Theorem poimirlem5 31945

Description: Lemma for poimir 31973 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.

Step	Hyp	Ref	Expression
1		poimirlem9.1	. . . 4 |- ( ph -> T e. S )
2		fveq2 5865	. . . . . . . . . . . 12 |- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) )
3	2	breq2d 4414	. . . . . . . . . . 11 |- ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) )
4	3	ifbid 3903	. . . . . . . . . 10 |- ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) )
5	4	csbeq1d 3370	. . . . . . . . 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 5865	. . . . . . . . . . . 12 |- ( t = T -> ( 1st ` t ) = ( 1st ` T ) )
7	6	fveq2d 5869	. . . . . . . . . . 11 |- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) )
8	6	fveq2d 5869	. . . . . . . . . . . . . 14 |- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) )
9	8	imaeq1d 5167	. . . . . . . . . . . . 13 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) )
10	9	xpeq1d 4857	. . . . . . . . . . . 12 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) )
11	8	imaeq1d 5167	. . . . . . . . . . . . 13 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) )
12	11	xpeq1d 4857	. . . . . . . . . . . 12 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
13	10, 12	uneq12d 3589	. . . . . . . . . . 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 } ) ) )
14	7, 13	oveq12d 6308	. . . . . . . . . 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 } ) ) ) )
15	14	csbeq2dv 3781	. . . . . . . . 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 } ) ) ) )
16	5, 15	eqtrd 2485	. . . . . . . 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 } ) ) ) )
17	16	mpteq2dv 4490	. . . . . . 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 } ) ) ) ) )
18	17	eqeq2d 2461	. . . . . 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 } ) ) ) ) }
20	18, 19	elrab2 3198	. . . . 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 } ) ) ) ) ) )
21	20	simprbi 466	. . . 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 } ) ) ) ) )
22	1, 21	syl 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 4405	. . . . . . 7 |- ( y = 0 -> ( y < ( 2nd ` T ) <-> 0 < ( 2nd ` T ) ) )
24		id 22	. . . . . . 7 |- ( y = 0 -> y = 0 )
25	23, 24	ifbieq1d 3904	. . . . . 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 ) )
27	26	iftrued 3889	. . . . . 6 |- ( ph -> if ( 0 < ( 2nd ` T ) , 0 , ( y + 1 ) ) = 0 )
28	25, 27	sylan9eqr 2507	. . . . 5 |- ( ( ph /\ y = 0 ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = 0 )
29	28	csbeq1d 3370	. . . 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 9637	. . . . . . 7 |- 0 e. _V
31		oveq2 6298	. . . . . . . . . . . . 13 |- ( j = 0 -> ( 1 ... j ) = ( 1 ... 0 ) )
32		fz10 11820	. . . . . . . . . . . . 13 |- ( 1 ... 0 ) = (/)
33	31, 32	syl6eq 2501	. . . . . . . . . . . 12 |- ( j = 0 -> ( 1 ... j ) = (/) )
34	33	imaeq2d 5168	. . . . . . . . . . 11 |- ( j = 0 -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " (/) ) )
35	34	xpeq1d 4857	. . . . . . . . . 10 |- ( j = 0 -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) )
36		oveq1 6297	. . . . . . . . . . . . . 14 |- ( j = 0 -> ( j + 1 ) = ( 0 + 1 ) )
37		0p1e1 10721	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
38	36, 37	syl6eq 2501	. . . . . . . . . . . . 13 |- ( j = 0 -> ( j + 1 ) = 1 )
39	38	oveq1d 6305	. . . . . . . . . . . 12 |- ( j = 0 -> ( ( j + 1 ) ... N ) = ( 1 ... N ) )
40	39	imaeq2d 5168	. . . . . . . . . . 11 |- ( j = 0 -> ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) )
41	40	xpeq1d 4857	. . . . . . . . . 10 |- ( j = 0 -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) )
42	35, 41	uneq12d 3589	. . . . . . . . 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 5183	. . . . . . . . . . . . 13 |- ( ( 2nd ` ( 1st ` T ) ) " (/) ) = (/)
44	43	xpeq1i 4854	. . . . . . . . . . . 12 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) = ( (/) X. { 1 } )
45		0xp 4915	. . . . . . . . . . . 12 |- ( (/) X. { 1 } ) = (/)
46	44, 45	eqtri 2473	. . . . . . . . . . 11 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) = (/)
47	46	uneq1i 3584	. . . . . . . . . 10 |- ( ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) ) = ( (/) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) )
48		uncom 3578	. . . . . . . . . 10 |- ( (/) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) u. (/) )
49		un0 3759	. . . . . . . . . 10 |- ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) u. (/) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } )
50	47, 48, 49	3eqtri 2477	. . . . . . . . 9 |- ( ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } )
51	42, 50	syl6eq 2501	. . . . . . . 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 } ) )
