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Theorem incexc2 15886

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 15885	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
2		hashcl 14405	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
3	2	ad2antrr 725	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (♯‘𝐴) ∈ ℕ₀)
4	3	nn0zd 12665	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (♯‘𝐴) ∈ ℤ)
5		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		elpwi 4629	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝒫 𝐴 → 𝑘 ⊆ 𝐴)
7		ssdomg 9060	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → (𝑘 ⊆ 𝐴 → 𝑘 ≼ 𝐴))
8	7	imp 406	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑘 ⊆ 𝐴) → 𝑘 ≼ 𝐴)
9	5, 6, 8	syl2an 595	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → 𝑘 ≼ 𝐴)
10		hashdomi 14429	. . . . . . . . . . 11 ⊢ (𝑘 ≼ 𝐴 → (♯‘𝑘) ≤ (♯‘𝐴))
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (♯‘𝑘) ≤ (♯‘𝐴))
12		fznn 13652	. . . . . . . . . . 11 ⊢ ((♯‘𝐴) ∈ ℤ → ((♯‘𝑘) ∈ (1...(♯‘𝐴)) ↔ ((♯‘𝑘) ∈ ℕ ∧ (♯‘𝑘) ≤ (♯‘𝐴))))
13	12	rbaibd 540	. . . . . . . . . 10 ⊢ (((♯‘𝐴) ∈ ℤ ∧ (♯‘𝑘) ≤ (♯‘𝐴)) → ((♯‘𝑘) ∈ (1...(♯‘𝐴)) ↔ (♯‘𝑘) ∈ ℕ))
14	4, 11, 13	syl2anc 583	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → ((♯‘𝑘) ∈ (1...(♯‘𝐴)) ↔ (♯‘𝑘) ∈ ℕ))
15		ssfi 9240	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ 𝑘 ⊆ 𝐴) → 𝑘 ∈ Fin)
16	5, 6, 15	syl2an 595	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → 𝑘 ∈ Fin)
17		hashnncl 14415	. . . . . . . . . 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 280	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (𝑘 ≠ ∅ ↔ (♯‘𝑘) ∈ (1...(♯‘𝐴))))
20		df-ne 2947	. . . . . . . 8 ⊢ (𝑘 ≠ ∅ ↔ ¬ 𝑘 = ∅)
21		risset 3239	. . . . . . . 8 ⊢ ((♯‘𝑘) ∈ (1...(♯‘𝐴)) ↔ ∃𝑛 ∈ (1...(♯‘𝐴))𝑛 = (♯‘𝑘))
22	19, 20, 21	3bitr3g 313	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (¬ 𝑘 = ∅ ↔ ∃𝑛 ∈ (1...(♯‘𝐴))𝑛 = (♯‘𝑘)))
23		velsn 4664	. . . . . . . 8 ⊢ (𝑘 ∈ {∅} ↔ 𝑘 = ∅)
24	23	notbii 320	. . . . . . 7 ⊢ (¬ 𝑘 ∈ {∅} ↔ ¬ 𝑘 = ∅)
25		eqcom 2747	. . . . . . . 8 ⊢ ((♯‘𝑘) = 𝑛 ↔ 𝑛 = (♯‘𝑘))
26	25	rexbii 3100	. . . . . . 7 ⊢ (∃𝑛 ∈ (1...(♯‘𝐴))(♯‘𝑘) = 𝑛 ↔ ∃𝑛 ∈ (1...(♯‘𝐴))𝑛 = (♯‘𝑘))
27	22, 24, 26	3bitr4g 314	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (¬ 𝑘 ∈ {∅} ↔ ∃𝑛 ∈ (1...(♯‘𝐴))(♯‘𝑘) = 𝑛))
28	27	rabbidva 3450	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → {𝑘 ∈ 𝒫 𝐴 ∣ ¬ 𝑘 ∈ {∅}} = {𝑘 ∈ 𝒫 𝐴 ∣ ∃𝑛 ∈ (1...(♯‘𝐴))(♯‘𝑘) = 𝑛})
29		dfdif2 3985	. . . . 5 ⊢ (𝒫 𝐴 ∖ {∅}) = {𝑘 ∈ 𝒫 𝐴 ∣ ¬ 𝑘 ∈ {∅}}
30		iunrab 5075	. . . . 5 ⊢ ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} = {𝑘 ∈ 𝒫 𝐴 ∣ ∃𝑛 ∈ (1...(♯‘𝐴))(♯‘𝑘) = 𝑛}
31	28, 29, 30	3eqtr4g 2805	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) = ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛})
32	31	sumeq1d 15748	. . 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 2780	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
34		fzfid 14024	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1...(♯‘𝐴)) ∈ Fin)
35		simpll 766	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝐴 ∈ Fin)
36		pwfi 9385	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
37	35, 36	sylib 218	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝒫 𝐴 ∈ Fin)
38		ssrab2 4103	. . . 4 ⊢ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ⊆ 𝒫 𝐴
39		ssfi 9240	. . . 4 ⊢ ((𝒫 𝐴 ∈ Fin ∧ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ⊆ 𝒫 𝐴) → {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ∈ Fin)
40	37, 38, 39	sylancl 585	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ∈ Fin)
41		fveqeq2 6929	. . . . . . . . 9 ⊢ (𝑘 = 𝑠 → ((♯‘𝑘) = 𝑛 ↔ (♯‘𝑠) = 𝑛))
42	41	elrab 3708	. . . . . . . 8 ⊢ (𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ↔ (𝑠 ∈ 𝒫 𝐴 ∧ (♯‘𝑠) = 𝑛))
