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Theorem incexc 15549

Description: The inclusion/exclusion principle for counting the elements of a finite union of finite sets. This is Metamath 100 proof #96. (Contributed by Mario Carneiro, 7-Aug-2017.)

**Assertion**
Ref	Expression
incexc	⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))

Distinct variable group: 𝐴,𝑠

**Proof of Theorem incexc**
Step	Hyp	Ref	Expression
1		unifi 9108	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
2		hashcl 14071	. . . 4 ⊢ (∪ 𝐴 ∈ Fin → (♯‘∪ 𝐴) ∈ ℕ₀)
3	2	nn0cnd 12295	. . 3 ⊢ (∪ 𝐴 ∈ Fin → (♯‘∪ 𝐴) ∈ ℂ)
4	1, 3	syl 17	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) ∈ ℂ)
5		simpl 483	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		pwfi 8961	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
7	5, 6	sylib 217	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 ∈ Fin)
8		diffi 8962	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
9	7, 8	syl 17	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
10		1cnd 10970	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 1 ∈ ℂ)
11	10	negcld 11319	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -1 ∈ ℂ)
12		eldifsni 4723	. . . . . . . 8 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ≠ ∅)
13	12	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ≠ ∅)
14		eldifi 4061	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ∈ 𝒫 𝐴)
15		elpwi 4542	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝐴 → 𝑠 ⊆ 𝐴)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ⊆ 𝐴)
17		ssfi 8956	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝑠 ⊆ 𝐴) → 𝑠 ∈ Fin)
18	5, 16, 17	syl2an 596	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ∈ Fin)
19		hashnncl 14081	. . . . . . . 8 ⊢ (𝑠 ∈ Fin → ((♯‘𝑠) ∈ ℕ ↔ 𝑠 ≠ ∅))
20	18, 19	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((♯‘𝑠) ∈ ℕ ↔ 𝑠 ≠ ∅))
21	13, 20	mpbird 256	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘𝑠) ∈ ℕ)
22		nnm1nn0 12274	. . . . . 6 ⊢ ((♯‘𝑠) ∈ ℕ → ((♯‘𝑠) − 1) ∈ ℕ₀)
23	21, 22	syl 17	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((♯‘𝑠) − 1) ∈ ℕ₀)
24	11, 23	expcld 13864	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑((♯‘𝑠) − 1)) ∈ ℂ)
25	16	adantl 482	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ⊆ 𝐴)
26		simplr 766	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝐴 ⊆ Fin)
27	25, 26	sstrd 3931	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ⊆ Fin)
28		unifi 9108	. . . . . . . 8 ⊢ ((𝑠 ∈ Fin ∧ 𝑠 ⊆ Fin) → ∪ 𝑠 ∈ Fin)
29	18, 27, 28	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ∈ Fin)
30		intssuni 4901	. . . . . . . 8 ⊢ (𝑠 ≠ ∅ → ∩ 𝑠 ⊆ ∪ 𝑠)
31	13, 30	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝑠)
32	29, 31	ssfid 9042	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ∈ Fin)
33		hashcl 14071	. . . . . 6 ⊢ (∩ 𝑠 ∈ Fin → (♯‘∩ 𝑠) ∈ ℕ₀)
34	32, 33	syl 17	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘∩ 𝑠) ∈ ℕ₀)
35	34	nn0cnd 12295	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘∩ 𝑠) ∈ ℂ)
36	24, 35	mulcld 10995	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
37	9, 36	fsumcl 15445	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
38		disjdif 4405	. . . . 5 ⊢ ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅
39	38	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅)
40		0elpw 5278	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 𝐴
41		snssi 4741	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 𝐴 → {∅} ⊆ 𝒫 𝐴)
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ {∅} ⊆ 𝒫 𝐴
43		undif 4415	. . . . . . 7 ⊢ ({∅} ⊆ 𝒫 𝐴 ↔ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴)
44	42, 43	mpbi 229	. . . . . 6 ⊢ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴
45	44	eqcomi 2747	. . . . 5 ⊢ 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅}))
46	45	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅})))
47		1cnd 10970	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 1 ∈ ℂ)
48	47	negcld 11319	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → -1 ∈ ℂ)
49	5, 15, 17	syl2an 596	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 𝑠 ∈ Fin)
50		hashcl 14071	. . . . . . 7 ⊢ (𝑠 ∈ Fin → (♯‘𝑠) ∈ ℕ₀)
51	49, 50	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (♯‘𝑠) ∈ ℕ₀)
52	48, 51	expcld 13864	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (-1↑(♯‘𝑠)) ∈ ℂ)
53	1	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ∪ 𝐴 ∈ Fin)
54		inss1 4162	. . . . . . . 8 ⊢ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴
55		ssfi 8956	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ Fin ∧ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin)
56	53, 54, 55	sylancl 586	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin)
57		hashcl 14071	. . . . . . 7 ⊢ ((∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℕ₀)
58	56, 57	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℕ₀)
59	58	nn0cnd 12295	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℂ)
60	52, 59	mulcld 10995	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) ∈ ℂ)
61	39, 46, 7, 60	fsumsplit 15453	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ 𝒫 𝐴((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (Σ𝑠 ∈ {∅} ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) + Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠)))))
62		inidm 4152	. . . . . . 7 ⊢ (∪ 𝐴 ∩ ∪ 𝐴) = ∪ 𝐴
63	62	fveq2i 6777	. . . . . 6 ⊢ (♯‘(∪ 𝐴 ∩ ∪ 𝐴)) = (♯‘∪ 𝐴)
64	63	oveq2i 7286	. . . . 5 ⊢ ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = ((♯‘∪ 𝐴) − (♯‘∪ 𝐴))
65	4	subidd 11320	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘∪ 𝐴)) = 0)
66	64, 65	eqtrid 2790	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = 0)
