hasheuni - Mathbox for Thierry Arnoux

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Theorem hasheuni 33083

Description: The cardinality of a disjoint union, not necessarily finite. cf. hashuni 15772. (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 5128	. . . . . . . 8 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝑥
2		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ Fin
3		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ⊆ Fin
4	1, 2, 3	nf3an 1905	. . . . . . 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 32018	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
9		simpl 484	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
10		dfss3 3971	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
11		hashcl 14316	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
12		nn0re 12481	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ ℕ₀ → (♯‘𝑥) ∈ ℝ)
13		nn0ge0 12497	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ ℕ₀ → 0 ≤ (♯‘𝑥))
14		elrege0 13431	. . . . . . . . . . . . . 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 3084	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ Fin → ∀𝑥 ∈ 𝐴 (♯‘𝑥) ∈ (0[,)+∞))
18	10, 17	sylbi 216	. . . . . . . . . 10 ⊢ (𝐴 ⊆ Fin → ∀𝑥 ∈ 𝐴 (♯‘𝑥) ∈ (0[,)+∞))
19	18	r19.21bi 3249	. . . . . . . . 9 ⊢ ((𝐴 ⊆ Fin ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,)+∞))
20	19	adantll 713	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,)+∞))
21	9, 20	esumpfinval 33073	. . . . . . 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 2776	. . . . 5 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
24	23	3adant1l 1177	. . . 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 7730	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
27	10	notbii 320	. . . . . . . . . 10 ⊢ (¬ 𝐴 ⊆ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
28		rexnal 3101	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
29	27, 28	bitr4i 278	. . . . . . . . 9 ⊢ (¬ 𝐴 ⊆ Fin ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin)
30		elssuni 4942	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
31		ssfi 9173	. . . . . . . . . . . . 13 ⊢ ((∪ 𝐴 ∈ Fin ∧ 𝑥 ⊆ ∪ 𝐴) → 𝑥 ∈ Fin)
32	31	expcom 415	. . . . . . . . . . . 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 3149	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
36	29, 35	sylbi 216	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ¬ ∪ 𝐴 ∈ Fin)
37		hashinf 14295	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ V ∧ ¬ ∪ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = +∞)
38	26, 36, 37	syl2an 597	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = +∞)
39		vex 3479	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
40		hashinf 14295	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ Fin) → (♯‘𝑥) = +∞)
41	39, 40	mpan 689	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ Fin → (♯‘𝑥) = +∞)
42	41	reximi 3085	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
43	29, 42	sylbi 216	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
44		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
45		nfre1 3283	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞
46	44, 45	nfan 1903	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
47		simpl 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → 𝐴 ∈ 𝑉)
48		hashf2 33082	. . . . . . . . . . 11 ⊢ ♯:V⟶(0[,]+∞)
49		ffvelcdm 7084	. . . . . . . . . . 11 ⊢ ((♯:V⟶(0[,]+∞) ∧ 𝑥 ∈ V) → (♯‘𝑥) ∈ (0[,]+∞))
50	48, 39, 49	mp2an 691	. . . . . . . . . 10 ⊢ (♯‘𝑥) ∈ (0[,]+∞)
51	50	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,]+∞))
52		simpr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
53	46, 47, 51, 52	esumpinfval 33071	. . . . . . . 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 2776	. . . . . 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 1178	. . . 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 812	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
60		pwfi 9178	. . . . . . 7 ⊢ (∪ 𝐴 ∈ Fin ↔ 𝒫 ∪ 𝐴 ∈ Fin)
61		pwuni 4950	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
62		ssfi 9173	. . . . . . . 8 ⊢ ((𝒫 ∪ 𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → 𝐴 ∈ Fin)
63	61, 62	mpan2 690	. . . . . . 7 ⊢ (𝒫 ∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin)
64	60, 63	sylbi 216	. . . . . 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 1807	. . . . . . . . 9 ⊢ Ⅎ𝑥⊤
68		unrab 4306	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)}
69		exmid 894	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)
70	69	rgenw 3066	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ 𝐴 ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)
71		rabid2 3465	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)} ↔ ∀𝑥 ∈ 𝐴 ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0))
72	70, 71	mpbir 230	. . . . . . . . . . 11 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)}
73	68, 72	eqtr4i 2764	. . . . . . . . . 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 33033	. . . . . . . 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 3452	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}
