Theorem ballotfilemfp1 13289

Description: If the J th ballot is for A, ( F ` C ) goes up 1. If the J th ballot is for B, ( F ` C ) goes down 1. (Contributed by Thierry Arnoux, 24-Nov-2016.)

Hypotheses

Ref	Expression
ballotth.m	|- M e. NN
ballotth.n	|- N e. NN
ballotfilem.o	|- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
ballotfilem.p	|- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
ballotth.f	|- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
ballotlemfp1.c	|- ( ph -> C e. O )
ballotlemfp1.j	|- ( ph -> J e. NN )

Assertion

Ref	Expression
ballotfilemfp1	|- ( ph -> ( ( -. J e. C -> ( ( F ` C ) ` J ) = ( ( ( F ` C ) ` ( J - 1 ) ) - 1 ) ) /\ ( J e. C -> ( ( F ` C ) ` J ) = ( ( ( F ` C ) ` ( J - 1 ) ) + 1 ) ) ) )

Distinct variable groups:   M, c   N, c   O, c   i, M   i, N   i, O, c   F, c, i   C, i   i, J   ph, i

Allowed substitution hints:   ph( x, c)   C( x, c)   P( x, i, c)   F( x)   J( x, c)   M( x)   N( x)   O( x)

Proof of Theorem ballotfilemfp1

Step	Hyp	Ref	Expression
1		ballotth.m	. . . . . 6 |- M e. NN
2		ballotth.n	. . . . . 6 |- N e. NN
3		ballotfilem.o	. . . . . 6 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
4		ballotfilem.p	. . . . . 6 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
5		ballotth.f	. . . . . 6 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
6		ballotlemfp1.c	. . . . . 6 |- ( ph -> C e. O )
7		ballotlemfp1.j	. . . . . . 7 |- ( ph -> J e. NN )
8	7	nnzd 9778	. . . . . 6 |- ( ph -> J e. ZZ )
9	1, 2, 3, 4, 5, 6, 8	ballotfilemfval 13287	. . . . 5 |- ( ph -> ( ( F ` C ) ` J ) = ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) )
10	9	adantr 276	. . . 4 |- ( ( ph /\ -. J e. C ) -> ( ( F ` C ) ` J ) = ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) )
11		peano2zm 9693	. . . . . . . . . . 11 |- ( J e. ZZ -> ( J - 1 ) e. ZZ )
12	8, 11	syl 14	. . . . . . . . . 10 |- ( ph -> ( J - 1 ) e. ZZ )
13	1, 2, 3, 6, 12	ballotfilemcinfi 13282	. . . . . . . . 9 |- ( ph -> ( ( 1 ... ( J - 1 ) ) i^i C ) e. Fin )
14		hashcl 11242	. . . . . . . . 9 |- ( ( ( 1 ... ( J - 1 ) ) i^i C ) e. Fin -> (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) e. NN0 )
15	13, 14	syl 14	. . . . . . . 8 |- ( ph -> (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) e. NN0 )
16	15	nn0cnd 9633	. . . . . . 7 |- ( ph -> (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) e. CC )
17	16	adantr 276	. . . . . 6 |- ( ( ph /\ -. J e. C ) -> (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) e. CC )
18	1, 2, 3, 6, 12	ballotfilemdifcfi 13283	. . . . . . . . 9 |- ( ph -> ( ( 1 ... ( J - 1 ) ) \ C ) e. Fin )
19		hashcl 11242	. . . . . . . . 9 |- ( ( ( 1 ... ( J - 1 ) ) \ C ) e. Fin -> (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) e. NN0 )
20	18, 19	syl 14	. . . . . . . 8 |- ( ph -> (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) e. NN0 )
21	20	nn0cnd 9633	. . . . . . 7 |- ( ph -> (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) e. CC )
22	21	adantr 276	. . . . . 6 |- ( ( ph /\ -. J e. C ) -> (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) e. CC )
23		1cnd 8348	. . . . . 6 |- ( ( ph /\ -. J e. C ) -> 1 e. CC )
24	17, 22, 23	subsub4d 8675	. . . . 5 |- ( ( ph /\ -. J e. C ) -> ( ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) ) - 1 ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) - ( (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) + 1 ) ) )
25		1zzd 9682	. . . . . . . . 9 |- ( ph -> 1 e. ZZ )
26	8, 25	zsubcld 9784	. . . . . . . 8 |- ( ph -> ( J - 1 ) e. ZZ )
27	1, 2, 3, 4, 5, 6, 26	ballotfilemfval 13287	. . . . . . 7 |- ( ph -> ( ( F ` C ) ` ( J - 1 ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) ) )
28	27	adantr 276	. . . . . 6 |- ( ( ph /\ -. J e. C ) -> ( ( F ` C ) ` ( J - 1 ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) ) )
