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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  poimirlem10 Structured version   Visualization version   Unicode version
Theorem poimirlem10 33419
Description: Lemma for poimir 33442 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 ovexd 6680 . 2 |- ( ph -> ( 1 ... N ) e. _V )
2 poimirlem22.1 . . . 4 |- ( ph -> F : ( 0 ... ( N - 1 ) ) --> ( ( 0 ... K ) ^m ( 1 ... N ) ) )
3 poimir.0 . . . . . 6 |- ( ph -> N e. NN )
4 nnm1nn0 11334 . . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
53, 4syl 17 . . . . 5 |- ( ph -> ( N - 1 ) e. NN0 )
6 nn0fz0 12437 . . . . 5 |- ( ( N - 1 ) e. NN0 <-> ( N - 1 ) e. ( 0 ... ( N - 1 ) ) )
75, 6sylib 208 . . . 4 |- ( ph -> ( N - 1 ) e. ( 0 ... ( N - 1 ) ) )
82, 7ffvelrnd 6360 . . 3 |- ( ph -> ( F ` ( N - 1 ) ) e. ( ( 0 ... K ) ^m ( 1 ... N ) ) )
9 elmapfn 7880 . . 3 |- ( ( F ` ( N - 1 ) ) e. ( ( 0 ... K ) ^m ( 1 ... N ) ) -> ( F ` ( N - 1 ) ) Fn ( 1 ... N ) )
108, 9syl 17 . 2 |- ( ph -> ( F ` ( N - 1 ) ) Fn ( 1 ... N ) )
11 1ex 10035 . . 3 |- 1 e. _V
12 fnconstg 6093 . . 3 |- ( 1 e. _V -> ( ( 1 ... N ) X. { 1 } ) Fn ( 1 ... N ) )
1311, 12mp1i 13 . 2 |- ( ph -> ( ( 1 ... N ) X. { 1 } ) Fn ( 1 ... N ) )
14 poimirlem12.2 . . . . . 6 |- ( ph -> T e. S )
15 elrabi 3359 . . . . . . 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 ) ) )
16 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 } ) ) ) ) }
1715, 16eleq2s 2719 . . . . . 6 |- ( T e. S -> T e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
1814, 17syl 17 . . . . 5 |- ( ph -> T e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
19 xp1st 7198 . . . . 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 ) } ) )
2018, 19syl 17 . . . 4 |- ( ph -> ( 1st ` T ) e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
21 xp1st 7198 . . . 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 ) ) )
2220, 21syl 17 . . 3 |- ( ph -> ( 1st ` ( 1st ` T ) ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) )
23 elmapfn 7880 . . 3 |- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) )
2422, 23syl 17 . 2 |- ( ph -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) )
25 fveq2 6191 . . . . . . . . . . . . . . 15 |- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) )
2625breq2d 4665 . . . . . . . . . . . . . 14 |- ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) )
2726ifbid 4108 . . . . . . . . . . . . 13 |- ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) )
2827csbeq1d 3540 . . . . . . . . . . . 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 } ) ) ) )
29 fveq2 6191 . . . . . . . . . . . . . . 15 |- ( t = T -> ( 1st ` t ) = ( 1st ` T ) )
3029fveq2d 6195 . . . . . . . . . . . . . 14 |- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) )
3129fveq2d 6195 . . . . . . . . . . . . . . . . 17 |- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) )
3231imaeq1d 5465 . . . . . . . . . . . . . . . 16 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) )
3332xpeq1d 5138 . . . . . . . . . . . . . . 15 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) )
3431imaeq1d 5465 . . . . . . . . . . . . . . . 16 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) )
3534xpeq1d 5138 . . . . . . . . . . . . . . 15 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
3633, 35uneq12d 3768 . . . . . . . . . . . . . 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 } ) ) )
3730, 36oveq12d 6668 . . . . . . . . . . . . 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 } ) ) ) )
3837csbeq2dv 3992 . . . . . . . . . . . 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 } ) ) ) )
3928, 38eqtrd 2656 . . . . . . . . . . 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 } ) ) ) )
4039mpteq2dv 4745 . . . . . . . . . 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 } ) ) ) ) )
4140eqeq2d 2632 . . . . . . . . 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 } ) ) ) ) ) )
4241, 16elrab2 3366 . . . . . . . 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 } ) ) ) ) ) )
4342simprbi 480 . . . . . . 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 } ) ) ) ) )
4414, 43syl 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 } ) ) ) ) )
