Theorem ballotfilemfp1 13152

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
ballotfi.o	|- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
ballotfi.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		ballotfi.o	. . . . . 6 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
4		ballotfi.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 9702	. . . . . 6 |- ( ph -> J e. ZZ )
9	1, 2, 3, 4, 5, 6, 8	ballotfilemfval 13150	. . . . 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 9617	. . . . . . . . . . 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 13147	. . . . . . . . 9 |- ( ph -> ( ( 1 ... ( J - 1 ) ) i^i C ) e. Fin )
14		hashcl 11148	. . . . . . . . 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 9557	. . . . . . 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 13148	. . . . . . . . 9 |- ( ph -> ( ( 1 ... ( J - 1 ) ) \ C ) e. Fin )
19		hashcl 11148	. . . . . . . . 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 9557	. . . . . . 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 8292	. . . . . 6 |- ( ( ph /\ -. J e. C ) -> 1 e. CC )
24	17, 22, 23	subsub4d 8617	. . . . 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 9606	. . . . . . . . 9 |- ( ph -> 1 e. ZZ )
26	8, 25	zsubcld 9708	. . . . . . . 8 |- ( ph -> ( J - 1 ) e. ZZ )
27	1, 2, 3, 4, 5, 6, 26	ballotfilemfval 13150	. . . . . . 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 6067	. . . . 5 |- ( ( ph /\ -. J e. C ) -> ( ( ( F ` C ) ` ( J - 1 ) ) - 1 ) = ( ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) ) - 1 ) )
30		elnnuz 9894	. . . . . . . . . . 11 |- ( J e. NN <-> J e. ( ZZ>= ` 1 ) )
31	7, 30	sylib 122	. . . . . . . . . 10 |- ( ph -> J e. ( ZZ>= ` 1 ) )
32		fzspl 10407	. . . . . . . . . . . 12 |- ( J e. ( ZZ>= ` 1 ) -> ( 1 ... J ) = ( ( 1 ... ( J - 1 ) ) u. { J } ) )
33	32	ineq1d 3423	. . . . . . . . . . 11 |- ( J e. ( ZZ>= ` 1 ) -> ( ( 1 ... J ) i^i C ) = ( ( ( 1 ... ( J - 1 ) ) u. { J } ) i^i C ) )
34		indir 3472	. . . . . . . . . . 11 |- ( ( ( 1 ... ( J - 1 ) ) u. { J } ) i^i C ) = ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. ( { J } i^i C ) )
35	33, 34	eqtrdi 2283	. . . . . . . . . 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 3753	. . . . . . . . . . . 12 |- ( ( C i^i { J } ) = (/) <-> -. J e. C )
39		ineqcom 3414	. . . . . . . . . . . 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 3375	. . . . . . . . 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 3544	. . . . . . . . 9 |- ( ( ( 1 ... ( J - 1 ) ) i^i C ) u. (/) ) = ( ( 1 ... ( J - 1 ) ) i^i C )
44	42, 43	eqtrdi 2283	. . . . . . . 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 2267	. . . . . . 7 |- ( ( ph /\ -. J e. C ) -> ( ( 1 ... J ) i^i C ) = ( ( 1 ... ( J - 1 ) ) i^i C ) )
46	45	fveq2d 5676	. . . . . 6 |- ( ( ph /\ -. J e. C ) -> (♯ ` ( ( 1 ... J ) i^i C ) ) = (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) )
47	32	difeq1d 3338	. . . . . . . . . . 11 |- ( J e. ( ZZ>= ` 1 ) -> ( ( 1 ... J ) \ C ) = ( ( ( 1 ... ( J - 1 ) ) u. { J } ) \ C ) )
48		difundir 3476	. . . . . . . . . . 11 |- ( ( ( 1 ... ( J - 1 ) ) u. { J } ) \ C ) = ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) )
49	47, 48	eqtrdi 2283	. . . . . . . . . 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 3563	. . . . . . . . . . . 12 |- ( ( { J } i^i C ) = (/) <-> { J } = ( { J } \ C ) )
52	40, 51	sylib 122	. . . . . . . . . . 11 |- ( -. J e. C -> { J } = ( { J } \ C ) )
53	52	eqcomd 2240	. . . . . . . . . 10 |- ( -. J e. C -> ( { J } \ C ) = { J } )
54	53	uneq2d 3375	. . . . . . . . 9 |- ( -. J e. C -> ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) ) = ( ( ( 1 ... ( J - 1 ) ) \ C ) u. { J } ) )
55	50, 54	sylan9eq 2287	. . . . . . . 8 |- ( ( ph /\ -. J e. C ) -> ( ( 1 ... J ) \ C ) = ( ( ( 1 ... ( J - 1 ) ) \ C ) u. { J } ) )
56	55	fveq2d 5676	. . . . . . 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 9871	. . . . . . . . . . . 12 |- ( J e. ZZ -> J e. ( ZZ>= ` J ) )
60		uznfz 10441	. . . . . . . . . . . 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 3343	. . . . . . . . . 10 |- ( J e. ( ( 1 ... ( J - 1 ) ) \ C ) -> J e. ( 1 ... ( J - 1 ) ) )
64	62, 63	nsyl 633	. . . . . . . . 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 11176	. . . . . . . 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 2267	. . . . . 6 |- ( ( ph /\ -. J e. C ) -> (♯ ` ( ( 1 ... J ) \ C ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) + 1 ) )
69	46, 68	oveq12d 6070	. . . . 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 2278	. . . 4 |- ( ( ph /\ -. J e. C ) -> ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) = ( ( ( F ` C ) ` ( J - 1 ) ) - 1 ) )
71	10, 70	eqtrd 2267	. . 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 8292	. . . . . 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 8607	. . . . 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 5676	. . . . . . . 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 3840	. . . . . . . . . . 11 |- ( J e. C -> { J } C_ C )
81		dfss2 3230	. . . . . . . . . . 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 3375	. . . . . . . . 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 5676	. . . . . . . 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 3407	. . . . . . . . . 10 |- ( J e. ( ( 1 ... ( J - 1 ) ) i^i C ) -> J e. ( 1 ... ( J - 1 ) ) )
91	89, 90	nsyl 633	. . . . . . . . 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 11176	. . . . . . . 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 2271	. . . . . 6 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( 1 ... J ) i^i C ) ) = ( (♯ ` ( ( 1 ... ( J - 1 ) ) i^i C ) ) + 1 ) )
96	50	fveq2d 5676	. . . . . . . 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 3575	. . . . . . . . . . 11 |- ( { J } C_ C -> ( { J } \ C ) = (/) )
99	80, 98	syl 14	. . . . . . . . . 10 |- ( J e. C -> ( { J } \ C ) = (/) )
100	99	uneq2d 3375	. . . . . . . . 9 |- ( J e. C -> ( ( ( 1 ... ( J - 1 ) ) \ C ) u. ( { J } \ C ) ) = ( ( ( 1 ... ( J - 1 ) ) \ C ) u. (/) ) )
101	100	fveq2d 5676	. . . . . . . 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 3544	. . . . . . . . 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 5676	. . . . . . 7 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( ( 1 ... ( J - 1 ) ) \ C ) u. (/) ) ) = (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) )
106	97, 102, 105	3eqtrd 2271	. . . . . 6 |- ( ( ph /\ J e. C ) -> (♯ ` ( ( 1 ... J ) \ C ) ) = (♯ ` ( ( 1 ... ( J - 1 ) ) \ C ) ) )
107	95, 106	oveq12d 6070	. . . . 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 6067	. . . . 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 2277	. . . 4 |- ( ( ph /\ J e. C ) -> ( (♯ ` ( ( 1 ... J ) i^i C ) ) - (♯ ` ( ( 1 ... J ) \ C ) ) ) = ( ( ( F ` C ) ` ( J - 1 ) ) + 1 ) )
111	73, 110	eqtrd 2267	. . 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 ) ) ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1398   e. wcel 2205   {crab 2526   \ cdif 3210   u. cun 3211   i^i cin 3212   C_ wss 3213   (/)c0 3510   ~Pcpw 3671   {csn 3691   |-> cmpt 4173   ` cfv 5354  (class class class)co 6052   Fincfn 6977   CCcc 8127   1c1 8130   + caddc 8132   - cmin 8446   / cdiv 8948   NNcn 9239   NN0cn0 9498   ZZcz 9579   ZZ>=cuz 9856   ...cfz 10345  ♯chash 11142

