Theorem indsumin 30084

Description: Finite sum of a product with the indicator function / Cartesian product with the indicator function. (Contributed by Thierry Arnoux, 11-Dec-2021.)

Hypotheses

Ref	Expression
indsumin.1	|- ( ph -> O e. V )
indsumin.2	|- ( ph -> A e. Fin )
indsumin.3	|- ( ph -> A C_ O )
indsumin.4	|- ( ph -> B C_ O )
indsumin.5	|- ( ( ph /\ k e. A ) -> C e. CC )

Assertion

Ref	Expression
indsumin	|- ( ph -> sum_ k e. A ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) = sum_ k e. ( A i^i B ) C )

Distinct variable groups:   A, k   B, k   k, O   ph, k

Allowed substitution hints:   C( k)   V( k)

Proof of Theorem indsumin

Step	Hyp	Ref	Expression
1		inindif 29353	. . . 4 |- ( ( A i^i B ) i^i ( A \ B ) ) = (/)
2	1	a1i 11	. . 3 |- ( ph -> ( ( A i^i B ) i^i ( A \ B ) ) = (/) )
3		inundif 4046	. . . . 5 |- ( ( A i^i B ) u. ( A \ B ) ) = A
4	3	eqcomi 2631	. . . 4 |- A = ( ( A i^i B ) u. ( A \ B ) )
5	4	a1i 11	. . 3 |- ( ph -> A = ( ( A i^i B ) u. ( A \ B ) ) )
6		indsumin.2	. . 3 |- ( ph -> A e. Fin )
7		pr01ssre 29570	. . . . . 6 |- { 0 , 1 } C_ RR
8		ax-resscn 9993	. . . . . 6 |- RR C_ CC
9	7, 8	sstri 3612	. . . . 5 |- { 0 , 1 } C_ CC
10		indsumin.1	. . . . . . . 8 |- ( ph -> O e. V )
11		indsumin.4	. . . . . . . 8 |- ( ph -> B C_ O )
12		indf 30077	. . . . . . . 8 |- ( ( O e. V /\ B C_ O ) -> ( (𝟭 ` O ) ` B ) : O --> { 0 , 1 } )
13	10, 11, 12	syl2anc 693	. . . . . . 7 |- ( ph -> ( (𝟭 ` O ) ` B ) : O --> { 0 , 1 } )
14	13	adantr 481	. . . . . 6 |- ( ( ph /\ k e. A ) -> ( (𝟭 ` O ) ` B ) : O --> { 0 , 1 } )
15		indsumin.3	. . . . . . 7 |- ( ph -> A C_ O )
16	15	sselda 3603	. . . . . 6 |- ( ( ph /\ k e. A ) -> k e. O )
17	14, 16	ffvelrnd 6360	. . . . 5 |- ( ( ph /\ k e. A ) -> ( ( (𝟭 ` O ) ` B ) ` k ) e. { 0 , 1 } )
18	9, 17	sseldi 3601	. . . 4 |- ( ( ph /\ k e. A ) -> ( ( (𝟭 ` O ) ` B ) ` k ) e. CC )
19		indsumin.5	. . . 4 |- ( ( ph /\ k e. A ) -> C e. CC )
20	18, 19	mulcld 10060	. . 3 |- ( ( ph /\ k e. A ) -> ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) e. CC )
21	2, 5, 6, 20	fsumsplit 14471	. 2 |- ( ph -> sum_ k e. A ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) = ( sum_ k e. ( A i^i B ) ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) + sum_ k e. ( A \ B ) ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) ) )
22	10	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. ( A i^i B ) ) -> O e. V )
23	11	adantr 481	. . . . . . 7 |- ( ( ph /\ k e. ( A i^i B ) ) -> B C_ O )
24		inss2 3834	. . . . . . . . 9 |- ( A i^i B ) C_ B
25	24	a1i 11	. . . . . . . 8 |- ( ph -> ( A i^i B ) C_ B )
26	25	sselda 3603	. . . . . . 7 |- ( ( ph /\ k e. ( A i^i B ) ) -> k e. B )
27		ind1 30079	. . . . . . 7 |- ( ( O e. V /\ B C_ O /\ k e. B ) -> ( ( (𝟭 ` O ) ` B ) ` k ) = 1 )
28	22, 23, 26, 27	syl3anc 1326	. . . . . 6 |- ( ( ph /\ k e. ( A i^i B ) ) -> ( ( (𝟭 ` O ) ` B ) ` k ) = 1 )
29	28	oveq1d 6665	. . . . 5 |- ( ( ph /\ k e. ( A i^i B ) ) -> ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) = ( 1 x. C ) )
30		inss1 3833	. . . . . . . . 9 |- ( A i^i B ) C_ A
31	30	a1i 11	. . . . . . . 8 |- ( ph -> ( A i^i B ) C_ A )
32	31	sselda 3603	. . . . . . 7 |- ( ( ph /\ k e. ( A i^i B ) ) -> k e. A )
33	32, 19	syldan 487	. . . . . 6 |- ( ( ph /\ k e. ( A i^i B ) ) -> C e. CC )
