hasheuni - Mathbox for Thierry Arnoux

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Theorem hasheuni 34262

Description: The cardinality of a disjoint union, not necessarily finite. cf. hashuni 15761. (Contributed by Thierry Arnoux, 19-Nov-2016.) (Revised by Thierry Arnoux, 2-Jan-2017.) (Revised by Thierry Arnoux, 20-Jun-2017.)

**Assertion**
Ref	Expression
hasheuni	⊢ ((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))

Distinct variable groups: 𝑥,𝐴 𝑥,𝑉

**Proof of Theorem hasheuni**
Step	Hyp	Ref	Expression
1		nfdisj1 5081	. . . . . . . 8 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝑥
2		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ Fin
3		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ⊆ Fin
4	1, 2, 3	nf3an 1903	. . . . . . 7 ⊢ Ⅎ𝑥(Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin)
5		simp2 1138	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		simp3 1139	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
7		simp1 1137	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Disj 𝑥 ∈ 𝐴 𝑥)
8	4, 5, 6, 7	hashunif 32896	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
9		simpl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
10		dfss3 3924	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
11		hashcl 14291	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
12		nn0re 12422	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ ℕ₀ → (♯‘𝑥) ∈ ℝ)
13		nn0ge0 12438	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ ℕ₀ → 0 ≤ (♯‘𝑥))
14		elrege0 13382	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ (0[,)+∞) ↔ ((♯‘𝑥) ∈ ℝ ∧ 0 ≤ (♯‘𝑥)))
15	12, 13, 14	sylanbrc 584	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) ∈ ℕ₀ → (♯‘𝑥) ∈ (0[,)+∞))
16	11, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ (0[,)+∞))
17	16	ralimi 3075	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ Fin → ∀𝑥 ∈ 𝐴 (♯‘𝑥) ∈ (0[,)+∞))
18	10, 17	sylbi 217	. . . . . . . . . 10 ⊢ (𝐴 ⊆ Fin → ∀𝑥 ∈ 𝐴 (♯‘𝑥) ∈ (0[,)+∞))
19	18	r19.21bi 3230	. . . . . . . . 9 ⊢ ((𝐴 ⊆ Fin ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,)+∞))
20	19	adantll 715	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,)+∞))
21	9, 20	esumpfinval 34252	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
22	21	3adant1 1131	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
23	8, 22	eqtr4d 2775	. . . . 5 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
24	23	3adant1l 1178	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
25	24	3expa 1119	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
26		uniexg 7695	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
27	10	notbii 320	. . . . . . . . . 10 ⊢ (¬ 𝐴 ⊆ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
28		rexnal 3090	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
29	27, 28	bitr4i 278	. . . . . . . . 9 ⊢ (¬ 𝐴 ⊆ Fin ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin)
30		elssuni 4896	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
31		ssfi 9109	. . . . . . . . . . . . 13 ⊢ ((∪ 𝐴 ∈ Fin ∧ 𝑥 ⊆ ∪ 𝐴) → 𝑥 ∈ Fin)
32	31	expcom 413	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ ∪ 𝐴 → (∪ 𝐴 ∈ Fin → 𝑥 ∈ Fin))
33	32	con3d 152	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ ∪ 𝐴 → (¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin))
34	30, 33	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin))
35	34	rexlimiv 3132	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
36	29, 35	sylbi 217	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ¬ ∪ 𝐴 ∈ Fin)
37		hashinf 14270	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ V ∧ ¬ ∪ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = +∞)
38	26, 36, 37	syl2an 597	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = +∞)
39		vex 3446	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
40		hashinf 14270	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ Fin) → (♯‘𝑥) = +∞)
41	39, 40	mpan 691	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ Fin → (♯‘𝑥) = +∞)
42	41	reximi 3076	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
43	29, 42	sylbi 217	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
44		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
45		nfre1 3263	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞
46	44, 45	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
47		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → 𝐴 ∈ 𝑉)
48		hashf2 34261	. . . . . . . . . . 11 ⊢ ♯:V⟶(0[,]+∞)
49		ffvelcdm 7035	. . . . . . . . . . 11 ⊢ ((♯:V⟶(0[,]+∞) ∧ 𝑥 ∈ V) → (♯‘𝑥) ∈ (0[,]+∞))
