Theorem ballotfilemdifcfi 13225

Description: Lemma for ballotfi . The portion of a counting representing votes for B up to a specified integer is finite. (Contributed by Jim Kingdon, 8-Jun-2026.)

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 }
ballotfilemc.c	|- ( ph -> C e. O )
ballotfilemc.j	|- ( ph -> J e. ZZ )

Assertion

Ref	Expression
ballotfilemdifcfi	|- ( ph -> ( ( 1 ... J ) \ C ) e. Fin )

Distinct variable groups:   M, c   N, c   O, c

Allowed substitution hints:   ph( c)   C( c)   J( c)

Proof of Theorem ballotfilemdifcfi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		1zzd 9671	. . 3 |- ( ph -> 1 e. ZZ )
2		ballotfilemc.j	. . 3 |- ( ph -> J e. ZZ )
3	1, 2	fzfigd 10868	. 2 |- ( ph -> ( 1 ... J ) e. Fin )
4		difssd 3356	. 2 |- ( ph -> ( ( 1 ... J ) \ C ) C_ ( 1 ... J ) )
5		elfzelz 10428	. . . . . . 7 |- ( x e. ( 1 ... J ) -> x e. ZZ )
6	5	adantl 277	. . . . . 6 |- ( ( ph /\ x e. ( 1 ... J ) ) -> x e. ZZ )
7		1zzd 9671	. . . . . 6 |- ( ( ph /\ x e. ( 1 ... J ) ) -> 1 e. ZZ )
8	2	adantr 276	. . . . . 6 |- ( ( ph /\ x e. ( 1 ... J ) ) -> J e. ZZ )
9		fzdcel 10444	. . . . . 6 |- ( ( x e. ZZ /\ 1 e. ZZ /\ J e. ZZ ) -> DECID x e. ( 1 ... J ) )
10	6, 7, 8, 9	syl3anc 1278	. . . . 5 |- ( ( ph /\ x e. ( 1 ... J ) ) -> DECID x e. ( 1 ... J ) )
11		ballotth.m	. . . . . . 7 |- M e. NN
12		ballotth.n	. . . . . . 7 |- N e. NN
13		ballotfilem.o	. . . . . . 7 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
14		ballotfilemc.c	. . . . . . . 8 |- ( ph -> C e. O )
15	14	adantr 276	. . . . . . 7 |- ( ( ph /\ x e. ( 1 ... J ) ) -> C e. O )
16	11, 12, 13, 15, 6	ballotfilemcdc 13223	. . . . . 6 |- ( ( ph /\ x e. ( 1 ... J ) ) -> DECID x e. C )
17		dcn 854	. . . . . 6 |- (DECID x e. C -> DECID -. x e. C )
18	16, 17	syl 14	. . . . 5 |- ( ( ph /\ x e. ( 1 ... J ) ) -> DECID -. x e. C )
19	10, 18	dcand 945	. . . 4 |- ( ( ph /\ x e. ( 1 ... J ) ) -> DECID ( x e. ( 1 ... J ) /\ -. x e. C ) )
20		eldif 3229	. . . . 5 |- ( x e. ( ( 1 ... J ) \ C ) <-> ( x e. ( 1 ... J ) /\ -. x e. C ) )
21	20	dcbii 852	. . . 4 |- (DECID x e. ( ( 1 ... J ) \ C ) <-> DECID ( x e. ( 1 ... J ) /\ -. x e. C ) )
22	19, 21	sylibr 134	. . 3 |- ( ( ph /\ x e. ( 1 ... J ) ) -> DECID x e. ( ( 1 ... J ) \ C ) )
23	22	ralrimiva 2623	. 2 |- ( ph -> A. x e. ( 1 ... J )DECID x e. ( ( 1 ... J ) \ C ) )
24		ssfidc 7245	. 2 |- ( ( ( 1 ... J ) e. Fin /\ ( ( 1 ... J ) \ C ) C_ ( 1 ... J ) /\ A. x e. ( 1 ... J )DECID x e. ( ( 1 ... J ) \ C ) ) -> ( ( 1 ... J ) \ C ) e. Fin )
25	3, 4, 23, 24	syl3anc 1278	1 |- ( ph -> ( ( 1 ... J ) \ C ) e. Fin )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 846   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   \ cdif 3217   i^i cin 3219   C_ wss 3220   ~Pcpw 3688   ` cfv 5377  (class class class)co 6085   Fincfn 7022   1c1 8180   + caddc 8182   NNcn 9304   ZZcz 9644   ...cfz 10411  ♯chash 11214

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 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-addass 8281  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295

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 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-recs 6576  df-frec 6662  df-1o 6687  df-er 6807  df-en 7023  df-fin 7025  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-inn 9305  df-n0 9564  df-z 9645  df-uz 9922  df-fz 10412

This theorem is used by:  ballotfilemfval  13229  ballotfilemfelz  13230  ballotfilemfp1  13231  ballotfilemfrc  13270