incexc - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > incexc		Structured version Visualization version GIF version

Theorem incexc 15853

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 9356	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
2		hashcl 14374	. . . 4 ⊢ (∪ 𝐴 ∈ Fin → (♯‘∪ 𝐴) ∈ ℕ₀)
3	2	nn0cnd 12564	. . 3 ⊢ (∪ 𝐴 ∈ Fin → (♯‘∪ 𝐴) ∈ ℂ)
4	1, 3	syl 17	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) ∈ ℂ)
5		simpl 482	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		pwfi 9329	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
7	5, 6	sylib 218	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 ∈ Fin)
8		diffi 9189	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
9	7, 8	syl 17	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
10		1cnd 11230	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 1 ∈ ℂ)
11	10	negcld 11581	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -1 ∈ ℂ)
12		eldifsni 4766	. . . . . . . 8 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ≠ ∅)
13	12	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ≠ ∅)
14		eldifi 4106	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ∈ 𝒫 𝐴)
15		elpwi 4582	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝐴 → 𝑠 ⊆ 𝐴)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ⊆ 𝐴)
17		ssfi 9187	. . . . . . . . 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 14384	. . . . . . . 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 257	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘𝑠) ∈ ℕ)
22		nnm1nn0 12542	. . . . . 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 14164	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑((♯‘𝑠) − 1)) ∈ ℂ)
25	16	adantl 481	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ⊆ 𝐴)
26		simplr 768	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝐴 ⊆ Fin)
27	25, 26	sstrd 3969	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ⊆ Fin)
28		unifi 9356	. . . . . . . 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 4946	. . . . . . . 8 ⊢ (𝑠 ≠ ∅ → ∩ 𝑠 ⊆ ∪ 𝑠)
31	13, 30	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝑠)
32	29, 31	ssfid 9273	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ∈ Fin)
33		hashcl 14374	. . . . . 6 ⊢ (∩ 𝑠 ∈ Fin → (♯‘∩ 𝑠) ∈ ℕ₀)
34	32, 33	syl 17	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘∩ 𝑠) ∈ ℕ₀)
35	34	nn0cnd 12564	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘∩ 𝑠) ∈ ℂ)
36	24, 35	mulcld 11255	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
37	9, 36	fsumcl 15749	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
38		disjdif 4447	. . . . 5 ⊢ ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅
39	38	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅)
40		0elpw 5326	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 𝐴
41		snssi 4784	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 𝐴 → {∅} ⊆ 𝒫 𝐴)
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ {∅} ⊆ 𝒫 𝐴
43		undif 4457	. . . . . . 7 ⊢ ({∅} ⊆ 𝒫 𝐴 ↔ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴)
44	42, 43	mpbi 230	. . . . . 6 ⊢ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴
45	44	eqcomi 2744	. . . . 5 ⊢ 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅}))
46	45	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅})))
47		1cnd 11230	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 1 ∈ ℂ)
48	47	negcld 11581	. . . . . 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 14374	. . . . . . 7 ⊢ (𝑠 ∈ Fin → (♯‘𝑠) ∈ ℕ₀)
51	49, 50	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (♯‘𝑠) ∈ ℕ₀)
52	48, 51	expcld 14164	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (-1↑(♯‘𝑠)) ∈ ℂ)
53	1	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ∪ 𝐴 ∈ Fin)
54		inss1 4212	. . . . . . . 8 ⊢ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴
55		ssfi 9187	. . . . . . . 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 14374	. . . . . . 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 12564	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℂ)
60	52, 59	mulcld 11255	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) ∈ ℂ)
61	39, 46, 7, 60	fsumsplit 15757	. . 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 4202	. . . . . . 7 ⊢ (∪ 𝐴 ∩ ∪ 𝐴) = ∪ 𝐴
63	62	fveq2i 6879	. . . . . 6 ⊢ (♯‘(∪ 𝐴 ∩ ∪ 𝐴)) = (♯‘∪ 𝐴)
64	63	oveq2i 7416	. . . . 5 ⊢ ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = ((♯‘∪ 𝐴) − (♯‘∪ 𝐴))
65	4	subidd 11582	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘∪ 𝐴)) = 0)
66	64, 65	eqtrid 2782	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = 0)
67		incexclem 15852	. . . . 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 2772	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 0 = Σ𝑠 ∈ 𝒫 𝐴((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
70	4, 37	negsubd 11600	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) + -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) = ((♯‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))))
71		0ex 5277	. . . . . . 7 ⊢ ∅ ∈ V
72		1cnd 11230	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 1 ∈ ℂ)
73	72, 4	mulcld 11255	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (♯‘∪ 𝐴)) ∈ ℂ)
74		fveq2 6876	. . . . . . . . . . . 12 ⊢ (𝑠 = ∅ → (♯‘𝑠) = (♯‘∅))
75		hash0 14385	. . . . . . . . . . . 12 ⊢ (♯‘∅) = 0
76	74, 75	eqtrdi 2786	. . . . . . . . . . 11 ⊢ (𝑠 = ∅ → (♯‘𝑠) = 0)
77	76	oveq2d 7421	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (-1↑(♯‘𝑠)) = (-1↑0))
78		neg1cn 12354	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
79		exp0 14083	. . . . . . . . . . 11 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
80	78, 79	ax-mp 5	. . . . . . . . . 10 ⊢ (-1↑0) = 1
81	77, 80	eqtrdi 2786	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (-1↑(♯‘𝑠)) = 1)
82		rint0 4964	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (∪ 𝐴 ∩ ∩ 𝑠) = ∪ 𝐴)
83	82	fveq2d 6880	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) = (♯‘∪ 𝐴))
84	81, 83	oveq12d 7423	. . . . . . . 8 ⊢ (𝑠 = ∅ → ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (♯‘∪ 𝐴)))
85	84	sumsn 15762	. . . . . . 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	mullidd 11253	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (♯‘∪ 𝐴)) = (♯‘∪ 𝐴))
88	86, 87	eqtr2d 2771	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ {∅} ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
89	9, 36	fsumneg 15803	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})-((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
90		expm1t 14108	. . . . . . . . . . 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 11256	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑((♯‘𝑠) − 1)) · -1) = (-1 · (-1↑((♯‘𝑠) − 1))))
93	24	mulm1d 11689	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1 · (-1↑((♯‘𝑠) − 1))) = -(-1↑((♯‘𝑠) − 1)))
94	91, 92, 93	3eqtrd 2774	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑(♯‘𝑠)) = -(-1↑((♯‘𝑠) − 1)))
95	25	unissd 4893	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ⊆ ∪ 𝐴)
96	31, 95	sstrd 3969	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝐴)
97		sseqin2 4198	. . . . . . . . . . 11 ⊢ (∩ 𝑠 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
98	96, 97	sylib 218	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
99	98	fveq2d 6880	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) = (♯‘∩ 𝑠))
100	94, 99	oveq12d 7423	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (-(-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
101	24, 35	mulneg1d 11690	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-(-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = -((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
102	100, 101	eqtr2d 2771	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
103	102	sumeq2dv 15718	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})-((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
104	89, 103	eqtr3d 2772	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
105	88, 104	oveq12d 7423	. . . 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 2772	. . 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 2781	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) = 0)
108	4, 37, 107	subeq0d 11602	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ≠ wne 2932 Vcvv 3459 ∖ cdif 3923 ∪ cun 3924 ∩ cin 3925 ⊆ wss 3926 ∅c0 4308 𝒫 cpw 4575 {csn 4601 ∪ cuni 4883 ∩ cint 4922 ‘cfv 6531 (class class class)co 7405 Fincfn 8959 ℂcc 11127 0cc0 11129 1c1 11130 + caddc 11132 · cmul 11134 − cmin 11466 -cneg 11467 ℕcn 12240 ℕ₀cn0 12501 ↑cexp 14079 ♯chash 14348 Σcsu 15702

