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Theorem poimirlem10 32014
Description: Lemma for poimir 32037 establishing the cube that yields the simplex that yields a face if the opposite vertex was first on the walk. (Contributed by Brendan Leahy, 21-Aug-2020.)
Hypotheses
Ref Expression
poimir.0  |-  ( ph  ->  N  e.  NN )
poimirlem22.s  |-  S  =  { t  e.  ( ( ( ( 0..^ K )  ^m  (
1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) )  |  F  =  ( y  e.  ( 0 ... ( N  - 
1 ) )  |->  [_ if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  /  j ]_ (
( 1st `  ( 1st `  t ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  t
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) ) }
poimirlem22.1  |-  ( ph  ->  F : ( 0 ... ( N  - 
1 ) ) --> ( ( 0 ... K
)  ^m  ( 1 ... N ) ) )
poimirlem12.2  |-  ( ph  ->  T  e.  S )
poimirlem11.3  |-  ( ph  ->  ( 2nd `  T
)  =  0 )
Assertion
Ref Expression
poimirlem10  |-  ( ph  ->  ( ( F `  ( N  -  1
) )  oF  -  ( ( 1 ... N )  X. 
{ 1 } ) )  =  ( 1st `  ( 1st `  T
) ) )
Distinct variable groups:    f, j,
t, y    ph, j, y   
j, F, y    j, N, y    T, j, y    ph, t    f, K, j, t    f, N, t    T, f    f, F, t   
t, T    S, j,
t, y
Allowed substitution hints:    ph( f)    S( f)    K( y)

Proof of Theorem poimirlem10
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 ovex 6336 . . 3  |-  ( 1 ... N )  e. 
_V
21a1i 11 . 2  |-  ( ph  ->  ( 1 ... N
)  e.  _V )
3 poimirlem22.1 . . . 4  |-  ( ph  ->  F : ( 0 ... ( N  - 
1 ) ) --> ( ( 0 ... K
)  ^m  ( 1 ... N ) ) )
4 poimir.0 . . . . . 6  |-  ( ph  ->  N  e.  NN )
5 nnm1nn0 10935 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
64, 5syl 17 . . . . 5  |-  ( ph  ->  ( N  -  1 )  e.  NN0 )
7 nn0fz0 11916 . . . . 5  |-  ( ( N  -  1 )  e.  NN0  <->  ( N  - 
1 )  e.  ( 0 ... ( N  -  1 ) ) )
86, 7sylib 201 . . . 4  |-  ( ph  ->  ( N  -  1 )  e.  ( 0 ... ( N  - 
1 ) ) )
93, 8ffvelrnd 6038 . . 3  |-  ( ph  ->  ( F `  ( N  -  1 ) )  e.  ( ( 0 ... K )  ^m  ( 1 ... N ) ) )
10 elmapfn 7512 . . 3  |-  ( ( F `  ( N  -  1 ) )  e.  ( ( 0 ... K )  ^m  ( 1 ... N
) )  ->  ( F `  ( N  -  1 ) )  Fn  ( 1 ... N ) )
119, 10syl 17 . 2  |-  ( ph  ->  ( F `  ( N  -  1 ) )  Fn  ( 1 ... N ) )
12 1ex 9656 . . 3  |-  1  e.  _V
13 fnconstg 5784 . . 3  |-  ( 1  e.  _V  ->  (
( 1 ... N
)  X.  { 1 } )  Fn  (
1 ... N ) )
1412, 13mp1i 13 . 2  |-  ( ph  ->  ( ( 1 ... N )  X.  {
1 } )  Fn  ( 1 ... N
) )
15 poimirlem12.2 . . . . . 6  |-  ( ph  ->  T  e.  S )
16 elrabi 3181 . . . . . . 7  |-  ( 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 ) ) )
17 poimirlem22.s . . . . . . 7  |-  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 } ) ) ) ) }
1816, 17eleq2s 2567 . . . . . 6  |-  ( T  e.  S  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
1915, 18syl 17 . . . . 5  |-  ( ph  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
20 xp1st 6842 . . . . 5  |-  ( 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 ) } ) )
2119, 20syl 17 . . . 4  |-  ( ph  ->  ( 1st `  T
)  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } ) )
22 xp1st 6842 . . . 4  |-  ( ( 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 ) ) )
2321, 22syl 17 . . 3  |-  ( ph  ->  ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
24 elmapfn 7512 . . 3  |-  ( ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  T ) )  Fn  ( 1 ... N ) )
2523, 24syl 17 . 2  |-  ( ph  ->  ( 1st `  ( 1st `  T ) )  Fn  ( 1 ... N ) )
26 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
2726breq2d 4407 . . . . . . . . . . . . . 14  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
