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Theorem poimirlem5 31859
Description: Lemma for poimir 31887 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 5878 . . . . . . . . . . . 12  |-  ( t  =  T  ->  ( 2nd `  t )  =  ( 2nd `  T
) )
32breq2d 4432 . . . . . . . . . . 11  |-  ( t  =  T  ->  (
y  <  ( 2nd `  t )  <->  y  <  ( 2nd `  T ) ) )
43ifbid 3931 . . . . . . . . . 10  |-  ( t  =  T  ->  if ( y  <  ( 2nd `  t ) ,  y ,  ( y  +  1 ) )  =  if ( y  <  ( 2nd `  T
) ,  y ,  ( y  +  1 ) ) )
54csbeq1d 3402 . . . . . . . . 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 5878 . . . . . . . . . . . 12  |-  ( t  =  T  ->  ( 1st `  t )  =  ( 1st `  T
) )
76fveq2d 5882 . . . . . . . . . . 11  |-  ( t  =  T  ->  ( 1st `  ( 1st `  t
) )  =  ( 1st `  ( 1st `  T ) ) )
86fveq2d 5882 . . . . . . . . . . . . . 14  |-  ( t  =  T  ->  ( 2nd `  ( 1st `  t
) )  =  ( 2nd `  ( 1st `  T ) ) )
98imaeq1d 5183 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) ) )
109xpeq1d 4873 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... j
) )  X.  {
1 } ) )
118imaeq1d 5183 . . . . . . . . . . . . 13  |-  ( t  =  T  ->  (
( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) ) )
1211xpeq1d 4873 . . . . . . . . . . . 12  |-  ( t  =  T  ->  (
( ( 2nd `  ( 1st `  t ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( ( j  +  1 ) ... N
) )  X.  {
0 } ) )
1310, 12uneq12d 3621 . . . . . . . . . . 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 6320 . . . . . . . . . 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 3809 . . . . . . . . 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 2463 . . . . . . . 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 4508 . . . . . . 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 2436 . . . . . 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 3231 . . . . 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 465 . . . 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 4423 . . . . . . 7  |-  ( y  =  0  ->  (
y  <  ( 2nd `  T )  <->  0  <  ( 2nd `  T ) ) )
24 id 23 . . . . . . 7  |-  ( y  =  0  ->  y  =  0 )
2523, 24ifbieq1d 3932 . . . . . 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 3917 . . . . . 6  |-  ( ph  ->  if ( 0  < 
( 2nd `  T
) ,  0 ,  ( y  +  1 ) )  =  0 )
2825, 27sylan9eqr 2485 . . . . 5  |-  ( (
ph  /\  y  = 
0 )  ->  if ( y  <  ( 2nd `  T ) ,  y ,  ( y  +  1 ) )  =  0 )
2928csbeq1d 3402 . . . 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 9638 . . . . . . 7  |-  0  e.  _V
31 oveq2 6310 . . . . . . . . . . . . 13  |-  ( j  =  0  ->  (
1 ... j )  =  ( 1 ... 0
) )
32 fz10 11821 . . . . . . . . . . . . 13  |-  ( 1 ... 0 )  =  (/)
3331, 32syl6eq 2479 . . . . . . . . . . . 12  |-  ( j  =  0  ->  (
1 ... j )  =  (/) )
3433imaeq2d 5184 . . . . . . . . . . 11  |-  ( j  =  0  ->  (
( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  =  ( ( 2nd `  ( 1st `  T ) )
" (/) ) )
3534xpeq1d 4873 . . . . . . . . . 10  |-  ( j  =  0  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( 1 ... j ) )  X. 
{ 1 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) " (/) )  X.  { 1 } ) )
36 oveq1 6309 . . . . . . . . . . . . . 14  |-  ( j  =  0  ->  (
j  +  1 )  =  ( 0  +  1 ) )
37 0p1e1 10722 . . . . . . . . . . . . . 14  |-  ( 0  +  1 )  =  1
3836, 37syl6eq 2479 . . . . . . . . . . . . 13  |-  ( j  =  0  ->  (
j  +  1 )  =  1 )
3938oveq1d 6317 . . . . . . . . . . . 12  |-  ( j  =  0  ->  (
( j  +  1 ) ... N )  =  ( 1 ... N ) )
4039imaeq2d 5184 . . . . . . . . . . 11  |-  ( j  =  0  ->  (
( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  =  ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) ) )
4140xpeq1d 4873 . . . . . . . . . 10  |-  ( j  =  0  ->  (
( ( 2nd `  ( 1st `  T ) )
" ( ( j  +  1 ) ... N ) )  X. 
