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Theorem poimirlem10 31908
Description: Lemma for poimir 31931 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 6331 . . 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 10913 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
64, 5syl 17 . . . . 5  |-  ( ph  ->  ( N  -  1 )  e.  NN0 )
7 nn0fz0 11892 . . . . 5  |-  ( ( N  -  1 )  e.  NN0  <->  ( N  - 
1 )  e.  ( 0 ... ( N  -  1 ) ) )
86, 7sylib 200 . . . 4  |-  ( ph  ->  ( N  -  1 )  e.  ( 0 ... ( N  - 
1 ) ) )
93, 8ffvelrnd 6036 . . 3  |-  ( ph  ->  ( F `  ( N  -  1 ) )  e.  ( ( 0 ... K )  ^m  ( 1 ... N ) ) )
10 elmapfn 7500 . . 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 9640 . . 3  |-  1  e.  _V
13 fnconstg 5786 . . 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 3227 . . . . . . 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 2531 . . . . . 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 6835 . . . . 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 6835 . . . 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 7500 . . 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 4433 . . . . . . . . . . . . . 14  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
2827ifbid 3932 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
2928csbeq1d 3403 . . . . . . . . . . . 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 5184 . . . . . . . . . . . . . . . 16  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
3433xpeq1d 4874 . . . . . . . . . . . . . . 15  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
3532imaeq1d 5184 . . . . . . . . . . . . . . . 16  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
3635xpeq1d 4874 . . . . . . . . . . . . . . 15  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
3734, 36uneq12d 3622 . . . . . . . . . . . . . 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 6321 . . . . . . . . . . . . 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 3810 . . . . . . . . . . . 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 2464 . . . . . . . . . . 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 4509 . . . . . . . . . 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 2437 . . . . . . . . 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 3232 . . . . . . . 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 466 . . . . . . 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 4426 . . . . . . . . . . . 12  |-  ( ( y  =  ( N  -  1 )  /\  ( 2nd `  T )  =  0 )  -> 
( y  <  ( 2nd `  T )  <->  ( N  -  1 )  <  0 ) )
4846, 47sylan2 477 . . . . . . . . . . 11  |-  ( ( y  =  ( N  -  1 )  /\  ph )  ->  ( y  <  ( 2nd `  T
)  <->  ( N  - 
1 )  <  0
) )
4948ancoms 455 . . . . . . . . . 10  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  (
y  <  ( 2nd `  T )  <->  ( N  -  1 )  <  0 ) )
50 oveq1 6310 . . . . . . . . . . 11  |-  ( y  =  ( N  - 
1 )  ->  (
y  +  1 )  =  ( ( N  -  1 )  +  1 ) )
514nncnd 10627 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
52 npcan1 10046 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
5351, 52syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
5450, 53sylan9eqr 2486 . . . . . . . . . 10  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  (
y  +  1 )  =  N )
5549, 54ifbieq2d 3935 . . . . . . . . 9  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  =  if ( ( N  -  1 )  <  0 ,  y ,  N ) )
566nn0ge0d 10930 . . . . . . . . . . . 12  |-  ( ph  ->  0  <_  ( N  -  1 ) )
57 0red 9646 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  RR )
586nn0red 10928 . . . . . . . . . . . . 13  |-  ( ph  ->  ( N  -  1 )  e.  RR )
5957, 58lenltd 9783 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0  <_  ( N  -  1 )  <->  -.  ( N  -  1 )  <  0 ) )
6056, 59mpbid 214 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( N  - 
1 )  <  0
)
6160iffalsed 3921 . . . . . . . . . 10  |-  ( ph  ->  if ( ( N  -  1 )  <  0 ,  y ,  N )  =  N )
6261adantr 467 . . . . . . . . 9  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  if ( ( N  - 
1 )  <  0 ,  y ,  N
)  =  N )
6355, 62eqtrd 2464 . . . . . . . 8  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  =  N )
6463csbeq1d 3403 . . . . . . 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 6311 . . . . . . . . . . . . . . 15  |-  ( j  =  N  ->  (
1 ... j )  =  ( 1 ... N
) )
6665imaeq2d 5185 . . . . . . . . . . . . . 14  |-  ( j  =  N  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) ) )
67 xp2nd 6836 . . . . . . . . . . . . . . . . 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 5820 . . . . . . . . . . . . . . . . 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 3219 . . . . . . . . . . . . . . . 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 200 . . . . . . . . . . . . . . 15  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
73 f1ofo 5836 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N )
-onto-> ( 1 ... N
) )
74 foima 5813 . . . . . . . . . . . . . . 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 2486 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  =  N )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  =  ( 1 ... N
) )
7776xpeq1d 4874 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  =  N )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( 1 ... N )  X. 