2827ifbid 3894 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
2928csbeq1d 3356 . . . . . . . . . . . 12  |-  ( 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 } ) ) ) )
30 fveq2 5879 . . . . . . . . . . . . . . 15  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
3130fveq2d 5883 . . . . . . . . . . . . . 14  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
3230fveq2d 5883 . . . . . . . . . . . . . . . . 17  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
3332imaeq1d 5173 . . . . . . . . . . . . . . . 16  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
3433xpeq1d 4862 . . . . . . . . . . . . . . 15  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
3532imaeq1d 5173 . . . . . . . . . . . . . . . 16  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
3635xpeq1d 4862 . . . . . . . . . . . . . . 15  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
3734, 36uneq12d 3580 . . . . . . . . . . . . . 14  |-  ( 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 } ) ) )
3831, 37oveq12d 6326 . . . . . . . . . . . . 13  |-  ( 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 } ) ) ) )
3938csbeq2dv 3785 . . . . . . . . . . . 12  |-  ( 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 } ) ) ) )
4029, 39eqtrd 2505 . . . . . . . . . . 11  |-  ( 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 } ) ) ) )
4140mpteq2dv 4483 . . . . . . . . . 10  |-  ( 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 } ) ) ) ) )
4241eqeq2d 2481 . . . . . . . . 9  |-  ( 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 } ) ) ) ) ) )
4342, 17elrab2 3186 . . . . . . . 8  |-  ( 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 } ) ) ) ) ) )
4443simprbi 471 . . . . . . 7  |-  ( 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 } ) ) ) ) )
4515, 44syl 17 . . . . . 6  |-  ( 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 } ) ) ) ) )
46 poimirlem11.3 . . . . . . . . . . . 12  |-  ( ph  ->  ( 2nd `  T
)  =  0 )
47 breq12 4400 . . . . . . . . . . . 12  |-  ( ( y  =  ( N  -  1 )  /\  ( 2nd `  T )  =  0 )  -> 
( y  <  ( 2nd `  T )  <->  ( N  -  1 )  <  0 ) )
4846, 47sylan2 482 . . . . . . . . . . 11  |-  ( ( y  =  ( N  -  1 )  /\  ph )  ->  ( y  <  ( 2nd `  T
)  <->  ( N  - 
1 )  <  0
) )
4948ancoms 460 . . . . . . . . . 10  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  (
y  <  ( 2nd `  T )  <->  ( N  -  1 )  <  0 ) )
50 oveq1 6315 . . . . . . . . . . 11  |-  ( y  =  ( N  - 
1 )  ->  (
y  +  1 )  =  ( ( N  -  1 )  +  1 ) )
514nncnd 10647 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
52 npcan1 10065 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
5351, 52syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
5450, 53sylan9eqr 2527 . . . . . . . . . 10  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  (
y  +  1 )  =  N )
5549, 54ifbieq2d 3897 . . . . . . . . 9  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  =  if ( ( N  -  1 )  <  0 ,  y ,  N ) )
566nn0ge0d 10952 . . . . . . . . . . . 12  |-  ( ph  ->  0  <_  ( N  -  1 ) )
57 0red 9662 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  RR )
586nn0red 10950 . . . . . . . . . . . . 13  |-  ( ph  ->  ( N  -  1 )  e.  RR )
5957, 58lenltd 9798 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0  <_  ( N  -  1 )  <->  -.  ( N  -  1 )  <  0 ) )
6056, 59mpbid 215 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( N  - 
1 )  <  0
)
6160iffalsed 3883 . . . . . . . . . 10  |-  ( ph  ->  if ( ( N  -  1 )  <  0 ,  y ,  N )  =  N )
