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Theorem incexc 15797

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 9248	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
2		hashcl 14313	. . . 4 ⊢ (∪ 𝐴 ∈ Fin → (♯‘∪ 𝐴) ∈ ℕ₀)
3	2	nn0cnd 12495	. . 3 ⊢ (∪ 𝐴 ∈ Fin → (♯‘∪ 𝐴) ∈ ℂ)
4	1, 3	syl 17	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) ∈ ℂ)
5		simpl 484	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		pwfi 9223	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
7	5, 6	sylib 220	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 ∈ Fin)
8		diffi 9103	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
9	7, 8	syl 17	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
10		1cnd 11134	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 1 ∈ ℂ)
11	10	negcld 11487	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -1 ∈ ℂ)
12		eldifsni 4726	. . . . . . . 8 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ≠ ∅)
13	12	adantl 483	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ≠ ∅)
14		eldifi 4064	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ∈ 𝒫 𝐴)
15		elpwi 4539	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝐴 → 𝑠 ⊆ 𝐴)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ⊆ 𝐴)
17		ssfi 9101	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝑠 ⊆ 𝐴) → 𝑠 ∈ Fin)
18	5, 16, 17	syl2an 603	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ∈ Fin)
19		hashnncl 14323	. . . . . . . 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 259	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘𝑠) ∈ ℕ)
22		nnm1nn0 12473	. . . . . 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 14103	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑((♯‘𝑠) − 1)) ∈ ℂ)
25	16	adantl 483	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ⊆ 𝐴)
26		simplr 775	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝐴 ⊆ Fin)
27	25, 26	sstrd 3927	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ⊆ Fin)
28		unifi 9248	. . . . . . . 8 ⊢ ((𝑠 ∈ Fin ∧ 𝑠 ⊆ Fin) → ∪ 𝑠 ∈ Fin)
29	18, 27, 28	syl2anc 591	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ∈ Fin)
30		intssuni 4903	. . . . . . . 8 ⊢ (𝑠 ≠ ∅ → ∩ 𝑠 ⊆ ∪ 𝑠)
31	13, 30	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝑠)
32	29, 31	ssfid 9173	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ∈ Fin)
33		hashcl 14313	. . . . . 6 ⊢ (∩ 𝑠 ∈ Fin → (♯‘∩ 𝑠) ∈ ℕ₀)
34	32, 33	syl 17	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘∩ 𝑠) ∈ ℕ₀)
35	34	nn0cnd 12495	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘∩ 𝑠) ∈ ℂ)
36	24, 35	mulcld 11160	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
37	9, 36	fsumcl 15690	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
38		disjdif 4403	. . . . 5 ⊢ ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅
39	38	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅)
40		0elpw 5287	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 𝐴
41		snssi 4720	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 𝐴 → {∅} ⊆ 𝒫 𝐴)
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ {∅} ⊆ 𝒫 𝐴
43		undif 4413	. . . . . . 7 ⊢ ({∅} ⊆ 𝒫 𝐴 ↔ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴)
44	42, 43	mpbi 232	. . . . . 6 ⊢ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴
45	44	eqcomi 2750	. . . . 5 ⊢ 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅}))
46	45	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅})))
47		1cnd 11134	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 1 ∈ ℂ)
48	47	negcld 11487	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → -1 ∈ ℂ)
49	5, 15, 17	syl2an 603	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 𝑠 ∈ Fin)
50		hashcl 14313	. . . . . . 7 ⊢ (𝑠 ∈ Fin → (♯‘𝑠) ∈ ℕ₀)
51	49, 50	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (♯‘𝑠) ∈ ℕ₀)
52	48, 51	expcld 14103	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (-1↑(♯‘𝑠)) ∈ ℂ)
53	1	adantr 482	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ∪ 𝐴 ∈ Fin)
54		inss1 4168	. . . . . . . 8 ⊢ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴
55		ssfi 9101	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ Fin ∧ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin)
56	53, 54, 55	sylancl 593	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin)
57		hashcl 14313	. . . . . . 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 12495	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℂ)
60	52, 59	mulcld 11160	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) ∈ ℂ)
61	39, 46, 7, 60	fsumsplit 15698	. . 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 4158	. . . . . . 7 ⊢ (∪ 𝐴 ∩ ∪ 𝐴) = ∪ 𝐴
63	62	fveq2i 6834	. . . . . 6 ⊢ (♯‘(∪ 𝐴 ∩ ∪ 𝐴)) = (♯‘∪ 𝐴)
64	63	oveq2i 7371	. . . . 5 ⊢ ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = ((♯‘∪ 𝐴) − (♯‘∪ 𝐴))
65	4	subidd 11488	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘∪ 𝐴)) = 0)
66	64, 65	eqtrid 2788	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = 0)
67		incexclem 15796	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∪ 𝐴 ∈ Fin) → ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
68	1, 67	syldan 598	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
69	66, 68	eqtr3d 2778	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 0 = Σ𝑠 ∈ 𝒫 𝐴((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
70	4, 37	negsubd 11506	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) + -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) = ((♯‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))))
71		0ex 5232	. . . . . . 7 ⊢ ∅ ∈ V
72		1cnd 11134	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 1 ∈ ℂ)
73	72, 4	mulcld 11160	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (♯‘∪ 𝐴)) ∈ ℂ)
74		fveq2 6831	. . . . . . . . . . . 12 ⊢ (𝑠 = ∅ → (♯‘𝑠) = (♯‘∅))
75		hash0 14324	. . . . . . . . . . . 12 ⊢ (♯‘∅) = 0
76	74, 75	eqtrdi 2792	. . . . . . . . . . 11 ⊢ (𝑠 = ∅ → (♯‘𝑠) = 0)
77	76	oveq2d 7376	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (-1↑(♯‘𝑠)) = (-1↑0))
78		neg1cn 12139	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
79		exp0 14022	. . . . . . . . . . 11 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
80	78, 79	ax-mp 5	. . . . . . . . . 10 ⊢ (-1↑0) = 1
81	77, 80	eqtrdi 2792	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (-1↑(♯‘𝑠)) = 1)
82		rint0 4921	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (∪ 𝐴 ∩ ∩ 𝑠) = ∪ 𝐴)
83	82	fveq2d 6835	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) = (♯‘∪ 𝐴))
84	81, 83	oveq12d 7378	. . . . . . . 8 ⊢ (𝑠 = ∅ → ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (♯‘∪ 𝐴)))
85	84	sumsn 15703	. . . . . . 7 ⊢ ((∅ ∈ V ∧ (1 · (♯‘∪ 𝐴)) ∈ ℂ) → Σ𝑠 ∈ {∅} ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (♯‘∪ 𝐴)))
86	71, 73, 85	sylancr 594	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ {∅} ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (♯‘∪ 𝐴)))
87	4	mullidd 11158	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (♯‘∪ 𝐴)) = (♯‘∪ 𝐴))
88	86, 87	eqtr2d 2777	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ {∅} ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
89	9, 36	fsumneg 15744	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})-((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
90		expm1t 14047	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (♯‘𝑠) ∈ ℕ) → (-1↑(♯‘𝑠)) = ((-1↑((♯‘𝑠) − 1)) · -1))
91	11, 21, 90	syl2anc 591	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑(♯‘𝑠)) = ((-1↑((♯‘𝑠) − 1)) · -1))
92	24, 11	mulcomd 11161	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑((♯‘𝑠) − 1)) · -1) = (-1 · (-1↑((♯‘𝑠) − 1))))
93	24	mulm1d 11597	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1 · (-1↑((♯‘𝑠) − 1))) = -(-1↑((♯‘𝑠) − 1)))
94	91, 92, 93	3eqtrd 2780	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑(♯‘𝑠)) = -(-1↑((♯‘𝑠) − 1)))
95	25	unissd 4851	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ⊆ ∪ 𝐴)
96	31, 95	sstrd 3927	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝐴)
97		sseqin2 4155	. . . . . . . . . . 11 ⊢ (∩ 𝑠 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
98	96, 97	sylib 220	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
99	98	fveq2d 6835	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) = (♯‘∩ 𝑠))
100	94, 99	oveq12d 7378	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (-(-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
101	24, 35	mulneg1d 11598	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-(-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = -((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
102	100, 101	eqtr2d 2777	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = ((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
103	102	sumeq2dv 15659	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})-((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
104	89, 103	eqtr3d 2778	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
105	88, 104	oveq12d 7378	. . . 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 2778	. . 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 2787	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) = 0)
108	4, 37, 107	subeq0d 11508	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 397 = wceq 1548 ∈ wcel 2121 ≠ wne 2936 Vcvv 3433 ∖ cdif 3882 ∪ cun 3883 ∩ cin 3884 ⊆ wss 3885 ∅c0 4264 𝒫 cpw 4532 {csn 4558 ∪ cuni 4841 ∩ cint 4880 ‘cfv 6489 (class class class)co 7360 Fincfn 8887 ℂcc 11031 0cc0 11033 1c1 11034 + caddc 11036 · cmul 11038 − cmin 11372 -cneg 11373 ℕcn 12169 ℕ₀cn0 12432 ↑cexp 14018 ♯chash 14287 Σcsu 15643

