Theorem ballotfilemgun 13270

Description: A property of the defined .^ operator. (Contributed by Thierry Arnoux, 26-Apr-2017.) (Revised by Jim Kingdon, 15-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 }
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 ) ) ) ) )
ballotth.e	|- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
ballotth.mgtn	|- N < M
ballotth.i	|- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
ballotth.s	|- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
ballotth.r	|- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
ballotlemg	|- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )
ballotfilemgun.1	|- ( ph -> U e. O )
ballotfilemgun.2	|- ( ph -> L e. ( J ... K ) )

Assertion

Ref	Expression
ballotfilemgun	|- ( ph -> ( U .^ ( J ... K ) ) = ( ( U .^ ( J ... ( L - 1 ) ) ) + ( U .^ ( L ... K ) ) ) )

Distinct variable groups:   M, c   N, c   O, c   i, M   i, N   i, O   k, M   k, N   k, O   i, c, F, k   i, E, k   k, I, c   E, c   i, I, c   k, J   S, k, i, c   R, i   v, u, I   u, J, v   u, R, v   u, S, v   u, U, v   u, O, v   u, K, v   u, L, v   k, L

Allowed substitution hints:   ph( x, v, u, i, k, c)   P( x, v, u, i, k, c)   R( x, k, c)   S( x)   U( x, i, k, c)   E( x, v, u)   .^ ( x, v, u, i, k, c)   F( x, v, u)   I( x)   J( x, i, c)   K( x, i, k, c)   L( x, i, c)   M( x, v, u)   N( x, v, u)   O( x)