6261adantr 472 . . . . . . . . 9  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  if ( ( N  - 
1 )  <  0 ,  y ,  N
)  =  N )
6355, 62eqtrd 2505 . . . . . . . 8  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  =  N )
6463csbeq1d 3356 . . . . . . 7  |-  ( (
ph  /\  y  =  ( 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 } ) ) )  =  [_ N  /  j ]_ (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
65 oveq2 6316 . . . . . . . . . . . . . . 15  |-  ( j  =  N  ->  (
1 ... j )  =  ( 1 ... N
) )
6665imaeq2d 5174 . . . . . . . . . . . . . 14  |-  ( j  =  N  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) ) )
67 xp2nd 6843 . . . . . . . . . . . . . . . . 17  |-  ( ( 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 ) } )
6821, 67syl 17 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
69 fvex 5889 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
70 f1oeq1 5818 . . . . . . . . . . . . . . . . 17  |-  ( f  =  ( 2nd `  ( 1st `  T ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
7169, 70elab 3173 . . . . . . . . . . . . . . . 16  |-  ( ( 2nd `  ( 1st `  T ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
7268, 71sylib 201 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
73 f1ofo 5835 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N )
-onto-> ( 1 ... N
) )
74 foima 5811 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
) -onto-> ( 1 ... N )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
7572, 73, 743syl 18 . . . . . . . . . . . . . 14  |-  ( ph  ->  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
7666, 75sylan9eqr 2527 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  =  N )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  =  ( 1 ... N
) )
7776xpeq1d 4862 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  =  N )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( 1 ... N )  X. 
{ 1 } ) )
78 oveq1 6315 . . . . . . . . . . . . . . . . 17  |-  ( j  =  N  ->  (
j  +  1 )  =  ( N  + 
1 ) )
7978oveq1d 6323 . . . . . . . . . . . . . . . 16  |-  ( j  =  N  ->  (
( j  +  1 ) ... N )  =  ( ( N  +  1 ) ... N ) )
804nnred 10646 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  RR )
8180ltp1d 10559 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  <  ( N  +  1 ) )
824nnzd 11062 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  ZZ )
8382peano2zd 11066 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
84 fzn 11841 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  +  1 )  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  ( N  +  1 )  <-> 
( ( N  + 
1 ) ... N
)  =  (/) ) )
8583, 82, 84syl2anc 673 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( N  <  ( N  +  1 )  <-> 
( ( N  + 
1 ) ... N
)  =  (/) ) )
8681, 85mpbid 215 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( N  + 
1 ) ... N
)  =  (/) )
8779, 86sylan9eqr 2527 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  j  =  N )  ->  (
( j  +  1 ) ... N )  =  (/) )
8887imaeq2d 5174 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  j  =  N )  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" (/) ) )
8988xpeq1d 4862 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  =  N )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) " (/) )  X.  { 0 } ) )
90 ima0 5189 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( 1st `  T ) ) " (/) )  =  (/)
9190xpeq1i 4859 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
0 } )  =  ( (/)  X.  { 0 } )
92 0xp 4920 . . . . . . . . . . . . . 14  |-  ( (/)  X. 