34	33	mulid2d 10058	. . . . 5 |- ( ( ph /\ k e. ( A i^i B ) ) -> ( 1 x. C ) = C )
35	29, 34	eqtrd 2656	. . . 4 |- ( ( ph /\ k e. ( A i^i B ) ) -> ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) = C )
36	35	sumeq2dv 14433	. . 3 |- ( ph -> sum_ k e. ( A i^i B ) ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) = sum_ k e. ( A i^i B ) C )
37	10	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. ( A \ B ) ) -> O e. V )
38	11	adantr 481	. . . . . . . 8 |- ( ( ph /\ k e. ( A \ B ) ) -> B C_ O )
39	15	ssdifd 3746	. . . . . . . . 9 |- ( ph -> ( A \ B ) C_ ( O \ B ) )
40	39	sselda 3603	. . . . . . . 8 |- ( ( ph /\ k e. ( A \ B ) ) -> k e. ( O \ B ) )
41		ind0 30080	. . . . . . . 8 |- ( ( O e. V /\ B C_ O /\ k e. ( O \ B ) ) -> ( ( (𝟭 ` O ) ` B ) ` k ) = 0 )
42	37, 38, 40, 41	syl3anc 1326	. . . . . . 7 |- ( ( ph /\ k e. ( A \ B ) ) -> ( ( (𝟭 ` O ) ` B ) ` k ) = 0 )
43	42	oveq1d 6665	. . . . . 6 |- ( ( ph /\ k e. ( A \ B ) ) -> ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) = ( 0 x. C ) )
44		difssd 3738	. . . . . . . . 9 |- ( ph -> ( A \ B ) C_ A )
45	44	sselda 3603	. . . . . . . 8 |- ( ( ph /\ k e. ( A \ B ) ) -> k e. A )
46	45, 19	syldan 487	. . . . . . 7 |- ( ( ph /\ k e. ( A \ B ) ) -> C e. CC )
47	46	mul02d 10234	. . . . . 6 |- ( ( ph /\ k e. ( A \ B ) ) -> ( 0 x. C ) = 0 )
48	43, 47	eqtrd 2656	. . . . 5 |- ( ( ph /\ k e. ( A \ B ) ) -> ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) = 0 )
49	48	sumeq2dv 14433	. . . 4 |- ( ph -> sum_ k e. ( A \ B ) ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) = sum_ k e. ( A \ B ) 0 )
50		diffi 8192	. . . . . 6 |- ( A e. Fin -> ( A \ B ) e. Fin )
51	6, 50	syl 17	. . . . 5 |- ( ph -> ( A \ B ) e. Fin )
52		sumz 14453	. . . . . 6 |- ( ( ( A \ B ) C_ ( ZZ>= ` 0 ) \/ ( A \ B ) e. Fin ) -> sum_ k e. ( A \ B ) 0 = 0 )
53	52	olcs 410	. . . . 5 |- ( ( A \ B ) e. Fin -> sum_ k e. ( A \ B ) 0 = 0 )
54	51, 53	syl 17	. . . 4 |- ( ph -> sum_ k e. ( A \ B ) 0 = 0 )
55	49, 54	eqtrd 2656	. . 3 |- ( ph -> sum_ k e. ( A \ B ) ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) = 0 )
56	36, 55	oveq12d 6668	. 2 |- ( ph -> ( sum_ k e. ( A i^i B ) ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) + sum_ k e. ( A \ B ) ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) ) = ( sum_ k e. ( A i^i B ) C + 0 ) )
57		infi 8184	. . . . 5 |- ( A e. Fin -> ( A i^i B ) e. Fin )
58	6, 57	syl 17	. . . 4 |- ( ph -> ( A i^i B ) e. Fin )
59	58, 33	fsumcl 14464	. . 3 |- ( ph -> sum_ k e. ( A i^i B ) C e. CC )
60	59	addid1d 10236	. 2 |- ( ph -> ( sum_ k e. ( A i^i B ) C + 0 ) = sum_ k e. ( A i^i B ) C )
61	21, 56, 60	3eqtrd 2660	1 |- ( ph -> sum_ k e. A ( ( ( (𝟭 ` O ) ` B ) ` k ) x. C ) = sum_ k e. ( A i^i B ) C )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   \ cdif 3571   u. cun 3572   i^i cin 3573   C_ wss 3574   (/)c0 3915   {cpr 4179   -->wf 5884   ` cfv 5888  (class class class)co 6650   Fincfn 7955   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   ZZ>=cuz 11687   sum_csu 14416  𝟭cind 30072

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-oadd 7564  df-er 7742  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-ind 30073

This theorem is referenced by:  breprexpnat  30712