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Theorem incexc2 15869

Description: The inclusion/exclusion principle for counting the elements of a finite union of finite sets. (Contributed by Mario Carneiro, 7-Aug-2017.)

**Assertion**
Ref	Expression
incexc2	⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑛 ∈ (1...(♯‘𝐴))((-1↑(𝑛 − 1)) · Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘∩ 𝑠)))

Distinct variable group: 𝑘,𝑛,𝑠,𝐴

**Proof of Theorem incexc2**
Step	Hyp	Ref	Expression
1		incexc 15868	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
2		hashcl 14370	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
3	2	ad2antrr 736	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (♯‘𝐴) ∈ ℕ₀)
4	3	nn0zd 12594	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (♯‘𝐴) ∈ ℤ)
5		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		elpwi 4563	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝒫 𝐴 → 𝑘 ⊆ 𝐴)
7		ssdomg 8982	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → (𝑘 ⊆ 𝐴 → 𝑘 ≼ 𝐴))
8	7	imp 410	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑘 ⊆ 𝐴) → 𝑘 ≼ 𝐴)
9	5, 6, 8	syl2an 605	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → 𝑘 ≼ 𝐴)
10		hashdomi 14394	. . . . . . . . . . 11 ⊢ (𝑘 ≼ 𝐴 → (♯‘𝑘) ≤ (♯‘𝐴))
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (♯‘𝑘) ≤ (♯‘𝐴))
12		fznn 13598	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℤ → ((♯‘𝑘) ∈ (1...(♯‘𝐴)) ↔ ((♯‘𝑘) ∈ ℕ ∧ (♯‘𝑘) ≤ (♯‘𝐴))))
13	12	rbaibd 548	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℤ ∧ (♯‘𝑘) ≤ (♯‘𝐴)) → ((♯‘𝑘) ∈ (1...(♯‘𝐴)) ↔ (♯‘𝑘) ∈ ℕ))
14	4, 11, 13	syl2anc 593	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → ((♯‘𝑘) ∈ (1...(♯‘𝐴)) ↔ (♯‘𝑘) ∈ ℕ))
15		ssfi 9142	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ 𝑘 ⊆ 𝐴) → 𝑘 ∈ Fin)
16	5, 6, 15	syl2an 605	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → 𝑘 ∈ Fin)
17		hashnncl 14380	. . . . . . . . . 10 ⊢ (𝑘 ∈ Fin → ((♯‘𝑘) ∈ ℕ ↔ 𝑘 ≠ ∅))
18	16, 17	syl 17	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → ((♯‘𝑘) ∈ ℕ ↔ 𝑘 ≠ ∅))
19	14, 18	bitr2d 282	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (𝑘 ≠ ∅ ↔ (♯‘𝑘) ∈ (1...(♯‘𝐴))))
20		df-ne 2959	. . . . . . . 8 ⊢ (𝑘 ≠ ∅ ↔ ¬ 𝑘 = ∅)
21		risset 3238	. . . . . . . 8 ⊢ ((♯‘𝑘) ∈ (1...(♯‘𝐴)) ↔ ∃𝑛 ∈ (1...(♯‘𝐴))𝑛 = (♯‘𝑘))
22	19, 20, 21	3bitr3g 315	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (¬ 𝑘 = ∅ ↔ ∃𝑛 ∈ (1...(♯‘𝐴))𝑛 = (♯‘𝑘)))
23		velsn 4599	. . . . . . . 8 ⊢ (𝑘 ∈ {∅} ↔ 𝑘 = ∅)
24	23	notbii 322	. . . . . . 7 ⊢ (¬ 𝑘 ∈ {∅} ↔ ¬ 𝑘 = ∅)
25		eqcom 2770	. . . . . . . 8 ⊢ ((♯‘𝑘) = 𝑛 ↔ 𝑛 = (♯‘𝑘))
26	25	rexbii 3110	. . . . . . 7 ⊢ (∃𝑛 ∈ (1...(♯‘𝐴))(♯‘𝑘) = 𝑛 ↔ ∃𝑛 ∈ (1...(♯‘𝐴))𝑛 = (♯‘𝑘))
27	22, 24, 26	3bitr4g 316	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (¬ 𝑘 ∈ {∅} ↔ ∃𝑛 ∈ (1...(♯‘𝐴))(♯‘𝑘) = 𝑛))
28	27	rabbidva 3421	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → {𝑘 ∈ 𝒫 𝐴 ∣ ¬ 𝑘 ∈ {∅}} = {𝑘 ∈ 𝒫 𝐴 ∣ ∃𝑛 ∈ (1...(♯‘𝐴))(♯‘𝑘) = 𝑛})
29		dfdif2 3914	. . . . 5 ⊢ (𝒫 𝐴 ∖ {∅}) = {𝑘 ∈ 𝒫 𝐴 ∣ ¬ 𝑘 ∈ {∅}}
30		iunrab 5011	. . . . 5 ⊢ ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} = {𝑘 ∈ 𝒫 𝐴 ∣ ∃𝑛 ∈ (1...(♯‘𝐴))(♯‘𝑘) = 𝑛}
31	28, 29, 30	3eqtr4g 2823	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) = ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛})
32	31	sumeq1d 15728	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑠 ∈ ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
33	1, 32	eqtrd 2798	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
34		fzfid 13987	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1...(♯‘𝐴)) ∈ Fin)
35		simpll 776	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝐴 ∈ Fin)
36		pwfi 9264	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
37	35, 36	sylib 220	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝒫 𝐴 ∈ Fin)
38		ssrab2 4034	. . . 4 ⊢ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ⊆ 𝒫 𝐴
39		ssfi 9142	. . . 4 ⊢ ((𝒫 𝐴 ∈ Fin ∧ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ⊆ 𝒫 𝐴) → {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ∈ Fin)
40	37, 38, 39	sylancl 595	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ∈ Fin)
41		fveqeq2 6877	. . . . . . . . 9 ⊢ (𝑘 = 𝑠 → ((♯‘𝑘) = 𝑛 ↔ (♯‘𝑠) = 𝑛))
42	41	elrab 3651	. . . . . . . 8 ⊢ (𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ↔ (𝑠 ∈ 𝒫 𝐴 ∧ (♯‘𝑠) = 𝑛))
43	42	simprbi 501	. . . . . . 7 ⊢ (𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} → (♯‘𝑠) = 𝑛)
44	43	adantl 485	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (♯‘𝑠) = 𝑛)
45	44	ralrimiva 3155	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘𝑠) = 𝑛)
46	45	ralrimiva 3155	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∀𝑛 ∈ (1...(♯‘𝐴))∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘𝑠) = 𝑛)
