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Theorem incexc 14408

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 8138	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
2		hashcl 13009	. . . 4 ⊢ (∪ 𝐴 ∈ Fin → (#‘∪ 𝐴) ∈ ℕ₀)
3	2	nn0cnd 11230	. . 3 ⊢ (∪ 𝐴 ∈ Fin → (#‘∪ 𝐴) ∈ ℂ)
4	1, 3	syl 17	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) ∈ ℂ)
5		simpl 472	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		pwfi 8144	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
7	5, 6	sylib 207	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 ∈ Fin)
8		diffi 8077	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
9	7, 8	syl 17	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
10		1cnd 9935	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 1 ∈ ℂ)
11	10	negcld 10258	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -1 ∈ ℂ)
12		eldifsni 4261	. . . . . . . 8 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ≠ ∅)
13	12	adantl 481	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ≠ ∅)
14		eldifi 3694	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ∈ 𝒫 𝐴)
15		elpwi 4117	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝐴 → 𝑠 ⊆ 𝐴)
16	14, 15	syl 17	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ⊆ 𝐴)
17		ssfi 8065	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝑠 ⊆ 𝐴) → 𝑠 ∈ Fin)
18	5, 16, 17	syl2an 493	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ∈ Fin)
19		hashnncl 13018	. . . . . . . 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 246	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (#‘𝑠) ∈ ℕ)
22		nnm1nn0 11211	. . . . . 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 12870	. . . 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 788	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝐴 ⊆ Fin)
27	25, 26	sstrd 3578	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ⊆ Fin)
28		unifi 8138	. . . . . . . 8 ⊢ ((𝑠 ∈ Fin ∧ 𝑠 ⊆ Fin) → ∪ 𝑠 ∈ Fin)
29	18, 27, 28	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ∈ Fin)
30		intssuni 4434	. . . . . . . 8 ⊢ (𝑠 ≠ ∅ → ∩ 𝑠 ⊆ ∪ 𝑠)
31	13, 30	syl 17	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝑠)
32		ssfi 8065	. . . . . . 7 ⊢ ((∪ 𝑠 ∈ Fin ∧ ∩ 𝑠 ⊆ ∪ 𝑠) → ∩ 𝑠 ∈ Fin)
33	29, 31, 32	syl2anc 691	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ∈ Fin)
34		hashcl 13009	. . . . . 6 ⊢ (∩ 𝑠 ∈ Fin → (#‘∩ 𝑠) ∈ ℕ₀)
35	33, 34	syl 17	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (#‘∩ 𝑠) ∈ ℕ₀)
36	35	nn0cnd 11230	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (#‘∩ 𝑠) ∈ ℂ)
37	24, 36	mulcld 9939	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) ∈ ℂ)
38	9, 37	fsumcl 14311	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) ∈ ℂ)
39		disjdif 3992	. . . . 5 ⊢ ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅
40	39	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅)
41		0elpw 4760	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 𝐴
42		snssi 4280	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 𝐴 → {∅} ⊆ 𝒫 𝐴)
43	41, 42	ax-mp 5	. . . . . . 7 ⊢ {∅} ⊆ 𝒫 𝐴
44		undif 4001	. . . . . . 7 ⊢ ({∅} ⊆ 𝒫 𝐴 ↔ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴)
45	43, 44	mpbi 219	. . . . . 6 ⊢ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴
46	45	eqcomi 2619	. . . . 5 ⊢ 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅}))
47	46	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅})))
48		1cnd 9935	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 1 ∈ ℂ)
49	48	negcld 10258	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → -1 ∈ ℂ)
50	5, 15, 17	syl2an 493	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 𝑠 ∈ Fin)
51		hashcl 13009	. . . . . . 7 ⊢ (𝑠 ∈ Fin → (#‘𝑠) ∈ ℕ₀)
52	50, 51	syl 17	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (#‘𝑠) ∈ ℕ₀)
53	49, 52	expcld 12870	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (-1↑(#‘𝑠)) ∈ ℂ)
54	1	adantr 480	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ∪ 𝐴 ∈ Fin)
55		inss1 3795	. . . . . . . 8 ⊢ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴
56		ssfi 8065	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ Fin ∧ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin)
57	54, 55, 56	sylancl 693	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin)
58		hashcl 13009	. . . . . . 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 11230	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (#‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℂ)
61	53, 60	mulcld 9939	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) ∈ ℂ)
62	40, 47, 7, 61	fsumsplit 14318	. . 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 3784	. . . . . . 7 ⊢ (∪ 𝐴 ∩ ∪ 𝐴) = ∪ 𝐴
64	63	fveq2i 6106	. . . . . 6 ⊢ (#‘(∪ 𝐴 ∩ ∪ 𝐴)) = (#‘∪ 𝐴)
65	64	oveq2i 6560	. . . . 5 ⊢ ((#‘∪ 𝐴) − (#‘(∪ 𝐴 ∩ ∪ 𝐴))) = ((#‘∪ 𝐴) − (#‘∪ 𝐴))
66	4	subidd 10259	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((#‘∪ 𝐴) − (#‘∪ 𝐴)) = 0)
67	65, 66	syl5eq 2656	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((#‘∪ 𝐴) − (#‘(∪ 𝐴 ∩ ∪ 𝐴))) = 0)