29	28	oveq1d 6103	. . . . 5 |- ( ( ph /\ -. J e. C ) -> ( ( ( F ` C ) ` ( J - 1 ) ) - 1 ) = ( ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) ) - 1 ) )
30		elnnuz 9975	. . . . . . . . . . 11 |- ( J e. NN <-> J e. ( ZZ>= ` 1 ) )
31	7, 30	sylib 122	. . . . . . . . . 10 |- ( ph -> J e. ( ZZ>= ` 1 ) )
32		fzspl 10493	. . . . . . . . . . . 12 |- ( J e. ( ZZ>= ` 1 ) -> ( 1 ... J ) = ( ( 1 ... ( J - 1 ) ) u. { J } ) )
33	32	ineq1d 3431	. . . . . . . . . . 11 |- ( J e. ( ZZ>= ` 1 ) -> ( ( 1 ... J ) i^i C ) = ( ( ( 1 ... ( J - 1 ) ) u. { J } ) i^i C ) )
34		indir 3480	. . . . . . . . . . 11 |- ( ( ( 1 ... ( J - 1 ) ) u. { J } ) i^i C ) = ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) )
35	33, 34	eqtrdi 2287	. . . . . . . . . 10 |- ( J e. ( ZZ>= ` 1 ) -> ( ( 1 ... J ) i^i C ) = ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) ) )
36	31, 35	syl 14	. . . . . . . . 9 |- ( ph -> ( ( 1 ... J ) i^i C ) = ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) ) )
37	36	adantr 276	. . . . . . . 8 |- ( ( ph /\ -. J e. C ) -> ( ( 1 ... J ) i^i C ) = ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) ) )
38		disjsn 3771	. . . . . . . . . . . 12 |- ( ( C i^i { J } ) = (/) <-> -. J e. C )
39		ineqcom 3422	. . . . . . . . . . . 12 |- ( ( C i^i { J } ) = (/) <-> ( { J } i^i C ) = (/) )
40	38, 39	sylbb1 137	. . . . . . . . . . 11 |- ( -. J e. C -> ( { J } i^i C ) = (/) )
41	40	adantl 277	. . . . . . . . . 10 |- ( ( ph /\ -. J e. C ) -> ( { J } i^i C ) = (/) )
42	41	uneq2d 3383	. . . . . . . . 9 |- ( ( ph /\ -. J e. C ) -> ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) ) = ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. (/) ) )
43		un0 3556	. . . . . . . . 9 |- ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. (/) ) = ( ( 1 ... ( J - 1 ) ) i^i C )
44	42, 43	eqtrdi 2287	. . . . . . . 8 |- ( ( ph /\ -. J e. C ) -> ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) ) = ( ( 1 ... ( J - 1 ) ) i^i C ) )
45	37, 44	eqtrd 2271	. . . . . . 7 |- ( ( ph /\ -. J e. C ) -> ( ( 1 ... J ) i^i C ) = ( ( 1 ... ( J - 1 ) ) i^i C ) )
46	45	fveq2d 5702	. . . . . 6 |- ( ( ph /\ -. J e. C ) -> (♯ ` ( ( 1 ... J ) i^i C ) ) = (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) )
47	32	difeq1d 3346	. . . . . . . . . . 11 |- ( J e. ( ZZ>= ` 1 ) -> ( ( 1 ... J ) \ C ) = ( ( ( 1 ... ( J - 1 ) ) u. { J } ) \ C ) )
48		difundir 3484	. . . . . . . . . . 11 |- ( ( ( 1 ... ( J - 1 ) ) u. { J } ) \ C ) = ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) )
49	47, 48	eqtrdi 2287	. . . . . . . . . 10 |- ( J e. ( ZZ>= ` 1 ) -> ( ( 1 ... J ) \ C ) = ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) ) )
50	31, 49	syl 14	. . . . . . . . 9 |- ( ph -> ( ( 1 ... J ) \ C ) = ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) ) )
51		disj3 3577	. . . . . . . . . . . 12 |- ( ( { J } i^i C ) = (/) <-> { J } = ( { J } \ C ) )
52	40, 51	sylib 122	. . . . . . . . . . 11 |- ( -. J e. C -> { J } = ( { J } \ C ) )
53	52	eqcomd 2244	. . . . . . . . . 10 |- ( -. J e. C -> ( { J } \ C ) = { J } )
54	53	uneq2d 3383	. . . . . . . . 9 |- ( -. J e. C -> ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) ) = ( ( ( 1 ... ( J - 1 ) ) \ C ) u. { J } ) )
55	50, 54	sylan9eq 2291	. . . . . . . 8 |- ( ( ph /\ -. J e. C ) -> ( ( 1 ... J ) \ C ) = ( ( ( 1 ... ( J - 1 ) ) \ C ) u. { J } ) )
56	55	fveq2d 5702	. . . . . . 7 |- ( ( ph /\ -. J e. C ) -> (♯ ` ( ( 1 ... J ) \ C ) ) = (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. { J } ) ) )
57	8	adantr 276	. . . . . . . 8 |- ( ( ph /\ -. J e. C ) -> J e. ZZ )
58	18	adantr 276	. . . . . . . . 9 |- ( ( ph /\ -. J e. C ) -> ( ( 1 ... ( J - 1 ) ) \ C ) e. Fin )
59		uzid 9952	. . . . . . . . . . . 12 |- ( J e. ZZ -> J e. ( ZZ>= ` J ) )
60		uznfz 10527	. . . . . . . . . . . 12 |- ( J e. ( ZZ>= ` J ) -> -. J e. ( 1 ... ( J - 1 ) ) )
