Theorem ballotlemfrc 30588

Description: Express the value of ( F ` ( R ` C ) ) in terms of the newly defined .^. (Contributed by Thierry Arnoux, 21-Apr-2017.)

Hypotheses

Ref	Expression
ballotth.m	|- M e. NN
ballotth.n	|- N e. NN
ballotth.o	|- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
ballotth.p	|- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
ballotth.f	|- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
ballotth.e	|- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
ballotth.mgtn	|- N < M
ballotth.i	|- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
ballotth.s	|- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
ballotth.r	|- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
ballotlemg	|- .^ = ( u e. Fin , v e. Fin |-> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) )

Assertion

Ref	Expression
ballotlemfrc	|- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( F ` ( R ` C ) ) ` J ) = ( C .^ ( ( ( S ` C ) ` J ) ... ( I ` C ) ) ) )

Distinct variable groups:   M, c   N, c   O, c   i, M   i, N   i, O   k, M   k, N   k, O   i, c, F, k   C, i, k   i, E, k   C, k   k, I, c   E, c   i, I, c   k, J   S, k, i, c   R, i   v, u, C   u, I, v   u, J, v   u, R, v   u, S, v   i, J

Allowed substitution hints:   C( x, c)   P( x, v, u, i, k, c)   R( x, k, c)   S( x)   E( x, v, u)   .^ ( x, v, u, i, k, c)   F( x, v, u)   I( x)   J( x, c)   M( x, v, u)   N( x, v, u)   O( x, v, u)