Proof of Theorem ballotfilemgun

Step	Hyp	Ref	Expression
1		indir 3480	. . . . . 6 |- ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) i^i U ) = ( ( ( J ... ( L - 1 ) ) i^i U ) u. ( ( L ... K ) i^i U ) )
2	1	fveq2i 5698	. . . . 5 |- (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) i^i U ) ) = (♯ ` ( ( ( J ... ( L - 1 ) ) i^i U ) u. ( ( L ... K ) i^i U ) ) )
3		ballotth.m	. . . . . . 7 |- M e. NN
4		ballotth.n	. . . . . . 7 |- N e. NN
5		ballotfilem.o	. . . . . . 7 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
6		ballotfilemgun.1	. . . . . . 7 |- ( ph -> U e. O )
7		ballotfilemgun.2	. . . . . . . 8 |- ( ph -> L e. ( J ... K ) )
8		elfzel1 10429	. . . . . . . 8 |- ( L e. ( J ... K ) -> J e. ZZ )
9	7, 8	syl 14	. . . . . . 7 |- ( ph -> J e. ZZ )
10	7	elfzelzd 10431	. . . . . . . 8 |- ( ph -> L e. ZZ )
11		peano2zm 9684	. . . . . . . 8 |- ( L e. ZZ -> ( L - 1 ) e. ZZ )
12	10, 11	syl 14	. . . . . . 7 |- ( ph -> ( L - 1 ) e. ZZ )
13	3, 4, 5, 6, 9, 12	ballotfilemcinfz 13228	. . . . . 6 |- ( ph -> ( ( J ... ( L - 1 ) ) i^i U ) e. Fin )
14		elfzel2 10428	. . . . . . . 8 |- ( L e. ( J ... K ) -> K e. ZZ )
15	7, 14	syl 14	. . . . . . 7 |- ( ph -> K e. ZZ )
16	3, 4, 5, 6, 10, 15	ballotfilemcinfz 13228	. . . . . 6 |- ( ph -> ( ( L ... K ) i^i U ) e. Fin )
17	10	zred 9770	. . . . . . . . . 10 |- ( ph -> L e. RR )
18	17	ltm1d 9263	. . . . . . . . 9 |- ( ph -> ( L - 1 ) < L )
19		fzdisj 10459	. . . . . . . . 9 |- ( ( L - 1 ) < L -> ( ( J ... ( L - 1 ) ) i^i ( L ... K ) ) = (/) )
20	18, 19	syl 14	. . . . . . . 8 |- ( ph -> ( ( J ... ( L - 1 ) ) i^i ( L ... K ) ) = (/) )
21	20	ineq1d 3431	. . . . . . 7 |- ( ph -> ( ( ( J ... ( L - 1 ) ) i^i ( L ... K ) ) i^i U ) = ( (/) i^i U ) )
22		inindir 3449	. . . . . . 7 |- ( ( ( J ... ( L - 1 ) ) i^i ( L ... K ) ) i^i U ) = ( ( ( J ... ( L - 1 ) ) i^i U ) i^i ( ( L ... K ) i^i U ) )
23		0in 3558	. . . . . . 7 |- ( (/) i^i U ) = (/)
24	21, 22, 23	3eqtr3g 2294	. . . . . 6 |- ( ph -> ( ( ( J ... ( L - 1 ) ) i^i U ) i^i ( ( L ... K ) i^i U ) ) = (/) )
25		hashun 11247	. . . . . 6 |- ( ( ( ( J ... ( L - 1 ) ) i^i U ) e. Fin /\ ( ( L ... K ) i^i U ) e. Fin /\ ( ( ( J ... ( L - 1 ) ) i^i U ) i^i ( ( L ... K ) i^i U ) ) = (/) ) -> (♯ ` ( ( ( J ... ( L - 1 ) ) i^i U ) u. ( ( L ... K ) i^i U ) ) ) = ( (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) + (♯ ` ( ( L ... K ) i^i U ) ) ) )
26	13, 16, 24, 25	syl3anc 1278	. . . . 5 |- ( ph -> (♯ ` ( ( ( J ... ( L - 1 ) ) i^i U ) u. ( ( L ... K ) i^i U ) ) ) = ( (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) + (♯ ` ( ( L ... K ) i^i U ) ) ) )
27	2, 26	eqtrid 2283	. . . 4 |- ( ph -> (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) i^i U ) ) = ( (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) + (♯ ` ( ( L ... K ) i^i U ) ) ) )
28		difundir 3484	. . . . . 6 |- ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) \ U ) = ( ( ( J ... ( L - 1 ) ) \ U ) u. ( ( L ... K ) \ U ) )
29	28	fveq2i 5698	. . . . 5 |- (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) \ U ) ) = (♯ ` ( ( ( J ... ( L - 1 ) ) \ U ) u. ( ( L ... K ) \ U ) ) )
30	3, 4, 5, 6, 9, 12	ballotfilemdifcfz 13229	. . . . . 6 |- ( ph -> ( ( J ... ( L - 1 ) ) \ U ) e. Fin )
31	3, 4, 5, 6, 10, 15	ballotfilemdifcfz 13229	. . . . . 6 |- ( ph -> ( ( L ... K ) \ U ) e. Fin )
32	20	difeq1d 3346	. . . . . . 7 |- ( ph -> ( ( ( J ... ( L - 1 ) ) i^i ( L ... K ) ) \ U ) = ( (/) \ U ) )
33		difindir 3486	. . . . . . 7 |- ( ( ( J ... ( L - 1 ) ) i^i ( L ... K ) ) \ U ) = ( ( ( J ... ( L - 1 ) ) \ U ) i^i ( ( L ... K ) \ U ) )
