Theorem ballotlemgun 30586

Description: A property of the defined .^ operator. (Contributed by Thierry Arnoux, 26-Apr-2017.)

Hypotheses

Ref	Expression
ballotth.m	|- M e. NN
ballotth.n	|- N e. NN
ballotth.o	|- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
ballotth.p	|- P = ( x e. ~P O |-> ( ( # ` 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. Fin , v e. Fin |-> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) )
ballotlemgun.1	|- ( ph -> U e. Fin )
ballotlemgun.2	|- ( ph -> V e. Fin )
ballotlemgun.3	|- ( ph -> W e. Fin )
ballotlemgun.4	|- ( ph -> ( V i^i W ) = (/) )

Assertion

Ref	Expression
ballotlemgun	|- ( ph -> ( U .^ ( V u. W ) ) = ( ( U .^ V ) + ( U .^ W ) ) )

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   S, k, i, c   R, i   v, u, I   u, R, v   u, S, v   u, U, v   u, V, v   u, W, v

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)   M( x, v, u)   N( x, v, u)   O( x, v, u)   V( x, i, k, c)   W( x, i, k, c)

Proof of Theorem ballotlemgun

Step	Hyp	Ref	Expression
1		indir 3875	. . . . . 6 |- ( ( V u. W ) i^i U ) = ( ( V i^i U ) u. ( W i^i U ) )
2	1	fveq2i 6194	. . . . 5 |- ( # ` ( ( V u. W ) i^i U ) ) = ( # ` ( ( V i^i U ) u. ( W i^i U ) ) )
3		ballotlemgun.2	. . . . . . 7 |- ( ph -> V e. Fin )
4		infi 8184	. . . . . . 7 |- ( V e. Fin -> ( V i^i U ) e. Fin )
5	3, 4	syl 17	. . . . . 6 |- ( ph -> ( V i^i U ) e. Fin )
6		ballotlemgun.3	. . . . . . 7 |- ( ph -> W e. Fin )
7		infi 8184	. . . . . . 7 |- ( W e. Fin -> ( W i^i U ) e. Fin )
8	6, 7	syl 17	. . . . . 6 |- ( ph -> ( W i^i U ) e. Fin )
9		ballotlemgun.4	. . . . . . . 8 |- ( ph -> ( V i^i W ) = (/) )
10	9	ineq1d 3813	. . . . . . 7 |- ( ph -> ( ( V i^i W ) i^i U ) = ( (/) i^i U ) )
11		inindir 3831	. . . . . . 7 |- ( ( V i^i W ) i^i U ) = ( ( V i^i U ) i^i ( W i^i U ) )
12		0in 3969	. . . . . . 7 |- ( (/) i^i U ) = (/)
13	10, 11, 12	3eqtr3g 2679	. . . . . 6 |- ( ph -> ( ( V i^i U ) i^i ( W i^i U ) ) = (/) )
14		hashun 13171	. . . . . 6 |- ( ( ( V i^i U ) e. Fin /\ ( W i^i U ) e. Fin /\ ( ( V i^i U ) i^i ( W i^i U ) ) = (/) ) -> ( # ` ( ( V i^i U ) u. ( W i^i U ) ) ) = ( ( # ` ( V i^i U ) ) + ( # ` ( W i^i U ) ) ) )
15	5, 8, 13, 14	syl3anc 1326	. . . . 5 |- ( ph -> ( # ` ( ( V i^i U ) u. ( W i^i U ) ) ) = ( ( # ` ( V i^i U ) ) + ( # ` ( W i^i U ) ) ) )
16	2, 15	syl5eq 2668	. . . 4 |- ( ph -> ( # ` ( ( V u. W ) i^i U ) ) = ( ( # ` ( V i^i U ) ) + ( # ` ( W i^i U ) ) ) )
17		difundir 3880	. . . . . 6 |- ( ( V u. W ) \ U ) = ( ( V \ U ) u. ( W \ U ) )
18	17	fveq2i 6194	. . . . 5 |- ( # ` ( ( V u. W ) \ U ) ) = ( # ` ( ( V \ U ) u. ( W \ U ) ) )
19		diffi 8192	. . . . . . 7 |- ( V e. Fin -> ( V \ U ) e. Fin )