61	8, 59, 60	3syl 17	. . . . . . . . . . 11 |- ( ph -> -. J e. ( 1 ... ( J - 1 ) ) )
62	61	adantr 276	. . . . . . . . . 10 |- ( ( ph /\ -. J e. C ) -> -. J e. ( 1 ... ( J - 1 ) ) )
63		eldifi 3351	. . . . . . . . . 10 |- ( J e. ( ( 1 ... ( J - 1 ) ) \ C ) -> J e. ( 1 ... ( J - 1 ) ) )
64	62, 63	nsyl 637	. . . . . . . . 9 |- ( ( ph /\ -. J e. C ) -> -. J e. ( ( 1 ... ( J - 1 ) ) \ C ) )
65	58, 64	jca 306	. . . . . . . 8 |- ( ( ph /\ -. J e. C ) -> ( ( ( 1 ... ( J - 1 ) ) \ C ) e. Fin /\ -. J e. ( ( 1 ... ( J - 1 ) ) \ C ) ) )
66		hashunsng 11270	. . . . . . . 8 |- ( J e. ZZ -> ( ( ( ( 1 ... ( J - 1 ) ) \ C ) e. Fin /\ -. J e. ( ( 1 ... ( J - 1 ) ) \ C ) ) -> (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. { J } ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) + 1 ) ) )
67	57, 65, 66	sylc 62	. . . . . . 7 |- ( ( ph /\ -. J e. C ) -> (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. { J } ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) + 1 ) )
68	56, 67	eqtrd 2271	. . . . . 6 |- ( ( ph /\ -. J e. C ) -> (♯ ` ( ( 1 ... J ) \ C ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) + 1 ) )
69	46, 68	oveq12d 6106	. . . . 5 |- ( ( ph /\ -. J e. C ) -> ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) - ( (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) + 1 ) ) )
70	24, 29, 69	3eqtr4rd 2282	. . . 4 |- ( ( ph /\ -. J e. C ) -> ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) = ( ( ( F ` C ) ` ( J - 1 ) ) - 1 ) )
71	10, 70	eqtrd 2271	. . 3 |- ( ( ph /\ -. J e. C ) -> ( ( F ` C ) ` J ) = ( ( ( F ` C ) ` ( J - 1 ) ) - 1 ) )
72	71	ex 115	. 2 |- ( ph -> ( -. J e. C -> ( ( F ` C ) ` J ) = ( ( ( F ` C ) ` ( J - 1 ) ) - 1 ) ) )
73	9	adantr 276	. . . 4 |- ( ( ph /\ J e. C ) -> ( ( F ` C ) ` J ) = ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) )
74	16	adantr 276	. . . . . 6 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) e. CC )
75		1cnd 8348	. . . . . 6 |- ( ( ph /\ J e. C ) -> 1 e. CC )
76	21	adantr 276	. . . . . 6 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) e. CC )
77	74, 75, 76	addsubd 8665	. . . . 5 |- ( ( ph /\ J e. C ) -> ( ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) + 1 ) - (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) ) = ( ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) ) + 1 ) )
78	36	fveq2d 5702	. . . . . . . 8 |- ( ph -> (♯ ` ( ( 1 ... J ) i^i C ) ) = (♯ ` ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) ) ) )
79	78	adantr 276	. . . . . . 7 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( 1 ... J ) i^i C ) ) = (♯ ` ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) ) ) )
80		snssi 3859	. . . . . . . . . . 11 |- ( J e. C -> { J } C_ C )
81		dfss2 3237	. . . . . . . . . . 11 |- ( { J } C_ C <-> ( { J } i^i C ) = { J } )
82	80, 81	sylib 122	. . . . . . . . . 10 |- ( J e. C -> ( { J } i^i C ) = { J } )
83	82	uneq2d 3383	. . . . . . . . 9 |- ( J e. C -> ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) ) = ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. { J } ) )
84	83	fveq2d 5702	. . . . . . . 8 |- ( J e. C -> (♯ ` ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) ) ) = (♯ ` ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. { J } ) ) )
85	84	adantl 277	. . . . . . 7 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) ) ) = (♯ ` ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. { J } ) ) )
86		simpr 110	. . . . . . . 8 |- ( ( ph /\ J e. C ) -> J e. C )
87	13	adantr 276	. . . . . . . . 9 |- ( ( ph /\ J e. C ) -> ( ( 1 ... ( J - 1 ) ) i^i C ) e. Fin )
88	8	adantr 276	. . . . . . . . . . 11 |- ( ( ph /\ J e. C ) -> J e. ZZ )
89	88, 59, 60	3syl 17	. . . . . . . . . 10 |- ( ( ph /\ J e. C ) -> -. J e. ( 1 ... ( J - 1 ) ) )
90		elinel1 3415	. . . . . . . . . 10 |- ( J e. ( ( 1 ... ( J - 1 ) ) i^i C ) -> J e. ( 1 ... ( J - 1 ) ) )
