nd `  T
)  -  1 ) )  \  ( { ( 2nd `  T
) }  u.  {
( ( 2nd `  T
)  +  1 ) } ) )
97 difundi 3686 . . . . . . . . . . . . 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 2498 . . . . . . . . . . . 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 3632 . . . . . . . . . . . 12  |-  ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  i^i  ( 1 ... ( ( 2nd `  T
)  -  1 ) ) )  =  ( 1 ... ( ( 2nd `  T )  -  1 ) )
10093, 98, 993eqtr3g 2528 . . . . . . . . . . 11  |-  ( ph  ->  ( ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  \  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )
10173, 100eqtrd 2505 . . . . . . . . . 10  |-  ( ph  ->  ( ( 1 ... ( ( 2nd `  T
)  +  1 ) )  \  { ( 2nd `  T ) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )
102 peano2re 9824 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 2nd `  T
)  +  1 )  e.  RR  ->  (
( ( 2nd `  T
)  +  1 )  +  1 )  e.  RR )
10384, 102syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( ( 2nd `  T )  +  1 )  +  1 )  e.  RR )
10484ltp1d 10559 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( 2nd `  T
)  +  1 )  <  ( ( ( 2nd `  T )  +  1 )  +  1 ) )
10574, 84, 103, 85, 104lttrd 9813 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 2nd `  T
)  <  ( (
( 2nd `  T
)  +  1 )  +  1 ) )
10674, 103ltnled 9799 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( 2nd `  T
)  <  ( (
( 2nd `  T
)  +  1 )  +  1 )  <->  -.  (
( ( 2nd `  T
)  +  1 )  +  1 )  <_ 
( 2nd `  T
) ) )
107105, 106mpbid 215 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  ( ( ( 2nd `  T )  +  1 )  +  1 )  <_  ( 2nd `  T ) )
108 elfzle1 11828 . . . . . . . . . . . . . 14  |-  ( ( 2nd `  T )  e.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  ->  (
( ( 2nd `  T
)  +  1 )  +  1 )  <_ 
( 2nd `  T
) )
109107, 108nsyl 125 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  ( 2nd `  T
)  e.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )
110 difsn 4097 . . . . . . . . . . . . 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 9799 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( ( ( 2nd `  T )  +  1 )  <  ( ( ( 2nd `  T
)  +  1 )  +  1 )  <->  -.  (
( ( 2nd `  T
)  +  1 )  +  1 )  <_ 
( ( 2nd `  T
)  +  1 ) ) )
113104, 112mpbid 215 . . . . . . . . . . . . . 14  |-  ( ph  ->  -.  ( ( ( 2nd `  T )  +  1 )  +  1 )  <_  (
( 2nd `  T
)  +  1 ) )
114 elfzle1 11828 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  T
)  +  1 )  e.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  ->  (
( ( 2nd `  T
)  +  1 )  +  1 )  <_ 
( ( 2nd `  T
)  +  1 ) )
115113, 114nsyl 125 . . . . . . . . . . . . 13  |-  ( ph  ->  -.  ( ( 2nd `  T )  +  1 )  e.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )
116 difsn 4097 . . . . . . . . . . . . 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 3626 . . . . . . . . . . 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 3537 . . . . . . . . . . . 12  |-  ( ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )  \  { ( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )  \ 
( { ( 2nd `  T ) }  u.  { ( ( 2nd `  T
)  +  1 ) } ) )
120 difundi 3686 . . . . . . . . . . . 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 2494 . . . . . . . . . . 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 3632 . . . . . . . . . . 11  |-  ( ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )  i^i  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  =  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N )
123118, 121, 1223eqtr3g 2528 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )
124101, 123uneq12d 3580 . . . . . . . . 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 2517 . . . . . . . 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 2505 . . . . . . 7  |-  ( ph  ->  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } )  =  ( ( 1 ... ( ( 2nd `  T )  -  1 ) )  u.  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) ) )
127126eleq2d 2534 . . . . . 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 3565 . . . . . 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 269 . . . . 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 492 . . . 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 5879 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
132131breq2d 4407 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
133132ifbid 3894 . . . . . . . . . . . . . . . . . . . 20  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
134133csbeq1d 3356 . . . . . . . . . . . . . . . . . . 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 5879 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
136135fveq2d 5883 . . . . . . . . . . . . . . . . . . . . 21  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
137135fveq2d 5883 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
138137imaeq1d 5173 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
139138xpeq1d 4862 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
140137imaeq1d 5173 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
141140xpeq1d 4862 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
