Theorem poimirlem10 31908

Description: Lemma for poimir 31931 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.

Step	Hyp	Ref	Expression
1		ovex 6331	. . 3 |- ( 1 ... N ) e. _V
2	1	a1i 11	. 2 |- ( ph -> ( 1 ... N ) e. _V )
3		poimirlem22.1	. . . 4 |- ( ph -> F : ( 0 ... ( N - 1 ) ) --> ( ( 0 ... K ) ^m ( 1 ... N ) ) )
4		poimir.0	. . . . . 6 |- ( ph -> N e. NN )
5		nnm1nn0 10913	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
6	4, 5	syl 17	. . . . 5 |- ( ph -> ( N - 1 ) e. NN0 )
7		nn0fz0 11892	. . . . 5 |- ( ( N - 1 ) e. NN0 <-> ( N - 1 ) e. ( 0 ... ( N - 1 ) ) )
8	6, 7	sylib 200	. . . 4 |- ( ph -> ( N - 1 ) e. ( 0 ... ( N - 1 ) ) )
9	3, 8	ffvelrnd 6036	. . 3 |- ( ph -> ( F ` ( N - 1 ) ) e. ( ( 0 ... K ) ^m ( 1 ... N ) ) )
10		elmapfn 7500	. . 3 |- ( ( F ` ( N - 1 ) ) e. ( ( 0 ... K ) ^m ( 1 ... N ) ) -> ( F ` ( N - 1 ) ) Fn ( 1 ... N ) )
11	9, 10	syl 17	. 2 |- ( ph -> ( F ` ( N - 1 ) ) Fn ( 1 ... N ) )
12		1ex 9640	. . 3 |- 1 e. _V
13		fnconstg 5786	. . 3 |- ( 1 e. _V -> ( ( 1 ... N ) X. { 1 } ) Fn ( 1 ... N ) )
14	12, 13	mp1i 13	. 2 |- ( ph -> ( ( 1 ... N ) X. { 1 } ) Fn ( 1 ... N ) )
15		poimirlem12.2	. . . . . 6 |- ( ph -> T e. S )
16		elrabi 3227	. . . . . . 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 ) ) )
17		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 } ) ) ) ) }
18	16, 17	eleq2s 2531	. . . . . 6 |- ( T e. S -> T e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
19	15, 18	syl 17	. . . . 5 |- ( ph -> T e. ( ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) X. ( 0 ... N ) ) )
20		xp1st 6835	. . . . 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 ) } ) )
21	19, 20	syl 17	. . . 4 |- ( ph -> ( 1st ` T ) e. ( ( ( 0..^ K ) ^m ( 1 ... N ) ) X. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } ) )
22		xp1st 6835	. . . 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 ) ) )
23	21, 22	syl 17	. . 3 |- ( ph -> ( 1st ` ( 1st ` T ) ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) )
24		elmapfn 7500	. . 3 |- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) )
25	23, 24	syl 17	. 2 |- ( ph -> ( 1st ` ( 1st ` T ) ) Fn ( 1 ... N ) )
26		fveq2 5879	. . . . . . . . . . . . . . 15 |- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) )
27	26	breq2d 4433	. . . . . . . . . . . . . 14 |- ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) )
28	27	ifbid 3932	. . . . . . . . . . . . 13 |- ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) )
29	28	csbeq1d 3403	. . . . . . . . . . . 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 } ) ) ) )
30		fveq2 5879	. . . . . . . . . . . . . . 15 |- ( t = T -> ( 1st ` t ) = ( 1st ` T ) )
31	30	fveq2d 5883	. . . . . . . . . . . . . 14 |- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) )
32	30	fveq2d 5883	. . . . . . . . . . . . . . . . 17 |- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) )
33	32	imaeq1d 5184	. . . . . . . . . . . . . . . 16 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) )
34	33	xpeq1d 4874	. . . . . . . . . . . . . . 15 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) )
35	32	imaeq1d 5184	. . . . . . . . . . . . . . . 16 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) )
36	35	xpeq1d 4874	. . . . . . . . . . . . . . 15 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
37	34, 36	uneq12d 3622	. . . . . . . . . . . . . 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 } ) ) )
38	31, 37	oveq12d 6321	. . . . . . . . . . . . 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 } ) ) ) )
39	38	csbeq2dv 3810	. . . . . . . . . . . 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 } ) ) ) )
40	29, 39	eqtrd 2464	. . . . . . . . . . 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 } ) ) ) )
