Metamath Proof Explorer


Theorem poimirlem5

Description: Lemma for poimir 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)

Ref Expression
Hypotheses 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 poimirlem5
|- ( ph -> ( F ` 0 ) = ( 1st ` ( 1st ` T ) ) )

Proof

Step Hyp Ref Expression
1 poimir.0
 |-  ( ph -> N e. NN )
2 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 } ) ) ) ) }
3 poimirlem9.1
 |-  ( ph -> T e. S )
4 poimirlem5.2
 |-  ( ph -> 0 < ( 2nd ` T ) )
5 fveq2
 |-  ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) )
6 5 breq2d
 |-  ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) )
7 6 ifbid
 |-  ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) )
8 7 csbeq1d
 |-  ( 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 } ) ) ) )
9 2fveq3
 |-  ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) )
10 2fveq3
 |-  ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) )
11 10 imaeq1d
 |-  ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) )
12 11 xpeq1d
 |-  ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) )
13 10 imaeq1d
 |-  ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) )
14 13 xpeq1d
 |-  ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
15 12 14 uneq12d
 |-  ( 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 } ) ) )
16 9 15 oveq12d
 |-  ( 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 } ) ) ) )
17 16 csbeq2dv
 |-  ( 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 } ) ) ) )
18 8 17 eqtrd
 |-  ( 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 } ) ) ) )
19 18 mpteq2dv
 |-  ( 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 } ) ) ) ) )
20 19 eqeq2d
 |-  ( 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 } ) ) ) ) ) )
21 20 2 elrab2
 |-  ( 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 } ) ) ) ) ) )
22 21 simprbi
 |-  ( 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 } ) ) ) ) )
23 3 22 syl
 |-  ( 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 } ) ) ) ) )
24 breq1
 |-  ( y = 0 -> ( y < ( 2nd ` T ) <-> 0 < ( 2nd ` T ) ) )
25 id
 |-  ( y = 0 -> y = 0 )
26 24 25 ifbieq1d
 |-  ( y = 0 -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = if ( 0 < ( 2nd ` T ) , 0 , ( y + 1 ) ) )
27 4 iftrued
 |-  ( ph -> if ( 0 < ( 2nd ` T ) , 0 , ( y + 1 ) ) = 0 )
28 26 27 sylan9eqr
 |-  ( ( ph /\ y = 0 ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = 0 )
29 28 csbeq1d
 |-  ( ( 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
 |-  0 e. _V
31 oveq2
 |-  ( j = 0 -> ( 1 ... j ) = ( 1 ... 0 ) )
32 fz10
 |-  ( 1 ... 0 ) = (/)
33 31 32 eqtrdi
 |-  ( j = 0 -> ( 1 ... j ) = (/) )
34 33 imaeq2d
 |-  ( j = 0 -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " (/) ) )
35 34 xpeq1d
 |-  ( j = 0 -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) )
36 oveq1
 |-  ( j = 0 -> ( j + 1 ) = ( 0 + 1 ) )
37 0p1e1
 |-  ( 0 + 1 ) = 1
38 36 37 eqtrdi
 |-  ( j = 0 -> ( j + 1 ) = 1 )
39 38 oveq1d
 |-  ( j = 0 -> ( ( j + 1 ) ... N ) = ( 1 ... N ) )
40 39 imaeq2d
 |-  ( j = 0 -> ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) )
41 40 xpeq1d
 |-  ( j = 0 -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) )
42 35 41 uneq12d
 |-  ( 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
 |-  ( ( 2nd ` ( 1st ` T ) ) " (/) ) = (/)
44 43 xpeq1i
 |-  ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) = ( (/) X. { 1 } )
45 0xp
 |-  ( (/) X. { 1 } ) = (/)
46 44 45 eqtri
 |-  ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) = (/)
47 46 uneq1i
 |-  ( ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) ) = ( (/) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) )
48 uncom
 |-  ( (/) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) u. (/) )
49 un0
 |-  ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) u. (/) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } )
50 47 48 49 3eqtri
