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Theorem indsumin 33020

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	⊢ (𝜑 → 𝑂 ∈ 𝑉)
indsumin.2	⊢ (𝜑 → 𝐴 ∈ Fin)
indsumin.3	⊢ (𝜑 → 𝐴 ⊆ 𝑂)
indsumin.4	⊢ (𝜑 → 𝐵 ⊆ 𝑂)
indsumin.5	⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)

**Assertion**
Ref	Expression
indsumin	⊢ (𝜑 → Σ𝑘 ∈ 𝐴 ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶)

Distinct variable groups: 𝐴,𝑘 𝐵,𝑘 𝑘,𝑂 𝜑,𝑘 Allowed substitution hints: 𝐶(𝑘) 𝑉(𝑘)

**Proof of Theorem indsumin**
Step	Hyp	Ref	Expression
1		inindif 31754	. . . 4 ⊢ ((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) = ∅
2	1	a1i 11	. . 3 ⊢ (𝜑 → ((𝐴 ∩ 𝐵) ∩ (𝐴 ∖ 𝐵)) = ∅)
3		inundif 4479	. . . . 5 ⊢ ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)) = 𝐴
4	3	eqcomi 2742	. . . 4 ⊢ 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵))
5	4	a1i 11	. . 3 ⊢ (𝜑 → 𝐴 = ((𝐴 ∩ 𝐵) ∪ (𝐴 ∖ 𝐵)))
6		indsumin.2	. . 3 ⊢ (𝜑 → 𝐴 ∈ Fin)
7		pr01ssre 32030	. . . . . 6 ⊢ {0, 1} ⊆ ℝ
8		ax-resscn 11167	. . . . . 6 ⊢ ℝ ⊆ ℂ
9	7, 8	sstri 3992	. . . . 5 ⊢ {0, 1} ⊆ ℂ
10		indsumin.1	. . . . . . . 8 ⊢ (𝜑 → 𝑂 ∈ 𝑉)
11		indsumin.4	. . . . . . . 8 ⊢ (𝜑 → 𝐵 ⊆ 𝑂)
12		indf 33013	. . . . . . . 8 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐵 ⊆ 𝑂) → ((𝟭‘𝑂)‘𝐵):𝑂⟶{0, 1})
13	10, 11, 12	syl2anc 585	. . . . . . 7 ⊢ (𝜑 → ((𝟭‘𝑂)‘𝐵):𝑂⟶{0, 1})
14	13	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((𝟭‘𝑂)‘𝐵):𝑂⟶{0, 1})
15		indsumin.3	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝑂)
16	15	sselda 3983	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝑘 ∈ 𝑂)
17	14, 16	ffvelcdmd 7088	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (((𝟭‘𝑂)‘𝐵)‘𝑘) ∈ {0, 1})
18	9, 17	sselid 3981	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → (((𝟭‘𝑂)‘𝐵)‘𝑘) ∈ ℂ)
19		indsumin.5	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → 𝐶 ∈ ℂ)
20	18, 19	mulcld 11234	. . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐴) → ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) ∈ ℂ)
21	2, 5, 6, 20	fsumsplit 15687	. 2 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = (Σ𝑘 ∈ (𝐴 ∩ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) + Σ𝑘 ∈ (𝐴 ∖ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶)))
22	10	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → 𝑂 ∈ 𝑉)
23	11	adantr 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → 𝐵 ⊆ 𝑂)
24		inss2 4230	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
25	24	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐵)
26	25	sselda 3983	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → 𝑘 ∈ 𝐵)
27		ind1 33015	. . . . . . 7 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐵 ⊆ 𝑂 ∧ 𝑘 ∈ 𝐵) → (((𝟭‘𝑂)‘𝐵)‘𝑘) = 1)
28	22, 23, 26, 27	syl3anc 1372	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → (((𝟭‘𝑂)‘𝐵)‘𝑘) = 1)
29	28	oveq1d 7424	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = (1 · 𝐶))
30		inss1 4229	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
31	30	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ⊆ 𝐴)
32	31	sselda 3983	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → 𝑘 ∈ 𝐴)
33	32, 19	syldan 592	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → 𝐶 ∈ ℂ)
34	33	mullidd 11232	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → (1 · 𝐶) = 𝐶)
