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Theorem incexc2 15745

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 15744	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
2		hashcl 14263	. . . . . . . . . . . 12 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ₀)
3	2	ad2antrr 726	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (♯‘𝐴) ∈ ℕ₀)
4	3	nn0zd 12497	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (♯‘𝐴) ∈ ℤ)
5		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		elpwi 4558	. . . . . . . . . . . 12 ⊢ (𝑘 ∈ 𝒫 𝐴 → 𝑘 ⊆ 𝐴)
7		ssdomg 8925	. . . . . . . . . . . . 13 ⊢ (𝐴 ∈ Fin → (𝑘 ⊆ 𝐴 → 𝑘 ≼ 𝐴))
8	7	imp 406	. . . . . . . . . . . 12 ⊢ ((𝐴 ∈ Fin ∧ 𝑘 ⊆ 𝐴) → 𝑘 ≼ 𝐴)
9	5, 6, 8	syl2an 596	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → 𝑘 ≼ 𝐴)
10		hashdomi 14287	. . . . . . . . . . 11 ⊢ (𝑘 ≼ 𝐴 → (♯‘𝑘) ≤ (♯‘𝐴))
11	9, 10	syl 17	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → (♯‘𝑘) ≤ (♯‘𝐴))
12		fznn 13495	. . . . . . . . . . 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 584	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → ((♯‘𝑘) ∈ (1...(♯‘𝐴)) ↔ (♯‘𝑘) ∈ ℕ))
15		ssfi 9087	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ Fin ∧ 𝑘 ⊆ 𝐴) → 𝑘 ∈ Fin)
16	5, 6, 15	syl2an 596	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑘 ∈ 𝒫 𝐴) → 𝑘 ∈ Fin)
17		hashnncl 14273	. . . . . . . . . 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 2926	. . . . . . . 8 ⊢ (𝑘 ≠ ∅ ↔ ¬ 𝑘 = ∅)
21		risset 3204	. . . . . . . 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 4593	. . . . . . . 8 ⊢ (𝑘 ∈ {∅} ↔ 𝑘 = ∅)
24	23	notbii 320	. . . . . . 7 ⊢ (¬ 𝑘 ∈ {∅} ↔ ¬ 𝑘 = ∅)
25		eqcom 2736	. . . . . . . 8 ⊢ ((♯‘𝑘) = 𝑛 ↔ 𝑛 = (♯‘𝑘))
26	25	rexbii 3076	. . . . . . 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 3401	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → {𝑘 ∈ 𝒫 𝐴 ∣ ¬ 𝑘 ∈ {∅}} = {𝑘 ∈ 𝒫 𝐴 ∣ ∃𝑛 ∈ (1...(♯‘𝐴))(♯‘𝑘) = 𝑛})
29		dfdif2 3912	. . . . 5 ⊢ (𝒫 𝐴 ∖ {∅}) = {𝑘 ∈ 𝒫 𝐴 ∣ ¬ 𝑘 ∈ {∅}}
30		iunrab 5001	. . . . 5 ⊢ ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} = {𝑘 ∈ 𝒫 𝐴 ∣ ∃𝑛 ∈ (1...(♯‘𝐴))(♯‘𝑘) = 𝑛}
31	28, 29, 30	3eqtr4g 2789	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) = ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛})
32	31	sumeq1d 15607	. . 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 2764	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ ∪ 𝑛 ∈ (1...(♯‘𝐴)){𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
34		fzfid 13880	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1...(♯‘𝐴)) ∈ Fin)
35		simpll 766	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝐴 ∈ Fin)
36		pwfi 9208	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
37	35, 36	sylib 218	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝒫 𝐴 ∈ Fin)
38		ssrab2 4031	. . . 4 ⊢ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ⊆ 𝒫 𝐴
39		ssfi 9087	. . . 4 ⊢ ((𝒫 𝐴 ∈ Fin ∧ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ⊆ 𝒫 𝐴) → {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ∈ Fin)
40	37, 38, 39	sylancl 586	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} ∈ Fin)
41		fveqeq2 6831	. . . . . . . . 9 ⊢ (𝑘 = 𝑠 → ((♯‘𝑘) = 𝑛 ↔ (♯‘𝑠) = 𝑛))
42	41	elrab 3648	. . . . . . . 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 3121	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → ∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘𝑠) = 𝑛)
46	45	ralrimiva 3121	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∀𝑛 ∈ (1...(♯‘𝐴))∀𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛} (♯‘𝑠) = 𝑛)
47		invdisj 5078	. . . 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 7364	. . . . . . 7 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((♯‘𝑠) − 1) = (𝑛 − 1))
50	49	oveq2d 7365	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (-1↑((♯‘𝑠) − 1)) = (-1↑(𝑛 − 1)))
51	50	oveq1d 7364	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = ((-1↑(𝑛 − 1)) · (♯‘∩ 𝑠)))
52		1cnd 11110	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 1 ∈ ℂ)
53	52	negcld 11462	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → -1 ∈ ℂ)
54		elfznn 13456	. . . . . . . . . 10 ⊢ (𝑛 ∈ (1...(♯‘𝐴)) → 𝑛 ∈ ℕ)
55	54	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) → 𝑛 ∈ ℕ)
56		nnm1nn0 12425	. . . . . . . . 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 14053	. . . . . . 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 9234	. . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
61	60	ad2antrr 726	. . . . . . . . 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 2828	. . . . . . . . . . . 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 3643	. . . . . . . . . . . . . . . 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 4558	. . . . . . . . . . . . . . 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 9158	. . . . . . . . . . . . 13 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → 𝑠 ∈ Fin)
70		hashnncl 14273	. . . . . . . . . . . . 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 4920	. . . . . . . . . . 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 4868	. . . . . . . . . 10 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∪ 𝑠 ⊆ ∪ 𝐴)
76	74, 75	sstrd 3946	. . . . . . . . 9 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∩ 𝑠 ⊆ ∪ 𝐴)
77	61, 76	ssfid 9158	. . . . . . . 8 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ∩ 𝑠 ∈ Fin)
78		hashcl 14263	. . . . . . . 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 12447	. . . . . 6 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → (♯‘∩ 𝑠) ∈ ℂ)
81	59, 80	mulcld 11135	. . . . 5 ⊢ ((((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑛 ∈ (1...(♯‘𝐴))) ∧ 𝑠 ∈ {𝑘 ∈ 𝒫 𝐴 ∣ (♯‘𝑘) = 𝑛}) → ((-1↑(𝑛 − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
82	51, 81	eqeltrd 2828	. . . 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 15728	. 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 15609	. . . 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 15691	. . . 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 2767	. . 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 15609	. 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 2768	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 1540 ∈ wcel 2109 ≠ wne 2925 ∀wral 3044 ∃wrex 3053 {crab 3394 ∖ cdif 3900 ⊆ wss 3903 ∅c0 4284 𝒫 cpw 4551 {csn 4577 ∪ cuni 4858 ∩ cint 4896 ∪ ciun 4941 Disj wdisj 5059 class class class wbr 5092 ‘cfv 6482 (class class class)co 7349 ≼ cdom 8870 Fincfn 8872 ℂcc 11007 1c1 11010 · cmul 11014 ≤ cle 11150 − cmin 11347 -cneg 11348 ℕcn 12128 ℕ₀cn0 12384 ℤcz 12471 ...cfz 13410 ↑cexp 13968 ♯chash 14237 Σcsu 15593

