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Theorem incexc2 14614

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 14613	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))
2		hashcl 13185	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ₀)
3	2	ad2antrr 762	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (#‘𝐴) ∈ ℕ₀)
4	3	nn0zd 11518	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (#‘𝐴) ∈ ℤ)
5		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		elpwi 4201	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝒫 𝐴 → 𝑘 ⊆ 𝐴)
7		ssdomg 8043	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → (𝑘 ⊆ 𝐴 → 𝑘 ≼ 𝐴))
8	7	imp 444	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑘 ⊆ 𝐴) → 𝑘 ≼ 𝐴)
9	5, 6, 8	syl2an 493	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → 𝑘 ≼ 𝐴)
10		hashdomi 13207	. . . . . . . . . . 11 ⊢ (𝑘 ≼ 𝐴 → (#‘𝑘) ≤ (#‘𝐴))
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (#‘𝑘) ≤ (#‘𝐴))
12		fznn 12446	. . . . . . . . . . 11 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝑘) ∈ (1...(#‘𝐴)) ↔ ((#‘𝑘) ∈ ℕ ∧ (#‘𝑘) ≤ (#‘𝐴))))
13	12	rbaibd 969	. . . . . . . . . 10 ⊢ (((#‘𝐴) ∈ ℤ ∧ (#‘𝑘) ≤ (#‘𝐴)) → ((#‘𝑘) ∈ (1...(#‘𝐴)) ↔ (#‘𝑘) ∈ ℕ))
14	4, 11, 13	syl2anc 694	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → ((#‘𝑘) ∈ (1...(#‘𝐴)) ↔ (#‘𝑘) ∈ ℕ))
15		ssfi 8221	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ 𝑘 ⊆ 𝐴) → 𝑘 ∈ Fin)
16	5, 6, 15	syl2an 493	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → 𝑘 ∈ Fin)
17		hashnncl 13195	. . . . . . . . . 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 269	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (𝑘 ≠ ∅ ↔ (#‘𝑘) ∈ (1...(#‘𝐴))))
20		df-ne 2824	. . . . . . . 8 ⊢ (𝑘 ≠ ∅ ↔ ¬ 𝑘 = ∅)
21		risset 3091	. . . . . . . 8 ⊢ ((#‘𝑘) ∈ (1...(#‘𝐴)) ↔ ∃𝑛 ∈ (1...(#‘𝐴))𝑛 = (#‘𝑘))
22	19, 20, 21	3bitr3g 302	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (¬ 𝑘 = ∅ ↔ ∃𝑛 ∈ (1...(#‘𝐴))𝑛 = (#‘𝑘)))
23		velsn 4226	. . . . . . . 8 ⊢ (𝑘 ∈ {∅} ↔ 𝑘 = ∅)
24	23	notbii 309	. . . . . . 7 ⊢ (¬ 𝑘 ∈ {∅} ↔ ¬ 𝑘 = ∅)
25		eqcom 2658	. . . . . . . 8 ⊢ ((#‘𝑘) = 𝑛 ↔ 𝑛 = (#‘𝑘))
26	25	rexbii 3070	. . . . . . 7 ⊢ (∃𝑛 ∈ (1...(#‘𝐴))(#‘𝑘) = 𝑛 ↔ ∃𝑛 ∈ (1...(#‘𝐴))𝑛 = (#‘𝑘))
27	22, 24, 26	3bitr4g 303	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (¬ 𝑘 ∈ {∅} ↔ ∃𝑛 ∈ (1...(#‘𝐴))(#‘𝑘) = 𝑛))
28	27	rabbidva 3219	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → {𝑘 ∈ 𝒫 𝐴 ∣ ¬ 𝑘 ∈ {∅}} = {𝑘 ∈ 𝒫 𝐴 ∣ ∃𝑛 ∈ (1...(#‘𝐴))(#‘𝑘) = 𝑛})
29		dfdif2 3616	. . . . 5 ⊢ (𝒫 𝐴 ∖ {∅}) = {𝑘 ∈ 𝒫 𝐴 ∣ ¬ 𝑘 ∈ {∅}}
30		iunrab 4599	. . . . 5 ⊢ ∪ 𝑛 ∈ (1...(#‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} = {𝑘 ∈ 𝒫 𝐴 ∣ ∃𝑛 ∈ (1...(#‘𝐴))(#‘𝑘) = 𝑛}
31	28, 29, 30	3eqtr4g 2710	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) = ∪ 𝑛 ∈ (1...(#‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛})
32	31	sumeq1d 14475	. . 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 2685	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ𝑠 ∈ ∪ 𝑛 ∈ (1...(#‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))
34		fzfid 12812	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1...(#‘𝐴)) ∈ Fin)
35		simpll 805	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 𝐴 ∈ Fin)
36		pwfi 8302	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
37	35, 36	sylib 208	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 𝒫 𝐴 ∈ Fin)