50	48, 39, 49	mp2an 693	. . . . . . . . . 10 ⊢ (♯‘𝑥) ∈ (0[,]+∞)
51	50	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,]+∞))
52		simpr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
53	46, 47, 51, 52	esumpinfval 34250	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = +∞)
54	43, 53	sylan2 594	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = +∞)
55	38, 54	eqtr4d 2775	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
56	55	3adant2 1132	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ Fin ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
57	56	3adant1r 1179	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
58	57	3expa 1119	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
59	25, 58	pm2.61dan 813	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
60		pwfi 9231	. . . . . . 7 ⊢ (∪ 𝐴 ∈ Fin ↔ 𝒫 ∪ 𝐴 ∈ Fin)
61		pwuni 4903	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
62		ssfi 9109	. . . . . . . 8 ⊢ ((𝒫 ∪ 𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → 𝐴 ∈ Fin)
63	61, 62	mpan2 692	. . . . . . 7 ⊢ (𝒫 ∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin)
64	60, 63	sylbi 217	. . . . . 6 ⊢ (∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin)
65	64	con3i 154	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
66	26, 65, 37	syl2an 597	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = +∞)
67		nftru 1806	. . . . . . . . 9 ⊢ Ⅎ𝑥⊤
68		unrab 4269	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)}
69		exmid 895	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)
70	69	rgenw 3056	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ 𝐴 ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)
71		rabid2 3434	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)} ↔ ∀𝑥 ∈ 𝐴 ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0))
72	70, 71	mpbir 231	. . . . . . . . . . 11 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)}
73	68, 72	eqtr4i 2763	. . . . . . . . . 10 ⊢ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) = 𝐴
74	73	a1i 11	. . . . . . . . 9 ⊢ (⊤ → ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) = 𝐴)
75	67, 74	esumeq1d 34212	. . . . . . . 8 ⊢ (⊤ → Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})(♯‘𝑥) = Σ^𝑥 ∈ 𝐴(♯‘𝑥))
76	75	mptru 1549	. . . . . . 7 ⊢ Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})(♯‘𝑥) = Σ^𝑥 ∈ 𝐴(♯‘𝑥)
77		nfrab1 3421	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}
78		nfrab1 3421	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}
79		rabexg 5284	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ V)
80		rabexg 5284	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ V)
81		rabnc 4345	. . . . . . . . 9 ⊢ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∩ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) = ∅
82	81	a1i 11	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∩ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) = ∅)
83	50	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) → (♯‘𝑥) ∈ (0[,]+∞))
84	50	a1i 11	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → (♯‘𝑥) ∈ (0[,]+∞))
85	44, 77, 78, 79, 80, 82, 83, 84	esumsplit 34230	. . . . . . 7 ⊢ (𝐴 ∈ 𝑉 → Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})(♯‘𝑥) = (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) +_𝑒 Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} (♯‘𝑥)))
86	76, 85	eqtr3id 2786	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → Σ^𝑥 ∈ 𝐴(♯‘𝑥) = (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) +_𝑒 Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} (♯‘𝑥)))
87	86	adantr 480	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^𝑥 ∈ 𝐴(♯‘𝑥) = (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) +_𝑒 Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} (♯‘𝑥)))
88		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin)
89	80	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ V)
90		simpr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
91		dfrab3 4273	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} = (𝐴 ∩ {𝑥 ∣ (♯‘𝑥) = 0})
92		hasheq0 14298	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
93	39, 92	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
94	93	abbii 2804	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∣ (♯‘𝑥) = 0} = {𝑥 ∣ 𝑥 = ∅}
95		df-sn 4583	. . . . . . . . . . . . . 14 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
96	94, 95	eqtr4i 2763	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ (♯‘𝑥) = 0} = {∅}
97	96	ineq2i 4171	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑥 ∣ (♯‘𝑥) = 0}) = (𝐴 ∩ {∅})