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5202 ax-sep 5221 ax-nul 5231 ax-pow 5297 ax-pr 5365 ax-un 7682 ax-inf2 9557 ax-cnex 11089 ax-resscn 11090 ax-1cn 11091 ax-icn 11092 ax-addcl 11093 ax-addrcl 11094 ax-mulcl 11095 ax-mulrcl 11096 ax-mulcom 11097 ax-addass 11098 ax-mulass 11099 ax-distr 11100 ax-i2m1 11101 ax-1ne0 11102 ax-1rid 11103 ax-rnegex 11104 ax-rrecex 11105 ax-cnre 11106 ax-pre-lttri 11107 ax-pre-lttrn 11108 ax-pre-ltadd 11109 ax-pre-mulgt0 11110 ax-pre-sup 11111

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3726 df-csb 3834 df-dif 3888 df-un 3890 df-in 3892 df-ss 3902 df-pss 3905 df-nul 4265 df-if 4458 df-pw 4534 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4842 df-int 4881 df-iun 4926 df-br 5076 df-opab 5138 df-mpt 5157 df-tr 5183 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-se 5575 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-isom 6498 df-riota 7317 df-ov 7363 df-oprab 7364 df-mpo 7365 df-om 7811 df-1st 7935 df-2nd 7936 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-1o 8399 df-oadd 8403 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-oi 9419 df-dju 9820 df-card 9858 df-pnf 11176 df-mnf 11177 df-xr 11178 df-ltxr 11179 df-le 11180 df-sub 11374 df-neg 11375 df-div 11803 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-rp 12938 df-fz 13457 df-fzo 13604 df-seq 13959 df-exp 14019 df-hash 14288 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-clim 15445 df-sum 15644

This theorem is referenced by: incexc2 15798

W3C validator