47		invdisj 5087	. . . 4 ⊢ (∀𝑛 ∈ (1...(♯‘𝐴))∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘𝑠) = 𝑛 → Disj 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛})
48	46, 47	syl 17	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Disj 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛})
49	44	oveq1d 7412	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((♯‘𝑠) − 1) = (𝑛 − 1))
50	49	oveq2d 7413	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (-1↑((♯‘𝑠) − 1)) = (-1↑(𝑛 − 1)))
51	50	oveq1d 7412	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = ((-1↑(𝑛 − 1)) · (♯‘∩ 𝑠)))
52		1cnd 11176	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 1 ∈ ℂ)
53	52	negcld 11530	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → -1 ∈ ℂ)
54		elfznn 13559	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ)
55	54	adantl 485	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑛 ∈ ℕ)
56		nnm1nn0 12523	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
57	55, 56	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (𝑛 − 1) ∈ ℕ₀)
58	53, 57	expcld 14160	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (-1↑(𝑛 − 1)) ∈ ℂ)
59	58	adantr 484	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (-1↑(𝑛 − 1)) ∈ ℂ)
60		unifi 9288	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
61	60	ad2antrr 736	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∪ 𝐴 ∈ Fin)
62	55	adantr 484	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝑛 ∈ ℕ)
63	44, 62	eqeltrd 2863	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (♯‘𝑠) ∈ ℕ)
64	35	adantr 484	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝐴 ∈ Fin)
65		elrabi 3647	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} → 𝑠 ∈ 𝒫 𝐴)
66	65	adantl 485	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝑠 ∈ 𝒫 𝐴)
67		elpwi 4563	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ 𝒫 𝐴 → 𝑠 ⊆ 𝐴)
68	66, 67	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝑠 ⊆ 𝐴)
69	64, 68	ssfid 9214	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝑠 ∈ Fin)
70		hashnncl 14380	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → ((♯‘𝑠) ∈ ℕ ↔ 𝑠 ≠ ∅))
71	69, 70	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((♯‘𝑠) ∈ ℕ ↔ 𝑠 ≠ ∅))
72	63, 71	mpbid 234	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝑠 ≠ ∅)
73		intssuni 4929	. . . . . . . . . . 11 ⊢ (𝑠 ≠ ∅ → ∩ 𝑠 ⊆ ∪ 𝑠)
74	72, 73	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∩ 𝑠 ⊆ ∪ 𝑠)
75	68	unissd 4876	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∪ 𝑠 ⊆ ∪ 𝐴)
76	74, 75	sstrd 3947	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∩ 𝑠 ⊆ ∪ 𝐴)
77	61, 76	ssfid 9214	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∩ 𝑠 ∈ Fin)
78		hashcl 14370	. . . . . . . 8 ⊢ (∩ 𝑠 ∈ Fin → (♯‘∩ 𝑠) ∈ ℕ₀)
79	77, 78	syl 17	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (♯‘∩ 𝑠) ∈ ℕ₀)
80	79	nn0cnd 12545	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (♯‘∩ 𝑠) ∈ ℂ)
81	59, 80	mulcld 11203	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((-1↑(𝑛 − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
82	51, 81	eqeltrd 2863	. . . 4 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
83	82	anasss 470	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ (𝑛 ∈ (1...(♯‘𝐴)) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛})) → ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
84	34, 40, 48, 83	fsumiun 15850	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑛 ∈ (1...(♯‘𝐴))Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
85	51	sumeq2dv 15730	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑(𝑛 − 1)) · (♯‘∩ 𝑠)))
86	40, 58, 80	fsummulc2 15812	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ((-1↑(𝑛 − 1)) · Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘∩ 𝑠)) = Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑(𝑛 − 1)) · (♯‘∩ 𝑠)))
87	85, 86	eqtr4d 2801	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = ((-1↑(𝑛 − 1)) · Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘∩ 𝑠)))
88	87	sumeq2dv 15730	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑛 ∈ (1...(♯‘𝐴))Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑛 ∈ (1...(♯‘𝐴))((-1↑(𝑛 − 1)) · Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘∩ 𝑠)))
89	33, 84, 88	3eqtrd 2802	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑛 ∈ (1...(♯‘𝐴))((-1↑(𝑛 − 1)) · Σ𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘∩ 𝑠)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1561 ∈ wcel 2143 ≠ wne 2958 ∀wral 3077 ∃wrex 3087 {crab 3415 ∖ cdif 3902 ⊆ wss 3905 ∅c0 4286 𝒫 cpw 4556 {csn 4583 ∪ cuni 4866 ∩ cint 4906 ∪ ciun 4950 Disj wdisj 5068 class class class wbr 5101 ‘cfv 6522 (class class class)co 7397 ≼ cdom 8926 Fincfn 8928 ℂcc 11072 1c1 11075 · cmul 11079 ≤ cle 11218 − cmin 11415 -cneg 11416 ℕcn 12211 ℕ₀cn0 12482 ℤcz 12569 ...cfz 13513 ↑cexp 14075 ♯chash 14344 Σcsu 15714