98	91, 97	eqtri 2760	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} = (𝐴 ∩ {∅})
99		snfi 8992	. . . . . . . . . . . 12 ⊢ {∅} ∈ Fin
100		inss2 4192	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {∅}) ⊆ {∅}
101		ssfi 9109	. . . . . . . . . . . 12 ⊢ (({∅} ∈ Fin ∧ (𝐴 ∩ {∅}) ⊆ {∅}) → (𝐴 ∩ {∅}) ∈ Fin)
102	99, 100, 101	mp2an 693	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {∅}) ∈ Fin
103	98, 102	eqeltri 2833	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin
104	103	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin)
105		difinf 9223	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin) → ¬ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) ∈ Fin)
106	90, 104, 105	syl2anc 585	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) ∈ Fin)
107		notrab 4276	. . . . . . . . 9 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}
108	107	eleq1i 2828	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ Fin)
109	106, 108	sylnib 328	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ Fin)
110	50	a1i 11	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → (♯‘𝑥) ∈ (0[,]+∞))
111	39	a1i 11	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 𝑥 ∈ V)
112		simpr 484	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})
113		rabid 3422	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ↔ (𝑥 ∈ 𝐴 ∧ ¬ (♯‘𝑥) = 0))
114	112, 113	sylib 218	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → (𝑥 ∈ 𝐴 ∧ ¬ (♯‘𝑥) = 0))
115	114	simprd 495	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → ¬ (♯‘𝑥) = 0)
116	93	biimpri 228	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
117	116	necon3bi 2959	. . . . . . . . 9 ⊢ (¬ (♯‘𝑥) = 0 → 𝑥 ≠ ∅)
118	115, 117	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 𝑥 ≠ ∅)
119		hashge1 14324	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → 1 ≤ (♯‘𝑥))
120	111, 118, 119	syl2anc 585	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 1 ≤ (♯‘𝑥))
121		1xr 11203	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
122	121	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 1 ∈ ℝ^*)
123		0lt1 11671	. . . . . . . 8 ⊢ 0 < 1
124	123	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 0 < 1)
125	88, 78, 89, 109, 110, 120, 122, 124	esumpinfsum 34254	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} (♯‘𝑥) = +∞)
126	125	oveq2d 7384	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) +_𝑒 Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} (♯‘𝑥)) = (Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) +_𝑒 +∞))
127		iccssxr 13358	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ^*
128	79	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ V)
129	50	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) → (♯‘𝑥) ∈ (0[,]+∞))
130	129	ralrimiva 3130	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞))
131	77	esumcl 34207	. . . . . . . 8 ⊢ (({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ V ∧ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞)) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞))
132	128, 130, 131	syl2anc 585	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞))
133	127, 132	sselid 3933	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ ℝ^)
134		xrge0neqmnf 13380	. . . . . . 7 ⊢ (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞) → Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ≠ -∞)
135	132, 134	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ≠ -∞)
136		xaddpnf1 13153	. . . . . 6 ⊢ ((Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ ℝ^ ∧ Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ≠ -∞) → (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) +_𝑒 +∞) = +∞)
137	133, 135, 136	syl2anc 585	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) +_𝑒 +∞) = +∞)
138	87, 126, 137	3eqtrd 2776	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = +∞)
139	66, 138	eqtr4d 2775	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
140	139	adantlr 716	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
141	59, 140	pm2.61dan 813	1 ⊢ ((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1542 ⊤wtru 1543 ∈ wcel 2114 {cab 2715 ≠ wne 2933 ∀wral 3052 ∃wrex 3062 {crab 3401 Vcvv 3442 ∖ cdif 3900 ∪ cun 3901 ∩ cin 3902 ⊆ wss 3903 ∅c0 4287 𝒫 cpw 4556 {csn 4582 ∪ cuni 4865 Disj wdisj 5067 class class class wbr 5100 ⟶wf 6496 ‘cfv 6500 (class class class)co 7368 Fincfn 8895 ℝcr 11037 0cc0 11038 1c1 11039 +∞cpnf 11175 -∞cmnf 11176 ℝ^*cxr 11177 < clt 11178 ≤ cle 11179 ℕ₀cn0 12413 +_𝑒 cxad 13036 [,)cico 13275 [,]cicc 13276 ♯chash 14265 Σcsu 15621 Σ^*cesum 34204

