Theorem ballotfilemfmpn 13212

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 9303	. . . . . 6 |- ( ( M e. NN /\ N e. NN ) -> ( M + N ) e. NN )
8	1, 2, 7	mp2an 430	. . . . 5 |- ( M + N ) e. NN
9	8	nnzi 9644	. . . 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 13207	. 2 |- ( C e. O -> ( ( F ` C ) ` ( M + N ) ) = ( (♯ ` ( ( 1 ... ( M + N ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( M + N ) ) \ C ) ) ) )
12	3	ssrab3 3334	. . . . . . . . 9 |- O C_ ( ~P ( 1 ... ( M + N ) ) i^i Fin )
13	12	sseli 3244	. . . . . . . 8 |- ( C e. O -> C e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) )
14	13	elin1d 3418	. . . . . . 7 |- ( C e. O -> C e. ~P ( 1 ... ( M + N ) ) )
15	14	elpwid 3696	. . . . . 6 |- ( C e. O -> C C_ ( 1 ... ( M + N ) ) )
16		sseqin2 3450	. . . . . 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 5694	. . . 4 |- ( C e. O -> (♯ ` ( ( 1 ... ( M + N ) ) i^i C ) ) = (♯ ` C ) )
19		rabssab 3337	. . . . . . 7 |- { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M } C_ { c | (♯ ` c ) = M }
20	19	sseli 3244	. . . . . 6 |- ( C e. { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M } -> C e. { c | (♯ ` c ) = M } )
21	20, 3	eleq2s 2333	. . . . 5 |- ( C e. O -> C e. { c | (♯ ` c ) = M } )
22		fveqeq2 5699	. . . . . 6 |- ( b = C -> ( (♯ ` b ) = M <-> (♯ ` C ) = M ) )
23		fveqeq2 5699	. . . . . . 7 |- ( c = b -> ( (♯ ` c ) = M <-> (♯ ` b ) = M ) )
24	23	cbvabv 2365	. . . . . 6 |- { c | (♯ ` c ) = M } = { b | (♯ ` b ) = M }
25	22, 24	elab2g 2973	. . . . 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 2271	. . 3 |- ( C e. O -> (♯ ` ( ( 1 ... ( M + N ) ) i^i C ) ) = M )
28		1z 9649	. . . . . 6 |- 1 e. ZZ
29		fzfig 10845	. . . . . 6 |- ( ( 1 e. ZZ /\ ( M + N ) e. ZZ ) -> ( 1 ... ( M + N ) ) e. Fin )
30	28, 9, 29	mp2an 430	. . . . 5 |- ( 1 ... ( M + N ) ) e. Fin
31	13	elin2d 3419	. . . . 5 |- ( C e. O -> C e. Fin )
32		fihashssdif 11237	. . . . 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 1383	. . . 4 |- ( C e. O -> (♯ ` ( ( 1 ... ( M + N ) ) \ C ) ) = ( (♯ ` ( 1 ... ( M + N ) ) ) - (♯ ` C ) ) )
34	8	nnnn0i 9550	. . . . . 6 |- ( M + N ) e. NN0
35		hashfz1 11200	. . . . . 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 6093	. . . 4 |- ( C e. O -> ( (♯ ` ( 1 ... ( M + N ) ) ) - (♯ ` C ) ) = ( ( M + N ) - M ) )
38	1	nncni 9293	. . . . . 6 |- M e. CC
39	2	nncni 9293	. . . . . 6 |- N e. CC
40		pncan2 8523	. . . . . 6 |- ( ( M e. CC /\ N e. CC ) -> ( ( M + N ) - M ) = N )
41	38, 39, 40	mp2an 430	. . . . 5 |- ( ( M + N ) - M ) = N
42	41	a1i 9	. . . 4 |- ( C e. O -> ( ( M + N ) - M ) = N )
43	33, 37, 42	3eqtrd 2275	. . 3 |- ( C e. O -> (♯ ` ( ( 1 ... ( M + N ) ) \ C ) ) = N )
44	27, 43	oveq12d 6093	. 2 |- ( C e. O -> ( (♯ ` ( ( 1 ... ( M + N ) ) i^i C ) ) - (♯ ` ( ( 1 ... ( M + N ) ) \ C ) ) ) = ( M - N ) )
45	11, 44	eqtrd 2271	1 |- ( C e. O -> ( ( F ` C ) ` ( M + N ) ) = ( M - N ) )

Syntax hints:   -> wi 4   = wceq 1402   e. wcel 2209   {cab 2224   {crab 2532   \ cdif 3217   i^i cin 3219   C_ wss 3220   ~Pcpw 3685   |-> cmpt 4187   ` cfv 5372  (class class class)co 6075   Fincfn 7012   CCcc 8167   1c1 8170   + caddc 8172   - cmin 8487   / cdiv 8992   NNcn 9283   NN0cn0 9542   ZZcz 9623   ...cfz 10390  ♯chash 11192

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-addcom 8269  ax-addass 8271  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-0id 8277  ax-rnegex 8278  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-en 7013  df-dom 7014  df-fin 7015  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-inn 9284  df-n0 9543  df-z 9624  df-uz 9901  df-fz 10391  df-ihash 11193

This theorem is referenced by:  ballotfilem5  13220