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Theorem incexc 16006

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 9333	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ∪ 𝐴 ∈ Fin)
2		hashcl 14500	. . . 4 ⊢ (∪ 𝐴 ∈ Fin → (♯‘∪ 𝐴) ∈ ℕ₀)
3	2	nn0cnd 12669	. . 3 ⊢ (∪ 𝐴 ∈ Fin → (♯‘∪ 𝐴) ∈ ℂ)
4	1, 3	syl 18	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) ∈ ℂ)
5		simpl 488	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		pwfi 9310	. . . . 5 ⊢ (𝐴 ∈ Fin ↔ 𝒫 𝐴 ∈ Fin)
7	5, 6	sylib 221	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 ∈ Fin)
8		diffi 9190	. . . 4 ⊢ (𝒫 𝐴 ∈ Fin → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
9	7, 8	syl 18	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (𝒫 𝐴 ∖ {∅}) ∈ Fin)
10		1cnd 11302	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 1 ∈ ℂ)
11	10	negcld 11656	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -1 ∈ ℂ)
12		eldifsni 4752	. . . . . . . 8 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ≠ ∅)
13	12	adantl 487	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ≠ ∅)
14		eldifi 4067	. . . . . . . . . 10 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ∈ 𝒫 𝐴)
15		elpwi 4563	. . . . . . . . . 10 ⊢ (𝑠 ∈ 𝒫 𝐴 → 𝑠 ⊆ 𝐴)
16	14, 15	syl 18	. . . . . . . . 9 ⊢ (𝑠 ∈ (𝒫 𝐴 ∖ {∅}) → 𝑠 ⊆ 𝐴)
17		ssfi 9188	. . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝑠 ⊆ 𝐴) → 𝑠 ∈ Fin)
18	5, 16, 17	syl2an 608	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ∈ Fin)
19		hashnncl 14510	. . . . . . . 8 ⊢ (𝑠 ∈ Fin → ((♯‘𝑠) ∈ ℕ ↔ 𝑠 ≠ ∅))
20	18, 19	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((♯‘𝑠) ∈ ℕ ↔ 𝑠 ≠ ∅))
21	13, 20	mpbird 260	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘𝑠) ∈ ℕ)
22		nnm1nn0 12647	. . . . . 6 ⊢ ((♯‘𝑠) ∈ ℕ → ((♯‘𝑠) − 1) ∈ ℕ₀)
23	21, 22	syl 18	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ((♯‘𝑠) − 1) ∈ ℕ₀)
24	11, 23	expcld 14289	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ( -1↑((♯‘𝑠) − 1)) ∈ ℂ)
25	16	adantl 487	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ⊆ 𝐴)
26		simplr 781	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝐴 ⊆ Fin)
27	25, 26	sstrd 3918	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → 𝑠 ⊆ Fin)
28		unifi 9333	. . . . . . . 8 ⊢ ((𝑠 ∈ Fin ∧ 𝑠 ⊆ Fin) → ∪ 𝑠 ∈ Fin)
29	18, 27, 28	syl2anc 596	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ∈ Fin)
30		intssuni 4929	. . . . . . . 8 ⊢ (𝑠 ≠ ∅ → ∩ 𝑠 ⊆ ∪ 𝑠)
31	13, 30	syl 18	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝑠)
32	29, 31	ssfid 9260	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ∈ Fin)
33		hashcl 14500	. . . . . 6 ⊢ (∩ 𝑠 ∈ Fin → (♯‘∩ 𝑠) ∈ ℕ₀)
34	32, 33	syl 18	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘∩ 𝑠) ∈ ℕ₀)
35	34	nn0cnd 12669	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘∩ 𝑠) ∈ ℂ)
36	24, 35	mulcld 11329	. . 3 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
37	9, 36	fsumcl 15899	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) ∈ ℂ)
38		disjdif 4425	. . . . 5 ⊢ ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅
39	38	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ({∅} ∩ (𝒫 𝐴 ∖ {∅})) = ∅)
40		0elpw 5316	. . . . . . . 8 ⊢ ∅ ∈ 𝒫 𝐴
41		snssi 4745	. . . . . . . 8 ⊢ (∅ ∈ 𝒫 𝐴 → {∅} ⊆ 𝒫 𝐴)
42	40, 41	ax-mp 5	. . . . . . 7 ⊢ {∅} ⊆ 𝒫 𝐴
43		undif 4437	. . . . . . 7 ⊢ ({∅} ⊆ 𝒫 𝐴 ↔ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴)
44	42, 43	mpbi 233	. . . . . 6 ⊢ ({∅} ∪ (𝒫 𝐴 ∖ {∅})) = 𝒫 𝐴
45	44	eqcomi 2769	. . . . 5 ⊢ 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅}))
46	45	a1i 11	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝒫 𝐴 = ({∅} ∪ (𝒫 𝐴 ∖ {∅})))
47		1cnd 11302	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 1 ∈ ℂ)
48	47	negcld 11656	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → -1 ∈ ℂ)
49	5, 15, 17	syl2an 608	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → 𝑠 ∈ Fin)
50		hashcl 14500	. . . . . . 7 ⊢ (𝑠 ∈ Fin → (♯‘𝑠) ∈ ℕ₀)
51	49, 50	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (♯‘𝑠) ∈ ℕ₀)
52	48, 51	expcld 14289	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ( -1↑(♯‘𝑠)) ∈ ℂ)
53	1	adantr 486	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → ∪ 𝐴 ∈ Fin)
54		inss1 4133	. . . . . . . 8 ⊢ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴
55		ssfi 9188	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ Fin ∧ (∪ 𝐴 ∩ ∩ 𝑠) ⊆ ∪ 𝐴) → (∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin)
56	53, 54, 55	sylancl 598	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin)
57		hashcl 14500	. . . . . . 7 ⊢ ((∪ 𝐴 ∩ ∩ 𝑠) ∈ Fin → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℕ₀)
58	56, 57	syl 18	. . . . . 6 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℕ₀)
59	58	nn0cnd 12669	. . . . 5 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) ∈ ℂ)
60	52, 59	mulcld 11329	. . . 4 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ 𝒫 𝐴) → (( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) ∈ ℂ)
61	39, 46, 7, 60	fsumsplit 15907	. . 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 4123	. . . . . . 7 ⊢ (∪ 𝐴 ∩ ∪ 𝐴) = ∪ 𝐴
63	62	fveq2i 6888	. . . . . 6 ⊢ (♯‘(∪ 𝐴 ∩ ∪ 𝐴)) = (♯‘∪ 𝐴)
64	63	oveq2i 7431	. . . . 5 ⊢ ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = ((♯‘∪ 𝐴) − (♯‘∪ 𝐴))
65	4	subidd 11657	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘∪ 𝐴)) = 0)
