Theorem ballotfilemfmpn 13178

Description: ( F ` C ) finishes counting at ( M - N ). (Contributed by Thierry Arnoux, 25-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 ) ) ) ) )

Assertion

Ref	Expression
ballotfilemfmpn	|- ( C e. O -> ( ( F ` C ) ` ( M + N ) ) = ( M - N ) )

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

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

Proof of Theorem ballotfilemfmpn

Dummy variable b is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ballotth.m	. . 3 |- M e. NN
2		ballotth.n	. . 3 |- N e. NN
3		ballotfilem.o	. . 3 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
4		ballotfilem.p	. . 3 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
5		ballotth.f	. . 3 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
6		id 19	. . 3 |- ( C e. O -> C e. O )
7		nnaddcl 9274	. . . . . 6 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
8	1, 2, 7	mp2an 426	. . . . 5 |- ( M + N ) e. NN
9	8	nnzi 9615	. . . 4 |- ( M + N ) e. ZZ
10	9	a1i 9	. . 3 |- ( C e. O -> ( M + N ) e. ZZ )
11	1, 2, 3, 4, 5, 6, 10	ballotfilemfval 13173	. 2 |- ( C e. O -> ( ( F ` C ) ` ( M + N ) ) = ( (♯ ` ( ( 1 ... ( M + N ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( M + N ) ) \ C ) ) ) )
12	3	ssrab3 3328	. . . . . . . . 9 |- O C_ ( ~P ( 1 ... ( M + N ) ) i^i Fin )
13	12	sseli 3238	. . . . . . . 8 |- ( C e. O -> C e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) )
14	13	elin1d 3412	. . . . . . 7 |- ( C e. O -> C e. ~P ( 1 ... ( M + N ) ) )
15	14	elpwid 3685	. . . . . 6 |- ( C e. O -> C C_ ( 1 ... ( M + N ) ) )
16		sseqin2 3444	. . . . . 6 |- ( C C_ ( 1 ... ( M + N ) ) <-> ( ( 1 ... ( M + N ) ) i^i C ) = C )
17	15, 16	sylib 122	. . . . 5 |- ( C e. O -> ( ( 1 ... ( M + N ) ) i^i C ) = C )
18	17	fveq2d 5679	. . . 4 |- ( C e. O -> (♯ ` ( ( 1 ... ( M + N ) ) i^i C ) ) = (♯ ` C ) )
19		rabssab 3331	. . . . . . 7 |- { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M } C_ { c | (♯ ` c ) = M }
20	19	sseli 3238	. . . . . 6 |- ( C e. { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M } -> C e. { c | (♯ ` c ) = M } )
21	20, 3	eleq2s 2329	. . . . 5 |- ( C e. O -> C e. { c | (♯ ` c ) = M } )
22		fveqeq2 5684	. . . . . 6 |- ( b = C -> ( (♯ ` b ) = M <-> (♯ ` C ) = M ) )
23		fveqeq2 5684	. . . . . . 7 |- ( c = b -> ( (♯ ` c ) = M <-> (♯ ` b ) = M ) )
24	23	cbvabv 2361	. . . . . 6 |- { c | (♯ ` c ) = M } = { b | (♯ ` b ) = M }
25	22, 24	elab2g 2967	. . . . 5 |- ( C e. O -> ( C e. { c | (♯ ` c ) = M } <-> (♯ ` C ) = M ) )
26	21, 25	mpbid 147	. . . 4 |- ( C e. O -> (♯ ` C ) = M )
27	18, 26	eqtrd 2267	. . 3 |- ( C e. O -> (♯ ` ( ( 1 ... ( M + N ) ) i^i C ) ) = M )
28		1z 9620	. . . . . 6 |- 1 e. ZZ
29		fzfig 10816	. . . . . 6 |- ( ( 1 e. ZZ /\ ( M + N ) e. ZZ ) -> ( 1 ... ( M + N ) ) e. Fin )
30	28, 9, 29	mp2an 426	. . . . 5 |- ( 1 ... ( M + N ) ) e. Fin
31	13	elin2d 3413	. . . . 5 |- ( C e. O -> C e. Fin )
32		fihashssdif 11208	. . . . 5 |- ( ( ( 1 ... ( M + N ) ) e. Fin /\ C e. Fin /\ C C_ ( 1 ... ( M + N ) ) ) -> (♯ ` ( ( 1 ... ( M + N ) ) \ C ) ) = ( (♯ ` ( 1 ... ( M + N ) ) ) - (♯ ` C ) ) )
33	30, 31, 15, 32	mp3an2i 1379	. . . 4 |- ( C e. O -> (♯ ` ( ( 1 ... ( M + N ) ) \ C ) ) = ( (♯ ` ( 1 ... ( M + N ) ) ) - (♯ ` C ) ) )
34	8	nnnn0i 9521	. . . . . 6 |- ( M + N ) e. NN0
35		hashfz1 11171	. . . . . 6 |- ( ( M + N ) e. NN0 -> (♯ ` ( 1 ... ( M + N ) ) ) = ( M + N ) )
36	34, 35	mp1i 10	. . . . 5 |- ( C e. O -> (♯ ` ( 1 ... ( M + N ) ) ) = ( M + N ) )
37	36, 26	oveq12d 6076	. . . 4 |- ( C e. O -> ( (♯ ` ( 1 ... ( M + N ) ) ) - (♯ ` C ) ) = ( ( M + N ) - M ) )
38	1	nncni 9264	. . . . . 6 |- M e. CC
39	2	nncni 9264	. . . . . 6 |- N e. CC
40		pncan2 8496	. . . . . 6 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - M ) = N )
41	38, 39, 40	mp2an 426	. . . . 5 |- ( ( M + N ) - M ) = N
42	41	a1i 9	. . . 4 |- ( C e. O -> ( ( M + N ) - M ) = N )
43	33, 37, 42	3eqtrd 2271	. . 3 |- ( C e. O -> (♯ ` ( ( 1 ... ( M + N ) ) \ C ) ) = N )
44	27, 43	oveq12d 6076	. 2 |- ( C e. O -> ( (♯ ` ( ( 1 ... ( M + N ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( M + N ) ) \ C ) ) ) = ( M - N ) )
45	11, 44	eqtrd 2267	1 |- ( C e. O -> ( ( F ` C ) ` ( M + N ) ) = ( M - N ) )

Syntax hints:   -> wi 4   = wceq 1398   e. wcel 2205   {cab 2220   {crab 2526   \ cdif 3211   i^i cin 3213   C_ wss 3214   ~Pcpw 3674   |-> cmpt 4176   ` cfv 5357  (class class class)co 6058   Fincfn 6988   CCcc 8141   1c1 8144   + caddc 8146   - cmin 8460   / cdiv 8963   NNcn 9254   NN0cn0 9513   ZZcz 9594   ...cfz 10361  ♯chash 11163

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 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-addass 8245  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259

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 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-irdg 6614  df-frec 6635  df-1o 6660  df-oadd 6664  df-er 6780  df-en 6989  df-dom 6990  df-fin 6991  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-inn 9255  df-n0 9514  df-z 9595  df-uz 9872  df-fz 10362  df-ihash 11164

This theorem is referenced by:  ballotfilem5  13186