Theorem poimirlem10 31950

Description: Lemma for poimir 31973 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 6318	. . 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 10911	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
6	4, 5	syl 17	. . . . 5 |- ( ph -> ( N - 1 ) e. NN0 )
7		nn0fz0 11890	. . . . 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 6023	. . 3 |- ( ph -> ( F ` ( N - 1 ) ) e. ( ( 0 ... K ) ^m ( 1 ... N ) ) )
10		elmapfn 7494	. . 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 9638	. . 3 |- 1 e. _V
13		fnconstg 5771	. . 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 3193	. . . . . . 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 2547	. . . . . 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 6823	. . . . 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 6823	. . . 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 7494	. . 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 5865	. . . . . . . . . . . . . . 15 |- ( t = T -> ( 2nd ` t ) = ( 2nd ` T ) )
27	26	breq2d 4414	. . . . . . . . . . . . . 14 |- ( t = T -> ( y < ( 2nd ` t ) <-> y < ( 2nd ` T ) ) )
28	27	ifbid 3903	. . . . . . . . . . . . 13 |- ( t = T -> if ( y < ( 2nd ` t ) , y , ( y + 1 ) ) = if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) )
29	28	csbeq1d 3370	. . . . . . . . . . . 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 5865	. . . . . . . . . . . . . . 15 |- ( t = T -> ( 1st ` t ) = ( 1st ` T ) )
31	30	fveq2d 5869	. . . . . . . . . . . . . 14 |- ( t = T -> ( 1st ` ( 1st ` t ) ) = ( 1st ` ( 1st ` T ) ) )
32	30	fveq2d 5869	. . . . . . . . . . . . . . . . 17 |- ( t = T -> ( 2nd ` ( 1st ` t ) ) = ( 2nd ` ( 1st ` T ) ) )
33	32	imaeq1d 5167	. . . . . . . . . . . . . . . 16 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) )
34	33	xpeq1d 4857	. . . . . . . . . . . . . . 15 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) )
35	32	imaeq1d 5167	. . . . . . . . . . . . . . . 16 |- ( t = T -> ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) )
36	35	xpeq1d 4857	. . . . . . . . . . . . . . 15 |- ( t = T -> ( ( ( 2nd ` ( 1st ` t ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) )
37	34, 36	uneq12d 3589	. . . . . . . . . . . . . 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 6308	. . . . . . . . . . . . 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 3781	. . . . . . . . . . . 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 2485	. . . . . . . . . . 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 4490	. . . . . . . . . 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 2461	. . . . . . . . 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 3198	. . . . . . . 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 4407	. . . . . . . . . . . 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 6297	. . . . . . . . . . 11 |- ( y = ( N - 1 ) -> ( y + 1 ) = ( ( N - 1 ) + 1 ) )
51	4	nncnd 10625	. . . . . . . . . . . 12 |- ( ph -> N e. CC )
52		npcan1 10044	. . . . . . . . . . . 12 |- ( N e. CC -> ( ( N - 1 ) + 1 ) = N )
53	51, 52	syl 17	. . . . . . . . . . 11 |- ( ph -> ( ( N - 1 ) + 1 ) = N )
54	50, 53	sylan9eqr 2507	. . . . . . . . . 10 |- ( ( ph /\ y = ( N - 1 ) ) -> ( y + 1 ) = N )
55	49, 54	ifbieq2d 3906	. . . . . . . . 9 |- ( ( ph /\ y = ( N - 1 ) ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = if ( ( N - 1 ) < 0 , y , N ) )
56	6	nn0ge0d 10928	. . . . . . . . . . . 12 |- ( ph -> 0 <_ ( N - 1 ) )
57		0red 9644	. . . . . . . . . . . . 13 |- ( ph -> 0 e. RR )
58	6	nn0red 10926	. . . . . . . . . . . . 13 |- ( ph -> ( N - 1 ) e. RR )
59	57, 58	lenltd 9781	. . . . . . . . . . . 12 |- ( ph -> ( 0 <_ ( N - 1 ) <-> -. ( N - 1 ) < 0 ) )
60	56, 59	mpbid 214	. . . . . . . . . . 11 |- ( ph -> -. ( N - 1 ) < 0 )
61	60	iffalsed 3892	. . . . . . . . . 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 2485	. . . . . . . 8 |- ( ( ph /\ y = ( N - 1 ) ) -> if ( y < ( 2nd ` T ) , y , ( y + 1 ) ) = N )
64	63	csbeq1d 3370	. . . . . . 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 6298	. . . . . . . . . . . . . . 15 |- ( j = N -> ( 1 ... j ) = ( 1 ... N ) )
66	65	imaeq2d 5168	. . . . . . . . . . . . . 14 |- ( j = N -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... N ) ) )
67		xp2nd 6824	. . . . . . . . . . . . . . . . 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 5875	. . . . . . . . . . . . . . . . 17 |- ( 2nd ` ( 1st ` T ) ) e. _V
