Theorem hashreprin 30698

Description: Express a sum of representations over an intersection using a product of the indicator function (Contributed by Thierry Arnoux, 11-Dec-2021.)

Hypotheses

Ref	Expression
reprval.a	|- ( ph -> A C_ NN )
reprval.m	|- ( ph -> M e. ZZ )
reprval.s	|- ( ph -> S e. NN0 )
hashreprin.b	|- ( ph -> B e. Fin )
hashreprin.1	|- ( ph -> B C_ NN )

Assertion

Ref	Expression
hashreprin	|- ( ph -> ( # ` ( ( A i^i B ) (repr ` S ) M ) ) = sum_ c e. ( B (repr ` S ) M ) prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) )

Distinct variable groups:   A, c   M, c   S, a, c   ph, c   A, a   B, a, c   M, a   ph, a

Proof of Theorem hashreprin

Step	Hyp	Ref	Expression
1		hashreprin.1	. . . . 5 |- ( ph -> B C_ NN )
2		reprval.m	. . . . 5 |- ( ph -> M e. ZZ )
3		reprval.s	. . . . 5 |- ( ph -> S e. NN0 )
4		hashreprin.b	. . . . 5 |- ( ph -> B e. Fin )
5	1, 2, 3, 4	reprfi 30694	. . . 4 |- ( ph -> ( B (repr ` S ) M ) e. Fin )
6		inss2 3834	. . . . . 6 |- ( A i^i B ) C_ B
7	6	a1i 11	. . . . 5 |- ( ph -> ( A i^i B ) C_ B )
8	1, 2, 3, 7	reprss 30695	. . . 4 |- ( ph -> ( ( A i^i B ) (repr ` S ) M ) C_ ( B (repr ` S ) M ) )
9	5, 8	ssfid 8183	. . 3 |- ( ph -> ( ( A i^i B ) (repr ` S ) M ) e. Fin )
10		1cnd 10056	. . 3 |- ( ph -> 1 e. CC )
11		fsumconst 14522	. . 3 |- ( ( ( ( A i^i B ) (repr ` S ) M ) e. Fin /\ 1 e. CC ) -> sum_ c e. ( ( A i^i B ) (repr ` S ) M ) 1 = ( ( # ` ( ( A i^i B ) (repr ` S ) M ) ) x. 1 ) )
12	9, 10, 11	syl2anc 693	. 2 |- ( ph -> sum_ c e. ( ( A i^i B ) (repr ` S ) M ) 1 = ( ( # ` ( ( A i^i B ) (repr ` S ) M ) ) x. 1 ) )
13	10	ralrimivw 2967	. . . 4 |- ( ph -> A. c e. ( ( A i^i B ) (repr ` S ) M ) 1 e. CC )
14	5	olcd 408	. . . 4 |- ( ph -> ( ( B (repr ` S ) M ) C_ ( ZZ>= ` 0 ) \/ ( B (repr ` S ) M ) e. Fin ) )
15		sumss2 14457	. . . 4 |- ( ( ( ( ( A i^i B ) (repr ` S ) M ) C_ ( B (repr ` S ) M ) /\ A. c e. ( ( A i^i B ) (repr ` S ) M ) 1 e. CC ) /\ ( ( B (repr ` S ) M ) C_ ( ZZ>= ` 0 ) \/ ( B (repr ` S ) M ) e. Fin ) ) -> sum_ c e. ( ( A i^i B ) (repr ` S ) M ) 1 = sum_ c e. ( B (repr ` S ) M ) if ( c e. ( ( A i^i B ) (repr ` S ) M ) , 1 , 0 ) )
16	8, 13, 14, 15	syl21anc 1325	. . 3 |- ( ph -> sum_ c e. ( ( A i^i B ) (repr ` S ) M ) 1 = sum_ c e. ( B (repr ` S ) M ) if ( c e. ( ( A i^i B ) (repr ` S ) M ) , 1 , 0 ) )
17	1, 2, 3	reprinrn 30696	. . . . . . . 8 |- ( ph -> ( c e. ( ( B i^i A ) (repr ` S ) M ) <-> ( c e. ( B (repr ` S ) M ) /\ ran c C_ A ) ) )
18		incom 3805	. . . . . . . . . . . 12 |- ( B i^i A ) = ( A i^i B )
19	18	oveq1i 6660	. . . . . . . . . . 11 |- ( ( B i^i A ) (repr ` S ) M ) = ( ( A i^i B ) (repr ` S ) M )
20	19	eleq2i 2693	. . . . . . . . . 10 |- ( c e. ( ( B i^i A ) (repr ` S ) M ) <-> c e. ( ( A i^i B ) (repr ` S ) M ) )
21	20	bibi1i 328	. . . . . . . . 9 |- ( ( c e. ( ( B i^i A ) (repr ` S ) M ) <-> ( c e. ( B (repr ` S ) M ) /\ ran c C_ A ) ) <-> ( c e. ( ( A i^i B ) (repr ` S ) M ) <-> ( c e. ( B (repr ` S ) M ) /\ ran c C_ A ) ) )
22	21	imbi2i 326	. . . . . . . 8 |- ( ( ph -> ( c e. ( ( B i^i A ) (repr ` S ) M ) <-> ( c e. ( B (repr ` S ) M ) /\ ran c C_ A ) ) ) <-> ( ph -> ( c e. ( ( A i^i B ) (repr ` S ) M ) <-> ( c e. ( B (repr ` S ) M ) /\ ran c C_ A ) ) ) )
23	17, 22	mpbi 220	. . . . . . 7 |- ( ph -> ( c e. ( ( A i^i B ) (repr ` S ) M ) <-> ( c e. ( B (repr ` S ) M ) /\ ran c C_ A ) ) )
24	23	baibd 948	. . . . . 6 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> ( c e. ( ( A i^i B ) (repr ` S ) M ) <-> ran c C_ A ) )
25	24	ifbid 4108	. . . . 5 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> if ( c e. ( ( A i^i B ) (repr ` S ) M ) , 1 , 0 ) = if ( ran c C_ A , 1 , 0 ) )
