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Theorem poimirlem5 32009
Description: Lemma for poimir 32037 to establish that, for the simplices defined by a walk along the edges of an  N-cube, if the starting vertex is not opposite a given face, it is the earliest vertex of the face 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 } ) ) ) ) }
poimirlem9.1  |-  ( ph  ->  T  e.  S )
poimirlem5.2  |-  ( ph  ->  0  <  ( 2nd `  T ) )
Assertion
Ref Expression
poimirlem5  |-  ( ph  ->  ( F `  0
)  =  ( 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 poimirlem5
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 poimirlem9.1 . . . 4  |-  ( ph  ->  T  e.  S )
2 fveq2 5879 . . . . . . . . . . . 12  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
32breq2d 4407 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
43ifbid 3894 . . . . . . . . . 10  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
54csbeq1d 3356 . . . . . . . . 9  |-  ( 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 } ) ) ) )
6 fveq2 5879 . . . . . . . . . . . 12  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
76fveq2d 5883 . . . . . . . . . . 11  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
86fveq2d 5883 . . . . . . . . . . . . . 14  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
98imaeq1d 5173 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
109xpeq1d 4862 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
118imaeq1d 5173 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
1211xpeq1d 4862 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
1310, 12uneq12d 3580 . . . . . . . . . . 11  |-  ( 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 } ) ) )
147, 13oveq12d 6326 . . . . . . . . . 10  |-  ( 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 } ) ) ) )
1514csbeq2dv 3785 . . . . . . . . 9  |-  ( 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 } ) ) ) )
165, 15eqtrd 2505 . . . . . . . 8  |-  ( 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 } ) ) ) )
1716mpteq2dv 4483 . . . . . . 7  |-  ( 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 } ) ) ) ) )
1817eqeq2d 2481 . . . . . 6  |-  ( 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 } ) ) ) ) ) )
19 poimirlem22.s . . . . . 6  |-  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 } ) ) ) ) }
2018, 19elrab2 3186 . . . . 5  |-  ( 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 } ) ) ) ) ) )
2120simprbi 471 . . . 4  |-  ( 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 } ) ) ) ) )
221, 21syl 17 . . 3  |-  ( 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 } ) ) ) ) )
23 breq1 4398 . . . . . . 7  |-  ( y  =  0  ->  (
y  <  ( 2nd `  T )  <->  0  <  ( 2nd `  T ) ) )
24 id 22 . . . . . . 7  |-  ( y  =  0  ->  y  =  0 )
2523, 24ifbieq1d 3895 . . . . . 6  |-  ( y  =  0  ->  if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  =  if ( 0  <  ( 2nd `  T
) ,  0 ,  ( y  +  1 ) ) )
26 poimirlem5.2 . . . . . . 7  |-  ( ph  ->  0  <  ( 2nd `  T ) )
2726iftrued 3880 . . . . . 6  |-  ( ph  ->  if ( 0  < 
( 2nd `  T
) ,  0 ,  ( y  +  1 ) )  =  0 )
2825, 27sylan9eqr 2527 . . . . 5  |-  ( (
ph  /\  y  = 
0 )  ->  if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  =  0 )
2928csbeq1d 3356 . . . 4  |-  ( (
ph  /\  y  = 
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 } ) ) )  =  [_
0  /  j ]_ ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) ) ) )
30 c0ex 9655 . . . . . . 7  |-  0  e.  _V
31 oveq2 6316 . . . . . . . . . . . . 13  |-  ( j  =  0  ->  (
1 ... j )  =  ( 1 ... 0
) )
32 fz10 11846 . . . . . . . . . . . . 13  |-  ( 1 ... 0 )  =  (/)
3331, 32syl6eq 2521 . . . . . . . . . . . 12  |-  ( j  =  0  ->  (
1 ... j )  =  (/) )
3433imaeq2d 5174 . . . . . . . . . . 11  |-  ( j  =  0  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" (/) ) )
3534xpeq1d 4862 . . . . . . . . . 10  |-  ( j  =  0  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) " (/) )  X.  { 1 } ) )
36 oveq1 6315 . . . . . . . . . . . . . 14  |-  ( j  =  0  ->  (
j  +  1 )  =  ( 0  +  1 ) )
37 0p1e1 10743 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
3836, 37syl6eq 2521 . . . . . . . . . . . . 13  |-  ( j  =  0  ->  (
j  +  1 )  =  1 )
3938oveq1d 6323 . . . . . . . . . . . 12  |-  ( j  =  0  ->  (
( j  +  1 ) ... N )  =  ( 1 ... N ) )
4039imaeq2d 5174 . . . . . . . . . . 11  |-  ( j  =  0  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) ) )
4140xpeq1d 4862 . . . . . . . . . 10  |-  ( j  =  0  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... N
) )  X.  {
0 } ) )
4235, 41uneq12d 3580 . . . . . . . . 9  |-  ( j  =  0  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) )  =  ( ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } ) ) )
43 ima0 5189 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( 1st `  T ) ) " (/) )  =  (/)
4443xpeq1i 4859 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
1 } )  =  ( (/)  X.  { 1 } )
45 0xp 4920 . . . . . . . . . . . 12  |-  ( (/)  X. 