38		ssrab2 3720	. . . 4 ⊢ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ⊆ 𝒫 𝐴
39		ssfi 8221	. . . 4 ⊢ ((𝒫 𝐴 ∈ Fin ∧ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ⊆ 𝒫 𝐴) → {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ∈ Fin)
40	37, 38, 39	sylancl 695	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ∈ Fin)
41		fveq2 6229	. . . . . . . . . 10 ⊢ (𝑘 = 𝑠 → (#‘𝑘) = (#‘𝑠))
42	41	eqeq1d 2653	. . . . . . . . 9 ⊢ (𝑘 = 𝑠 → ((#‘𝑘) = 𝑛 ↔ (#‘𝑠) = 𝑛))
43	42	elrab 3396	. . . . . . . 8 ⊢ (𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ↔ (𝑠 ∈ 𝒫 𝐴 ∧ (#‘𝑠) = 𝑛))
44	43	simprbi 479	. . . . . . 7 ⊢ (𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} → (#‘𝑠) = 𝑛)
45	44	adantl 481	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → (#‘𝑠) = 𝑛)
46	45	ralrimiva 2995	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} (#‘𝑠) = 𝑛)
47	46	ralrimiva 2995	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∀𝑛 ∈ (1...(#‘𝐴))∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} (#‘𝑠) = 𝑛)
48		invdisj 4670	. . . 4 ⊢ (∀𝑛 ∈ (1...(#‘𝐴))∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} (#‘𝑠) = 𝑛 → Disj 𝑛 ∈ (1...(#‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛})
49	47, 48	syl 17	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Disj 𝑛 ∈ (1...(#‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛})
50	45	oveq1d 6705	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ((#‘𝑠) − 1) = (𝑛 − 1))
51	50	oveq2d 6706	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → (-1↑((#‘𝑠) − 1)) = (-1↑(𝑛 − 1)))
52	51	oveq1d 6705	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = ((-1↑(𝑛 − 1)) · (#‘∩ 𝑠)))
53		1cnd 10094	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 1 ∈ ℂ)
54	53	negcld 10417	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → -1 ∈ ℂ)
55		elfznn 12408	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(#‘𝐴)) → 𝑛 ∈ ℕ)
56	55	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 𝑛 ∈ ℕ)
57		nnm1nn0 11372	. . . . . . . . 9 ⊢ (𝑛 ∈ ℕ → (𝑛 − 1) ∈ ℕ₀)
58	56, 57	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (𝑛 − 1) ∈ ℕ₀)
59	54, 58	expcld 13048	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → (-1↑(𝑛 − 1)) ∈ ℂ)
60	59	adantr 480	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → (-1↑(𝑛 − 1)) ∈ ℂ)
61		unifi 8296	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
62	61	ad2antrr 762	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ∪ 𝐴 ∈ Fin)
63	56	adantr 480	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → 𝑛 ∈ ℕ)
64	45, 63	eqeltrd 2730	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → (#‘𝑠) ∈ ℕ)
65	35	adantr 480	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → 𝐴 ∈ Fin)
66		elrabi 3391	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} → 𝑠 ∈ 𝒫 𝐴)
67	66	adantl 481	. . . . . . . . . . . . . . 15 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → 𝑠 ∈ 𝒫 𝐴)
68		elpwi 4201	. . . . . . . . . . . . . . 15 ⊢ (𝑠 ∈ 𝒫 𝐴 → 𝑠 ⊆ 𝐴)
69	67, 68	syl 17	. . . . . . . . . . . . . 14 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → 𝑠 ⊆ 𝐴)
70		ssfi 8221	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Fin ∧ 𝑠 ⊆ 𝐴) → 𝑠 ∈ Fin)
71	65, 69, 70	syl2anc 694	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → 𝑠 ∈ Fin)