91	89, 90	nsyl 637	. . . . . . . . 9 |- ( ( ph /\ J e. C ) -> -. J e. ( ( 1 ... ( J - 1 ) ) i^i C ) )
92	87, 91	jca 306	. . . . . . . 8 |- ( ( ph /\ J e. C ) -> ( ( ( 1 ... ( J - 1 ) ) i^i C ) e. Fin /\ -. J e. ( ( 1 ... ( J - 1 ) ) i^i C ) ) )
93		hashunsng 11270	. . . . . . . 8 |- ( J e. C -> ( ( ( ( 1 ... ( J - 1 ) ) i^i C ) e. Fin /\ -. J e. ( ( 1 ... ( J - 1 ) ) i^i C ) ) -> (♯ ` ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. { J } ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) + 1 ) ) )
94	86, 92, 93	sylc 62	. . . . . . 7 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. { J } ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) + 1 ) )
95	79, 85, 94	3eqtrd 2275	. . . . . 6 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( 1 ... J ) i^i C ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) + 1 ) )
96	50	fveq2d 5702	. . . . . . . 8 |- ( ph -> (♯ ` ( ( 1 ... J ) \ C ) ) = (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) ) ) )
97	96	adantr 276	. . . . . . 7 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( 1 ... J ) \ C ) ) = (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) ) ) )
98		ssdif0im 3589	. . . . . . . . . . 11 |- ( { J } C_ C -> ( { J } \ C ) = (/) )
99	80, 98	syl 14	. . . . . . . . . 10 |- ( J e. C -> ( { J } \ C ) = (/) )
100	99	uneq2d 3383	. . . . . . . . 9 |- ( J e. C -> ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) ) = ( ( ( 1 ... ( J - 1 ) ) \ C ) u. (/) ) )
101	100	fveq2d 5702	. . . . . . . 8 |- ( J e. C -> (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) ) ) = (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. (/) ) ) )
102	101	adantl 277	. . . . . . 7 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) ) ) = (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. (/) ) ) )
103		un0 3556	. . . . . . . . 9 |- ( ( ( 1 ... ( J - 1 ) ) \ C ) u. (/) ) = ( ( 1 ... ( J - 1 ) ) \ C )
104	103	a1i 9	. . . . . . . 8 |- ( ( ph /\ J e. C ) -> ( ( ( 1 ... ( J - 1 ) ) \ C ) u. (/) ) = ( ( 1 ... ( J - 1 ) ) \ C ) )
105	104	fveq2d 5702	. . . . . . 7 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. (/) ) ) = (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) )
106	97, 102, 105	3eqtrd 2275	. . . . . 6 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( 1 ... J ) \ C ) ) = (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) )
107	95, 106	oveq12d 6106	. . . . 5 |- ( ( ph /\ J e. C ) -> ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) = ( ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) + 1 ) - (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) ) )
108	27	adantr 276	. . . . . 6 |- ( ( ph /\ J e. C ) -> ( ( F ` C ) ` ( J - 1 ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) ) )
109	108	oveq1d 6103	. . . . 5 |- ( ( ph /\ J e. C ) -> ( ( ( F ` C ) ` ( J - 1 ) ) + 1 ) = ( ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) ) + 1 ) )
110	77, 107, 109	3eqtr4d 2281	. . . 4 |- ( ( ph /\ J e. C ) -> ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) = ( ( ( F ` C ) ` ( J - 1 ) ) + 1 ) )
111	73, 110	eqtrd 2271	. . 3 |- ( ( ph /\ J e. C ) -> ( ( F ` C ) ` J ) = ( ( ( F ` C ) ` ( J - 1 ) ) + 1 ) )
112	111	ex 115	. 2 |- ( ph -> ( J e. C -> ( ( F ` C ) ` J ) = ( ( ( F ` C ) ` ( J - 1 ) ) + 1 ) ) )
113	72, 112	jca 306	1 |- ( ph -> ( ( -. J e. C -> ( ( F ` C ) ` J ) = ( ( ( F ` C ) ` ( J - 1 ) ) - 1 ) ) /\ ( J e. C -> ( ( F ` C ) ` J ) = ( ( ( F ` C ) ` ( J - 1 ) ) + 1 ) ) ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {crab 2532   \ cdif 3217   u. cun 3218   i^i cin 3219   C_ wss 3220   (/)c0 3520   ~Pcpw 3688   {csn 3709   |-> cmpt 4193   ` cfv 5380  (class class class)co 6088   Fincfn 7026   CCcc 8183   1c1 8186   + caddc 8188   - cmin 8504   / cdiv 9010   NNcn 9313   NN0cn0 9574   ZZcz 9655   ZZ>=cuz 9937   ...cfz 10428  ♯chash 11236