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5218 ax-sep 5235 ax-nul 5245 ax-pow 5304 ax-pr 5371 ax-un 7671 ax-inf2 9537 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 ax-pre-sup 11087

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3343 df-reu 3344 df-rab 3395 df-v 3438 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4285 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4859 df-int 4897 df-iun 4943 df-disj 5060 df-br 5093 df-opab 5155 df-mpt 5174 df-tr 5200 df-id 5514 df-eprel 5519 df-po 5527 df-so 5528 df-fr 5572 df-se 5573 df-we 5574 df-xp 5625 df-rel 5626 df-cnv 5627 df-co 5628 df-dm 5629 df-rn 5630 df-res 5631 df-ima 5632 df-pred 6249 df-ord 6310 df-on 6311 df-lim 6312 df-suc 6313 df-iota 6438 df-fun 6484 df-fn 6485 df-f 6486 df-f1 6487 df-fo 6488 df-f1o 6489 df-fv 6490 df-isom 6491 df-riota 7306 df-ov 7352 df-oprab 7353 df-mpo 7354 df-om 7800 df-1st 7924 df-2nd 7925 df-frecs 8214 df-wrecs 8245 df-recs 8294 df-rdg 8332 df-1o 8388 df-oadd 8392 df-er 8625 df-en 8873 df-dom 8874 df-sdom 8875 df-fin 8876 df-sup 9332 df-oi 9402 df-dju 9797 df-card 9835 df-pnf 11151 df-mnf 11152 df-xr 11153 df-ltxr 11154 df-le 11155 df-sub 11349 df-neg 11350 df-div 11778 df-nn 12129 df-2 12191 df-3 12192 df-n0 12385 df-xnn0 12458 df-z 12472 df-uz 12736 df-rp 12894 df-fz 13411 df-fzo 13558 df-seq 13909 df-exp 13969 df-hash 14238 df-cj 15006 df-re 15007 df-im 15008 df-sqrt 15142 df-abs 15143 df-clim 15395 df-sum 15594

This theorem is referenced by: (None)

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