Theorem ballotfilemcdc 13146

Description: Lemma for ballotfi . It is decidable whether a given integer is an element of a particular element of O. (Contributed by Jim Kingdon, 7-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 )
ballotfilemcdc.dc	|- ( ph -> K e. ZZ )

Assertion

Ref	Expression
ballotfilemcdc	|- ( ph -> DECID K e. C )

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

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

Proof of Theorem ballotfilemcdc

Dummy variables w y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq2 2298	. . 3 |- ( w = (/) -> ( K e. w <-> K e. (/) ) )
2	1	dcbid 846	. 2 |- ( w = (/) -> (DECID K e. w <-> DECID K e. (/) ) )
3		eleq2 2298	. . 3 |- ( w = y -> ( K e. w <-> K e. y ) )
4	3	dcbid 846	. 2 |- ( w = y -> (DECID K e. w <-> DECID K e. y ) )
5		eleq2 2298	. . 3 |- ( w = ( y u. { z } ) -> ( K e. w <-> K e. ( y u. { z } ) ) )
6	5	dcbid 846	. 2 |- ( w = ( y u. { z } ) -> (DECID K e. w <-> DECID K e. ( y u. { z } ) ) )
7		eleq2 2298	. . 3 |- ( w = C -> ( K e. w <-> K e. C ) )
8	7	dcbid 846	. 2 |- ( w = C -> (DECID K e. w <-> DECID K e. C ) )
9		noel 3514	. . . . 5 |- -. K e. (/)
10	9	olci 740	. . . 4 |- ( K e. (/) \/ -. K e. (/) )
11		df-dc 843	. . . 4 |- (DECID K e. (/) <-> ( K e. (/) \/ -. K e. (/) ) )
12	10, 11	mpbir 146	. . 3 |- DECID K e. (/)
13	12	a1i 9	. 2 |- ( ph -> DECID K e. (/) )
14		simpr 110	. . . 4 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> DECID K e. y )
15		ballotfilemcdc.dc	. . . . . . 7 |- ( ph -> K e. ZZ )
16	15	ad3antrrr 492	. . . . . 6 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> K e. ZZ )
17		ballotfilemc.c	. . . . . . . . . . 11 |- ( ph -> C e. O )
18		ballotth.m	. . . . . . . . . . . 12 |- M e. NN
19		ballotth.n	. . . . . . . . . . . 12 |- N e. NN
20		ballotfi.o	. . . . . . . . . . . 12 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
21	18, 19, 20	ballotfilemelo 13145	. . . . . . . . . . 11 |- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin /\ (♯ ` C ) = M ) )
22	17, 21	sylib 122	. . . . . . . . . 10 |- ( ph -> ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin /\ (♯ ` C ) = M ) )
23	22	simp1d 1036	. . . . . . . . 9 |- ( ph -> C C_ ( 1 ... ( M + N ) ) )
24	23	ad3antrrr 492	. . . . . . . 8 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> C C_ ( 1 ... ( M + N ) ) )
25		simplrr 538	. . . . . . . . 9 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> z e. ( C \ y ) )
26	25	eldifad 3224	. . . . . . . 8 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> z e. C )
27	24, 26	sseldd 3241	. . . . . . 7 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> z e. ( 1 ... ( M + N ) ) )
28	27	elfzelzd 10363	. . . . . 6 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> z e. ZZ )
29		zdceq 9655	. . . . . 6 |- ( ( K e. ZZ /\ z e. ZZ ) -> DECID K = z )
30	16, 28, 29	syl2anc 411	. . . . 5 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> DECID K = z )
31		vex 2818	. . . . . . 7 |- z e. _V
32	31	elsn2 3725	. . . . . 6 |- ( K e. { z } <-> K = z )
33	32	dcbii 848	. . . . 5 |- (DECID K e. { z } <-> DECID K = z )
34	30, 33	sylibr 134	. . . 4 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> DECID K e. { z } )
35	14, 34	dcun 3621	. . 3 |- ( ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) /\ DECID K e. y ) -> DECID K e. ( y u. { z } ) )
36	35	ex 115	. 2 |- ( ( ( ph /\ y e. Fin ) /\ ( y C_ C /\ z e. ( C \ y ) ) ) -> (DECID K e. y -> DECID K e. ( y u. { z } ) ) )
37	22	simp2d 1037	. 2 |- ( ph -> C e. Fin )
38	2, 4, 6, 8, 13, 36, 37	findcard2sd 7151	1 |- ( ph -> DECID K e. C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   e. wcel 2205   {crab 2526   \ cdif 3210   u. cun 3211   i^i cin 3212   C_ wss 3213   (/)c0 3510   ~Pcpw 3671   {csn 3691   ` 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-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-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-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:  ballotfilemcinfi  13147  ballotfilemdifcfi  13148