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4248  ax-sep 4251  ax-nul 4261  ax-pow 4313  ax-pr 4348  ax-un 4580  ax-setind 4686  ax-iinf 4737  ax-cnex 8276  ax-resscn 8277  ax-1cn 8278  ax-1re 8279  ax-icn 8280  ax-addcl 8281  ax-addrcl 8282  ax-mulcl 8283  ax-addcom 8285  ax-addass 8287  ax-distr 8289  ax-i2m1 8290  ax-0lt1 8291  ax-0id 8293  ax-rnegex 8294  ax-cnre 8296  ax-pre-ltirr 8297  ax-pre-ltwlin 8298  ax-pre-lttrn 8299  ax-pre-apti 8300  ax-pre-ltadd 8301

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-br 4132  df-opab 4194  df-mpt 4195  df-tr 4232  df-id 4440  df-iord 4513  df-on 4515  df-ilim 4516  df-suc 4518  df-iom 4740  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-res 4788  df-ima 4789  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-riota 6041  df-ov 6091  df-oprab 6092  df-mpo 6093  df-1st 6378  df-2nd 6379  df-recs 6580  df-irdg 6645  df-frec 6666  df-1o 6691  df-oadd 6695  df-er 6811  df-en 7027  df-dom 7028  df-fin 7029  df-pnf 8368  df-mnf 8369  df-xr 8370  df-ltxr 8371  df-le 8372  df-sub 8506  df-neg 8507  df-inn 9314  df-n0 9575  df-z 9656  df-uz 9938  df-fz 10429  df-ihash 11237

This theorem is used by:  ballotfilemfc0  13290  ballotfilemfcc  13291  ballotfilem4  13299  ballotfilemi1  13303  ballotfilemii  13304  ballotfilemic  13308  ballotfilem1c  13309