26		nnex 11026	. . . . . . . . 9 |- NN e. _V
27	26	a1i 11	. . . . . . . 8 |- ( ph -> NN e. _V )
28	27	ralrimivw 2967	. . . . . . 7 |- ( ph -> A. c e. ( B (repr ` S ) M ) NN e. _V )
29	28	r19.21bi 2932	. . . . . 6 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> NN e. _V )
30		fzofi 12773	. . . . . . 7 |- ( 0..^ S ) e. Fin
31	30	a1i 11	. . . . . 6 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> ( 0..^ S ) e. Fin )
32		reprval.a	. . . . . . 7 |- ( ph -> A C_ NN )
33	32	adantr 481	. . . . . 6 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> A C_ NN )
34	1	adantr 481	. . . . . . . 8 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> B C_ NN )
35	2	adantr 481	. . . . . . . 8 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> M e. ZZ )
36	3	adantr 481	. . . . . . . 8 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> S e. NN0 )
37		simpr 477	. . . . . . . 8 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> c e. ( B (repr ` S ) M ) )
38	34, 35, 36, 37	reprf 30690	. . . . . . 7 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> c : ( 0..^ S ) --> B )
39	38, 34	fssd 6057	. . . . . 6 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> c : ( 0..^ S ) --> NN )
40	29, 31, 33, 39	prodindf 30085	. . . . 5 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) = if ( ran c C_ A , 1 , 0 ) )
41	25, 40	eqtr4d 2659	. . . 4 |- ( ( ph /\ c e. ( B (repr ` S ) M ) ) -> if ( c e. ( ( A i^i B ) (repr ` S ) M ) , 1 , 0 ) = prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) )
42	41	sumeq2dv 14433	. . 3 |- ( ph -> sum_ c e. ( B (repr ` S ) M ) if ( c e. ( ( A i^i B ) (repr ` S ) M ) , 1 , 0 ) = sum_ c e. ( B (repr ` S ) M ) prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) )
43	16, 42	eqtrd 2656	. 2 |- ( ph -> sum_ c e. ( ( A i^i B ) (repr ` S ) M ) 1 = sum_ c e. ( B (repr ` S ) M ) prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) )
44		hashcl 13147	. . . . 5 |- ( ( ( A i^i B ) (repr ` S ) M ) e. Fin -> ( # ` ( ( A i^i B ) (repr ` S ) M ) ) e. NN0 )
45	9, 44	syl 17	. . . 4 |- ( ph -> ( # ` ( ( A i^i B ) (repr ` S ) M ) ) e. NN0 )
46	45	nn0cnd 11353	. . 3 |- ( ph -> ( # ` ( ( A i^i B ) (repr ` S ) M ) ) e. CC )
47	46	mulid1d 10057	. 2 |- ( ph -> ( ( # ` ( ( A i^i B ) (repr ` S ) M ) ) x. 1 ) = ( # ` ( ( A i^i B ) (repr ` S ) M ) ) )
48	12, 43, 47	3eqtr3rd 2665	1 |- ( ph -> ( # ` ( ( A i^i B ) (repr ` S ) M ) ) = sum_ c e. ( B (repr ` S ) M ) prod_ a e. ( 0..^ S ) ( ( (𝟭 ` NN ) ` A ) ` ( c ` a ) ) )

Syntax hints:   -> wi 4   <-> wb 196   \/ wo 383   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   _Vcvv 3200   i^i cin 3573   C_ wss 3574   ifcif 4086   ran crn 5115   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   0cc0 9936   1c1 9937   x. cmul 9941   NNcn 11020   NN0cn0 11292   ZZcz 11377   ZZ>=cuz 11687  ..^cfzo 12465   #chash 13117   sum_csu 14416   prod_cprod 14635  𝟭cind 30072  reprcrepr 30686

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-inf2 8538  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  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  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-se 5074  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-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-oi 8415  df-card 8765  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-rp 11833  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-clim 14219  df-sum 14417  df-prod 14636  df-ind 30073  df-repr 30687

This theorem is referenced by:  hashrepr  30703  breprexpnat  30712