20	3, 19	syl 17	. . . . . 6 |- ( ph -> ( V \ U ) e. Fin )
21		diffi 8192	. . . . . . 7 |- ( W e. Fin -> ( W \ U ) e. Fin )
22	6, 21	syl 17	. . . . . 6 |- ( ph -> ( W \ U ) e. Fin )
23	9	difeq1d 3727	. . . . . . 7 |- ( ph -> ( ( V i^i W ) \ U ) = ( (/) \ U ) )
24		difindir 3882	. . . . . . 7 |- ( ( V i^i W ) \ U ) = ( ( V \ U ) i^i ( W \ U ) )
25		0dif 3977	. . . . . . 7 |- ( (/) \ U ) = (/)
26	23, 24, 25	3eqtr3g 2679	. . . . . 6 |- ( ph -> ( ( V \ U ) i^i ( W \ U ) ) = (/) )
27		hashun 13171	. . . . . 6 |- ( ( ( V \ U ) e. Fin /\ ( W \ U ) e. Fin /\ ( ( V \ U ) i^i ( W \ U ) ) = (/) ) -> ( # ` ( ( V \ U ) u. ( W \ U ) ) ) = ( ( # ` ( V \ U ) ) + ( # ` ( W \ U ) ) ) )
28	20, 22, 26, 27	syl3anc 1326	. . . . 5 |- ( ph -> ( # ` ( ( V \ U ) u. ( W \ U ) ) ) = ( ( # ` ( V \ U ) ) + ( # ` ( W \ U ) ) ) )
29	18, 28	syl5eq 2668	. . . 4 |- ( ph -> ( # ` ( ( V u. W ) \ U ) ) = ( ( # ` ( V \ U ) ) + ( # ` ( W \ U ) ) ) )
30	16, 29	oveq12d 6668	. . 3 |- ( ph -> ( ( # ` ( ( V u. W ) i^i U ) ) - ( # ` ( ( V u. W ) \ U ) ) ) = ( ( ( # ` ( V i^i U ) ) + ( # ` ( W i^i U ) ) ) - ( ( # ` ( V \ U ) ) + ( # ` ( W \ U ) ) ) ) )
31		hashcl 13147	. . . . . 6 |- ( ( V i^i U ) e. Fin -> ( # ` ( V i^i U ) ) e. NN0 )
32	3, 4, 31	3syl 18	. . . . 5 |- ( ph -> ( # ` ( V i^i U ) ) e. NN0 )
33	32	nn0cnd 11353	. . . 4 |- ( ph -> ( # ` ( V i^i U ) ) e. CC )
34		hashcl 13147	. . . . . 6 |- ( ( W i^i U ) e. Fin -> ( # ` ( W i^i U ) ) e. NN0 )
35	6, 7, 34	3syl 18	. . . . 5 |- ( ph -> ( # ` ( W i^i U ) ) e. NN0 )
36	35	nn0cnd 11353	. . . 4 |- ( ph -> ( # ` ( W i^i U ) ) e. CC )
37		hashcl 13147	. . . . . 6 |- ( ( V \ U ) e. Fin -> ( # ` ( V \ U ) ) e. NN0 )
38	3, 19, 37	3syl 18	. . . . 5 |- ( ph -> ( # ` ( V \ U ) ) e. NN0 )
39	38	nn0cnd 11353	. . . 4 |- ( ph -> ( # ` ( V \ U ) ) e. CC )
40		hashcl 13147	. . . . . 6 |- ( ( W \ U ) e. Fin -> ( # ` ( W \ U ) ) e. NN0 )
41	6, 21, 40	3syl 18	. . . . 5 |- ( ph -> ( # ` ( W \ U ) ) e. NN0 )
42	41	nn0cnd 11353	. . . 4 |- ( ph -> ( # ` ( W \ U ) ) e. CC )
43	33, 36, 39, 42	addsub4d 10439	. . 3 |- ( ph -> ( ( ( # ` ( V i^i U ) ) + ( # ` ( W i^i U ) ) ) - ( ( # ` ( V \ U ) ) + ( # ` ( W \ U ) ) ) ) = ( ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) + ( ( # ` ( W i^i U ) ) - ( # ` ( W \ U ) ) ) ) )
44	30, 43	eqtrd 2656	. 2 |- ( ph -> ( ( # ` ( ( V u. W ) i^i U ) ) - ( # ` ( ( V u. W ) \ U ) ) ) = ( ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) + ( ( # ` ( W i^i U ) ) - ( # ` ( W \ U ) ) ) ) )
45		ballotlemgun.1	. . 3 |- ( ph -> U e. Fin )
46		unfi 8227	. . . 4 |- ( ( V e. Fin /\ W e. Fin ) -> ( V u. W ) e. Fin )
47	3, 6, 46	syl2anc 693	. . 3 |- ( ph -> ( V u. W ) e. Fin )