Proof of Theorem ballotlemfrc

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . . . . 9 |- M e. NN
2		ballotth.n	. . . . . . . . 9 |- N e. NN
3		ballotth.o	. . . . . . . . 9 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4		ballotth.p	. . . . . . . . 9 |- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
5		ballotth.f	. . . . . . . . 9 |- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
6		ballotth.e	. . . . . . . . 9 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
7		ballotth.mgtn	. . . . . . . . 9 |- N < M
8		ballotth.i	. . . . . . . . 9 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
9		ballotth.s	. . . . . . . . 9 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
10	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsf1o 30575	. . . . . . . 8 |- ( C e. ( O \ E ) -> ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) /\ `' ( S ` C ) = ( S ` C ) ) )
11	10	simpld 475	. . . . . . 7 |- ( C e. ( O \ E ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) )
12		f1of1 6136	. . . . . . 7 |- ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) )
13	11, 12	syl 17	. . . . . 6 |- ( C e. ( O \ E ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) )
14	13	adantr 481	. . . . 5 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) )
15	1, 2, 3, 4, 5, 6, 7, 8	ballotlemiex 30563	. . . . . . . . . . 11 |- ( C e. ( O \ E ) -> ( ( I ` C ) e. ( 1 ... ( M + N ) ) /\ ( ( F ` C ) ` ( I ` C ) ) = 0 ) )
16	15	simpld 475	. . . . . . . . . 10 |- ( C e. ( O \ E ) -> ( I ` C ) e. ( 1 ... ( M + N ) ) )
17	16	adantr 481	. . . . . . . . 9 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( I ` C ) e. ( 1 ... ( M + N ) ) )
18		elfzuz3 12339	. . . . . . . . 9 |- ( ( I ` C ) e. ( 1 ... ( M + N ) ) -> ( M + N ) e. ( ZZ>= ` ( I ` C ) ) )
19	17, 18	syl 17	. . . . . . . 8 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( M + N ) e. ( ZZ>= ` ( I ` C ) ) )
20		elfzuz3 12339	. . . . . . . . 9 |- ( J e. ( 1 ... ( I ` C ) ) -> ( I ` C ) e. ( ZZ>= ` J ) )
21	20	adantl 482	. . . . . . . 8 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( I ` C ) e. ( ZZ>= ` J ) )
22		uztrn 11704	. . . . . . . 8 |- ( ( ( M + N ) e. ( ZZ>= ` ( I ` C ) ) /\ ( I ` C ) e. ( ZZ>= ` J ) ) -> ( M + N ) e. ( ZZ>= ` J ) )
23	19, 21, 22	syl2anc 693	. . . . . . 7 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( M + N ) e. ( ZZ>= ` J ) )
24		fzss2 12381	. . . . . . 7 |- ( ( M + N ) e. ( ZZ>= ` J ) -> ( 1 ... J ) C_ ( 1 ... ( M + N ) ) )
25	23, 24	syl 17	. . . . . 6 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( 1 ... J ) C_ ( 1 ... ( M + N ) ) )
26		ssinss1 3841	. . . . . 6 |- ( ( 1 ... J ) C_ ( 1 ... ( M + N ) ) -> ( ( 1 ... J ) i^i ( R ` C ) ) C_ ( 1 ... ( M + N ) ) )
27	25, 26	syl 17	. . . . 5 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( 1 ... J ) i^i ( R ` C ) ) C_ ( 1 ... ( M + N ) ) )
28		f1ores 6151	. . . . 5 |- ( ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) /\ ( ( 1 ... J ) i^i ( R ` C ) ) C_ ( 1 ... ( M + N ) ) ) -> ( ( S ` C ) |` ( ( 1 ... J ) i^i ( R ` C ) ) ) : ( ( 1 ... J ) i^i ( R ` C ) ) -1-1-onto-> ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) )
29	14, 27, 28	syl2anc 693	. . . 4 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( S ` C ) |` ( ( 1 ... J ) i^i ( R ` C ) ) ) : ( ( 1 ... J ) i^i ( R ` C ) ) -1-1-onto-> ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) )
30		ovex 6678	. . . . . 6 |- ( 1 ... J ) e. _V
31	30	inex1 4799	. . . . 5 |- ( ( 1 ... J ) i^i ( R ` C ) ) e. _V
32	31	f1oen 7976	. . . 4 |- ( ( ( S ` C ) |` ( ( 1 ... J ) i^i ( R ` C ) ) ) : ( ( 1 ... J ) i^i ( R ` C ) ) -1-1-onto-> ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) -> ( ( 1 ... J ) i^i ( R ` C ) ) ~~ ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) )
33		hasheni 13136	. . . 4 |- ( ( ( 1 ... J ) i^i ( R ` C ) ) ~~ ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) -> ( # ` ( ( 1 ... J ) i^i ( R ` C ) ) ) = ( # ` ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) ) )
34	29, 32, 33	3syl 18	. . 3 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( # ` ( ( 1 ... J ) i^i ( R ` C ) ) ) = ( # ` ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) ) )