34		0dif 3597	. . . . . . 7 |- ( (/) \ U ) = (/)
35	32, 33, 34	3eqtr3g 2294	. . . . . 6 |- ( ph -> ( ( ( J ... ( L - 1 ) ) \ U ) i^i ( ( L ... K ) \ U ) ) = (/) )
36		hashun 11247	. . . . . 6 |- ( ( ( ( J ... ( L - 1 ) ) \ U ) e. Fin /\ ( ( L ... K ) \ U ) e. Fin /\ ( ( ( J ... ( L - 1 ) ) \ U ) i^i ( ( L ... K ) \ U ) ) = (/) ) -> (♯ ` ( ( ( J ... ( L - 1 ) ) \ U ) u. ( ( L ... K ) \ U ) ) ) = ( (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) + (♯ ` ( ( L ... K ) \ U ) ) ) )
37	30, 31, 35, 36	syl3anc 1278	. . . . 5 |- ( ph -> (♯ ` ( ( ( J ... ( L - 1 ) ) \ U ) u. ( ( L ... K ) \ U ) ) ) = ( (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) + (♯ ` ( ( L ... K ) \ U ) ) ) )
38	29, 37	eqtrid 2283	. . . 4 |- ( ph -> (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) \ U ) ) = ( (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) + (♯ ` ( ( L ... K ) \ U ) ) ) )
39	27, 38	oveq12d 6103	. . 3 |- ( ph -> ( (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) i^i U ) ) - (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) \ U ) ) ) = ( ( (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) + (♯ ` ( ( L ... K ) i^i U ) ) ) - ( (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) + (♯ ` ( ( L ... K ) \ U ) ) ) ) )
40		hashcl 11222	. . . . . 6 |- ( ( ( J ... ( L - 1 ) ) i^i U ) e. Fin -> (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) e. NN0 )
41	13, 40	syl 14	. . . . 5 |- ( ph -> (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) e. NN0 )
42	41	nn0cnd 9624	. . . 4 |- ( ph -> (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) e. CC )
43		hashcl 11222	. . . . . 6 |- ( ( ( L ... K ) i^i U ) e. Fin -> (♯ ` ( ( L ... K ) i^i U ) ) e. NN0 )
44	16, 43	syl 14	. . . . 5 |- ( ph -> (♯ ` ( ( L ... K ) i^i U ) ) e. NN0 )
45	44	nn0cnd 9624	. . . 4 |- ( ph -> (♯ ` ( ( L ... K ) i^i U ) ) e. CC )
46		hashcl 11222	. . . . . 6 |- ( ( ( J ... ( L - 1 ) ) \ U ) e. Fin -> (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) e. NN0 )
47	30, 46	syl 14	. . . . 5 |- ( ph -> (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) e. NN0 )
48	47	nn0cnd 9624	. . . 4 |- ( ph -> (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) e. CC )
49		hashcl 11222	. . . . . 6 |- ( ( ( L ... K ) \ U ) e. Fin -> (♯ ` ( ( L ... K ) \ U ) ) e. NN0 )
50	31, 49	syl 14	. . . . 5 |- ( ph -> (♯ ` ( ( L ... K ) \ U ) ) e. NN0 )
51	50	nn0cnd 9624	. . . 4 |- ( ph -> (♯ ` ( ( L ... K ) \ U ) ) e. CC )
52	42, 45, 48, 51	addsub4d 8684	. . 3 |- ( ph -> ( ( (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) + (♯ ` ( ( L ... K ) i^i U ) ) ) - ( (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) + (♯ ` ( ( L ... K ) \ U ) ) ) ) = ( ( (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) - (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) ) + ( (♯ ` ( ( L ... K ) i^i U ) ) - (♯ ` ( ( L ... K ) \ U ) ) ) ) )
53	39, 52	eqtrd 2271	. 2 |- ( ph -> ( (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) i^i U ) ) - (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) \ U ) ) ) = ( ( (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) - (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) ) + ( (♯ ` ( ( L ... K ) i^i U ) ) - (♯ ` ( ( L ... K ) \ U ) ) ) ) )
54		ballotfilem.p	. . . 4 |- P = ( x e. ( ~P O i^i Fin ) |-> ( (♯ ` x ) / (♯ ` O ) ) )
55		ballotth.f	. . . 4 |- F = ( c e. O |-> ( i e. ZZ |-> ( (♯ ` ( ( 1 ... i ) i^i c ) ) - (♯ ` ( ( 1 ... i ) \ c ) ) ) ) )
56		ballotth.e	. . . 4 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