{ 0 } )  =  ( ( ( 2nd `  ( 1st `  T ) ) "
( 1 ... N
) )  X.  {
0 } ) )
4235, 41uneq12d 3621 . . . . . . . . 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 5199 . . . . . . . . . . . . 13  |-  ( ( 2nd `  ( 1st `  T ) ) " (/) )  =  (/)
4443xpeq1i 4870 . . . . . . . . . . . 12  |-  ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
1 } )  =  ( (/)  X.  { 1 } )
45 0xp 4931 . . . . . . . . . . . 12  |-  ( (/)  X. 
{ 1 } )  =  (/)
4644, 45eqtri 2451 . . . . . . . . . . 11  |-  ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
1 } )  =  (/)
4746uneq1i 3616 . . . . . . . . . 10  |-  ( ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } ) )  =  (
(/)  u.  ( (
( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } ) )
48 uncom 3610 . . . . . . . . . 10  |-  ( (/)  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } ) )  =  ( ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } )  u.  (/) )
49 un0 3787 . . . . . . . . . 10  |-  ( ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } )  u.  (/) )  =  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } )
5047, 48, 493eqtri 2455 . . . . . . . . 9  |-  ( ( ( ( 2nd `  ( 1st `  T ) )
" (/) )  X.  {
1 } )  u.  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } ) )  =  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } )
5142, 50syl6eq 2479 . . . . . . . 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 6318 . . . . . . 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 3421 . . . . . 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 3226 . . . . . . . . . . . . . . 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 2530 . . . . . . . . . . . . . 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 6834 . . . . . . . . . . . . 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 6835 . . . . . . . . . . . 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 5888 . . . . . . . . . . . 12  |-  ( 2nd `  ( 1st `  T
) )  e.  _V
62 f1oeq1 5819 . . . . . . . . . . . 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 3218 . . . . . . . . . . 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 199 . . . . . . . . . 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 5812 . . . . . . . . 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 4873 . . . . . . 7  |-  ( ph  ->  ( ( ( 2nd `  ( 1st `  T
) ) " (
1 ... N ) )  X.  { 0 } )  =  ( ( 1 ... N )  X.  { 0 } ) )
7069oveq2d 6318 . . . . . 6  |-  ( ph  ->  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( ( 2nd `  ( 1st `  T ) )
" ( 1 ... N ) )  X. 
{ 0 } ) )  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N )  X.  { 0 } ) ) )
7153, 70syl5eq 2475 . . . . 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 466 . . . 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 2463 . . 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 10912 . . . . 5  |-  ( N  e.  NN  ->  ( N  -  1 )  e.  NN0 )
7674, 75syl 17 . . . 4  |-  ( ph  ->  ( N  -  1 )  e.  NN0 )
77 0elfz 11890 . . . 4  |-  ( ( N  -  1 )  e.  NN0  ->  0  e.  ( 0 ... ( N  -  1 ) ) )
7876, 77syl 17 . . 3  |-  ( ph  ->  0  e.  ( 0 ... ( N  - 
1 ) ) )
79 ovex 6330 . . . 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 5967 . 2  |-  ( ph  ->  ( F `  0
)  =  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N )  X.  { 0 } ) ) )
82 ovex 6330 . . . 4  |-  ( 1 ... N )  e. 
