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Theorem poimirlem10 31950
Description: Lemma for poimir 31973 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 6318 . . 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 10911 . . . . . 6  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
64, 5syl 17 . . . . 5  |-  ( ph  ->  ( N  -  1 )  e.  NN0 )
7 nn0fz0 11890 . . . . 5  |-  ( ( N  -  1 )  e.  NN0  <->  ( N  - 
1 )  e.  ( 0 ... ( N  -  1 ) ) )
86, 7sylib 200 . . . 4  |-  ( ph  ->  ( N  -  1 )  e.  ( 0 ... ( N  - 
1 ) ) )
93, 8ffvelrnd 6023 . . 3  |-  ( ph  ->  ( F `  ( N  -  1 ) )  e.  ( ( 0 ... K )  ^m  ( 1 ... N ) ) )
10 elmapfn 7494 . . 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 9638 . . 3  |-  1  e.  _V
13 fnconstg 5771 . . 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 3193 . . . . . . 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 2547 . . . . . 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 6823 . . . . 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 6823 . . . 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 7494 . . 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 5865 . . . . . . . . . . . . . . 15  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
2726breq2d 4414 . . . . . . . . . . . . . 14  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
2827ifbid 3903 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
2928csbeq1d 3370 . . . . . . . . . . . 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 5865 . . . . . . . . . . . . . . 15  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
3130fveq2d 5869 . . . . . . . . . . . . . 14  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
3230fveq2d 5869 . . . . . . . . . . . . . . . . 17  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
3332imaeq1d 5167 . . . . . . . . . . . . . . . 16  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
3433xpeq1d 4857 . . . . . . . . . . . . . . 15  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
3532imaeq1d 5167 . . . . . . . . . . . . . . . 16  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
3635xpeq1d 4857 . . . . . . . . . . . . . . 15  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
3734, 36uneq12d 3589 . . . . . . . . . . . . . 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 6308 . . . . . . . . . . . . 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 3781 . . . . . . . . . . . 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 2485 . . . . . . . . . . 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 4490 . . . . . . . . . 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 2461 . . . . . . . . 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 3198 . . . . . . . 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 4407 . . . . . . . . . . . 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 6297 . . . . . . . . . . 11  |-  ( y  =  ( N  - 
1 )  ->  (
y  +  1 )  =  ( ( N  -  1 )  +  1 ) )
514nncnd 10625 . . . . . . . . . . . 12  |-  ( ph  ->  N  e.  CC )
52 npcan1 10044 . . . . . . . . . . . 12  |-  ( N  e.  CC  ->  (
( N  -  1 )  +  1 )  =  N )
5351, 52syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( ( N  - 
1 )  +  1 )  =  N )
5450, 53sylan9eqr 2507 . . . . . . . . . 10  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  (
y  +  1 )  =  N )
5549, 54ifbieq2d 3906 . . . . . . . . 9  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  =  if ( ( N  -  1 )  <  0 ,  y ,  N ) )
566nn0ge0d 10928 . . . . . . . . . . . 12  |-  ( ph  ->  0  <_  ( N  -  1 ) )
57 0red 9644 . . . . . . . . . . . . 13  |-  ( ph  ->  0  e.  RR )
586nn0red 10926 . . . . . . . . . . . . 13  |-  ( ph  ->  ( N  -  1 )  e.  RR )
5957, 58lenltd 9781 . . . . . . . . . . . 12  |-  ( ph  ->  ( 0  <_  ( N  -  1 )  <->  -.  ( N  -  1 )  <  0 ) )
6056, 59mpbid 214 . . . . . . . . . . 11  |-  ( ph  ->  -.  ( N  - 
1 )  <  0
)
6160iffalsed 3892 . . . . . . . . . 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 2485 . . . . . . . 8  |-  ( (
ph  /\  y  =  ( N  -  1
) )  ->  if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  =  N )
6463csbeq1d 3370 . . . . . . 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 6298 . . . . . . . . . . . . . . 15  |-  ( j  =  N  ->  (
1 ... j )  =  ( 1 ... N
) )
6665imaeq2d 5168 . . . . . . . . . . . . . 14  |-  ( j  =  N  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) ) )
67 xp2nd 6824 . . . . . . . . . . . . . . . . 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 5875 . . . . . . . . . . . . . . . . 17  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
70 f1oeq1 5805 . . . . . . . . . . . . . . . . 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 3185 . . . . . . . . . . . . . . . 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 5821 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N )
-onto-> ( 1 ... N
) )
74 foima 5798 . . . . . . . . . . . . . . 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 2507 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  =  N )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  =  ( 1 ... N
) )
7776xpeq1d 4857 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  =  N )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( 1 ... N )  X. 