66	64, 65	eqtrid 2807	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = 0)
67		incexclem 16005	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ ∪ 𝐴 ∈ Fin) → ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴(( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
68	1, 67	syldan 603	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − (♯‘(∪ 𝐴 ∩ ∪ 𝐴))) = Σ𝑠 ∈ 𝒫 𝐴(( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
69	66, 68	eqtr3d 2797	. . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 0 = Σ𝑠 ∈ 𝒫 𝐴(( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
70	4, 37	negsubd 11675	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) + -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) = ((♯‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))))
71		0ex 5260	. . . . . . 7 ⊢ ∅ ∈ V
72		1cnd 11302	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 1 ∈ ℂ)
73	72, 4	mulcld 11329	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (♯‘∪ 𝐴)) ∈ ℂ)
74		fveq2 6885	. . . . . . . . . . . 12 ⊢ (𝑠 = ∅ → (♯‘𝑠) = (♯‘∅))
75		hash0 14511	. . . . . . . . . . . 12 ⊢ (♯‘∅) = 0
76	74, 75	eqtrdi 2811	. . . . . . . . . . 11 ⊢ (𝑠 = ∅ → (♯‘𝑠) = 0)
77	76	oveq2d 7436	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → ( -1↑(♯‘𝑠)) = ( -1↑0))
78		neg1cn 12305	. . . . . . . . . . 11 ⊢ -1 ∈ ℂ
79		exp0 14208	. . . . . . . . . . 11 ⊢ ( -1 ∈ ℂ → ( -1↑0) = 1)
80	78, 79	ax-mp 5	. . . . . . . . . 10 ⊢ ( -1↑0) = 1
81	77, 80	eqtrdi 2811	. . . . . . . . 9 ⊢ (𝑠 = ∅ → ( -1↑(♯‘𝑠)) = 1)
82		rint0 4947	. . . . . . . . . 10 ⊢ (𝑠 = ∅ → (∪ 𝐴 ∩ ∩ 𝑠) = ∪ 𝐴)
83	82	fveq2d 6889	. . . . . . . . 9 ⊢ (𝑠 = ∅ → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) = (♯‘∪ 𝐴))
84	81, 83	oveq12d 7438	. . . . . . . 8 ⊢ (𝑠 = ∅ → (( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (♯‘∪ 𝐴)))
85	84	sumsn 15912	. . . . . . 7 ⊢ ((∅ ∈ V ∧ (1 · (♯‘∪ 𝐴)) ∈ ℂ) → Σ𝑠 ∈ {∅} (( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (♯‘∪ 𝐴)))
86	71, 73, 85	sylancr 599	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ {∅} (( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = (1 · (♯‘∪ 𝐴)))
87	4	mullidd 11327	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (1 · (♯‘∪ 𝐴)) = (♯‘∪ 𝐴))
88	86, 87	eqtr2d 2796	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ {∅} (( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
89	9, 36	fsumneg 15953	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅}) -(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
90		expm1t 14233	. . . . . . . . . . 11 ⊢ (( -1 ∈ ℂ ∧ (♯‘𝑠) ∈ ℕ) → ( -1↑(♯‘𝑠)) = (( -1↑((♯‘𝑠) − 1)) · -1))
91	11, 21, 90	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ( -1↑(♯‘𝑠)) = (( -1↑((♯‘𝑠) − 1)) · -1))
92	24, 11	mulcomd 11330	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (( -1↑((♯‘𝑠) − 1)) · -1) = ( -1 · ( -1↑((♯‘𝑠) − 1))))
93	24	mulm1d 11768	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ( -1 · ( -1↑((♯‘𝑠) − 1))) = -( -1↑((♯‘𝑠) − 1)))
94	91, 92, 93	3eqtrd 2799	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ( -1↑(♯‘𝑠)) = -( -1↑((♯‘𝑠) − 1)))
95	25	unissd 4876	. . . . . . . . . . . 12 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∪ 𝑠 ⊆ ∪ 𝐴)
96	31, 95	sstrd 3918	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ∩ 𝑠 ⊆ ∪ 𝐴)
97		sseqin2 4120	. . . . . . . . . . 11 ⊢ (∩ 𝑠 ⊆ ∪ 𝐴 ↔ (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
98	96, 97	sylib 221	. . . . . . . . . 10 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (∪ 𝐴 ∩ ∩ 𝑠) = ∩ 𝑠)
99	98	fveq2d 6889	. . . . . . . . 9 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (♯‘(∪ 𝐴 ∩ ∩ 𝑠)) = (♯‘∩ 𝑠))
100	94, 99	oveq12d 7438	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → (( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))) = ( -( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
101	24, 35	mulneg1d 11769	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → ( -( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = -(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))
102	100, 101	eqtr2d 2796	. . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑠 ∈ (𝒫 𝐴 ∖ {∅})) → -(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = (( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
103	102	sumeq2dv 15869	. . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅}) -(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})(( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
104	89, 103	eqtr3d 2797	. . . . 5 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → -Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})(( -1↑(♯‘𝑠)) · (♯‘(∪ 𝐴 ∩ ∩ 𝑠))))
105	88, 104	oveq12d 7438	. . . 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 2797	. . 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 2806	. 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → ((♯‘∪ 𝐴) − Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠))) = 0)
108	4, 37, 107	subeq0d 11678	1 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑠 ∈ (𝒫 𝐴 ∖ {∅})(( -1↑((♯‘𝑠) − 1)) · (♯‘∩ 𝑠)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1569 ∈ wcel 2144 ≠ wne 2955 Vcvv 3450 ⊆ wss 3895 ∖ cdif 4045 ∩ cin 4089 ∪ cun 4149 ∅c0 4278 𝒫 cpw 4556 {csn 4583 ∪ cuni 4866 ∩ cint 4906 ‘cfv 6538 (class class class)co 7420 Fincfn 8973 ℂcc 11198 0cc0 11200 1c1 11201 + caddc 11203 · cmul 11205 − cmin 11541 -cneg 11542 ℕcn 12335 ℕ₀cn0 12606 ↑cexp 14204 ♯chash 14474 Σcsu 15853