67		incexclem 15548	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∪ 𝐴 ∈ Fin) → ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
68	1, 67	syldan 591	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
69	66, 68	eqtr3d 2780	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 0 = Σ𝑠 ∈ 𝒫 𝐴((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
70	4, 37	negsubd 11338	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) + -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) = ((♯‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))))
71		0ex 5231	. . . . . . 7 ⊢ ∅ ∈ V
72		1cnd 10970	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 1 ∈ ℂ)
73	72, 4	mulcld 10995	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (♯‘∪ 𝐴)) ∈ ℂ)
74		fveq2 6774	. . . . . . . . . . . 12 ⊢ (𝑠 = ∅ → (♯‘𝑠) = (♯‘∅))
75		hash0 14082	. . . . . . . . . . . 12 ⊢ (♯‘∅) = 0
76	74, 75	eqtrdi 2794	. . . . . . . . . . 11 ⊢ (𝑠 = ∅ → (♯‘𝑠) = 0)
77	76	oveq2d 7291	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (-1↑(♯‘𝑠)) = (-1↑0))
78		neg1cn 12087	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
79		exp0 13786	. . . . . . . . . . 11 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
80	78, 79	ax-mp 5	. . . . . . . . . 10 ⊢ (-1↑0) = 1
81	77, 80	eqtrdi 2794	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (-1↑(♯‘𝑠)) = 1)
82		rint0 4921	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (∪ 𝐴 ∩ ∩ 𝑠) = ∪ 𝐴)
83	82	fveq2d 6778	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) = (♯‘∪ 𝐴))
84	81, 83	oveq12d 7293	. . . . . . . 8 ⊢ (𝑠 = ∅ → ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (♯‘∪ 𝐴)))
85	84	sumsn 15458	. . . . . . 7 ⊢ ((∅ ∈ V ∧ (1 · (♯‘∪ 𝐴)) ∈ ℂ) → Σ𝑠 ∈ {∅} ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (♯‘∪ 𝐴)))
86	71, 73, 85	sylancr 587	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ {∅} ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (♯‘∪ 𝐴)))
87	4	mulid2d 10993	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (♯‘∪ 𝐴)) = (♯‘∪ 𝐴))
88	86, 87	eqtr2d 2779	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ {∅} ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
89	9, 36	fsumneg 15499	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})-((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
90		expm1t 13811	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (♯‘𝑠) ∈ ℕ) → (-1↑(♯‘𝑠)) = ((-1↑((♯‘𝑠) − 1)) · -1))
91	11, 21, 90	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑(♯‘𝑠)) = ((-1↑((♯‘𝑠) − 1)) · -1))
92	24, 11	mulcomd 10996	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑((♯‘𝑠) − 1)) · -1) = (-1 · (-1↑((♯‘𝑠) − 1))))
93	24	mulm1d 11427	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1 · (-1↑((♯‘𝑠) − 1))) = -(-1↑((♯‘𝑠) − 1)))
94	91, 92, 93	3eqtrd 2782	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑(♯‘𝑠)) = -(-1↑((♯‘𝑠) − 1)))
95	25	unissd 4849	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ⊆ ∪ 𝐴)
96	31, 95	sstrd 3931	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝐴)
97		sseqin2 4149	. . . . . . . . . . 11 ⊢ (∩ 𝑠 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
98	96, 97	sylib 217	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
99	98	fveq2d 6778	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) = (♯‘∩ 𝑠))
100	94, 99	oveq12d 7293	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (-(-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
101	24, 35	mulneg1d 11428	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-(-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = -((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
102	100, 101	eqtr2d 2779	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
103	102	sumeq2dv 15415	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})-((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
104	89, 103	eqtr3d 2780	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
105	88, 104	oveq12d 7293	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) + -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) = (Σ𝑠 ∈ {∅} ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) + Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠)))))
106	70, 105	eqtr3d 2780	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) = (Σ𝑠 ∈ {∅} ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) + Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠)))))
107	61, 69, 106	3eqtr4rd 2789	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) = 0)
108	4, 37, 107	subeq0d 11340	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 Vcvv 3432 ∖ cdif 3884 ∪ cun 3885 ∩ cin 3886 ⊆ wss 3887 ∅c0 4256 𝒫 cpw 4533 {csn 4561 ∪ cuni 4839 ∩ cint 4879 ‘cfv 6433 (class class class)co 7275 Fincfn 8733 ℂcc 10869 0cc0 10871 1c1 10872 + caddc 10874 · cmul 10876 − cmin 11205 -cneg 11206 ℕcn 11973 ℕ₀cn0 12233 ↑cexp 13782 ♯chash 14044 Σcsu 15397

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-inf2 9399 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 ax-pre-sup 10949

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-se 5545 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-isom 6442 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-oadd 8301 df-er 8498 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-sup 9201 df-oi 9269 df-dju 9659 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-div 11633 df-nn 11974 df-2 12036 df-3 12037 df-n0 12234 df-z 12320 df-uz 12583 df-rp 12731 df-fz 13240 df-fzo 13383 df-seq 13722 df-exp 13783 df-hash 14045 df-cj 14810 df-re 14811 df-im 14812 df-sqrt 14946 df-abs 14947 df-clim 15197 df-sum 15398

This theorem is referenced by: incexc2 15550

W3C validator