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Theorem incexc 14569

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 8255	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
2		hashcl 13147	. . . 4 ⊢ (∪ 𝐴 ∈ Fin → (#‘∪ 𝐴) ∈ ℕ₀)
3	2	nn0cnd 11353	. . 3 ⊢ (∪ 𝐴 ∈ Fin → (#‘∪ 𝐴) ∈ ℂ)
4	1, 3	syl 17	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) ∈ ℂ)
5		simpl 473	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		pwfi 8261	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
7	5, 6	sylib 208	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 ∈ Fin)
8		diffi 8192	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
9	7, 8	syl 17	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
10		1cnd 10056	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 1 ∈ ℂ)
11	10	negcld 10379	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -1 ∈ ℂ)
12		eldifsni 4320	. . . . . . . 8 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ≠ ∅)
13	12	adantl 482	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ≠ ∅)
14		eldifi 3732	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ∈ 𝒫 𝐴)
15		elpwi 4168	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝐴 → 𝑠 ⊆ 𝐴)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ⊆ 𝐴)
17		ssfi 8180	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝑠 ⊆ 𝐴) → 𝑠 ∈ Fin)
18	5, 16, 17	syl2an 494	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ∈ Fin)
19		hashnncl 13157	. . . . . . . 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 247	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (#‘𝑠) ∈ ℕ)
22		nnm1nn0 11334	. . . . . 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 13008	. . . 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 792	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝐴 ⊆ Fin)
27	25, 26	sstrd 3613	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ⊆ Fin)
28		unifi 8255	. . . . . . . 8 ⊢ ((𝑠 ∈ Fin ∧ 𝑠 ⊆ Fin) → ∪ 𝑠 ∈ Fin)
29	18, 27, 28	syl2anc 693	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ∈ Fin)
30		intssuni 4499	. . . . . . . 8 ⊢ (𝑠 ≠ ∅ → ∩ 𝑠 ⊆ ∪ 𝑠)
31	13, 30	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝑠)
32		ssfi 8180	. . . . . . 7 ⊢ ((∪ 𝑠 ∈ Fin ∧ ∩ 𝑠 ⊆ ∪ 𝑠) → ∩ 𝑠 ∈ Fin)
33	29, 31, 32	syl2anc 693	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ∈ Fin)
34		hashcl 13147	. . . . . 6 ⊢ (∩ 𝑠 ∈ Fin → (#‘∩ 𝑠) ∈ ℕ₀)
35	33, 34	syl 17	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (#‘∩ 𝑠) ∈ ℕ₀)
36	35	nn0cnd 11353	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (#‘∩ 𝑠) ∈ ℂ)
37	24, 36	mulcld 10060	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) ∈ ℂ)
38	9, 37	fsumcl 14464	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) ∈ ℂ)
39		disjdif 4040	. . . . 5 ⊢ ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅
40	39	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅)
41		0elpw 4834	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 𝐴
42		snssi 4339	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 𝐴 → {∅} ⊆ 𝒫 𝐴)
43	41, 42	ax-mp 5	. . . . . . 7 ⊢ {∅} ⊆ 𝒫 𝐴
44		undif 4049	. . . . . . 7 ⊢ ({∅} ⊆ 𝒫 𝐴 ↔ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴)
45	43, 44	mpbi 220	. . . . . 6 ⊢ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴
46	45	eqcomi 2631	. . . . 5 ⊢ 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅}))
47	46	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅})))
48		1cnd 10056	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 1 ∈ ℂ)
49	48	negcld 10379	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → -1 ∈ ℂ)
50	5, 15, 17	syl2an 494	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 𝑠 ∈ Fin)
51		hashcl 13147	. . . . . . 7 ⊢ (𝑠 ∈ Fin → (#‘𝑠) ∈ ℕ₀)
52	50, 51	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (#‘𝑠) ∈ ℕ₀)
53	49, 52	expcld 13008	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (-1↑(#‘𝑠)) ∈ ℂ)
54	1	adantr 481	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ∪ 𝐴 ∈ Fin)
55		inss1 3833	. . . . . . . 8 ⊢ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴
56		ssfi 8180	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ Fin ∧ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin)
57	54, 55, 56	sylancl 694	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin)
58		hashcl 13147	. . . . . . 7 ⊢ ((∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin → (#‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℕ₀)
59	57, 58	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (#‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℕ₀)
60	59	nn0cnd 11353	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (#‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℂ)
61	53, 60	mulcld 10060	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) ∈ ℂ)
62	40, 47, 7, 61	fsumsplit 14471	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ 𝒫 𝐴((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) = (Σ𝑠 ∈ {∅} ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) + Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠)))))
63		inidm 3822	. . . . . . 7 ⊢ (∪ 𝐴 ∩ ∪ 𝐴) = ∪ 𝐴
64	63	fveq2i 6194	. . . . . 6 ⊢ (#‘(∪ 𝐴 ∩ ∪ 𝐴)) = (#‘∪ 𝐴)
65	64	oveq2i 6661	. . . . 5 ⊢ ((#‘∪ 𝐴) − (#‘(∪ 𝐴 ∩ ∪ 𝐴))) = ((#‘∪ 𝐴) − (#‘∪ 𝐴))