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-rep 5231 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-inf2 9642 ax-cnex 11256 ax-resscn 11257 ax-1cn 11258 ax-icn 11259 ax-addcl 11260 ax-addrcl 11261 ax-mulcl 11262 ax-mulrcl 11263 ax-mulcom 11264 ax-addass 11265 ax-mulass 11266 ax-distr 11267 ax-i2m1 11268 ax-1ne0 11269 ax-1rid 11270 ax-rnegex 11271 ax-rrecex 11272 ax-cnre 11273 ax-pre-lttri 11274 ax-pre-lttrn 11275 ax-pre-ltadd 11276 ax-pre-mulgt0 11277 ax-pre-sup 11278

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rmo 3365 df-reu 3366 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-pss 4014 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-int 4907 df-iun 4952 df-br 5103 df-opab 5167 df-mpt 5186 df-tr 5212 df-id 5545 df-eprel 5550 df-po 5558 df-so 5559 df-fr 5603 df-se 5604 df-we 5605 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7377 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7878 df-1st 8001 df-2nd 8002 df-frecs 8299 df-wrecs 8330 df-recs 8379 df-rdg 8418 df-1o 8476 df-oadd 8480 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-fin 8977 df-sup 9434 df-oi 9504 df-dju 9982 df-card 10020 df-pnf 11345 df-mnf 11346 df-xr 11347 df-ltxr 11348 df-le 11349 df-sub 11543 df-neg 11544 df-div 11974 df-nn 12336 df-2 12405 df-3 12406 df-n0 12607 df-z 12694 df-uz 12966 df-rp 13121 df-fz 13640 df-fzo 13789 df-seq 14145 df-exp 14205 df-hash 14475 df-cj 15266 df-re 15267 df-im 15268 df-sqrt 15402 df-abs 15403 df-clim 15655 df-sum 15854

This theorem is used by: incexc2 16007

W3C validator