48		ballotth.m	. . . 4 |- M e. NN
49		ballotth.n	. . . 4 |- N e. NN
50		ballotth.o	. . . 4 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
51		ballotth.p	. . . 4 |- P = ( x e. ~P O |-> ( ( # ` x ) / ( # ` O ) ) )
52		ballotth.f	. . . 4 |- F = ( c e. O |-> ( i e. ZZ |-> ( ( # ` ( ( 1 ... i ) i^i c ) ) - ( # ` ( ( 1 ... i ) \ c ) ) ) ) )
53		ballotth.e	. . . 4 |- E = { c e. O | A. i e. ( 1 ... ( M + N ) ) 0 < ( ( F ` c ) ` i ) }
54		ballotth.mgtn	. . . 4 |- N < M
55		ballotth.i	. . . 4 |- I = ( c e. ( O \ E ) |-> inf ( { k e. ( 1 ... ( M + N ) ) | ( ( F ` c ) ` k ) = 0 } , RR , < ) )
56		ballotth.s	. . . 4 |- S = ( c e. ( O \ E ) |-> ( i e. ( 1 ... ( M + N ) ) |-> if ( i <_ ( I ` c ) , ( ( ( I ` c ) + 1 ) - i ) , i ) ) )
57		ballotth.r	. . . 4 |- R = ( c e. ( O \ E ) |-> ( ( S ` c ) " c ) )
58		ballotlemg	. . . 4 |- .^ = ( u e. Fin , v e. Fin |-> ( ( # ` ( v i^i u ) ) - ( # ` ( v \ u ) ) ) )
59	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 30585	. . 3 |- ( ( U e. Fin /\ ( V u. W ) e. Fin ) -> ( U .^ ( V u. W ) ) = ( ( # ` ( ( V u. W ) i^i U ) ) - ( # ` ( ( V u. W ) \ U ) ) ) )
60	45, 47, 59	syl2anc 693	. 2 |- ( ph -> ( U .^ ( V u. W ) ) = ( ( # ` ( ( V u. W ) i^i U ) ) - ( # ` ( ( V u. W ) \ U ) ) ) )
61	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 30585	. . . 4 |- ( ( U e. Fin /\ V e. Fin ) -> ( U .^ V ) = ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) )
62	45, 3, 61	syl2anc 693	. . 3 |- ( ph -> ( U .^ V ) = ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) )
63	48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58	ballotlemgval 30585	. . . 4 |- ( ( U e. Fin /\ W e. Fin ) -> ( U .^ W ) = ( ( # ` ( W i^i U ) ) - ( # ` ( W \ U ) ) ) )
64	45, 6, 63	syl2anc 693	. . 3 |- ( ph -> ( U .^ W ) = ( ( # ` ( W i^i U ) ) - ( # ` ( W \ U ) ) ) )
65	62, 64	oveq12d 6668	. 2 |- ( ph -> ( ( U .^ V ) + ( U .^ W ) ) = ( ( ( # ` ( V i^i U ) ) - ( # ` ( V \ U ) ) ) + ( ( # ` ( W i^i U ) ) - ( # ` ( W \ U ) ) ) ) )
66	44, 60, 65	3eqtr4d 2666	1 |- ( ph -> ( U .^ ( V u. W ) ) = ( ( U .^ V ) + ( U .^ W ) ) )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   {crab 2916   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   ifcif 4086   ~Pcpw 4158   class class class wbr 4653   |-> cmpt 4729   "cima 5117   ` cfv 5888  (class class class)co 6650   |-> cmpt2 6652   Fincfn 7955  infcinf 8347   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   < clt 10074   <_ cle 10075   - cmin 10266   / cdiv 10684   NNcn 11020   NN0cn0 11292   ZZcz 11377   ...cfz 12326   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-hash 13118

This theorem is referenced by:  ballotlemfrceq  30590