41	40	mpteq2dv 4509	. . . . . . . . . 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 } ) ) ) ) )
42	41	eqeq2d 2437	. . . . . . . . 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 } ) ) ) ) ) )
43	42, 17	elrab2 3232	. . . . . . . 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 } ) ) ) ) ) )
44	43	simprbi 466	. . . . . . 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 } ) ) ) ) )
45	15, 44	syl 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 } ) ) ) ) )
46		poimirlem11.3	. . . . . . . . . . . 12 |- ( ph -> ( 2nd ` T ) = 0 )
47		breq12 4426	. . . . . . . . . . . 12 |- ( ( y = ( N - 1 ) /\ ( 2nd ` T ) = 0 ) -> ( y < ( 2nd ` T ) <-> ( N - 1 ) < 0 ) )
48	46, 47	sylan2 477	. . . . . . . . . . 11 |- ( ( y = ( N - 1 ) /\ ph ) -> ( y < ( 2nd ` T ) <-> ( N - 1 ) < 0 ) )
49	48	ancoms 455	. . . . . . . . . 10 |- ( ( ph /\ y = ( N - 1 ) ) -> ( y < ( 2nd ` T ) <-> ( N - 1 ) < 0 ) )
50		oveq1 6310	. . . . . . . . . . 11 |- ( y = ( N - 1 ) -> ( y + 1 ) = ( ( N - 1 ) + 1 ) )
51	4	nncnd 10627	. . . . . . . . . . . 12 |- ( ph -> N e. CC )
52		npcan1 10046	. . . . . . . . . . . 12 |- ( N e. CC -> ( ( N - 1 ) + 1 ) = N )
53	51, 52	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
54	50, 53	sylan9eqr 2486	. . . . . . . . . 10 |- ( ( ph /\ y = ( N - 1 ) ) -> ( y + 1 ) = N )
55	49, 54	ifbieq2d 3935	. . . . . . . . 9 |- ( ( ph /\ y = ( N - 1 ) ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = if ( ( N - 1 ) < 0 , y , N ) )
56	6	nn0ge0d 10930	. . . . . . . . . . . 12 |- ( ph -> 0 <_ ( N - 1 ) )
57		0red 9646	. . . . . . . . . . . . 13 |- ( ph -> 0 e. RR )
58	6	nn0red 10928	. . . . . . . . . . . . 13 |- ( ph -> ( N - 1 ) e. RR )
59	57, 58	lenltd 9783	. . . . . . . . . . . 12 |- ( ph -> ( 0 <_ ( N - 1 ) <-> -. ( N - 1 ) < 0 ) )
60	56, 59	mpbid 214	. . . . . . . . . . 11 |- ( ph -> -. ( N - 1 ) < 0 )
61	60	iffalsed 3921	. . . . . . . . . 10 |- ( ph -> if ( ( N - 1 ) < 0 , y , N ) = N )
62	61	adantr 467	. . . . . . . . 9 |- ( ( ph /\ y = ( N - 1 ) ) -> if ( ( N - 1 ) < 0 , y , N ) = N )
63	55, 62	eqtrd 2464	. . . . . . . 8 |- ( ( ph /\ y = ( N - 1 ) ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = N )
64	63	csbeq1d 3403	. . . . . . 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 } ) ) ) )
65		oveq2 6311	. . . . . . . . . . . . . . 15 |- ( j = N -> ( 1 ... j ) = ( 1 ... N ) )
66	65	imaeq2d 5185	. . . . . . . . . . . . . 14 |- ( j = N -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) )
67		xp2nd 6836	. . . . . . . . . . . . . . . . 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 ) } )
68	21, 67	syl 17	. . . . . . . . . . . . . . . 16 |- ( ph -> ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } )
69		fvex 5889	. . . . . . . . . . . . . . . . 17 |- ( 2nd ` ( 1st ` T ) ) e. _V
70		f1oeq1 5820	. . . . . . . . . . . . . . . . 17 |- ( f = ( 2nd ` ( 1st ` T ) ) -> ( f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) ) )
71	69, 70	elab 3219	. . . . . . . . . . . . . . . 16 |- ( ( 2nd ` ( 1st ` T ) ) e. { f | f : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) } <-> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
72	68, 71	sylib 200	. . . . . . . . . . . . . . 15 |- ( ph -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) )
73		f1ofo 5836	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) )
74		foima 5813	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) )
75	72, 73, 74	3syl 18	. . . . . . . . . . . . . 14 |- ( ph -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) = ( 1 ... N ) )
76	66, 75	sylan9eqr 2486	. . . . . . . . . . . . 13 |- ( ( ph /\ j = N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( 1 ... N ) )