72		hashnncl 13195	. . . . . . . . . . . . 13 ⊢ (𝑠 ∈ Fin → ((#‘𝑠) ∈ ℕ ↔ 𝑠 ≠ ∅))
73	71, 72	syl 17	. . . . . . . . . . . 12 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ((#‘𝑠) ∈ ℕ ↔ 𝑠 ≠ ∅))
74	64, 73	mpbid 222	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → 𝑠 ≠ ∅)
75		intssuni 4531	. . . . . . . . . . 11 ⊢ (𝑠 ≠ ∅ → ∩ 𝑠 ⊆ ∪ 𝑠)
76	74, 75	syl 17	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ∩ 𝑠 ⊆ ∪ 𝑠)
77	69	unissd 4494	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ∪ 𝑠 ⊆ ∪ 𝐴)
78	76, 77	sstrd 3646	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ∩ 𝑠 ⊆ ∪ 𝐴)
79		ssfi 8221	. . . . . . . . 9 ⊢ ((∪ 𝐴 ∈ Fin ∧ ∩ 𝑠 ⊆ ∪ 𝐴) → ∩ 𝑠 ∈ Fin)
80	62, 78, 79	syl2anc 694	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ∩ 𝑠 ∈ Fin)
81		hashcl 13185	. . . . . . . 8 ⊢ (∩ 𝑠 ∈ Fin → (#‘∩ 𝑠) ∈ ℕ₀)
82	80, 81	syl 17	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → (#‘∩ 𝑠) ∈ ℕ₀)
83	82	nn0cnd 11391	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → (#‘∩ 𝑠) ∈ ℂ)
84	60, 83	mulcld 10098	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ((-1↑(𝑛 − 1)) · (#‘∩ 𝑠)) ∈ ℂ)
85	52, 84	eqeltrd 2730	. . . 4 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) ∈ ℂ)
86	85	anasss 680	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ (𝑛 ∈ (1...(#‘𝐴)) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛})) → ((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) ∈ ℂ)
87	34, 40, 49, 86	fsumiun 14597	. 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)))
88	52	sumeq2dv 14477	. . . 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)))
89	40, 59, 83	fsummulc2 14560	. . . 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)))
90	88, 89	eqtr4d 2688	. . 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)))
91	90	sumeq2dv 14477	. 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)))
92	33, 87, 91	3eqtrd 2689	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 196 ∧ wa 383 = wceq 1523 ∈ wcel 2030 ≠ wne 2823 ∀wral 2941 ∃wrex 2942 {crab 2945 ∖ cdif 3604 ⊆ wss 3607 ∅c0 3948 𝒫 cpw 4191 {csn 4210 ∪ cuni 4468 ∩ cint 4507 ∪ ciun 4552 Disj wdisj 4652 class class class wbr 4685 ‘cfv 5926 (class class class)co 6690 ≼ cdom 7995 Fincfn 7997 ℂcc 9972 1c1 9975 · cmul 9979 ≤ cle 10113 − cmin 10304 -cneg 10305 ℕcn 11058 ℕ₀cn0 11330 ℤcz 11415 ...cfz 12364 ↑cexp 12900 #chash 13157 Σcsu 14460

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-inf2 8576 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 ax-pre-sup 10052

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-fal 1529 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-disj 4653 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-se 5103 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-isom 5935 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-1o 7605 df-2o 7606 df-oadd 7609 df-er 7787 df-map 7901 df-en 7998 df-dom 7999 df-sdom 8000 df-fin 8001 df-sup 8389 df-oi 8456 df-card 8803 df-cda 9028 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-n0 11331 df-xnn0 11402 df-z 11416 df-uz 11726 df-rp 11871 df-fz 12365 df-fzo 12505 df-seq 12842 df-exp 12901 df-hash 13158 df-cj 13883 df-re 13884 df-im 13885 df-sqrt 14019 df-abs 14020 df-clim 14263 df-sum 14461

This theorem is referenced by: (None)

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