142139, 141uneq12d 3580 . . . . . . . . . . . . . . . . . . . . 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 6326 . . . . . . . . . . . . . . . . . . . 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 3785 . . . . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . . . . 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 4483 . . . . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . 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 7511 . . . . . . . . . . . . . . 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 11947 . . . . . . . . . . . . . . 15  |-  ( n  e.  ( 0..^ K )  ->  n  e.  ZZ )
156155ssriv 3422 . . . . . . . . . . . . . 14  |-  ( 0..^ K )  C_  ZZ
157 fss 5749 . . . . . . . . . . . . . 14  |-  ( ( ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K )  /\  (
0..^ K )  C_  ZZ )  ->  ( 1st `  ( 1st `  T
) ) : ( 1 ... N ) --> ZZ )
158154, 156, 157sylancl 675 . . . . . . . . . . . . 13  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ZZ )
15938, 150, 158, 30, 35poimirlem1 32005 . . . . . . . . . . . 12  |-  ( ph  ->  -.  E* n  e.  ( 1 ... N
) ( ( F `
 ( ( 2nd `  T )  -  1 ) ) `  n
)  =/=  ( ( F `  ( 2nd `  T ) ) `  n ) )
16038adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  N  e.  NN )
161 fveq2 5879 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  U  ->  ( 2nd `  t )  =  ( 2nd `  U
) )
162161breq2d 4407 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  U  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  U ) ) )
163162ifbid 3894 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( t  =  U  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  U
) ,  y ,  ( y  +  1 ) ) )
164163csbeq1d 3356 . . . . . . . . . . . . . . . . . . . . . 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 5879 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  U  ->  ( 1st `  t )  =  ( 1st `  U
) )
166165fveq2d 5883 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( t  =  U  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  U ) ) )
167165fveq2d 5883 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( t  =  U  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  U ) ) )
168167imaeq1d 5173 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  U  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  U ) )
" ( 1 ... j ) ) )
169168xpeq1d 4862 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  U  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  U ) ) "
( 1 ... j
) )  X.  {
1 } ) )
170167imaeq1d 5173 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( t  =  U  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  U ) )
" ( ( j  +  1 ) ... N ) ) )
171170xpeq1d 4862 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( t  =  U  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  U ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
172169, 171uneq12d 3580 . . . . . . . . . . . . . . . . . . . . . . . 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 6326 . . . . . . . . . . . . . . . . . . . . . . 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 3785 . . . . . . . . . . . . . . . . . . . . . 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 2505 . . . . . . . . . . . . . . . . . . . . 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 4483 . . . . . . . . . . . . . . . . . . . 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 2481 . . . . . . . . . . . . . . . . . . 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 3186 . . . . . . . . . . . . . . . . . 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 471 . . . . . . . . . . . . . . . . 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 472 . . . . . . . . . . . . . . 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 6842 . . . . . . . . . . . . . . . . . . 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 7511 . . . . . . . . . . . . . . . . . 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 5749 . . . . . . . . . . . . . . . . 17  |-  ( ( ( 1st `  ( 1st `  U ) ) : ( 1 ... N ) --> ( 0..^ K )  /\  (
0..^ K )  C_  ZZ )  ->  ( 1st `  ( 1st `  U
) ) : ( 1 ... N ) --> ZZ )
187185, 156, 186sylancl 675 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 1st `  ( 1st `  U ) ) : ( 1 ... N ) --> ZZ )
188187adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  ( 1st `  ( 1st `  U ) ) : ( 1 ... N ) --> ZZ )
18913adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  ( 2nd `  ( 1st `  U ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
19035adantr 472 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  ( 2nd `  T
)  e.  ( 1 ... ( N  - 
1 ) ) )
191 xp2nd 6843 . . . . . . . . . . . . . . . . 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 4088 . . . . . . . . . . . . . . . . 17  |-  ( ( 2nd `  U )  e.  ( ( 0 ... N )  \  { ( 2nd `  T
) } )  <->  ( ( 2nd `  U )  e.  ( 0 ... N
)  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) ) )
194193biimpri 211 . . . . . . . . . . . . . . . 16  |-  ( ( ( 2nd `  U
)  e.  ( 0 ... N )  /\  ( 2nd `  U )  =/=  ( 2nd `  T
) )  ->  ( 2nd `  U )  e.  ( ( 0 ... N )  \  {
( 2nd `  T
) } ) )
195192, 194sylan 479 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  ( 2nd `  U
)  e.  ( ( 0 ... N ) 
\  { ( 2nd `  T ) } ) )
196160, 181, 188, 189, 190, 195poimirlem2 32006 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( 2nd `  U )  =/=  ( 2nd `  T ) )  ->  E* n  e.  ( 1 ... N
) ( ( F `
