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Theorem incexc2 14358

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 14357	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))
2		hashcl 12964	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (#‘𝐴) ∈ ℕ₀)
3	2	ad2antrr 758	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (#‘𝐴) ∈ ℕ₀)
4	3	nn0zd 11315	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (#‘𝐴) ∈ ℤ)
5		simpl 472	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		elpwi 4117	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝒫 𝐴 → 𝑘 ⊆ 𝐴)
7		ssdomg 7865	. . . . . . . . . . . . 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 12985	. . . . . . . . . . 11 ⊢ (𝑘 ≼ 𝐴 → (#‘𝑘) ≤ (#‘𝐴))
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (#‘𝑘) ≤ (#‘𝐴))
12		fznn 12236	. . . . . . . . . . 11 ⊢ ((#‘𝐴) ∈ ℤ → ((#‘𝑘) ∈ (1...(#‘𝐴)) ↔ ((#‘𝑘) ∈ ℕ ∧ (#‘𝑘) ≤ (#‘𝐴))))
13	12	rbaibd 947	. . . . . . . . . 10 ⊢ (((#‘𝐴) ∈ ℤ ∧ (#‘𝑘) ≤ (#‘𝐴)) → ((#‘𝑘) ∈ (1...(#‘𝐴)) ↔ (#‘𝑘) ∈ ℕ))
14	4, 11, 13	syl2anc 691	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → ((#‘𝑘) ∈ (1...(#‘𝐴)) ↔ (#‘𝑘) ∈ ℕ))
15		ssfi 8043	. . . . . . . . . . 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 12973	. . . . . . . . . 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 268	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (𝑘 ≠ ∅ ↔ (#‘𝑘) ∈ (1...(#‘𝐴))))
20		df-ne 2782	. . . . . . . 8 ⊢ (𝑘 ≠ ∅ ↔ ¬ 𝑘 = ∅)
21		risset 3044	. . . . . . . 8 ⊢ ((#‘𝑘) ∈ (1...(#‘𝐴)) ↔ ∃𝑛 ∈ (1...(#‘𝐴))𝑛 = (#‘𝑘))
22	19, 20, 21	3bitr3g 301	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (¬ 𝑘 = ∅ ↔ ∃𝑛 ∈ (1...(#‘𝐴))𝑛 = (#‘𝑘)))
23		velsn 4141	. . . . . . . 8 ⊢ (𝑘 ∈ {∅} ↔ 𝑘 = ∅)
24	23	notbii 309	. . . . . . 7 ⊢ (¬ 𝑘 ∈ {∅} ↔ ¬ 𝑘 = ∅)
25		eqcom 2617	. . . . . . . 8 ⊢ ((#‘𝑘) = 𝑛 ↔ 𝑛 = (#‘𝑘))
26	25	rexbii 3023	. . . . . . 7 ⊢ (∃𝑛 ∈ (1...(#‘𝐴))(#‘𝑘) = 𝑛 ↔ ∃𝑛 ∈ (1...(#‘𝐴))𝑛 = (#‘𝑘))
27	22, 24, 26	3bitr4g 302	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (¬ 𝑘 ∈ {∅} ↔ ∃𝑛 ∈ (1...(#‘𝐴))(#‘𝑘) = 𝑛))
28	27	rabbidva 3163	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → {𝑘 ∈ 𝒫 𝐴 ∣ ¬ 𝑘 ∈ {∅}} = {𝑘 ∈ 𝒫 𝐴 ∣ ∃𝑛 ∈ (1...(#‘𝐴))(#‘𝑘) = 𝑛})
29		dfdif2 3549	. . . . 5 ⊢ (𝒫 𝐴 ∖ {∅}) = {𝑘 ∈ 𝒫 𝐴 ∣ ¬ 𝑘 ∈ {∅}}
30		iunrab 4498	. . . . 5 ⊢ ∪ 𝑛 ∈ (1...(#‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} = {𝑘 ∈ 𝒫 𝐴 ∣ ∃𝑛 ∈ (1...(#‘𝐴))(#‘𝑘) = 𝑛}
31	28, 29, 30	3eqtr4g 2669	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) = ∪ 𝑛 ∈ (1...(#‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛})
32	31	sumeq1d 14228	. . 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 2644	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ𝑠 ∈ ∪ 𝑛 ∈ (1...(#‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))
34		fzfid 12592	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1...(#‘𝐴)) ∈ Fin)
35		simpll 786	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 𝐴 ∈ Fin)
36		pwfi 8122	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
37	35, 36	sylib 207	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 𝒫 𝐴 ∈ Fin)
38		ssrab2 3650	. . . 4 ⊢ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ⊆ 𝒫 𝐴
39		ssfi 8043	. . . 4 ⊢ ((𝒫 𝐴 ∈ Fin ∧ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ⊆ 𝒫 𝐴) → {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ∈ Fin)
40	37, 38, 39	sylancl 693	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} ∈ Fin)
41		fveq2 6088	. . . . . . . . . 10 ⊢ (𝑘 = 𝑠 → (#‘𝑘) = (#‘𝑠))
42	41	eqeq1d 2612	. . . . . . . . 9 ⊢ (𝑘 = 𝑠 → ((#‘𝑘) = 𝑛 ↔ (#‘𝑠) = 𝑛))
43	42	elrab 3331	. . . . . . . 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 2949	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → ∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} (#‘𝑠) = 𝑛)
47	46	ralrimiva 2949	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∀𝑛 ∈ (1...(#‘𝐴))∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛} (#‘𝑠) = 𝑛)
48		invdisj 4566	. . . 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 6542	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ((#‘𝑠) − 1) = (𝑛 − 1))
51	50	oveq2d 6543	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → (-1↑((#‘𝑠) − 1)) = (-1↑(𝑛 − 1)))
52	51	oveq1d 6542	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = ((-1↑(𝑛 − 1)) · (#‘∩ 𝑠)))
53		1cnd 9913	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 1 ∈ ℂ)
54	53	negcld 10231	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → -1 ∈ ℂ)
55		elfznn 12199	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(#‘𝐴)) → 𝑛 ∈ ℕ)
56	55	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) → 𝑛 ∈ ℕ)
57		nnm1nn0 11184	. . . . . . . . 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 12828	. . . . . . 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 8116	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
62	61	ad2antrr 758	. . . . . . . . 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 2688	. . . . . . . . . . . 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 3328	. . . . . . . . . . . . . . . 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 4117	. . . . . . . . . . . . . . 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 8043	. . . . . . . . . . . . . 14 ⊢ ((𝐴 ∈ Fin ∧ 𝑠 ⊆ 𝐴) → 𝑠 ∈ Fin)
71	65, 69, 70	syl2anc 691	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → 𝑠 ∈ Fin)
72		hashnncl 12973	. . . . . . . . . . . . 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 221	. . . . . . . . . . 11 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → 𝑠 ≠ ∅)
75		intssuni 4429	. . . . . . . . . . 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 4393	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ∪ 𝑠 ⊆ ∪ 𝐴)
78	76, 77	sstrd 3578	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ∩ 𝑠 ⊆ ∪ 𝐴)
79		ssfi 8043	. . . . . . . . 9 ⊢ ((∪ 𝐴 ∈ Fin ∧ ∩ 𝑠 ⊆ ∪ 𝐴) → ∩ 𝑠 ∈ Fin)
80	62, 78, 79	syl2anc 691	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ∩ 𝑠 ∈ Fin)
81		hashcl 12964	. . . . . . . 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 11203	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → (#‘∩ 𝑠) ∈ ℂ)
84	60, 83	mulcld 9917	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ((-1↑(𝑛 − 1)) · (#‘∩ 𝑠)) ∈ ℂ)
85	52, 84	eqeltrd 2688	. . . 4 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(#‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (#‘𝑘) = 𝑛}) → ((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) ∈ ℂ)
86	85	anasss 677	. . 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 14343	. 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 14230	. . . 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 14307	. . . 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 2647	. . 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 14230	. 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 2648	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 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 ∖ cdif 3537 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ∪ cuni 4367 ∩ cint 4405 ∪ ciun 4450 Disj wdisj 4548 class class class wbr 4578 ‘cfv 5790 (class class class)co 6527 ≼ cdom 7817 Fincfn 7819 ℂcc 9791 1c1 9794 · cmul 9798 ≤ cle 9932 − cmin 10118 -cneg 10119 ℕcn 10870 ℕ₀cn0 11142 ℤcz 11213 ...cfz 12155 ↑cexp 12680 #chash 12937 Σcsu 14213