70		f1oeq1 5805	. . . . . . . . . . . . . . . . 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 3185	. . . . . . . . . . . . . . . 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 5821	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -1-1-onto-> ( 1 ... N ) -> ( 2nd ` ( 1st ` T ) ) : ( 1 ... N ) -onto-> ( 1 ... N ) )
74		foima 5798	. . . . . . . . . . . . . . 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 2507	. . . . . . . . . . . . 13 |- ( ( ph /\ j = N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) = ( 1 ... N ) )
77	76	xpeq1d 4857	. . . . . . . . . . . 12 |- ( ( ph /\ j = N ) -> ( ( ( 2nd ` ( 1st ` T ) ) " ( 1 ... j ) ) X. { 1 } ) = ( ( 1 ... N ) X. { 1 } ) )
78		oveq1 6297	. . . . . . . . . . . . . . . . 17 |- ( j = N -> ( j + 1 ) = ( N + 1 ) )
79	78	oveq1d 6305	. . . . . . . . . . . . . . . 16 |- ( j = N -> ( ( j + 1 ) ... N ) = ( ( N + 1 ) ... N ) )
80	4	nnred 10624	. . . . . . . . . . . . . . . . . 18 |- ( ph -> N e. RR )
81	80	ltp1d 10537	. . . . . . . . . . . . . . . . 17 |- ( ph -> N < ( N + 1 ) )
82	4	nnzd 11039	. . . . . . . . . . . . . . . . . . 19 |- ( ph -> N e. ZZ )
83	82	peano2zd 11043	. . . . . . . . . . . . . . . . . 18 |- ( ph -> ( N + 1 ) e. ZZ )
84		fzn 11815	. . . . . . . . . . . . . . . . . 18 |- ( ( ( N + 1 ) e. ZZ /\ N e. ZZ ) -> ( N < ( N + 1 ) <-> ( ( N + 1 ) ... N ) = (/) ) )
85	83, 82, 84	syl2anc 667	. . . . . . . . . . . . . . . . 17 |- ( ph -> ( N < ( N + 1 ) <-> ( ( N + 1 ) ... N ) = (/) ) )
86	81, 85	mpbid 214	. . . . . . . . . . . . . . . 16 |- ( ph -> ( ( N + 1 ) ... N ) = (/) )
87	79, 86	sylan9eqr 2507	. . . . . . . . . . . . . . 15 |- ( ( ph /\ j = N ) -> ( ( j + 1 ) ... N ) = (/) )
88	87	imaeq2d 5168	. . . . . . . . . . . . . 14 |- ( ( ph /\ j = N ) -> ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) = ( ( 2nd ` ( 1st ` T ) ) " (/) ) )
89	88	xpeq1d 4857	. . . . . . . . . . . . 13 |- ( ( ph /\ j = N ) -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 0 } ) )
90		ima0 5183	. . . . . . . . . . . . . . 15 |- ( ( 2nd ` ( 1st ` T ) ) " (/) ) = (/)
91	90	xpeq1i 4854	. . . . . . . . . . . . . 14 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 0 } ) = ( (/) X. { 0 } )
92		0xp 4915	. . . . . . . . . . . . . 14 |- ( (/) X. { 0 } ) = (/)
93	91, 92	eqtri 2473	. . . . . . . . . . . . 13 |- ( ( ( 2nd ` ( 1st ` T ) ) " (/) ) X. { 0 } ) = (/)
94	89, 93	syl6eq 2501	. . . . . . . . . . . 12 |- ( ( ph /\ j = N ) -> ( ( ( 2nd ` ( 1st ` T ) ) " ( ( j + 1 ) ... N ) ) X. { 0 } ) = (/) )
95	77, 94	uneq12d 3589	. . . . . . . . . . 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 3759	. . . . . . . . . . 11 |- ( ( ( 1 ... N ) X. { 1 } ) u. (/) ) = ( ( 1 ... N ) X. { 1 } )
97	95, 96	syl6eq 2501	. . . . . . . . . 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 6306	. . . . . . . . 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 3390	. . . . . . . 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 2485	. . . . . 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 6318	. . . . . . 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 5954	. . . . 5 |- ( ph -> ( F ` ( N - 1 ) ) = ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) )
105	104	fveq1d 5867	. . . 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 3641	. . . 4 |- ( ( 1 ... N ) i^i ( 1 ... N ) ) = ( 1 ... N )
108		eqidd 2452	. . . 4 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) = ( ( 1st ` ( 1st ` T ) ) ` n ) )
109	12	fvconst2 6120	. . . . 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 6540	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( ( 1st ` ( 1st ` T ) ) oF + ( ( 1 ... N ) X. { 1 } ) ) ` n ) = ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 1 ) )
112	106, 111	eqtrd 2485	. 2 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( F ` ( N - 1 ) ) ` n ) = ( ( ( 1st ` ( 1st ` T ) ) ` n ) + 1 ) )
113		elmapi 7493	. . . . . . 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 6022	. . . . 5 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. ( 0..^ K ) )
116		elfzonn0 11960	. . . . 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 10927	. . 3 |- ( ( ph /\ n e. ( 1 ... N ) ) -> ( ( 1st ` ( 1st ` T ) ) ` n ) e. CC )
119		pncan1 10043	. . 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 6552	1 |- ( ph -> ( ( F ` ( N - 1 ) ) oF - ( ( 1 ... N ) X. { 1 } ) ) = ( 1st ` ( 1st ` T ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 188   /\ 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   CCcc 9537   0cc0 9539   1c1 9540   + caddc 9542   < clt 9675   <_ cle 9676   - cmin 9860   NNcn 10609   NN0cn0 10869   ZZcz 10937   ...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:  poimirlem11  31951  poimirlem13  31953