35	29, 34	eqtrd 2773	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∩ 𝐵)) → ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = 𝐶)
36	35	sumeq2dv 15649	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∩ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶)
37	10	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝑂 ∈ 𝑉)
38	11	adantr 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝐵 ⊆ 𝑂)
39	15	ssdifd 4141	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ (𝑂 ∖ 𝐵))
40	39	sselda 3983	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝑘 ∈ (𝑂 ∖ 𝐵))
41		ind0 33016	. . . . . . . 8 ⊢ ((𝑂 ∈ 𝑉 ∧ 𝐵 ⊆ 𝑂 ∧ 𝑘 ∈ (𝑂 ∖ 𝐵)) → (((𝟭‘𝑂)‘𝐵)‘𝑘) = 0)
42	37, 38, 40, 41	syl3anc 1372	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → (((𝟭‘𝑂)‘𝐵)‘𝑘) = 0)
43	42	oveq1d 7424	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = (0 · 𝐶))
44		difssd 4133	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ⊆ 𝐴)
45	44	sselda 3983	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝑘 ∈ 𝐴)
46	45, 19	syldan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → 𝐶 ∈ ℂ)
47	46	mul02d 11412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → (0 · 𝐶) = 0)
48	43, 47	eqtrd 2773	. . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ (𝐴 ∖ 𝐵)) → ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = 0)
49	48	sumeq2dv 15649	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∖ 𝐵)0)
50		diffi 9179	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 ∖ 𝐵) ∈ Fin)
51	6, 50	syl 17	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ 𝐵) ∈ Fin)
52		sumz 15668	. . . . . 6 ⊢ (((𝐴 ∖ 𝐵) ⊆ (ℤ_≥‘0) ∨ (𝐴 ∖ 𝐵) ∈ Fin) → Σ𝑘 ∈ (𝐴 ∖ 𝐵)0 = 0)
53	52	olcs 875	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ∈ Fin → Σ𝑘 ∈ (𝐴 ∖ 𝐵)0 = 0)
54	51, 53	syl 17	. . . 4 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ 𝐵)0 = 0)
55	49, 54	eqtrd 2773	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∖ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = 0)
56	36, 55	oveq12d 7427	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∩ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) + Σ𝑘 ∈ (𝐴 ∖ 𝐵)((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶)) = (Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶 + 0))
57		infi 9268	. . . . 5 ⊢ (𝐴 ∈ Fin → (𝐴 ∩ 𝐵) ∈ Fin)
58	6, 57	syl 17	. . . 4 ⊢ (𝜑 → (𝐴 ∩ 𝐵) ∈ Fin)
59	58, 33	fsumcl 15679	. . 3 ⊢ (𝜑 → Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶 ∈ ℂ)
60	59	addridd 11414	. 2 ⊢ (𝜑 → (Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶 + 0) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶)
61	21, 56, 60	3eqtrd 2777	1 ⊢ (𝜑 → Σ𝑘 ∈ 𝐴 ((((𝟭‘𝑂)‘𝐵)‘𝑘) · 𝐶) = Σ𝑘 ∈ (𝐴 ∩ 𝐵)𝐶)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 {cpr 4631 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℂcc 11108 ℝcr 11109 0cc0 11110 1c1 11111 + caddc 11113 · cmul 11115 ℤ_≥cuz 12822 Σcsu 15632 𝟭cind 33008

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-1st 7975 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-sup 9437 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fz 13485 df-fzo 13628 df-seq 13967 df-exp 14028 df-hash 14291 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-sum 15633 df-ind 33009

This theorem is referenced by: breprexpnat 33646

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