{ 1 } ) )
78 oveq1 6297 . . . . . . . . . . . . . . . . 17  |-  ( j  =  N  ->  (
j  +  1 )  =  ( N  + 
1 ) )
7978oveq1d 6305 . . . . . . . . . . . . . . . 16  |-  ( j  =  N  ->  (
( j  +  1 ) ... N )  =  ( ( N  +  1 ) ... N ) )
804nnred 10624 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  N  e.  RR )
8180ltp1d 10537 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  N  <  ( N  +  1 ) )
824nnzd 11039 . . . . . . . . . . . . . . . . . . 19  |-  ( ph  ->  N  e.  ZZ )
8382peano2zd 11043 . . . . . . . . . . . . . . . . . 18  |-  ( ph  ->  ( N  +  1 )  e.  ZZ )
84 fzn 11815 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( N  +  1 )  e.  ZZ  /\  N  e.  ZZ )  ->  ( N  <  ( N  +  1 )  <-> 
( ( N  + 
1 ) ... N
)  =  (/) ) )
8583, 82, 84syl2anc 667 . . . . . . . . . . . . . . . . 17  |-  ( ph  ->  ( N  <  ( N  +  1 )  <-> 
( ( N  + 
1 ) ... N
)  =  (/) ) )
8681, 85mpbid 214 . . . . . . . . . . . . . . . 16  |-  ( ph  ->  ( ( N  + 
1 ) ... N
)  =  (/) )
8779, 86sylan9eqr 2507 . . . . . . . . . . . . . . 15  |-  ( (
ph  /\  j  =  N )  ->  (
( j  +  1 ) ... N )  =  (/) )
8887imaeq2d 5168 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  j  =  N )  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" (/) ) )
8988xpeq1d 4857 . . . . . . . . . . . . 13  |-  ( (
ph  /\  j  =  N )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) " (/) )  X.  { 0 } ) )
90 ima0 5183 . . . . . . . . . . . . . . 15  |-  ( ( 2nd `  ( 1st `  T ) ) " (/) )  =  (/)
9190xpeq1i 4854 . . . . . . . . . . . . . 14  |-  ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
0 } )  =  ( (/)  X.  { 0 } )
92 0xp 4915 . . . . . . . . . . . . . 14  |-  ( (/)  X. 
{ 0 } )  =  (/)
9391, 92eqtri 2473 . . . . . . . . . . . . 13  |-  ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
0 } )  =  (/)
9489, 93syl6eq 2501 . . . . . . . . . . . 12  |-  ( (
ph  /\  j  =  N )  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  (/) )
9577, 94uneq12d 3589 . . . . . . . . . . 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 3759 . . . . . . . . . . 11  |-  ( ( ( 1 ... N
)  X.  { 1 } )  u.  (/) )  =  ( ( 1 ... N )  X.  {
1 } )
9795, 96syl6eq 2501 . . . . . . . . . 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 6306 . . . . . . . . 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 3390 . . . . . . . 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 2485 . . . . . 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 6318 . . . . . . 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 5954 . . . . 5  |-  ( ph  ->  ( F `  ( N  -  1 ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N )  X.  { 1 } ) ) )
105104fveq1d 5867 . . . 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 3641 . . . 4  |-  ( ( 1 ... N )  i^i  ( 1 ... N ) )  =  ( 1 ... N
)
108 eqidd 2452 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  =  ( ( 1st `  ( 1st `  T ) ) `
 n ) )
10912fvconst2 6120 . . . . 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 6540 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 1 } ) ) `  n )  =  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  +  1 ) )
112106, 111eqtrd 2485 . 2  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( F `  ( N  -  1 ) ) `  n )  =  ( ( ( 1st `  ( 1st `  T ) ) `  n )  +  1 ) )
113 elmapi 7493 . . . . . . 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 6022 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  ( 0..^ K ) )
116 elfzonn0 11960 . . . . 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 10927 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  CC )
119 pncan1 10043 . . 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 6552 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 1444    e. wcel 1887   {cab 2437   {crab 2741   _Vcvv 3045   [_csb 3363    u. cun 3402   (/)c0 3731   ifcif 3881   {csn 3968   class class class wbr 4402    |-> cmpt 4461    X. cxp 4832   "cima 4837    Fn wfn 5577   -->wf 5578   -onto->wfo 5580   -1-1-onto->wf1o 5581   ` cfv 5582  (class class class)co 6290    oFcof 6529   1stc1st 6791   2ndc2nd 6792    ^m cmap 7472   CCcc 9537   0cc0 9539   1c1 9540    + caddc 9542    < clt 9675    <_ cle 9676    - cmin 9860   NNcn 10609   NN0cn0 10869   ZZcz 10937   ...cfz 11784  ..^cfzo 11915
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583  ax-cnex 9595  ax-resscn 9596  ax-1cn 9597  ax-icn 9598  ax-addcl 9599  ax-addrcl 9600  ax-mulcl 9601  ax-mulrcl 9602  ax-mulcom 9603  ax-addass 9604  ax-mulass 9605  ax-distr 9606  ax-i2m1 9607  ax-1ne0 9608  ax-1rid 9609  ax-rnegex 9610  ax-rrecex 9611  ax-cnre 9612  ax-pre-lttri 9613  ax-pre-lttrn 9614  ax-pre-ltadd 9615  ax-pre-mulgt0 9616
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 986  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-pss 3420  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-tp 3973  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-tr 4498  df-eprel 4745  df-id 4749  df-po 4755  df-so 4756  df-fr 4793  df-we 4795  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-pred 5380  df-ord 5426  df-on 5427  df-lim 5428  df-suc 5429  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-riota 6252  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-of 6531  df-om 6693  df-1st 6793  df-2nd 6794  df-wrecs 7028  df-recs 7090  df-rdg 7128  df-er 7363  df-map 7474  df-en 7570  df-dom 7571  df-sdom 7572  df-pnf 9677  df-mnf 9678  df-xr 9679  df-ltxr 9680  df-le 9681  df-sub 9862  df-neg 9863  df-nn 10610  df-n0 10870  df-z 10938  df-uz 11160  df-fz 11785  df-fzo 11916
This theorem is referenced by:  poimirlem11  31951  poimirlem13  31953
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