43	42	simprbi 496	. . . . . . 7 ⊢ (𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} → (♯‘𝑠) = 𝑛)
44	43	adantl 481	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (♯‘𝑠) = 𝑛)
45	44	ralrimiva 3152	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘𝑠) = 𝑛)
46	45	ralrimiva 3152	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∀𝑛 ∈ (1...(♯‘𝐴))∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘𝑠) = 𝑛)
47		invdisj 5152	. . . 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 7463	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((♯‘𝑠) − 1) = (𝑛 − 1))
50	49	oveq2d 7464	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (-1↑((♯‘𝑠) − 1)) = (-1↑(𝑛 − 1)))
51	50	oveq1d 7463	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = ((-1↑(𝑛 − 1)) · (♯‘∩ 𝑠)))
52		1cnd 11285	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 1 ∈ ℂ)
53	52	negcld 11634	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → -1 ∈ ℂ)
54		elfznn 13613	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ)
55	54	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑛 ∈ ℕ)
56		nnm1nn0 12594	. . . . . . . . 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 14196	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → (-1↑(𝑛 − 1)) ∈ ℂ)
59	58	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (-1↑(𝑛 − 1)) ∈ ℂ)
60		unifi 9412	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
61	60	ad2antrr 725	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∪ 𝐴 ∈ Fin)
62	55	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝑛 ∈ ℕ)
63	44, 62	eqeltrd 2844	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (♯‘𝑠) ∈ ℕ)
64	35	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝐴 ∈ Fin)
65		elrabi 3703	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} → 𝑠 ∈ 𝒫 𝐴)
66	65	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝑠 ∈ 𝒫 𝐴)
67		elpwi 4629	. . . . . . . . . . . . . . 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 9329	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝑠 ∈ Fin)
70		hashnncl 14415	. . . . . . . . . . . . 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 232	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝑠 ≠ ∅)
73		intssuni 4994	. . . . . . . . . . 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 4941	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∪ 𝑠 ⊆ ∪ 𝐴)
76	74, 75	sstrd 4019	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∩ 𝑠 ⊆ ∪ 𝐴)
77	61, 76	ssfid 9329	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∩ 𝑠 ∈ Fin)
78		hashcl 14405	. . . . . . . 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 12615	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (♯‘∩ 𝑠) ∈ ℂ)
81	59, 80	mulcld 11310	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((-1↑(𝑛 − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
82	51, 81	eqeltrd 2844	. . . 4 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
83	82	anasss 466	. . 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 15869	. 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 15750	. . . 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 15832	. . . 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 2783	. . 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 15750	. 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 2784	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 206 ∧ wa 395 = wceq 1537 ∈ wcel 2108 ≠ wne 2946 ∀wral 3067 ∃wrex 3076 {crab 3443 ∖ cdif 3973 ⊆ wss 3976 ∅c0 4352 𝒫 cpw 4622 {csn 4648 ∪ cuni 4931 ∩ cint 4970 ∪ ciun 5015 Disj wdisj 5133 class class class wbr 5166 ‘cfv 6573 (class class class)co 7448 ≼ cdom 9001 Fincfn 9003 ℂcc 11182 1c1 11185 · cmul 11189 ≤ cle 11325 − cmin 11520 -cneg 11521 ℕcn 12293 ℕ₀cn0 12553 ℤcz 12639 ...cfz 13567 ↑cexp 14112 ♯chash 14379 Σcsu 15734