77	76	xpeq1d 4874	. . . . . . . . . . . 12 |- ( ( ph /\ j = N ) -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( 1 ... N ) X. { 1 } ) )
78		oveq1 6310	. . . . . . . . . . . . . . . . 17 |- ( j = N -> ( j + 1 ) = ( N + 1 ) )
79	78	oveq1d 6318	. . . . . . . . . . . . . . . 16 |- ( j = N -> ( ( j + 1 ) ... N ) = ( ( N + 1 ) ... N ) )
80	4	nnred 10626	. . . . . . . . . . . . . . . . . 18 |- ( ph -> N e. RR )
81	80	ltp1d 10539	. . . . . . . . . . . . . . . . 17 |- ( ph -> N < ( N + 1 ) )
82	4	nnzd 11041	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> N e. ZZ )
83	82	peano2zd 11045	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( N + 1 ) e. ZZ )
84		fzn 11817	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N + 1 ) e. ZZ /\ N e. ZZ ) -> ( N < ( N + 1 ) <-> ( ( N + 1 ) ... N ) = (/) ) )
85	83, 82, 84	syl2anc 666	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( N < ( N + 1 ) <-> ( ( N + 1 ) ... N ) = (/) ) )
86	81, 85	mpbid 214	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( N + 1 ) ... N ) = (/) )
87	79, 86	sylan9eqr 2486	. . . . . . . . . . . . . . 15 |- ( ( ph /\ j = N ) -> ( ( j + 1 ) ... N ) = (/) )
88	87	imaeq2d 5185	. . . . . . . . . . . . . 14 |- ( ( ph /\ j = N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " (/) ) )
89	88	xpeq1d 4874	. . . . . . . . . . . . 13 |- ( ( ph /\ j = N ) -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 0 } ) )
90		ima0 5200	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` ( 1st ` T ) ) " (/) ) = (/)
91	90	xpeq1i 4871	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 0 } ) = ( (/) X. { 0 } )
92		0xp 4932	. . . . . . . . . . . . . 14 |- ( (/) X. { 0 } ) = (/)
93	91, 92	eqtri 2452	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 0 } ) = (/)
94	89, 93	syl6eq 2480	. . . . . . . . . . . 12 |- ( ( ph /\ j = N ) -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = (/) )
95	77, 94	uneq12d 3622	. . . . . . . . . . 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. (/) ) )
96		un0 3788	. . . . . . . . . . 11 |- ( ( ( 1 ... N ) X. { 1 } ) u. (/) ) = ( ( 1 ... N ) X. { 1 } )
97	95, 96	syl6eq 2480	. . . . . . . . . 10 |- ( ( ph /\ j = N ) -> ( ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) u. ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) ) = ( ( 1 ... N ) X. { 1 } ) )
98	97	oveq2d 6319	. . . . . . . . 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 } ) ) )
99	4, 98	csbied 3423	. . . . . . . 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 } ) ) )
100	99	adantr 467	. . . . . . 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 } ) ) )
101	64, 100	eqtrd 2464	. . . . . 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 } ) ) )
102		ovex 6331	. . . . . . 7 |- ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) e. _V
103	102	a1i 11	. . . . . 6 |- ( ph -> ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) e. _V )
104	45, 101, 8, 103	fvmptd 5968	. . . . 5 |- ( ph -> ( F ` ( N - 1 ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) )
105	104	fveq1d 5881	. . . 4 |- ( ph -> ( ( F ` ( N - 1 ) ) ` n ) = ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) ` n ) )
106	105	adantr 467	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( F ` ( N - 1 ) ) ` n ) = ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) ` n ) )
107		inidm 3672	. . . 4 |- ( ( 1 ... N ) i^i ( 1 ... N ) ) = ( 1 ... N )
108		eqidd 2424	. . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
109	12	fvconst2 6133	. . . . 5 |- ( n e. ( 1 ... N ) -> ( ( ( 1 ... N ) X. { 1 } ) ` n ) = 1 )
110	109	adantl 468	. . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1 ... N ) X. { 1 } ) ` n ) = 1 )