52	51	oveq2d 6306	. . . . . . 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 } ) ) )
53	30, 52	csbie 3389	. . . . . 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 3193	. . . . . . . . . . . . . . 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 ) ) )
55	54, 19	eleq2s 2547	. . . . . . . . . . . . . 14 |- ( T e. S -> T e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
56	1, 55	syl 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 6823	. . . . . . . . . . . . 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 ) } ) )
58	56, 57	syl 17	. . . . . . . . . . . 12 |- ( ph -> ( 1st ` T ) e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
59		xp2nd 6824	. . . . . . . . . . . 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 ) } )
60	58, 59	syl 17	. . . . . . . . . . 11 |- ( ph -> ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
61		fvex 5875	. . . . . . . . . . . 12 |- ( 2nd ` ( 1st ` T ) ) e. _V
62		f1oeq1 5805	. . . . . . . . . . . 12 |- ( 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 3185	. . . . . . . . . . 11 |- ( ( 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 200	. . . . . . . . . 10 |- ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
65		f1ofo 5821	. . . . . . . . . 10 |- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) )
66	64, 65	syl 17	. . . . . . . . 9 |- ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) )
67		foima 5798	. . . . . . . . 9 |- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) )
68	66, 67	syl 17	. . . . . . . 8 |- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) )
69	68	xpeq1d 4857	. . . . . . 7 |- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) = ( ( 1 ... N ) X. { 0 } ) )
70	69	oveq2d 6306	. . . . . 6 |- ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) )
71	53, 70	syl5eq 2497	. . . . 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 } ) ) )
72	71	adantr 467	. . . 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 } ) ) )
73	29, 72	eqtrd 2485	. . 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 10911	. . . . 5 |- ( N e. NN -> ( N - 1 ) e. NN0 )
76	74, 75	syl 17	. . . 4 |- ( ph -> ( N - 1 ) e. NN0 )
77		0elfz 11889	. . . 4 |- ( ( N - 1 ) e. NN0 -> 0 e. ( 0 ... ( N - 1 ) ) )
78	76, 77	syl 17	. . 3 |- ( ph -> 0 e. ( 0 ... ( N - 1 ) ) )
79		ovex 6318	. . . 4 |- ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) e. _V
80	79	a1i 11	. . 3 |- ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) e. _V )
81	22, 73, 78, 80	fvmptd 5954	. 2 |- ( ph -> ( F ` 0 ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) )
82		ovex 6318	. . . 4 |- ( 1 ... N ) e. _V
83	82	a1i 11	. . 3 |- ( ph -> ( 1 ... N ) e. _V )
84		xp1st 6823	. . . . 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 ) ) )
85	58, 84	syl 17	. . . 4 |- ( ph -> ( 1st ` ( 1st ` T ) ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) )
86		elmapfn 7494	. . . 4 |- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) )
87	85, 86	syl 17	. . 3 |- ( ph -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) )
88		fnconstg 5771	. . . 4 |- ( 0 e. _V -> ( ( 1 ... N ) X. { 0 } ) Fn ( 1 ... N ) )
89	30, 88	mp1i 13	. . 3 |- ( ph -> ( ( 1 ... N ) X. { 0 } ) Fn ( 1 ... N ) )
90		eqidd 2452	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
91	30	fvconst2 6120	. . . 4 |- ( n e. ( 1 ... N ) -> ( ( ( 1 ... N ) X. { 0 } ) ` n ) = 0 )
92	91	adantl 468	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1 ... N ) X. { 0 } ) ` n ) = 0 )
93		elmapi 7493	. . . . . . . 8 |- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
94	85, 93	syl 17	. . . . . . 7 |- ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
95	94	ffvelrnda 6022	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. ( 0..^ K ) )
96		elfzonn0 11960	. . . . . 6 |- ( ( ( 1st ` ( 1st ` T ) ) ` n ) e. ( 0..^ K ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. NN0 )
97	95, 96	syl 17	. . . . 5 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. NN0 )
98	97	nn0cnd 10927	. . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. CC )
99	98	addid1d 9833	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 0 ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
100	83, 87, 89, 87, 90, 92, 99	offveq 6552	. 2 |- ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) = ( 1st ` ( 1st ` T ) ) )
101	81, 100	eqtrd 2485	1 |- ( ph -> ( F ` 0 ) = ( 1st ` ( 1st ` T ) ) )

Syntax hints:   -> wi 4   /\ 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   0cc0 9539   1c1 9540   + caddc 9542   < clt 9675   - cmin 9860   NNcn 10609   NN0cn0 10869   ...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:  poimirlem12  31952  poimirlem14  31954