Theorem poimirlem5 32009

Description: Lemma for poimir 32037 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 5879	. . . . . . . . . . . 12 |- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) )
3	2	breq2d 4407	. . . . . . . . . . 11 |- ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) )
4	3	ifbid 3894	. . . . . . . . . 10 |- ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) )
5	4	csbeq1d 3356	. . . . . . . . 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 5879	. . . . . . . . . . . 12 |- ( t = T -> ( 1st ` t ) = ( 1st ` T ) )
7	6	fveq2d 5883	. . . . . . . . . . 11 |- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) )
8	6	fveq2d 5883	. . . . . . . . . . . . . 14 |- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) )
9	8	imaeq1d 5173	. . . . . . . . . . . . 13 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) )
10	9	xpeq1d 4862	. . . . . . . . . . . 12 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) )
11	8	imaeq1d 5173	. . . . . . . . . . . . 13 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) )
12	11	xpeq1d 4862	. . . . . . . . . . . 12 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
13	10, 12	uneq12d 3580	. . . . . . . . . . 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 6326	. . . . . . . . . 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 3785	. . . . . . . . 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 2505	. . . . . . . 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 4483	. . . . . . 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 2481	. . . . . 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 3186	. . . . 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 471	. . . 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 4398	. . . . . . 7 |- ( y = 0 -> ( y < ( 2nd ` T ) <-> 0 < ( 2nd ` T ) ) )
24		id 22	. . . . . . 7 |- ( y = 0 -> y = 0 )
25	23, 24	ifbieq1d 3895	. . . . . 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 3880	. . . . . 6 |- ( ph -> if ( 0 < ( 2nd ` T ) , 0 , ( y + 1 ) ) = 0 )
28	25, 27	sylan9eqr 2527	. . . . 5 |- ( ( ph /\ y = 0 ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = 0 )
29	28	csbeq1d 3356	. . . 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 9655	. . . . . . 7 |- 0 e. _V
31		oveq2 6316	. . . . . . . . . . . . 13 |- ( j = 0 -> ( 1 ... j ) = ( 1 ... 0 ) )
32		fz10 11846	. . . . . . . . . . . . 13 |- ( 1 ... 0 ) = (/)
33	31, 32	syl6eq 2521	. . . . . . . . . . . 12 |- ( j = 0 -> ( 1 ... j ) = (/) )
34	33	imaeq2d 5174	. . . . . . . . . . 11 |- ( j = 0 -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " (/) ) )
35	34	xpeq1d 4862	. . . . . . . . . 10 |- ( j = 0 -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) )
36		oveq1 6315	. . . . . . . . . . . . . 14 |- ( j = 0 -> ( j + 1 ) = ( 0 + 1 ) )
37		0p1e1 10743	. . . . . . . . . . . . . 14 |- ( 0 + 1 ) = 1
38	36, 37	syl6eq 2521	. . . . . . . . . . . . 13 |- ( j = 0 -> ( j + 1 ) = 1 )
39	38	oveq1d 6323	. . . . . . . . . . . 12 |- ( j = 0 -> ( ( j + 1 ) ... N ) = ( 1 ... N ) )
40	39	imaeq2d 5174	. . . . . . . . . . 11 |- ( j = 0 -> ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) )
41	40	xpeq1d 4862	. . . . . . . . . 10 |- ( j = 0 -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) )
42	35, 41	uneq12d 3580	. . . . . . . . 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 5189	. . . . . . . . . . . . 13 |- ( ( 2nd ` ( 1st ` T ) ) " (/) ) = (/)
44	43	xpeq1i 4859	. . . . . . . . . . . 12 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) = ( (/) X. { 1 } )
45		0xp 4920	. . . . . . . . . . . 12 |- ( (/) X. { 1 } ) = (/)
46	44, 45	eqtri 2493	. . . . . . . . . . 11 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) = (/)
47	46	uneq1i 3575	. . . . . . . . . 10 |- ( ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) ) = ( (/) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) )
48		uncom 3569	. . . . . . . . . 10 |- ( (/) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) ) = ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) u. (/) )
49		un0 3762	. . . . . . . . . 10 |- ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) u. (/) ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } )
50	47, 48, 49	3eqtri 2497	. . . . . . . . 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 2521	. . . . . . . 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 6324	. . . . . . 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 3375	. . . . . 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 3181	. . . . . . . . . . . . . . 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 2567	. . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . 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 6843	. . . . . . . . . . . 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 5889	. . . . . . . . . . . 12 |- ( 2nd ` ( 1st ` T ) ) e. _V
62		f1oeq1 5818	. . . . . . . . . . . 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 3173	. . . . . . . . . . 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 201	. . . . . . . . . 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 5811	. . . . . . . . 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 4862	. . . . . . 7 |- ( ph -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) X. { 0 } ) = ( ( 1 ... N ) X. { 0 } ) )
70	69	oveq2d 6324	. . . . . 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 2517	. . . . 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 472	. . . 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 2505	. . 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 10935	. . . . 5 |- ( N e. NN -> ( N - 1 ) e. NN0 )
76	74, 75	syl 17	. . . 4 |- ( ph -> ( N - 1 ) e. NN0 )
77		0elfz 11915	. . . 4 |- ( ( N - 1 ) e. NN0 -> 0 e. ( 0 ... ( N - 1 ) ) )
78	76, 77	syl 17	. . 3 |- ( ph -> 0 e. ( 0 ... ( N - 1 ) ) )
79		ovex 6336	. . . 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 5969	. 2 |- ( ph -> ( F ` 0 ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) )
82		ovex 6336	. . . 4 |- ( 1 ... N ) e. _V
83	82	a1i 11	. . 3 |- ( ph -> ( 1 ... N ) e. _V )
84		xp1st 6842	. . . . 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 7512	. . . 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 5784	. . . 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 2472	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
91	30	fvconst2 6136	. . . 4 |- ( n e. ( 1 ... N ) -> ( ( ( 1 ... N ) X. { 0 } ) ` n ) = 0 )
92	91	adantl 473	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1 ... N ) X. { 0 } ) ` n ) = 0 )
93		elmapi 7511	. . . . . . . 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 6037	. . . . . 6 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. ( 0..^ K ) )
96		elfzonn0 11988	. . . . . 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 10951	. . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. CC )
99	98	addid1d 9851	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 0 ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
100	83, 87, 89, 87, 90, 92, 99	offveq 6571	. 2 |- ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 0 } ) ) = ( 1st ` ( 1st ` T ) ) )
101	81, 100	eqtrd 2505	1 |- ( ph -> ( F ` 0 ) = ( 1st ` ( 1st ` T ) ) )

Syntax hints:   -> wi 4   /\ wa 376   = wceq 1452   e. wcel 1904   {cab 2457   {crab 2760   _Vcvv 3031   [_csb 3349   u. cun 3388   (/)c0 3722   ifcif 3872   {csn 3959   class class class wbr 4395   |-> cmpt 4454   X. cxp 4837   "cima 4842   Fn wfn 5584   -->wf 5585   -onto->wfo 5587   -1-1-onto->wf1o 5588   ` cfv 5589  (class class class)co 6308   oFcof 6548   1stc1st 6810   2ndc2nd 6811   ^m cmap 7490   0cc0 9557   1c1 9558   + caddc 9560   < clt 9693   - cmin 9880   NNcn 10631   NN0cn0 10893   ...cfz 11810  ..^cfzo 11942

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-of 6550  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-map 7492  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-fzo 11943

This theorem is referenced by:  poimirlem12  32016  poimirlem14  32018