{ 1 } ) )
78 oveq1 6310 . . . . . . . . . . . . . . . . 17  |-  ( j  =  N  ->  (
j  +  1 )  =  ( N  + 
1 ) )
7978oveq1d 6318 . . . . . . . . . . . . . . . 16  |-  ( j  =  N  ->  (
( j  +  1 ) ... N )  =  ( ( N  +  1 ) ... N ) )
804nnred 10626 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  RR )
8180ltp1d 10539 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  <  ( N  +  1 ) )
824nnzd 11041 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  ZZ )
8382peano2zd 11045 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
84 fzn 11817 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  +  1 )  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  ( N  +  1 )  <-> 
( ( N  + 
1 ) ... N
)  =  (/) ) )
8583, 82, 84syl2anc 666 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( N  <  ( N  +  1 )  <-> 
( ( N  + 
1 ) ... N
)  =  (/) ) )
8681, 85mpbid 214 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( N  + 
1 ) ... N
)  =  (/) )
8779, 86sylan9eqr 2486 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  j  =  N )  ->  (
( j  +  1 ) ... N )  =  (/) )
8887imaeq2d 5185 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  j  =  N )  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" (/) ) )
8988xpeq1d 4874 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  =  N )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) " (/) )  X.  { 0 } ) )
90 ima0 5200 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( 1st `  T ) ) " (/) )  =  (/)
9190xpeq1i 4871 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
0 } )  =  ( (/)  X.  { 0 } )
92 0xp 4932 . . . . . . . . . . . . . 14  |-  ( (/)  X. 
{ 0 } )  =  (/)
9391, 92eqtri 2452 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
0 } )  =  (/)
9489, 93syl6eq 2480 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  =  N )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  (/) )
9577, 94uneq12d 3622 . . . . . . . . . . 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 3788 . . . . . . . . . . 11  |-  ( ( ( 1 ... N
)  X.  { 1 } )  u.  (/) )  =  ( ( 1 ... N )  X.  {
1 } )
9795, 96syl6eq 2480 . . . . . . . . . 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 6319 . . . . . . . . 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 3423 . . . . . . . 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 467 . . . . . . 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 2464 . . . . . 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 6331 . . . . . . 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 5968 . . . . 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 467 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  ( N  -  1 ) ) `  n )  =  ( ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N )  X.  { 1 } ) ) `  n
) )
107 inidm 3672 . . . 4  |-  ( ( 1 ... N )  i^i  ( 1 ... N ) )  =  ( 1 ... N
)
108 eqidd 2424 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  =  ( ( 1st `  ( 1st `  T ) ) `
 n ) )
10912fvconst2 6133 . . . . 5  |-  ( n  e.  ( 1 ... N )  ->  (
( ( 1 ... N )  X.  {
1 } ) `  n )  =  1 )
110109adantl 468 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1 ... N )  X.  {
1 } ) `  n )  =  1 )
11125, 14, 2, 2, 107, 108, 110ofval 6552 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 1 } ) ) `  n )  =  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  +  1 ) )
112106, 111eqtrd 2464 . 2  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  ( N  -  1 ) ) `  n )  =  ( ( ( 1st `  ( 1st `  T ) ) `  n )  +  1 ) )
113 elmapi 7499 . . . . . . 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 6035 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  ( 0..^ K ) )
116 elfzonn0 11962 . . . . 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 10929 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  CC )
119 pncan1 10045 . . 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 6564 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 188    /\ wa 371    = wceq 1438    e. wcel 1869   {cab 2408   {crab 2780   _Vcvv 3082   [_csb 3396    u. cun 3435   (/)c0 3762   ifcif 3910   {csn 3997   class class class wbr 4421    |-> cmpt 4480    X. cxp 4849   "cima 4854    Fn wfn 5594   -->wf 5595   -onto->wfo 5597   -1-1-onto->wf1o 5598   ` cfv 5599  (class class class)co 6303    oFcof 6541   1stc1st 6803   2ndc2nd 6804    ^m cmap 7478   CCcc 9539   0cc0 9541   1c1 9542    + caddc 9544    < clt 9677    <_ cle 9678    - cmin 9862   NNcn 10611   NN0cn0 10871   ZZcz 10939   ...cfz 11786  ..^cfzo 11917
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-fzo 11918
This theorem is referenced by:  poimirlem11  31909  poimirlem13  31911
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