{ 0 } )  =  (/)
9391, 92eqtri 2493 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
0 } )  =  (/)
9489, 93syl6eq 2521 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  =  N )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  (/) )
9577, 94uneq12d 3580 . . . . . . . . . . 11  |-  ( (
ph  /\  j  =  N )  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) )  =  ( ( ( 1 ... N
)  X.  { 1 } )  u.  (/) ) )
96 un0 3762 . . . . . . . . . . 11  |-  ( ( ( 1 ... N
)  X.  { 1 } )  u.  (/) )  =  ( ( 1 ... N )  X.  {
1 } )
9795, 96syl6eq 2521 . . . . . . . . . 10  |-  ( (
ph  /\  j  =  N )  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) )  =  ( ( 1 ... N )  X.  { 1 } ) )
9897oveq2d 6324 . . . . . . . . 9  |-  ( (
ph  /\  j  =  N )  ->  (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 1 } ) ) )
994, 98csbied 3376 . . . . . . . 8  |-  ( ph  ->  [_ N  /  j ]_ ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 1 } ) ) )
10099adantr 472 . . . . . . 7  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  [_ N  /  j ]_ (
( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 1 } ) ) )
10164, 100eqtrd 2505 . . . . . 6  |-  ( (
ph  /\  y  =  ( 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 } ) ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 1 } ) ) )
102 ovex 6336 . . . . . . 7  |-  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N )  X.  { 1 } ) )  e.  _V
103102a1i 11 . . . . . 6  |-  ( ph  ->  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 1 } ) )  e. 
_V )
10445, 101, 8, 103fvmptd 5969 . . . . 5  |-  ( ph  ->  ( F `  ( N  -  1 ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N )  X.  { 1 } ) ) )
105104fveq1d 5881 . . . 4  |-  ( ph  ->  ( ( F `  ( N  -  1
) ) `  n
)  =  ( ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 1 } ) ) `  n ) )
106105adantr 472 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  ( N  -  1 ) ) `  n )  =  ( ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N )  X.  { 1 } ) ) `  n
) )
107 inidm 3632 . . . 4  |-  ( ( 1 ... N )  i^i  ( 1 ... N ) )  =  ( 1 ... N
)
108 eqidd 2472 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  =  ( ( 1st `  ( 1st `  T ) ) `
 n ) )
10912fvconst2 6136 . . . . 5  |-  ( n  e.  ( 1 ... N )  ->  (
( ( 1 ... N )  X.  {
1 } ) `  n )  =  1 )
110109adantl 473 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1 ... N )  X.  {
1 } ) `  n )  =  1 )
11125, 14, 2, 2, 107, 108, 110ofval 6559 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 1 } ) ) `  n )  =  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  +  1 ) )
112106, 111eqtrd 2505 . 2  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  ( N  -  1 ) ) `  n )  =  ( ( ( 1st `  ( 1st `  T ) ) `  n )  +  1 ) )
113 elmapi 7511 . . . . . . 7  |-  ( ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
11423, 113syl 17 . . . . . 6  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
115114ffvelrnda 6037 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  ( 0..^ K ) )
116 elfzonn0 11988 . . . . 5  |-  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  e.  ( 0..^ K )  ->  ( ( 1st `  ( 1st `  T
) ) `  n
)  e.  NN0 )
117115, 116syl 17 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e. 
NN0 )
118117nn0cnd 10951 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  CC )
119 pncan1 10064 . . 3  |-  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  e.  CC  ->  ( (
( ( 1st `  ( 1st `  T ) ) `
 n )  +  1 )  -  1 )  =  ( ( 1st `  ( 1st `  T ) ) `  n ) )
120118, 119syl 17 . 2  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( ( ( 1st `  ( 1st `  T
) ) `  n
)  +  1 )  -  1 )  =  ( ( 1st `  ( 1st `  T ) ) `
 n ) )
1212, 11, 14, 25, 112, 110, 120offveq 6571 1  |-  ( ph  ->  ( ( F `  ( N  -  1
) )  oF  -  ( ( 1 ... N )  X. 
{ 1 } ) )  =  ( 1st `  ( 1st `  T
) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    = wceq 1452    e. wcel 1904   {cab 2457   {crab 2760   _Vcvv 3031   [_csb 3349    u. cun 3388   (/)c0 3722   ifcif 3872   {csn 3959   class class class wbr 4395    |-> cmpt 4454    X. cxp 4837   "cima 4842    Fn wfn 5584   -->wf 5585   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308    oFcof 6548   1stc1st 6810   2ndc2nd 6811    ^m cmap 7490   CCcc 9555   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    <_ cle 9694    - cmin 9880   NNcn 10631   NN0cn0 10893   ZZcz 10961   ...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-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943
This theorem is referenced by:  poimirlem11  32015  poimirlem13  32017
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