45 poimirlem11.3 . . . . . . . . . . . 12 |- ( ph -> ( 2nd ` T ) = 0 )
46 breq12 4658 . . . . . . . . . . . 12 |- ( ( y = ( N - 1 ) /\ ( 2nd ` T ) = 0 ) -> ( y < ( 2nd ` T ) <-> ( N - 1 ) < 0 ) )
4745, 46sylan2 491 . . . . . . . . . . 11 |- ( ( y = ( N - 1 ) /\ ph ) -> ( y < ( 2nd ` T ) <-> ( N - 1 ) < 0 ) )
4847ancoms 469 . . . . . . . . . 10 |- ( ( ph /\ y = ( N - 1 ) ) -> ( y < ( 2nd ` T ) <-> ( N - 1 ) < 0 ) )
49 oveq1 6657 . . . . . . . . . . 11 |- ( y = ( N - 1 ) -> ( y + 1 ) = ( ( N - 1 ) + 1 ) )
503nncnd 11036 . . . . . . . . . . . 12 |- ( ph -> N e. CC )
51 npcan1 10455 . . . . . . . . . . . 12 |- ( N e. CC -> ( ( N - 1 ) + 1 ) = N )
5250, 51syl 17 . . . . . . . . . . 11 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
5349, 52sylan9eqr 2678 . . . . . . . . . 10 |- ( ( ph /\ y = ( N - 1 ) ) -> ( y + 1 ) = N )
5448, 53ifbieq2d 4111 . . . . . . . . 9 |- ( ( ph /\ y = ( N - 1 ) ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = if ( ( N - 1 ) < 0 , y , N ) )
555nn0ge0d 11354 . . . . . . . . . . . 12 |- ( ph -> 0 <_ ( N - 1 ) )
56 0red 10041 . . . . . . . . . . . . 13 |- ( ph -> 0 e. RR )
575nn0red 11352 . . . . . . . . . . . . 13 |- ( ph -> ( N - 1 ) e. RR )
5856, 57lenltd 10183 . . . . . . . . . . . 12 |- ( ph -> ( 0 <_ ( N - 1 ) <-> -. ( N - 1 ) < 0 ) )
5955, 58mpbid 222 . . . . . . . . . . 11 |- ( ph -> -. ( N - 1 ) < 0 )
6059iffalsed 4097 . . . . . . . . . 10 |- ( ph -> if ( ( N - 1 ) < 0 , y , N ) = N )
6160adantr 481 . . . . . . . . 9 |- ( ( ph /\ y = ( N - 1 ) ) -> if ( ( N - 1 ) < 0 , y , N ) = N )
6254, 61eqtrd 2656 . . . . . . . 8 |- ( ( ph /\ y = ( N - 1 ) ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = N )
6362csbeq1d 3540 . . . . . . 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 } ) ) ) )
64 oveq2 6658 . . . . . . . . . . . . . . 15 |- ( j = N -> ( 1 ... j ) = ( 1 ... N ) )
6564imaeq2d 5466 . . . . . . . . . . . . . 14 |- ( j = N -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) )
66 xp2nd 7199 . . . . . . . . . . . . . . . . 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 ) } )
6720, 66syl 17 . . . . . . . . . . . . . . . 16 |- ( ph -> ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
68 fvex 6201 . . . . . . . . . . . . . . . . 17 |- ( 2nd ` ( 1st ` T ) ) e. _V
69 f1oeq1 6127 . . . . . . . . . . . . . . . . 17 |- ( f = ( 2nd ` ( 1st ` T ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) )
7068, 69elab 3350 . . . . . . . . . . . . . . . 16 |- ( ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
7167, 70sylib 208 . . . . . . . . . . . . . . 15 |- ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
72 f1ofo 6144 . . . . . . . . . . . . . . 15 |- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) )
73 foima 6120 . . . . . . . . . . . . . . 15 |- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) )
7471, 72, 733syl 18 . . . . . . . . . . . . . 14 |- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) )
7565, 74sylan9eqr 2678 . . . . . . . . . . . . 13 |- ( ( ph /\ j = N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( 1 ... N ) )
7675xpeq1d 5138 . . . . . . . . . . . 12 |- ( ( ph /\ j = N ) -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( 1 ... N ) X. { 1 } ) )
77 oveq1 6657 . . . . . . . . . . . . . . . . 17 |- ( j = N -> ( j + 1 ) = ( N + 1 ) )
7877oveq1d 6665 . . . . . . . . . . . . . . . 16 |- ( j = N -> ( ( j + 1 ) ... N ) = ( ( N + 1 ) ... N ) )
793nnred 11035 . . . . . . . . . . . . . . . . . 18 |- ( ph -> N e. RR )
8079ltp1d 10954 . . . . . . . . . . . . . . . . 17 |- ( ph -> N < ( N + 1 ) )
813nnzd 11481 . . . . . . . . . . . . . . . . . . 19 |- ( ph -> N e. ZZ )
8281peano2zd 11485 . . . . . . . . . . . . . . . . . 18 |- ( ph -> ( N + 1 ) e. ZZ )
83 fzn 12357 . . . . . . . . . . . . . . . . . 18 |- ( ( ( N + 1 ) e. ZZ /\ N e. ZZ ) -> ( N < ( N + 1 ) <-> ( ( N + 1 ) ... N ) = (/) ) )
8482, 81, 83syl2anc 693 . . . . . . . . . . . . . . . . 17 |- ( ph -> ( N < ( N + 1 ) <-> ( ( N + 1 ) ... N ) = (/) ) )
8580, 84mpbid 222 . . . . . . . . . . . . . . . 16 |- ( ph -> ( ( N + 1 ) ... N ) = (/) )
8678, 85sylan9eqr 2678 . . . . . . . . . . . . . . 15 |- ( ( ph /\ j = N ) -> ( ( j + 1 ) ... N ) = (/) )
8786imaeq2d 5466 . . . . . . . . . . . . . 14 |- ( ( ph /\ j = N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " (/) ) )