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2158 ax-12 2178 ax-ext 2711 ax-rep 5303 ax-sep 5317 ax-nul 5324 ax-pow 5383 ax-pr 5447 ax-un 7770 ax-inf2 9710 ax-cnex 11240 ax-resscn 11241 ax-1cn 11242 ax-icn 11243 ax-addcl 11244 ax-addrcl 11245 ax-mulcl 11246 ax-mulrcl 11247 ax-mulcom 11248 ax-addass 11249 ax-mulass 11250 ax-distr 11251 ax-i2m1 11252 ax-1ne0 11253 ax-1rid 11254 ax-rnegex 11255 ax-rrecex 11256 ax-cnre 11257 ax-pre-lttri 11258 ax-pre-lttrn 11259 ax-pre-ltadd 11260 ax-pre-mulgt0 11261 ax-pre-sup 11262

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3or 1088 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-nf 1782 df-sb 2065 df-mo 2543 df-eu 2572 df-clab 2718 df-cleq 2732 df-clel 2819 df-nfc 2895 df-ne 2947 df-nel 3053 df-ral 3068 df-rex 3077 df-rmo 3388 df-reu 3389 df-rab 3444 df-v 3490 df-sbc 3805 df-csb 3922 df-dif 3979 df-un 3981 df-in 3983 df-ss 3993 df-pss 3996 df-nul 4353 df-if 4549 df-pw 4624 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 df-int 4971 df-iun 5017 df-disj 5134 df-br 5167 df-opab 5229 df-mpt 5250 df-tr 5284 df-id 5593 df-eprel 5599 df-po 5607 df-so 5608 df-fr 5652 df-se 5653 df-we 5654 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-res 5712 df-ima 5713 df-pred 6332 df-ord 6398 df-on 6399 df-lim 6400 df-suc 6401 df-iota 6525 df-fun 6575 df-fn 6576 df-f 6577 df-f1 6578 df-fo 6579 df-f1o 6580 df-fv 6581 df-isom 6582 df-riota 7404 df-ov 7451 df-oprab 7452 df-mpo 7453 df-om 7904 df-1st 8030 df-2nd 8031 df-frecs 8322 df-wrecs 8353 df-recs 8427 df-rdg 8466 df-1o 8522 df-oadd 8526 df-er 8763 df-en 9004 df-dom 9005 df-sdom 9006 df-fin 9007 df-sup 9511 df-oi 9579 df-dju 9970 df-card 10008 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11522 df-neg 11523 df-div 11948 df-nn 12294 df-2 12356 df-3 12357 df-n0 12554 df-xnn0 12626 df-z 12640 df-uz 12904 df-rp 13058 df-fz 13568 df-fzo 13712 df-seq 14053 df-exp 14113 df-hash 14380 df-cj 15148 df-re 15149 df-im 15150 df-sqrt 15284 df-abs 15285 df-clim 15534 df-sum 15735

This theorem is referenced by: (None)

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