_V
8382a1i 11 . . 3  |-  ( ph  ->  ( 1 ... N
)  e.  _V )
84 xp1st 6834 . . . . 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 7499 . . . 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 5785 . . . 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 2423 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  =  ( ( 1st `  ( 1st `  T ) ) `
 n ) )
9130fvconst2 6132 . . . 4  |-  ( n  e.  ( 1 ... N )  ->  (
( ( 1 ... N )  X.  {
0 } ) `  n )  =  0 )
9291adantl 467 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1 ... N )  X.  {
0 } ) `  n )  =  0 )
93 elmapi 7498 . . . . . . . 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 6034 . . . . . 6  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  ( 0..^ K ) )
96 elfzonn0 11961 . . . . . 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 10928 . . . 4  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( 1st `  ( 1st `  T ) ) `
 n )  e.  CC )
9998addid1d 9834 . . 3  |-  ( (
ph  /\  n  e.  ( 1 ... N
) )  ->  (
( ( 1st `  ( 1st `  T ) ) `
 n )  +  0 )  =  ( ( 1st `  ( 1st `  T ) ) `
 n ) )
10083, 87, 89, 87, 90, 92, 99offveq 6563 . 2  |-  ( ph  ->  ( ( 1st `  ( 1st `  T ) )  oF  +  ( ( 1 ... N
)  X.  { 0 } ) )  =  ( 1st `  ( 1st `  T ) ) )
10181, 100eqtrd 2463 1  |-  ( ph  ->  ( F `  0
)  =  ( 1st `  ( 1st `  T
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1868   {cab 2407   {crab 2779   _Vcvv 3081   [_csb 3395    u. cun 3434   (/)c0 3761   ifcif 3909   {csn 3996   class class class wbr 4420    |-> cmpt 4479    X. cxp 4848   "cima 4853    Fn wfn 5593   -->wf 5594   -onto->wfo 5596   -1-1-onto->wf1o 5597   ` cfv 5598  (class class class)co 6302    oFcof 6540   1stc1st 6802   2ndc2nd 6803    ^m cmap 7477   0cc0 9540   1c1 9541    + caddc 9543    < clt 9676    - cmin 9861   NNcn 10610   NN0cn0 10870   ...cfz 11785  ..^cfzo 11916
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1839  ax-8 1870  ax-9 1872  ax-10 1887  ax-11 1892  ax-12 1905  ax-13 2053  ax-ext 2400  ax-rep 4533  ax-sep 4543  ax-nul 4552  ax-pow 4599  ax-pr 4657  ax-un 6594  ax-cnex 9596  ax-resscn 9597  ax-1cn 9598  ax-icn 9599  ax-addcl 9600  ax-addrcl 9601  ax-mulcl 9602  ax-mulrcl 9603  ax-mulcom 9604  ax-addass 9605  ax-mulass 9606  ax-distr 9607  ax-i2m1 9608  ax-1ne0 9609  ax-1rid 9610  ax-rnegex 9611  ax-rrecex 9612  ax-cnre 9613  ax-pre-lttri 9614  ax-pre-lttrn 9615  ax-pre-ltadd 9616  ax-pre-mulgt0 9617
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2269  df-mo 2270  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2572  df-ne 2620  df-nel 2621  df-ral 2780  df-rex 2781  df-reu 2782  df-rab 2784  df-v 3083  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3910  df-pw 3981  df-sn 3997  df-pr 3999  df-tp 4001  df-op 4003  df-uni 4217  df-iun 4298  df-br 4421  df-opab 4480  df-mpt 4481  df-tr 4516  df-eprel 4761  df-id 4765  df-po 4771  df-so 4772  df-fr 4809  df-we 4811  df-xp 4856  df-rel 4857  df-cnv 4858  df-co 4859  df-dm 4860  df-rn 4861  df-res 4862  df-ima 4863  df-pred 5396  df-ord 5442  df-on 5443  df-lim 5444  df-suc 5445  df-iota 5562  df-fun 5600  df-fn 5601  df-f 5602  df-f1 5603  df-fo 5604  df-f1o 5605  df-fv 5606  df-riota 6264  df-ov 6305  df-oprab 6306  df-mpt2 6307  df-of 6542  df-om 6704  df-1st 6804  df-2nd 6805  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-er 7368  df-map 7479  df-en 7575  df-dom 7576  df-sdom 7577  df-pnf 9678  df-mnf 9679  df-xr 9680  df-ltxr 9681  df-le 9682  df-sub 9863  df-neg 9864  df-nn 10611  df-n0 10871  df-z 10939  df-uz 11161  df-fz 11786  df-fzo 11917
This theorem is referenced by:  poimirlem12  31866  poimirlem14  31868
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