8887xpeq1d 5138 . . . . . . . . . . . . 13 |- ( ( ph /\ j = N ) -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 0 } ) )
89 ima0 5481 . . . . . . . . . . . . . . 15 |- ( ( 2nd ` ( 1st ` T ) ) " (/) ) = (/)
9089xpeq1i 5135 . . . . . . . . . . . . . 14 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 0 } ) = ( (/) X. { 0 } )
91 0xp 5199 . . . . . . . . . . . . . 14 |- ( (/) X. { 0 } ) = (/)
9290, 91eqtri 2644 . . . . . . . . . . . . 13 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 0 } ) = (/)
9388, 92syl6eq 2672 . . . . . . . . . . . 12 |- ( ( ph /\ j = N ) -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = (/) )
9476, 93uneq12d 3768 . . . . . . . . . . 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. (/) ) )
95 un0 3967 . . . . . . . . . . 11 |- ( ( ( 1 ... N ) X. { 1 } ) u. (/) ) = ( ( 1 ... N ) X. { 1 } )
9694, 95syl6eq 2672 . . . . . . . . . 10 |- ( ( ph /\ j = N ) -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( 1 ... N ) X. { 1 } ) )
9796oveq2d 6666 . . . . . . . . 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 } ) ) )
983, 97csbied 3560 . . . . . . . 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 } ) ) )
9998adantr 481 . . . . . . 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 } ) ) )
10063, 99eqtrd 2656 . . . . . 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 } ) ) )
101 ovexd 6680 . . . . . 6 |- ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) e. _V )
10244, 100, 7, 101fvmptd 6288 . . . . 5 |- ( ph -> ( F ` ( N - 1 ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) )
103102fveq1d 6193 . . . 4 |- ( ph -> ( ( F ` ( N - 1 ) ) ` n ) = ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) ` n ) )
104103adantr 481 . . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( F ` ( N - 1 ) ) ` n ) = ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) ` n ) )
105 inidm 3822 . . . 4 |- ( ( 1 ... N ) i^i ( 1 ... N ) ) = ( 1 ... N )
106 eqidd 2623 . . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
10711fvconst2 6469 . . . . 5 |- ( n e. ( 1 ... N ) -> ( ( ( 1 ... N ) X. { 1 } ) ` n ) = 1 )
108107adantl 482 . . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1 ... N ) X. { 1 } ) ` n ) = 1 )
10924, 13, 1, 1, 105, 106, 108ofval 6906 . . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) ` n ) = ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 1 ) )
110104, 109eqtrd 2656 . 2 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( F ` ( N - 1 ) ) ` n ) = ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 1 ) )
111 elmapi 7879 . . . . . . 7 |- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
11222, 111syl 17 . . . . . 6 |- ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
113112ffvelrnda 6359 . . . . 5 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. ( 0..^ K ) )
114 elfzonn0 12512 . . . . 5 |- ( ( ( 1st ` ( 1st ` T ) ) ` n ) e. ( 0..^ K ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. NN0 )
115113, 114syl 17 . . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. NN0 )
116115nn0cnd 11353 . . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. CC )
117 pncan1 10454 . . 3 |- ( ( ( 1st ` ( 1st ` T ) ) ` n ) e. CC -> ( ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 1 ) - 1 ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
118116, 117syl 17 . 2 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 1 ) - 1 ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
1191, 10, 13, 24, 110, 108, 118offveq 6918 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 196   /\ wa 384   = wceq 1483   e. wcel 1990   {cab 2608   {crab 2916   _Vcvv 3200   [_csb 3533   u. cun 3572   (/)c0 3915   ifcif 4086   {csn 4177   class class class wbr 4653   |-> cmpt 4729   X. cxp 5112   "cima 5117   Fn wfn 5883   -->wf 5884   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   oFcof 6895   1stc1st 7166   2ndc2nd 7167   ^m cmap 7857   CCcc 9934   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326  ..^cfzo 12465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-fz 12327  df-fzo 12466
This theorem is referenced by:  poimirlem11  33420  poimirlem13  33422
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