35	25	ssdifssd 3748	. . . . 5 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( 1 ... J ) \ ( R ` C ) ) C_ ( 1 ... ( M + N ) ) )
36		f1ores 6151	. . . . 5 |- ( ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-> ( 1 ... ( M + N ) ) /\ ( ( 1 ... J ) \ ( R ` C ) ) C_ ( 1 ... ( M + N ) ) ) -> ( ( S ` C ) |` ( ( 1 ... J ) \ ( R ` C ) ) ) : ( ( 1 ... J ) \ ( R ` C ) ) -1-1-onto-> ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) )
37	14, 35, 36	syl2anc 693	. . . 4 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( S ` C ) |` ( ( 1 ... J ) \ ( R ` C ) ) ) : ( ( 1 ... J ) \ ( R ` C ) ) -1-1-onto-> ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) )
38		difexg 4808	. . . . . 6 |- ( ( 1 ... J ) e. _V -> ( ( 1 ... J ) \ ( R ` C ) ) e. _V )
39	30, 38	ax-mp 5	. . . . 5 |- ( ( 1 ... J ) \ ( R ` C ) ) e. _V
40	39	f1oen 7976	. . . 4 |- ( ( ( S ` C ) |` ( ( 1 ... J ) \ ( R ` C ) ) ) : ( ( 1 ... J ) \ ( R ` C ) ) -1-1-onto-> ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) -> ( ( 1 ... J ) \ ( R ` C ) ) ~~ ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) )
41		hasheni 13136	. . . 4 |- ( ( ( 1 ... J ) \ ( R ` C ) ) ~~ ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) -> ( # ` ( ( 1 ... J ) \ ( R ` C ) ) ) = ( # ` ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) ) )
42	37, 40, 41	3syl 18	. . 3 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( # ` ( ( 1 ... J ) \ ( R ` C ) ) ) = ( # ` ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) ) )
43	34, 42	oveq12d 6668	. 2 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( # ` ( ( 1 ... J ) i^i ( R ` C ) ) ) - ( # ` ( ( 1 ... J ) \ ( R ` C ) ) ) ) = ( ( # ` ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) ) - ( # ` ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) ) ) )
44		ballotth.r	. . . . 5 |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
45	1, 2, 3, 4, 5, 6, 7, 8, 9, 44	ballotlemro 30584	. . . 4 |- ( C e. ( O \ E ) -> ( R ` C ) e. O )
46	45	adantr 481	. . 3 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( R ` C ) e. O )
47		elfzelz 12342	. . . 4 |- ( J e. ( 1 ... ( I ` C ) ) -> J e. ZZ )
48	47	adantl 482	. . 3 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> J e. ZZ )
49	1, 2, 3, 4, 5, 46, 48	ballotlemfval 30551	. 2 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( F ` ( R ` C ) ) ` J ) = ( ( # ` ( ( 1 ... J ) i^i ( R ` C ) ) ) - ( # ` ( ( 1 ... J ) \ ( R ` C ) ) ) ) )
50		fzfi 12771	. . . . 5 |- ( 1 ... ( M + N ) ) e. Fin
51		eldifi 3732	. . . . . . 7 |- ( C e. ( O \ E ) -> C e. O )
52	1, 2, 3	ballotlemelo 30549	. . . . . . . 8 |- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
53	52	simplbi 476	. . . . . . 7 |- ( C e. O -> C C_ ( 1 ... ( M + N ) ) )
54	51, 53	syl 17	. . . . . 6 |- ( C e. ( O \ E ) -> C C_ ( 1 ... ( M + N ) ) )
55	54	adantr 481	. . . . 5 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> C C_ ( 1 ... ( M + N ) ) )
56		ssfi 8180	. . . . 5 |- ( ( ( 1 ... ( M + N ) ) e. Fin /\ C C_ ( 1 ... ( M + N ) ) ) -> C e. Fin )
57	50, 55, 56	sylancr 695	. . . 4 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> C e. Fin )
58		fzfid 12772	. . . 4 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( ( S ` C ) ` J ) ... ( I ` C ) ) e. Fin )
59		ballotlemg	. . . . 5 |- .^ = ( u e. Fin , v e. Fin |-> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) )
60	1, 2, 3, 4, 5, 6, 7, 8, 9, 44, 59	ballotlemgval 30585	. . . 4 |- ( ( C e. Fin /\ ( ( ( S ` C ) ` J ) ... ( I ` C ) ) e. Fin ) -> ( C .^ ( ( ( S ` C ) ` J ) ... ( I ` C ) ) ) = ( ( # ` ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) i^i C ) ) - ( # ` ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) \ C ) ) ) )
61	57, 58, 60	syl2anc 693	. . 3 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( C .^ ( ( ( S ` C ) ` J ) ... ( I ` C ) ) ) = ( ( # ` ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) i^i C ) ) - ( # ` ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) \ C ) ) ) )
62		dff1o3 6143	. . . . . . . . 9 |- ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) <-> ( ( S ` C ) : ( 1 ... ( M + N ) ) -onto-> ( 1 ... ( M + N ) ) /\ Fun `' ( S ` C ) ) )