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4227  ax-sep 4230  ax-nul 4238  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-iinf 4712  ax-cnex 8220  ax-resscn 8221  ax-1cn 8222  ax-1re 8223  ax-icn 8224  ax-addcl 8225  ax-addrcl 8226  ax-mulcl 8227  ax-addcom 8229  ax-addass 8231  ax-distr 8233  ax-i2m1 8234  ax-0lt1 8235  ax-0id 8237  ax-rnegex 8238  ax-cnre 8240  ax-pre-ltirr 8241  ax-pre-ltwlin 8242  ax-pre-lttrn 8243  ax-pre-apti 8244  ax-pre-ltadd 8245

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3045  df-csb 3141  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-nul 3511  df-if 3623  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-int 3952  df-iun 3995  df-br 4112  df-opab 4174  df-mpt 4175  df-tr 4211  df-id 4416  df-iord 4489  df-on 4491  df-ilim 4492  df-suc 4494  df-iom 4715  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-iota 5314  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-fv 5362  df-riota 6005  df-ov 6055  df-oprab 6056  df-mpo 6057  df-1st 6336  df-2nd 6337  df-recs 6538  df-irdg 6603  df-frec 6624  df-1o 6649  df-oadd 6653  df-er 6769  df-en 6978  df-dom 6979  df-fin 6980  df-pnf 8312  df-mnf 8313  df-xr 8314  df-ltxr 8315  df-le 8316  df-sub 8448  df-neg 8449  df-inn 9240  df-n0 9499  df-z 9580  df-uz 9857  df-fz 10346  df-ihash 11143

This theorem is referenced by:  ballotfilemfc0  13153  ballotfilemfcc  13154  ballotfilem4  13159