Theorem ballotfilemfmpn 13155

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
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 ) ) ) ) )

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		ballotfi.o	. . 3 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
4		ballotfi.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 9259	. . . . . 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 9600	. . . 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 13150	. 2 |- ( C e. O -> ( ( F ` C ) ` ( M + N ) ) = ( (♯ ` ( ( 1 ... ( M + N ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( M + N ) ) \ C ) ) ) )
12	3	ssrab3 3326	. . . . . . . . 9 |- O C_ ( ~P ( 1 ... ( M + N ) ) i^i Fin )
13	12	sseli 3236	. . . . . . . 8 |- ( C e. O -> C e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) )
14	13	elin1d 3410	. . . . . . 7 |- ( C e. O -> C e. ~P ( 1 ... ( M + N ) ) )
15	14	elpwid 3682	. . . . . 6 |- ( C e. O -> C C_ ( 1 ... ( M + N ) ) )
16		sseqin2 3442	. . . . . 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 5676	. . . 4 |- ( C e. O -> (♯ ` ( ( 1 ... ( M + N ) ) i^i C ) ) = (♯ ` C ) )
19		rabssab 3329	. . . . . . 7 |- { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M } C_ { c | (♯ ` c ) = M }
20	19	sseli 3236	. . . . . 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 5681	. . . . . 6 |- ( b = C -> ( (♯ ` b ) = M <-> (♯ ` C ) = M ) )
23		fveqeq2 5681	. . . . . . 7 |- ( c = b -> ( (♯ ` c ) = M <-> (♯ ` b ) = M ) )
24	23	cbvabv 2361	. . . . . 6 |- { c | (♯ ` c ) = M } = { b | (♯ ` b ) = M }
25	22, 24	elab2g 2966	. . . . 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 9605	. . . . . 6 |- 1 e. ZZ
29		fzfig 10796	. . . . . 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 3411	. . . . 5 |- ( C e. O -> C e. Fin )
32		fihashssdif 11187	. . . . 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 9506	. . . . . 6 |- ( M + N ) e. NN0
35		hashfz1 11150	. . . . . 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 6070	. . . 4 |- ( C e. O -> ( (♯ ` ( 1 ... ( M + N ) ) ) - (♯ ` C ) ) = ( ( M + N ) - M ) )
38	1	nncni 9249	. . . . . 6 |- M e. CC
39	2	nncni 9249	. . . . . 6 |- N e. CC
40		pncan2 8482	. . . . . 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 6070	. 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 3210   i^i cin 3212   C_ wss 3213   ~Pcpw 3671   |-> 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   ...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: (None)