{ 1 } )  =  (/)
4644, 45eqtri 2493 . . . . . . . . . . 11  |-  ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
1 } )  =  (/)
4746uneq1i 3575 . . . . . . . . . 10  |-  ( ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } ) )  =  (
(/)  u.  ( (
( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } ) )
48 uncom 3569 . . . . . . . . . 10  |-  ( (/)  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } ) )  =  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } )  u.  (/) )
49 un0 3762 . . . . . . . . . 10  |-  ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } )  u.  (/) )  =  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } )
5047, 48, 493eqtri 2497 . . . . . . . . 9  |-  ( ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } ) )  =  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } )
5142, 50syl6eq 2521 . . . . . . . 8  |-  ( j  =  0  ->  (
( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... j ) )  X.  { 1 } )  u.  ( ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } ) )  =  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } ) )
5251oveq2d 6324 . . . . . . 7  |-  ( j  =  0  ->  (
( 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 ... N ) )  X. 
{ 0 } ) ) )
5330, 52csbie 3375 . . . . . 6  |-  [_ 0  /  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  +  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } ) )
54 elrabi 3181 . . . . . . . . . . . . . . 15  |-  ( 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 ) ) )
5554, 19eleq2s 2567 . . . . . . . . . . . . . 14  |-  ( T  e.  S  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N
) )  X.  {
f  |  f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
) } )  X.  ( 0 ... N
) ) )
561, 55syl 17 . . . . . . . . . . . . 13  |-  ( ph  ->  T  e.  ( ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )  X.  ( 0 ... N ) ) )
57 xp1st 6842 . . . . . . . . . . . . 13  |-  ( 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 ) } ) )
5856, 57syl 17 . . . . . . . . . . . 12  |-  ( ph  ->  ( 1st `  T
)  e.  ( ( ( 0..^ K )  ^m  ( 1 ... N ) )  X. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) } ) )
59 xp2nd 6843 . . . . . . . . . . . 12  |-  ( ( 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 ) } )
6058, 59syl 17 . . . . . . . . . . 11  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) )  e.  { f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
61 fvex 5889 . . . . . . . . . . . 12  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
62 f1oeq1 5818 . . . . . . . . . . . 12  |-  ( f  =  ( 2nd `  ( 1st `  T ) )  ->  ( f : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  <->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) ) )
6361, 62elab 3173 . . . . . . . . . . 11  |-  ( ( 2nd `  ( 1st `  T ) )  e. 
