Theorem ballotfilemdifcfi 13148

Description: Lemma for ballotfi . The portion of an integer range which is not part of a particular element of O is finite. (Contributed by Jim Kingdon, 8-Jun-2026.)

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 }
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 9606	. . 3 |- ( ph -> 1 e. ZZ )
2		ballotfilemc.j	. . 3 |- ( ph -> J e. ZZ )
3	1, 2	fzfigd 10797	. 2 |- ( ph -> ( 1 ... J ) e. Fin )
4		difssd 3348	. 2 |- ( ph -> ( ( 1 ... J ) \ C ) C_ ( 1 ... J ) )
5		elfzelz 10362	. . . . . . 7 |- ( x e. ( 1 ... J ) -> x e. ZZ )
6	5	adantl 277	. . . . . 6 |- ( ( ph /\ x e. ( 1 ... J ) ) -> x e. ZZ )
7		1zzd 9606	. . . . . 6 |- ( ( ph /\ x e. ( 1 ... J ) ) -> 1 e. ZZ )
8	2	adantr 276	. . . . . 6 |- ( ( ph /\ x e. ( 1 ... J ) ) -> J e. ZZ )
9		fzdcel 10377	. . . . . 6 |- ( ( x e. ZZ /\ 1 e. ZZ /\ J e. ZZ ) -> DECID x e. ( 1 ... J ) )
10	6, 7, 8, 9	syl3anc 1274	. . . . 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		ballotfi.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 13146	. . . . . 6 |- ( ( ph /\ x e. ( 1 ... J ) ) -> DECID x e. C )
17		dcn 850	. . . . . 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 941	. . . 4 |- ( ( ph /\ x e. ( 1 ... J ) ) -> DECID ( x e. ( 1 ... J ) /\ -. x e. C ) )
20		eldif 3222	. . . . 5 |- ( x e. ( ( 1 ... J ) \ C ) <-> ( x e. ( 1 ... J ) /\ -. x e. C ) )
21	20	dcbii 848	. . . 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 2617	. 2 |- ( ph -> A. x e. ( 1 ... J )DECID x e. ( ( 1 ... J ) \ C ) )
24		ssfidc 7200	. 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 1274	1 |- ( ph -> ( ( 1 ... J ) \ C ) e. Fin )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104  DECID wdc 842   = wceq 1398   e. wcel 2205   A.wral 2522   {crab 2526   \ cdif 3210   i^i cin 3212   C_ wss 3213   ~Pcpw 3671   ` cfv 5354  (class class class)co 6052   Fincfn 6977   1c1 8130   + caddc 8132   NNcn 9239   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-frec 6624  df-1o 6649  df-er 6769  df-en 6978  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

This theorem is referenced by:  ballotfilemfval  13150  ballotfilemfelz  13151  ballotfilemfp1  13152