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.

Step	Hyp	Ref	Expression
1		poimirlem9.1	. . . 4 |- ( ph -> T e. S )
2		fveq2 5878	. . . . . . . . . . . 12 |- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) )
3	2	breq2d 4432	. . . . . . . . . . 11 |- ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) )
4	3	ifbid 3931	. . . . . . . . . 10 |- ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) )
5	4	csbeq1d 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 ) )
7	6	fveq2d 5882	. . . . . . . . . . 11 |- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) )
8	6	fveq2d 5882	. . . . . . . . . . . . . 14 |- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) )
9	8	imaeq1d 5183	. . . . . . . . . . . . 13 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) )
10	9	xpeq1d 4873	. . . . . . . . . . . 12 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) )
11	8	imaeq1d 5183	. . . . . . . . . . . . 13 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) )
12	11	xpeq1d 4873	. . . . . . . . . . . 12 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
13	10, 12	uneq12d 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 } ) ) )
14	7, 13	oveq12d 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 } ) ) ) )
15	14	csbeq2dv 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 } ) ) ) )
16	5, 15	eqtrd 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 } ) ) ) )
17	16	mpteq2dv 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 } ) ) ) ) )
18	17	eqeq2d 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 } ) ) ) ) }
20	18, 19	elrab2 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 } ) ) ) ) ) )
21	20	simprbi 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 } ) ) ) ) )
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 4423	. . . . . . 7 |- ( y = 0 -> ( y < ( 2nd ` T ) <-> 0 < ( 2nd ` T ) ) )
24		id 23	. . . . . . 7 |- ( y = 0 -> y = 0 )
25	23, 24	ifbieq1d 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 ) )
27	26	iftrued 3917	. . . . . 6 |- ( ph -> if ( 0 < ( 2nd ` T ) , 0 , ( y + 1 ) ) = 0 )
28	25, 27	sylan9eqr 2485	. . . . 5 |- ( ( ph /\ y = 0 ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = 0 )
29	28	csbeq1d 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 ) = (/)
33	31, 32	syl6eq 2479	. . . . . . . . . . . 12 |- ( j = 0 -> ( 1 ... j ) = (/) )
34	33	imaeq2d 5184	. . . . . . . . . . 11 |- ( j = 0 -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " (/) ) )
35	34	xpeq1d 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
38	36, 37	syl6eq 2479	. . . . . . . . . . . . 13 |- ( j = 0 -> ( j + 1 ) = 1 )
39	38	oveq1d 6317	. . . . . . . . . . . 12 |- ( j = 0 -> ( ( j + 1 ) ... N ) = ( 1 ... N ) )
40	39	imaeq2d 5184	. . . . . . . . . . 11 |- ( j = 0 -> ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) )
41	40	xpeq1d 4873	. . . . . . . . . 10 |- ( j = 0 -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) )
42	35, 41	uneq12d 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 ) ) " (/) ) = (/)
44	43	xpeq1i 4870	. . . . . . . . . . . 12 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) = ( (/) X. { 1 } )
45		0xp 4931	. . . . . . . . . . . 12 |- ( (/) X. { 1 } ) = (/)
46	44, 45	eqtri 2451	. . . . . . . . . . 11 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) = (/)
47	46	uneq1i 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 } )
50	47, 48, 49	3eqtri 2455	. . . . . . . . 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 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 } ) )
52	51	oveq2d 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 } ) ) )
53	30, 52	csbie 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 ) ) )
55	54, 19	eleq2s 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 ) ) )
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 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 ) } ) )
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 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 ) } )
60	58, 59	syl 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 ) ) )
63	61, 62	elab 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 ) )
64	60, 63	sylib 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 ) )
66	64, 65	syl 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 ) )
68	66, 67	syl 17	. . . . . . . 8 |- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) )
69	68	xpeq1d 4873	. . . . . . 7 |- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) = ( ( 1 ... N ) X. { 0 } ) )
70	69	oveq2d 6318	. . . . . 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 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 } ) ) )
72	71	adantr 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 } ) ) )
73	29, 72	eqtrd 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 )
76	74, 75	syl 17	. . . 4 |- ( ph -> ( N - 1 ) e. NN0 )
77		0elfz 11890	. . . 4 |- ( ( N - 1 ) e. NN0 -> 0 e. ( 0 ... ( N - 1 ) ) )
78	76, 77	syl 17	. . 3 |- ( ph -> 0 e. ( 0 ... ( N - 1 ) ) )
79		ovex 6330	. . . 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 5967	. 2 |- ( ph -> ( F ` 0 ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) )
82		ovex 6330	. . . 4 |- ( 1 ... N ) e. _V
83	82	a1i 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 ) ) )
85	58, 84	syl 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 ) )
87	85, 86	syl 17	. . 3 |- ( ph -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) )
88		fnconstg 5785	. . . 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 2423	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
91	30	fvconst2 6132	. . . 4 |- ( n e. ( 1 ... N ) -> ( ( ( 1 ... N ) X. { 0 } ) ` n ) = 0 )
92	91	adantl 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 ) )
94	85, 93	syl 17	. . . . . . 7 |- ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
95	94	ffvelrnda 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 )
97	95, 96	syl 17	. . . . 5 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. NN0 )
98	97	nn0cnd 10928	. . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. CC )
99	98	addid1d 9834	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 0 ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
100	83, 87, 89, 87, 90, 92, 99	offveq 6563	. 2 |- ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) = ( 1st ` ( 1st ` T ) ) )
101	81, 100	eqtrd 2463	1 |- ( ph -> ( F ` 0 ) = ( 1st ` ( 1st ` T ) ) )

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