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 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-inf2 9655 ax-cnex 11185 ax-resscn 11186 ax-1cn 11187 ax-icn 11188 ax-addcl 11189 ax-addrcl 11190 ax-mulcl 11191 ax-mulrcl 11192 ax-mulcom 11193 ax-addass 11194 ax-mulass 11195 ax-distr 11196 ax-i2m1 11197 ax-1ne0 11198 ax-1rid 11199 ax-rnegex 11200 ax-rrecex 11201 ax-cnre 11202 ax-pre-lttri 11203 ax-pre-lttrn 11204 ax-pre-ltadd 11205 ax-pre-mulgt0 11206 ax-pre-sup 11207

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 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rmo 3359 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-se 5607 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-pred 6290 df-ord 6355 df-on 6356 df-lim 6357 df-suc 6358 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-isom 6540 df-riota 7362 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7862 df-1st 7988 df-2nd 7989 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-fin 8963 df-sup 9454 df-oi 9524 df-dju 9915 df-card 9953 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275 df-sub 11468 df-neg 11469 df-div 11895 df-nn 12241 df-2 12303 df-3 12304 df-n0 12502 df-z 12589 df-uz 12853 df-rp 13009 df-fz 13525 df-fzo 13672 df-seq 14020 df-exp 14080 df-hash 14349 df-cj 15118 df-re 15119 df-im 15120 df-sqrt 15254 df-abs 15255 df-clim 15504 df-sum 15703

This theorem is referenced by: incexc2 15854

W3C validator