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1816 ax-4 1830 ax-5 1931 ax-6 1988 ax-7 2029 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5228 ax-sep 5247 ax-nul 5257 ax-pow 5323 ax-pr 5391 ax-un 7719 ax-inf2 9597 ax-cnex 11130 ax-resscn 11131 ax-1cn 11132 ax-icn 11133 ax-addcl 11134 ax-addrcl 11135 ax-mulcl 11136 ax-mulrcl 11137 ax-mulcom 11138 ax-addass 11139 ax-mulass 11140 ax-distr 11141 ax-i2m1 11142 ax-1ne0 11143 ax-1rid 11144 ax-rnegex 11145 ax-rrecex 11146 ax-cnre 11147 ax-pre-lttri 11148 ax-pre-lttrn 11149 ax-pre-ltadd 11150 ax-pre-mulgt0 11151 ax-pre-sup 11152

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1100 df-3an 1101 df-tru 1564 df-fal 1574 df-ex 1801 df-nf 1805 df-sb 2092 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3368 df-reu 3369 df-rab 3416 df-v 3457 df-sbc 3746 df-csb 3854 df-dif 3908 df-un 3910 df-in 3912 df-ss 3922 df-pss 3925 df-nul 4287 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-disj 5069 df-br 5102 df-opab 5164 df-mpt 5183 df-tr 5209 df-id 5543 df-eprel 5548 df-po 5556 df-so 5557 df-fr 5601 df-se 5602 df-we 5603 df-xp 5654 df-rel 5655 df-cnv 5656 df-co 5657 df-dm 5658 df-rn 5659 df-res 5660 df-ima 5661 df-pred 6289 df-ord 6350 df-on 6351 df-lim 6352 df-suc 6353 df-iota 6478 df-fun 6524 df-fn 6525 df-f 6526 df-f1 6527 df-fo 6528 df-f1o 6529 df-fv 6530 df-isom 6531 df-riota 7354 df-ov 7400 df-oprab 7401 df-mpo 7402 df-om 7848 df-1st 7971 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8382 df-1o 8438 df-oadd 8442 df-er 8679 df-en 8929 df-dom 8930 df-sdom 8931 df-fin 8932 df-sup 9389 df-oi 9459 df-dju 9860 df-card 9898 df-pnf 11219 df-mnf 11220 df-xr 11221 df-ltxr 11222 df-le 11223 df-sub 11417 df-neg 11418 df-div 11846 df-nn 12212 df-2 12281 df-3 12282 df-n0 12483 df-xnn0 12556 df-z 12570 df-uz 12841 df-rp 12995 df-fz 13514 df-fzo 13661 df-seq 14016 df-exp 14076 df-hash 14345 df-cj 15127 df-re 15128 df-im 15129 df-sqrt 15263 df-abs 15264 df-clim 15516 df-sum 15715

This theorem is referenced by: (None)

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