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-inf2 9562 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115 ax-pre-sup 11116 ax-addf 11117 ax-mulf 11118

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rmo 3352 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-tp 4587 df-op 4589 df-uni 4866 df-int 4905 df-iun 4950 df-iin 4951 df-disj 5068 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-se 5586 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-isom 6509 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-of 7632 df-om 7819 df-1st 7943 df-2nd 7944 df-supp 8113 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-1o 8407 df-2o 8408 df-oadd 8411 df-er 8645 df-map 8777 df-pm 8778 df-ixp 8848 df-en 8896 df-dom 8897 df-sdom 8898 df-fin 8899 df-fsupp 9277 df-fi 9326 df-sup 9357 df-inf 9358 df-oi 9427 df-card 9863 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-div 11807 df-nn 12158 df-2 12220 df-3 12221 df-4 12222 df-5 12223 df-6 12224 df-7 12225 df-8 12226 df-9 12227 df-n0 12414 df-xnn0 12487 df-z 12501 df-dec 12620 df-uz 12764 df-q 12874 df-rp 12918 df-xneg 13038 df-xadd 13039 df-xmul 13040 df-ioo 13277 df-ioc 13278 df-ico 13279 df-icc 13280 df-fz 13436 df-fzo 13583 df-fl 13724 df-mod 13802 df-seq 13937 df-exp 13997 df-fac 14209 df-bc 14238 df-hash 14266 df-shft 15002 df-cj 15034 df-re 15035 df-im 15036 df-sqrt 15170 df-abs 15171 df-limsup 15406 df-clim 15423 df-rlim 15424 df-sum 15622 df-ef 16002 df-sin 16004 df-cos 16005 df-pi 16007 df-struct 17086 df-sets 17103 df-slot 17121 df-ndx 17133 df-base 17149 df-ress 17170 df-plusg 17202 df-mulr 17203 df-starv 17204 df-sca 17205 df-vsca 17206 df-ip 17207 df-tset 17208 df-ple 17209 df-ds 17211 df-unif 17212 df-hom 17213 df-cco 17214 df-rest 17354 df-topn 17355 df-0g 17373 df-gsum 17374 df-topgen 17375 df-pt 17376 df-prds 17379 df-ordt 17434 df-xrs 17435 df-qtop 17440 df-imas 17441 df-xps 17443 df-mre 17517 df-mrc 17518 df-acs 17520 df-ps 18501 df-tsr 18502 df-plusf 18576 df-mgm 18577 df-sgrp 18656 df-mnd 18672 df-mhm 18720 df-submnd 18721 df-grp 18878 df-minusg 18879 df-sbg 18880 df-mulg 19010 df-subg 19065 df-cntz 19258 df-cmn 19723 df-abl 19724 df-mgp 20088 df-rng 20100 df-ur 20129 df-ring 20182 df-cring 20183 df-subrng 20491 df-subrg 20515 df-abv 20754 df-lmod 20825 df-scaf 20826 df-sra 21137 df-rgmod 21138 df-psmet 21313 df-xmet 21314 df-met 21315 df-bl 21316 df-mopn 21317 df-fbas 21318 df-fg 21319 df-cnfld 21322 df-top 22850 df-topon 22867 df-topsp 22889 df-bases 22902 df-cld 22975 df-ntr 22976 df-cls 22977 df-nei 23054 df-lp 23092 df-perf 23093 df-cn 23183 df-cnp 23184 df-haus 23271 df-tx 23518 df-hmeo 23711 df-fil 23802 df-fm 23894 df-flim 23895 df-flf 23896 df-tmd 24028 df-tgp 24029 df-tsms 24083 df-trg 24116 df-xms 24276 df-ms 24277 df-tms 24278 df-nm 24538 df-ngp 24539 df-nrg 24541 df-nlm 24542 df-ii 24838 df-cncf 24839 df-limc 25835 df-dv 25836 df-log 26533 df-esum 34205

This theorem is referenced by: cntmeas 34403

W3C validator