{ f  |  f : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) }  <->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
6460, 63sylib 201 . . . . . . . . . 10  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N
) )
65 f1ofo 5835 . . . . . . . . . 10  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
)
-1-1-onto-> ( 1 ... N
)  ->  ( 2nd `  ( 1st `  T
) ) : ( 1 ... N )
-onto-> ( 1 ... N
) )
6664, 65syl 17 . . . . . . . . 9  |-  ( ph  ->  ( 2nd `  ( 1st `  T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) )
67 foima 5811 . . . . . . . . 9  |-  ( ( 2nd `  ( 1st `  T ) ) : ( 1 ... N
) -onto-> ( 1 ... N )  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
6866, 67syl 17 . . . . . . . 8  |-  ( ph  ->  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  =  ( 1 ... N
) )
6968xpeq1d 4862 . . . . . . 7  |-  ( ph  ->  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } )  =  ( ( 1 ... N )  X.  { 0 } ) )
7069oveq2d 6324 . . . . . 6  |-  ( ph  ->  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N )  X.  { 0 } ) ) )
7153, 70syl5eq 2517 . . . . 5  |-  ( ph  ->  [_ 0  /  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.  { 0 } ) ) )
7271adantr 472 . . . 4  |-  ( (
ph  /\  y  = 
0 )  ->  [_ 0  /  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.  { 0 } ) ) )
7329, 72eqtrd 2505 . . 3  |-  ( (
ph  /\  y  = 
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 } ) ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 0 } ) ) )
74 poimir.0 . . . . 5  |-  ( ph  ->  N  e.  NN )
75 nnm1nn0 10935 . . . . 5  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
7674, 75syl 17 . . . 4  |-  ( ph  ->  ( N  -  1 )  e.  NN0 )
77 0elfz 11915 . . . 4  |-  ( ( N  -  1 )  e.  NN0  ->  0  e.  ( 0 ... ( N  -  1 ) ) )
7876, 77syl 17 . . 3  |-  ( ph  ->  0  e.  ( 0 ... ( N  - 
1 ) ) )
79 ovex 6336 . . . 4  |-  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N )  X.  { 0 } ) )  e.  _V
8079a1i 11 . . 3  |-  ( ph  ->  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 0 } ) )  e. 
_V )
8122, 73, 78, 80fvmptd 5969 . 2  |-  ( ph  ->  ( F `  0
)  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N )  X.  { 0 } ) ) )
82 ovex 6336 . . . 4  |-  ( 1 ... N )  e. 
_V
8382a1i 11 . . 3  |-  ( ph  ->  ( 1 ... N
)  e.  _V )
84 xp1st 6842 . . . . 5  |-  ( ( 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 ) ) )
8558, 84syl 17 . . . 4  |-  ( ph  ->  ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  (
1 ... N ) ) )
86 elmapfn 7512 . . . 4  |-  ( ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  T ) )  Fn  ( 1 ... N ) )
8785, 86syl 17 . . 3  |-  ( ph  ->  ( 1st `  ( 1st `  T ) )  Fn  ( 1 ... N ) )
88 fnconstg 5784 . . . 4  |-  ( 0  e.  _V  ->  (
( 1 ... N
)  X.  { 0 } )  Fn  (
1 ... N ) )
8930, 88mp1i 13 . . 3  |-  ( ph  ->  ( ( 1 ... N )  X.  {
0 } )  Fn  ( 1 ... N
) )
90 eqidd 2472 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  =  ( ( 1st `  ( 1st `  T ) ) `
 n ) )
9130fvconst2 6136 . . . 4  |-  ( n  e.  ( 1 ... N )  ->  (
( ( 1 ... N )  X.  {
0 } ) `  n )  =  0 )
9291adantl 473 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1 ... N )  X.  {
0 } ) `  n )  =  0 )
93 elmapi 7511 . . . . . . . 8  |-  ( ( 1st `  ( 1st `  T ) )  e.  ( ( 0..^ K )  ^m  ( 1 ... N ) )  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
9485, 93syl 17 . . . . . . 7  |-  ( ph  ->  ( 1st `  ( 1st `  T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
9594ffvelrnda 6037 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  ( 0..^ K ) )
96 elfzonn0 11988 . . . . . 6  |-  ( ( ( 1st `  ( 1st `  T ) ) `
 n )  e.  ( 0..^ K )  ->  ( ( 1st `  ( 1st `  T
) ) `  n
)  e.  NN0 )
9795, 96syl 17 . . . . 5  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e. 
NN0 )
9897nn0cnd 10951 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  CC )
9998addid1d 9851 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1st `  ( 1st `  T ) ) `
 n )  +  0 )  =  ( ( 1st `  ( 1st `  T ) ) `
 n ) )
10083, 87, 89, 87, 90, 92, 99offveq 6571 . 2  |-  ( ph  ->  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 0 } ) )  =  ( 1st `  ( 1st `  T ) ) )
10181, 100eqtrd 2505 1  |-  ( ph  ->  ( F `  0
)  =  ( 1st `  ( 1st `  T
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ 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   0cc0 9557   1c1 9558    + caddc 9560    < clt 9693    - cmin 9880   NNcn 10631   NN0cn0 10893   ...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:  poimirlem12  32016  poimirlem14  32018
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