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4694 ax-sep 4704 ax-nul 4712 ax-pow 4764 ax-pr 4828 ax-un 6825 ax-inf2 8399 ax-cnex 9849 ax-resscn 9850 ax-1cn 9851 ax-icn 9852 ax-addcl 9853 ax-addrcl 9854 ax-mulcl 9855 ax-mulrcl 9856 ax-mulcom 9857 ax-addass 9858 ax-mulass 9859 ax-distr 9860 ax-i2m1 9861 ax-1ne0 9862 ax-1rid 9863 ax-rnegex 9864 ax-rrecex 9865 ax-cnre 9866 ax-pre-lttri 9867 ax-pre-lttrn 9868 ax-pre-ltadd 9869 ax-pre-mulgt0 9870 ax-pre-sup 9871

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4368 df-int 4406 df-iun 4452 df-disj 4549 df-br 4579 df-opab 4639 df-mpt 4640 df-tr 4676 df-eprel 4939 df-id 4943 df-po 4949 df-so 4950 df-fr 4987 df-se 4988 df-we 4989 df-xp 5034 df-rel 5035 df-cnv 5036 df-co 5037 df-dm 5038 df-rn 5039 df-res 5040 df-ima 5041 df-pred 5583 df-ord 5629 df-on 5630 df-lim 5631 df-suc 5632 df-iota 5754 df-fun 5792 df-fn 5793 df-f 5794 df-f1 5795 df-fo 5796 df-f1o 5797 df-fv 5798 df-isom 5799 df-riota 6489 df-ov 6530 df-oprab 6531 df-mpt2 6532 df-om 6936 df-1st 7037 df-2nd 7038 df-wrecs 7272 df-recs 7333 df-rdg 7371 df-1o 7425 df-2o 7426 df-oadd 7429 df-er 7607 df-map 7724 df-en 7820 df-dom 7821 df-sdom 7822 df-fin 7823 df-sup 8209 df-oi 8276 df-card 8626 df-cda 8851 df-pnf 9933 df-mnf 9934 df-xr 9935 df-ltxr 9936 df-le 9937 df-sub 10120 df-neg 10121 df-div 10537 df-nn 10871 df-2 10929 df-3 10930 df-n0 11143 df-z 11214 df-uz 11523 df-rp 11668 df-fz 12156 df-fzo 12293 df-seq 12622 df-exp 12681 df-hash 12938 df-cj 13636 df-re 13637 df-im 13638 df-sqrt 13772 df-abs 13773 df-clim 14016 df-sum 14214

This theorem is referenced by: (None)

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