 |-  ( ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } )
51 42 50 eqtrdi
 |-  ( 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 } ) )
52 51 oveq2d
 |-  ( 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 } ) ) )
53 30 52 csbie
 |-  [_ 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
 |-  ( 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 ) ) )
55 54 2 eleq2s
 |-  ( T e. S -> T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
56 3 55 syl
 |-  ( ph -> T e. ( ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
57 xp1st
 |-  ( 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 ) } ) )
58 56 57 syl
 |-  ( ph -> ( 1st ` T ) e. ( ( ( 0 ..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
59 xp2nd
 |-  ( ( 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 ) } )
60 58 59 syl
 |-  ( ph -> ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
61 fvex
 |-  ( 2nd ` ( 1st ` T ) ) e. _V
62 f1oeq1
 |-  ( f = ( 2nd ` ( 1st ` T ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) )
63 61 62 elab
 |-  ( ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
64 60 63 sylib
 |-  ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
65 f1ofo
 |-  ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) )
66 64 65 syl
 |-  ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) )
67 foima
 |-  ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) )
68 66 67 syl
 |-  ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) )
69 68 xpeq1d
 |-  ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) = ( ( 1 ... N ) X. { 0 } ) )
70 69 oveq2d
 |-  ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) )
71 53 70 syl5eq
 |-  ( 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 } ) ) )
72 71 adantr
 |-  ( ( 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 } ) ) )
73 29 72 eqtrd
 |-  ( ( 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 nnm1nn0
 |-  ( N e. NN -> ( N - 1 ) e. NN0 )
75 1 74 syl
 |-  ( ph -> ( N - 1 ) e. NN0 )
76 0elfz
 |-  ( ( N - 1 ) e. NN0 -> 0 e. ( 0 ... ( N - 1 ) ) )
77 75 76 syl
 |-  ( ph -> 0 e. ( 0 ... ( N - 1 ) ) )
78 ovexd
 |-  ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) e. _V )
79 23 73 77 78 fvmptd
 |-  ( ph -> ( F ` 0 ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) )
80 ovexd
 |-  ( ph -> ( 1 ... N ) e. _V )
81 xp1st
 |-  ( ( 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 ) ) )
82 58 81 syl
 |-  ( ph -> ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) )
83 elmapfn
 |-  ( ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) )
84 82 83 syl
 |-  ( ph -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) )
85 fnconstg
 |-  ( 0 e. _V -> ( ( 1 ... N ) X. { 0 } ) Fn ( 1 ... N ) )
86 30 85 mp1i
 |-  ( ph -> ( ( 1 ... N ) X. { 0 } ) Fn ( 1 ... N ) )
87 eqidd
 |-  ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
88 30 fvconst2
 |-  ( n e. ( 1 ... N ) -> ( ( ( 1 ... N ) X. { 0 } ) ` n ) = 0 )
89 88 adantl
 |-  ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1 ... N ) X. { 0 } ) ` n ) = 0 )
90 elmapi
 |-  ( ( 1st ` ( 1st ` T ) ) e. ( ( 0 ..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) )
91 82 90 syl
 |-  ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0 ..^ K ) )
92 91 ffvelrnda
 |-  ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. ( 0 ..^ K ) )
93 elfzonn0
 |-  ( ( ( 1st ` ( 1st ` T ) ) ` n ) e. ( 0 ..^ K ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. NN0 )
94 92 93 syl
 |-  ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. NN0 )
95 94 nn0cnd
 |-  ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. CC )
96 95 addid1d
 |-  ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 0 ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
97 80 84 86 84 87 89 96 offveq
 |-  ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) = ( 1st ` ( 1st ` T ) ) )
98 79 97 eqtrd
 |-  ( ph -> ( F ` 0 ) = ( 1st ` ( 1st ` T ) ) )