68		incexclem 14407	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∪ 𝐴 ∈ Fin) → ((#‘∪ 𝐴) − (#‘(∪ 𝐴 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
69	1, 68	syldan 486	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((#‘∪ 𝐴) − (#‘(∪ 𝐴 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
70	67, 69	eqtr3d 2646	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 0 = Σ𝑠 ∈ 𝒫 𝐴((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
71	4, 38	negsubd 10277	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((#‘∪ 𝐴) + -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠))) = ((#‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠))))
72		0ex 4718	. . . . . . 7 ⊢ ∅ ∈ V
73		1cnd 9935	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 1 ∈ ℂ)
74	73, 4	mulcld 9939	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (#‘∪ 𝐴)) ∈ ℂ)
75		fveq2 6103	. . . . . . . . . . . 12 ⊢ (𝑠 = ∅ → (#‘𝑠) = (#‘∅))
76		hash0 13019	. . . . . . . . . . . 12 ⊢ (#‘∅) = 0
77	75, 76	syl6eq 2660	. . . . . . . . . . 11 ⊢ (𝑠 = ∅ → (#‘𝑠) = 0)
78	77	oveq2d 6565	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (-1↑(#‘𝑠)) = (-1↑0))
79		neg1cn 11001	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
80		exp0 12726	. . . . . . . . . . 11 ⊢ (-1 ∈ ℂ → (-1↑0) = 1)
81	79, 80	ax-mp 5	. . . . . . . . . 10 ⊢ (-1↑0) = 1
82	78, 81	syl6eq 2660	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (-1↑(#‘𝑠)) = 1)
83		rint0 4452	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (∪ 𝐴 ∩ ∩ 𝑠) = ∪ 𝐴)
84	83	fveq2d 6107	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (#‘(∪ 𝐴 ∩ ∩ 𝑠)) = (#‘∪ 𝐴))
85	82, 84	oveq12d 6567	. . . . . . . 8 ⊢ (𝑠 = ∅ → ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (#‘∪ 𝐴)))
86	85	sumsn 14319	. . . . . . 7 ⊢ ((∅ ∈ V ∧ (1 · (#‘∪ 𝐴)) ∈ ℂ) → Σ𝑠 ∈ {∅} ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (#‘∪ 𝐴)))
87	72, 74, 86	sylancr 694	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ {∅} ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (#‘∪ 𝐴)))
88	4	mulid2d 9937	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (#‘∪ 𝐴)) = (#‘∪ 𝐴))
89	87, 88	eqtr2d 2645	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ𝑠 ∈ {∅} ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
90	9, 37	fsumneg 14361	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})-((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))
91		expm1t 12750	. . . . . . . . . . 11 ⊢ ((-1 ∈ ℂ ∧ (#‘𝑠) ∈ ℕ) → (-1↑(#‘𝑠)) = ((-1↑((#‘𝑠) − 1)) · -1))
92	11, 21, 91	syl2anc 691	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑(#‘𝑠)) = ((-1↑((#‘𝑠) − 1)) · -1))
93	24, 11	mulcomd 9940	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑((#‘𝑠) − 1)) · -1) = (-1 · (-1↑((#‘𝑠) − 1))))
94	24	mulm1d 10361	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1 · (-1↑((#‘𝑠) − 1))) = -(-1↑((#‘𝑠) − 1)))
95	92, 93, 94	3eqtrd 2648	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-1↑(#‘𝑠)) = -(-1↑((#‘𝑠) − 1)))
96	25	unissd 4398	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ⊆ ∪ 𝐴)
97	31, 96	sstrd 3578	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝐴)
98		sseqin2 3779	. . . . . . . . . . 11 ⊢ (∩ 𝑠 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
99	97, 98	sylib 207	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
100	99	fveq2d 6107	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (#‘(∪ 𝐴 ∩ ∩ 𝑠)) = (#‘∩ 𝑠))
101	95, 100	oveq12d 6567	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))) = (-(-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))
102	24, 36	mulneg1d 10362	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (-(-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = -((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))
103	101, 102	eqtr2d 2645	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = ((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
104	103	sumeq2dv 14281	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})-((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
105	90, 104	eqtr3d 2646	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑(#‘𝑠)) · (#‘(∪ 𝐴 ∩ ∩ 𝑠))))
106	89, 105	oveq12d 6567	. . . 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 2646	. . 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 2655	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((#‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠))) = 0)
109	4, 38, 108	subeq0d 10279	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})((-1↑((#‘𝑠) − 1)) · (#‘∩ 𝑠)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 ∩ cint 4410 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℂcc 9813 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 − cmin 10145 -cneg 10146 ℕcn 10897 ℕ₀cn0 11169 ↑cexp 12722 #chash 12979 Σcsu 14264

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-n0 11170 df-z 11255 df-uz 11564 df-rp 11709 df-fz 12198 df-fzo 12335 df-seq 12664 df-exp 12723 df-hash 12980 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-clim 14067 df-sum 14265

This theorem is referenced by: incexc2 14409

W3C validator