 ( ( 2nd `  T )  -  1 ) ) `  n
)  =/=  ( ( F `  ( 2nd `  T ) ) `  n ) )
197196ex 441 . . . . . . . . . . . . 13  |-  ( ph  ->  ( ( 2nd `  U
)  =/=  ( 2nd `  T )  ->  E* n  e.  ( 1 ... N ) ( ( F `  (
( 2nd `  T
)  -  1 ) ) `  n )  =/=  ( ( F `
 ( 2nd `  T
) ) `  n
) ) )
198197necon1bd 2661 . . . . . . . . . . . 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 6323 . . . . . . . . . 10  |-  ( ph  ->  ( ( 2nd `  U
)  -  1 )  =  ( ( 2nd `  T )  -  1 ) )
201200oveq2d 6324 . . . . . . . . 9  |-  ( ph  ->  ( 1 ... (
( 2nd `  U
)  -  1 ) )  =  ( 1 ... ( ( 2nd `  T )  -  1 ) ) )
202201eleq2d 2534 . . . . . . . 8  |-  ( ph  ->  ( k  e.  ( 1 ... ( ( 2nd `  U )  -  1 ) )  <-> 
k  e.  ( 1 ... ( ( 2nd `  T )  -  1 ) ) ) )
203202biimpar 493 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )
20438adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )  ->  N  e.  NN )
2051adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )  ->  U  e.  S )
206199, 35eqeltrd 2549 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  U
)  e.  ( 1 ... ( N  - 
1 ) ) )
207206adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )  ->  ( 2nd `  U )  e.  ( 1 ... ( N  -  1 ) ) )
208 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )  ->  k  e.  ( 1 ... (
( 2nd `  U
)  -  1 ) ) )
209204, 3, 205, 207, 208poimirlem6 32010 . . . . . . 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 478 . . . . . 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 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  N  e.  NN )
21219adantr 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  T  e.  S )
21335adantr 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  ( 2nd `  T )  e.  ( 1 ... ( N  -  1 ) ) )
214 simpr 468 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )
215211, 3, 212, 213, 214poimirlem6 32010 . . . . . 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 2507 . . . . 5  |-  ( (
ph  /\  k  e.  ( 1 ... (
( 2nd `  T
)  -  1 ) ) )  ->  (
( 2nd `  ( 1st `  U ) ) `
 k )  =  ( ( 2nd `  ( 1st `  T ) ) `
 k ) )
217199oveq1d 6323 . . . . . . . . . . 11  |-  ( ph  ->  ( ( 2nd `  U
)  +  1 )  =  ( ( 2nd `  T )  +  1 ) )
218217oveq1d 6323 . . . . . . . . . 10  |-  ( ph  ->  ( ( ( 2nd `  U )  +  1 )  +  1 )  =  ( ( ( 2nd `  T )  +  1 )  +  1 ) )
219218oveq1d 6323 . . . . . . . . 9  |-  ( ph  ->  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
)  =  ( ( ( ( 2nd `  T
)  +  1 )  +  1 ) ... N ) )
220219eleq2d 2534 . . . . . . . 8  |-  ( ph  ->  ( k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N )  <->  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) ) )
221220biimpar 493 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )
22238adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )  ->  N  e.  NN )
2231adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )  ->  U  e.  S )
224206adantr 472 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )  ->  ( 2nd `  U )  e.  ( 1 ... ( N  -  1 ) ) )
225 simpr 468 . . . . . . . 8  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )  ->  k  e.  ( ( ( ( 2nd `  U )  +  1 )  +  1 ) ... N
) )
226222, 3, 223, 224, 225poimirlem7 32011 . . . . . . 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 478 . . . . . 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 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  N  e.  NN )
22919adantr 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  T  e.  S )
23035adantr 472 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  ( 2nd `  T )  e.  ( 1 ... ( N  -  1 ) ) )
231 simpr 468 . . . . . . 7  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )
232228, 3, 229, 230, 231poimirlem7 32011 . . . . . 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 2507 . . . . 5  |-  ( (
ph  /\  k  e.  ( ( ( ( 2nd `  T )  +  1 )  +  1 ) ... N
) )  ->  (
( 2nd `  ( 1st `  U ) ) `
 k )  =  ( ( 2nd `  ( 1st `  T ) ) `
 k ) )
234216, 233jaodan 802 . . . 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 478 . . 3  |-  ( (
ph  /\  k  e.  ( ( 1 ... N )  \  {
( 2nd `  T
) ,  ( ( 2nd `  T )  +  1 ) } ) )  ->  (
( 2nd `  ( 1st `  U ) ) `
 k )  =  ( ( 2nd `  ( 1st `  T ) ) `
 k ) )
236 fvres 5893 . . . 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 473 . . 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 5893 . . . 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 473 . . 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 2515 . 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 375    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457    =/= wne 2641   E*wrmo 2759   {crab 2760   [_csb 3349    \ cdif 3387    u. cun 3388    i^i cin 3389    C_ wss 3390   ifcif 3872   {csn 3959   {cpr 3961   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837    |` cres 4841   "cima 4842    Fn wfn 5584   -->wf 5585   -1-1-onto->wf1o 5588   ` cfv 5589   iota_crio 6269  (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   2c2 10681   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-1o 7200  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  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-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943
This theorem is referenced by:  poimirlem9  32013
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