66	4	subidd 10380	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((#‘∪ 𝐴) − (#‘∪ 𝐴)) = 0)
67	65, 66	syl5eq 2668	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((#‘∪ 𝐴) − (#‘(∪ 𝐴 ∩ ∪ 𝐴))) = 0)
68		incexclem 14568	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∪ 𝐴 ∈ Fin) → ((#‘∪ 𝐴) − (#‘(∪ 𝐴 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
69	1, 68	syldan 487	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((#‘∪ 𝐴) − (#‘(∪ 𝐴 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
70	67, 69	eqtr3d 2658	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 0 = Σ𝑠 ∈ 𝒫 𝐴((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
71	4, 38	negsubd 10398	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((#‘∪ 𝐴) + -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠))) = ((#‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠))))
72		0ex 4790	. . . . . . 7 ⊢ ∅ ∈ V
73		1cnd 10056	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 1 ∈ ℂ)
74	73, 4	mulcld 10060	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (#‘∪ 𝐴)) ∈ ℂ)
75		fveq2 6191	. . . . . . . . . . . 12 ⊢ (𝑠 = ∅ → (#‘𝑠) = (#‘∅))
76		hash0 13158	. . . . . . . . . . . 12 ⊢ (#‘∅) = 0
77	75, 76	syl6eq 2672	. . . . . . . . . . 11 ⊢ (𝑠 = ∅ → (#‘𝑠) = 0)
78	77	oveq2d 6666	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (-1↑(#‘𝑠)) = (-1↑0))
79		neg1cn 11124	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
80		exp0 12864	. . . . . . . . . . 11 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
81	79, 80	ax-mp 5	. . . . . . . . . 10 ⊢ (-1↑0) = 1
82	78, 81	syl6eq 2672	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (-1↑(#‘𝑠)) = 1)
83		rint0 4517	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (∪ 𝐴 ∩ ∩ 𝑠) = ∪ 𝐴)
84	83	fveq2d 6195	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (#‘(∪ 𝐴 ∩ ∩ 𝑠)) = (#‘∪ 𝐴))
85	82, 84	oveq12d 6668	. . . . . . . 8 ⊢ (𝑠 = ∅ → ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (#‘∪ 𝐴)))
86	85	sumsn 14475	. . . . . . 7 ⊢ ((∅ ∈ V ∧ (1 · (#‘∪ 𝐴)) ∈ ℂ) → Σ𝑠 ∈ {∅} ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (#‘∪ 𝐴)))
87	72, 74, 86	sylancr 695	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ {∅} ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (#‘∪ 𝐴)))
88	4	mulid2d 10058	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (#‘∪ 𝐴)) = (#‘∪ 𝐴))
89	87, 88	eqtr2d 2657	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ𝑠 ∈ {∅} ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
90	9, 37	fsumneg 14519	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})-((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))
91		expm1t 12888	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (#‘𝑠) ∈ ℕ) → (-1↑(#‘𝑠)) = ((-1↑((#‘𝑠) − 1)) · -1))
92	11, 21, 91	syl2anc 693	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑(#‘𝑠)) = ((-1↑((#‘𝑠) − 1)) · -1))
93	24, 11	mulcomd 10061	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑((#‘𝑠) − 1)) · -1) = (-1 · (-1↑((#‘𝑠) − 1))))
94	24	mulm1d 10482	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1 · (-1↑((#‘𝑠) − 1))) = -(-1↑((#‘𝑠) − 1)))
95	92, 93, 94	3eqtrd 2660	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑(#‘𝑠)) = -(-1↑((#‘𝑠) − 1)))
96	25	unissd 4462	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ⊆ ∪ 𝐴)
97	31, 96	sstrd 3613	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝐴)
98		sseqin2 3817	. . . . . . . . . . 11 ⊢ (∩ 𝑠 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
99	97, 98	sylib 208	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
100	99	fveq2d 6195	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (#‘(∪ 𝐴 ∩ ∩ 𝑠)) = (#‘∩ 𝑠))
101	95, 100	oveq12d 6668	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) = (-(-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))
102	24, 36	mulneg1d 10483	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-(-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = -((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))
103	101, 102	eqtr2d 2657	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
104	103	sumeq2dv 14433	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})-((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
105	90, 104	eqtr3d 2658	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
106	89, 105	oveq12d 6668	. . . 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)))))
107	71, 106	eqtr3d 2658	. . 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)))))
108	62, 70, 107	3eqtr4rd 2667	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((#‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠))) = 0)
109	4, 38, 108	subeq0d 10400	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ≠ wne 2794 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ∪ cuni 4436 ∩ cint 4475 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℂcc 9934 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 − cmin 10266 -cneg 10267 ℕcn 11020 ℕ₀cn0 11292 ↑cexp 12860 #chash 13117 Σcsu 14416

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-rp 11833 df-fz 12327 df-fzo 12466 df-seq 12802 df-exp 12861 df-hash 13118 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-clim 14219 df-sum 14417

This theorem is referenced by: incexc2 14570

W3C validator