63	62	simprbi 480	. . . . . . . 8 |- ( ( S ` C ) : ( 1 ... ( M + N ) ) -1-1-onto-> ( 1 ... ( M + N ) ) -> Fun `' ( S ` C ) )
64		imain 5974	. . . . . . . 8 |- ( Fun `' ( S ` C ) -> ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) = ( ( ( S ` C ) " ( 1 ... J ) ) i^i ( ( S ` C ) " ( R ` C ) ) ) )
65	11, 63, 64	3syl 18	. . . . . . 7 |- ( C e. ( O \ E ) -> ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) = ( ( ( S ` C ) " ( 1 ... J ) ) i^i ( ( S ` C ) " ( R ` C ) ) ) )
66	65	adantr 481	. . . . . 6 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) = ( ( ( S ` C ) " ( 1 ... J ) ) i^i ( ( S ` C ) " ( R ` C ) ) ) )
67	1, 2, 3, 4, 5, 6, 7, 8, 9	ballotlemsima 30577	. . . . . . 7 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( S ` C ) " ( 1 ... J ) ) = ( ( ( S ` C ) ` J ) ... ( I ` C ) ) )
68	1, 2, 3, 4, 5, 6, 7, 8, 9, 44	ballotlemscr 30580	. . . . . . . 8 |- ( C e. ( O \ E ) -> ( ( S ` C ) " ( R ` C ) ) = C )
69	68	adantr 481	. . . . . . 7 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( S ` C ) " ( R ` C ) ) = C )
70	67, 69	ineq12d 3815	. . . . . 6 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( ( S ` C ) " ( 1 ... J ) ) i^i ( ( S ` C ) " ( R ` C ) ) ) = ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) i^i C ) )
71	66, 70	eqtrd 2656	. . . . 5 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) = ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) i^i C ) )
72	71	fveq2d 6195	. . . 4 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( # ` ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) ) = ( # ` ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) i^i C ) ) )
73		imadif 5973	. . . . . . . 8 |- ( Fun `' ( S ` C ) -> ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) = ( ( ( S ` C ) " ( 1 ... J ) ) \ ( ( S ` C ) " ( R ` C ) ) ) )
74	11, 63, 73	3syl 18	. . . . . . 7 |- ( C e. ( O \ E ) -> ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) = ( ( ( S ` C ) " ( 1 ... J ) ) \ ( ( S ` C ) " ( R ` C ) ) ) )
75	74	adantr 481	. . . . . 6 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) = ( ( ( S ` C ) " ( 1 ... J ) ) \ ( ( S ` C ) " ( R ` C ) ) ) )
76	67, 69	difeq12d 3729	. . . . . 6 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( ( S ` C ) " ( 1 ... J ) ) \ ( ( S ` C ) " ( R ` C ) ) ) = ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) \ C ) )
77	75, 76	eqtrd 2656	. . . . 5 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) = ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) \ C ) )
78	77	fveq2d 6195	. . . 4 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( # ` ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) ) = ( # ` ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) \ C ) ) )
79	72, 78	oveq12d 6668	. . 3 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( # ` ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) ) - ( # ` ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) ) ) = ( ( # ` ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) i^i C ) ) - ( # ` ( ( ( ( S ` C ) ` J ) ... ( I ` C ) ) \ C ) ) ) )
80	61, 79	eqtr4d 2659	. 2 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( C .^ ( ( ( S ` C ) ` J ) ... ( I ` C ) ) ) = ( ( # ` ( ( S ` C ) " ( ( 1 ... J ) i^i ( R ` C ) ) ) ) - ( # ` ( ( S ` C ) " ( ( 1 ... J ) \ ( R ` C ) ) ) ) ) )
81	43, 49, 80	3eqtr4d 2666	1 |- ( ( C e. ( O \ E ) /\ J e. ( 1 ... ( I ` C ) ) ) -> ( ( F ` ( R ` C ) ) ` J ) = ( C .^ ( ( ( S ` C ) ` J ) ... ( I ` C ) ) ) )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   _Vcvv 3200   \ cdif 3571   i^i cin 3573   C_ wss 3574   ifcif 4086   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   `'ccnv 5113   |` cres 5116   "cima 5117   Fun wfun 5882   -1-1->wf1 5885   -onto->wfo 5886   -1-1-onto->wf1o 5887   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   ~~ cen 7952   Fincfn 7955  infcinf 8347   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   ZZcz 11377   ZZ>=cuz 11687   ...cfz 12326   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-2 11079  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-hash 13118

This theorem is referenced by:  ballotlemfrci  30589  ballotlemfrceq  30590