111	25, 14, 2, 2, 107, 108, 110	ofval 6552	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) ` n ) = ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 1 ) )
112	106, 111	eqtrd 2464	. 2 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( F ` ( N - 1 ) ) ` n ) = ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 1 ) )
113		elmapi 7499	. . . . . . 7 |- ( ( 1st ` ( 1st ` T ) ) e. ( ( 0..^ K ) ^m ( 1 ... N ) ) -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
114	23, 113	syl 17	. . . . . 6 |- ( ph -> ( 1st ` ( 1st ` T ) ) : ( 1 ... N ) --> ( 0..^ K ) )
115	114	ffvelrnda 6035	. . . . 5 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. ( 0..^ K ) )
116		elfzonn0 11962	. . . . 5 |- ( ( ( 1st ` ( 1st ` T ) ) ` n ) e. ( 0..^ K ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. NN0 )
117	115, 116	syl 17	. . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. NN0 )
118	117	nn0cnd 10929	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. CC )
119		pncan1 10045	. . 3 |- ( ( ( 1st ` ( 1st ` T ) ) ` n ) e. CC -> ( ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 1 ) - 1 ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
120	118, 119	syl 17	. 2 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 1 ) - 1 ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
121	2, 11, 14, 25, 112, 110, 120	offveq 6564	1 |- ( ph -> ( ( F ` ( N - 1 ) ) oF - ( ( 1 ... N ) X. { 1 } ) ) = ( 1st ` ( 1st ` T ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ wa 371   = wceq 1438   e. wcel 1869   {cab 2408   {crab 2780   _Vcvv 3082   [_csb 3396   u. cun 3435   (/)c0 3762   ifcif 3910   {csn 3997   class class class wbr 4421   |-> cmpt 4480   X. cxp 4849   "cima 4854   Fn wfn 5594   -->wf 5595   -onto->wfo 5597   -1-1-onto->wf1o 5598   ` cfv 5599  (class class class)co 6303   oFcof 6541   1stc1st 6803   2ndc2nd 6804   ^m cmap 7478   CCcc 9539   0cc0 9541   1c1 9542   + caddc 9544   < clt 9677   <_ cle 9678   - cmin 9862   NNcn 10611   NN0cn0 10871   ZZcz 10939   ...cfz 11786  ..^cfzo 11917

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-8 1871  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-rep 4534  ax-sep 4544  ax-nul 4553  ax-pow 4600  ax-pr 4658  ax-un 6595  ax-cnex 9597  ax-resscn 9598  ax-1cn 9599  ax-icn 9600  ax-addcl 9601  ax-addrcl 9602  ax-mulcl 9603  ax-mulrcl 9604  ax-mulcom 9605  ax-addass 9606  ax-mulass 9607  ax-distr 9608  ax-i2m1 9609  ax-1ne0 9610  ax-1rid 9611  ax-rnegex 9612  ax-rrecex 9613  ax-cnre 9614  ax-pre-lttri 9615  ax-pre-lttrn 9616  ax-pre-ltadd 9617  ax-pre-mulgt0 9618

This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 984  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-nel 2622  df-ral 2781  df-rex 2782  df-reu 2783  df-rab 2785  df-v 3084  df-sbc 3301  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3763  df-if 3911  df-pw 3982  df-sn 3998  df-pr 4000  df-tp 4002  df-op 4004  df-uni 4218  df-iun 4299  df-br 4422  df-opab 4481  df-mpt 4482  df-tr 4517  df-eprel 4762  df-id 4766  df-po 4772  df-so 4773  df-fr 4810  df-we 4812  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864  df-pred 5397  df-ord 5443  df-on 5444  df-lim 5445  df-suc 5446  df-iota 5563  df-fun 5601  df-fn 5602  df-f 5603  df-f1 5604  df-fo 5605  df-f1o 5606  df-fv 5607  df-riota 6265  df-ov 6306  df-oprab 6307  df-mpt2 6308  df-of 6543  df-om 6705  df-1st 6805  df-2nd 6806  df-wrecs 7034  df-recs 7096  df-rdg 7134  df-er 7369  df-map 7480  df-en 7576  df-dom 7577  df-sdom 7578  df-pnf 9679  df-mnf 9680  df-xr 9681  df-ltxr 9682  df-le 9683  df-sub 9864  df-neg 9865  df-nn 10612  df-n0 10872  df-z 10940  df-uz 11162  df-fz 11787  df-fzo 11918

This theorem is referenced by:  poimirlem11  31909  poimirlem13  31911