57		ballotth.mgtn	. . . 4 |- N < M
58		ballotth.i	. . . 4 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
59		ballotth.s	. . . 4 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
60		ballotth.r	. . . 4 |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
61		ballotlemg	. . . 4 |- .^ = ( u e. O , v e. Fin |-> ( (♯ ` ( v i^i u ) ) - (♯ ` ( v \ u ) ) ) )
62		eqidd 2239	. . . 4 |- ( ph -> ( J ... K ) = ( J ... K ) )
63	3, 4, 5, 54, 55, 56, 57, 58, 59, 60, 61, 6, 9, 15, 62	ballotfilemgval 13269	. . 3 |- ( ph -> ( U .^ ( J ... K ) ) = ( (♯ ` ( ( J ... K ) i^i U ) ) - (♯ ` ( ( J ... K ) \ U ) ) ) )
64		fzsplit3 10460	. . . . . . 7 |- ( L e. ( J ... K ) -> ( J ... K ) = ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) )
65	7, 64	syl 14	. . . . . 6 |- ( ph -> ( J ... K ) = ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) )
66	65	ineq1d 3431	. . . . 5 |- ( ph -> ( ( J ... K ) i^i U ) = ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) i^i U ) )
67	66	fveq2d 5699	. . . 4 |- ( ph -> (♯ ` ( ( J ... K ) i^i U ) ) = (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) i^i U ) ) )
68	65	difeq1d 3346	. . . . 5 |- ( ph -> ( ( J ... K ) \ U ) = ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) \ U ) )
69	68	fveq2d 5699	. . . 4 |- ( ph -> (♯ ` ( ( J ... K ) \ U ) ) = (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) \ U ) ) )
70	67, 69	oveq12d 6103	. . 3 |- ( ph -> ( (♯ ` ( ( J ... K ) i^i U ) ) - (♯ ` ( ( J ... K ) \ U ) ) ) = ( (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) i^i U ) ) - (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) \ U ) ) ) )
71	63, 70	eqtrd 2271	. 2 |- ( ph -> ( U .^ ( J ... K ) ) = ( (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) i^i U ) ) - (♯ ` ( ( ( J ... ( L - 1 ) ) u. ( L ... K ) ) \ U ) ) ) )
72		eqidd 2239	. . . 4 |- ( ph -> ( J ... ( L - 1 ) ) = ( J ... ( L - 1 ) ) )
73	3, 4, 5, 54, 55, 56, 57, 58, 59, 60, 61, 6, 9, 12, 72	ballotfilemgval 13269	. . 3 |- ( ph -> ( U .^ ( J ... ( L - 1 ) ) ) = ( (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) - (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) ) )
74		eqidd 2239	. . . 4 |- ( ph -> ( L ... K ) = ( L ... K ) )
75	3, 4, 5, 54, 55, 56, 57, 58, 59, 60, 61, 6, 10, 15, 74	ballotfilemgval 13269	. . 3 |- ( ph -> ( U .^ ( L ... K ) ) = ( (♯ ` ( ( L ... K ) i^i U ) ) - (♯ ` ( ( L ... K ) \ U ) ) ) )
76	73, 75	oveq12d 6103	. 2 |- ( ph -> ( ( U .^ ( J ... ( L - 1 ) ) ) + ( U .^ ( L ... K ) ) ) = ( ( (♯ ` ( ( J ... ( L - 1 ) ) i^i U ) ) - (♯ ` ( ( J ... ( L - 1 ) ) \ U ) ) ) + ( (♯ ` ( ( L ... K ) i^i U ) ) - (♯ ` ( ( L ... K ) \ U ) ) ) ) )
77	53, 71, 76	3eqtr4d 2281	1 |- ( ph -> ( U .^ ( J ... K ) ) = ( ( U .^ ( J ... ( L - 1 ) ) ) + ( U .^ ( L ... K ) ) ) )

This proof depends on syntax axioms:   -> wi 4   = wceq 1402   e. wcel 2209   A.wral 2528   {crab 2532   \ cdif 3217   u. cun 3218   i^i cin 3219   (/)c0 3520   ifcif 3638   ~Pcpw 3688   class class class wbr 4130   |-> cmpt 4192   "cima 4777   ` cfv 5377  (class class class)co 6085   e. cmpo 6087   Fincfn 7022  infcinf 7323   RRcr 8178   0cc0 8179   1c1 8180   + caddc 8182   < clt 8360   <_ cle 8361   - cmin 8497   / cdiv 9003   NNcn 9305   NN0cn0 9565   ZZcz 9646   ...cfz 10413  ♯chash 11216

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-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-en 7023  df-dom 7024  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 9306  df-n0 9566  df-z 9647  df-uz 9924  df-fz 10414  df-ihash 11217

This theorem is used by:  ballotfilemfrceq  13274