78		nfrab1 3452	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}
79		rabexg 5332	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ V)
80		rabexg 5332	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ V)
81		rabnc 4388	. . . . . . . . 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 33051	. . . . . . 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 2787	. . . . . 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 482	. . . . 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 1918	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin)
89	80	adantr 482	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ V)
90		simpr 486	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
91		dfrab3 4310	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} = (𝐴 ∩ {𝑥 ∣ (♯‘𝑥) = 0})
92		hasheq0 14323	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
93	39, 92	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
94	93	abbii 2803	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∣ (♯‘𝑥) = 0} = {𝑥 ∣ 𝑥 = ∅}
95		df-sn 4630	. . . . . . . . . . . . . 14 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
96	94, 95	eqtr4i 2764	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ (♯‘𝑥) = 0} = {∅}
97	96	ineq2i 4210	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑥 ∣ (♯‘𝑥) = 0}) = (𝐴 ∩ {∅})
98	91, 97	eqtri 2761	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} = (𝐴 ∩ {∅})
99		snfi 9044	. . . . . . . . . . . 12 ⊢ {∅} ∈ Fin
100		inss2 4230	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {∅}) ⊆ {∅}
101		ssfi 9173	. . . . . . . . . . . 12 ⊢ (({∅} ∈ Fin ∧ (𝐴 ∩ {∅}) ⊆ {∅}) → (𝐴 ∩ {∅}) ∈ Fin)
102	99, 100, 101	mp2an 691	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {∅}) ∈ Fin
103	98, 102	eqeltri 2830	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin
104	103	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin)
105		difinf 9316	. . . . . . . . 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 4312	. . . . . . . . 9 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}
108	107	eleq1i 2825	. . . . . . . 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 486	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})
113		rabid 3453	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ↔ (𝑥 ∈ 𝐴 ∧ ¬ (♯‘𝑥) = 0))
114	112, 113	sylib 217	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → (𝑥 ∈ 𝐴 ∧ ¬ (♯‘𝑥) = 0))
115	114	simprd 497	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → ¬ (♯‘𝑥) = 0)
116	93	biimpri 227	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
117	116	necon3bi 2968	. . . . . . . . 9 ⊢ (¬ (♯‘𝑥) = 0 → 𝑥 ≠ ∅)
118	115, 117	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 𝑥 ≠ ∅)
119		hashge1 14349	. . . . . . . 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 11273	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
122	121	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 1 ∈ ℝ^*)
123		0lt1 11736	. . . . . . . 8 ⊢ 0 < 1
124	123	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 0 < 1)
125	88, 78, 89, 109, 110, 120, 122, 124	esumpinfsum 33075	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} (♯‘𝑥) = +∞)
126	125	oveq2d 7425	. . . . 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 13407	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ^*
128	79	adantr 482	. . . . . . . 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 3147	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞))
131	77	esumcl 33028	. . . . . . . 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 3981	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ ℝ^)
134		xrge0neqmnf 13429	. . . . . . 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 13205	. . . . . 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 2777	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = +∞)
139	66, 138	eqtr4d 2776	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
140	139	adantlr 714	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
141	59, 140	pm2.61dan 812	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 205 ∧ wa 397 ∨ wo 846 ∧ w3a 1088 = wceq 1542 ⊤wtru 1543 ∈ wcel 2107 {cab 2710 ≠ wne 2941 ∀wral 3062 ∃wrex 3071 {crab 3433 Vcvv 3475 ∖ cdif 3946 ∪ cun 3947 ∩ cin 3948 ⊆ wss 3949 ∅c0 4323 𝒫 cpw 4603 {csn 4629 ∪ cuni 4909 Disj wdisj 5114 class class class wbr 5149 ⟶wf 6540 ‘cfv 6544 (class class class)co 7409 Fincfn 8939 ℝcr 11109 0cc0 11110 1c1 11111 +∞cpnf 11245 -∞cmnf 11246 ℝ^*cxr 11247 < clt 11248 ≤ cle 11249 ℕ₀cn0 12472 +_𝑒 cxad 13090 [,)cico 13326 [,]cicc 13327 ♯chash 14290 Σcsu 15632 Σ^*cesum 33025

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-inf2 9636 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188 ax-addf 11189 ax-mulf 11190

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-tp 4634 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-iin 5001 df-disj 5115 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-se 5633 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-isom 6553 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-of 7670 df-om 7856 df-1st 7975 df-2nd 7976 df-supp 8147 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-2o 8467 df-oadd 8470 df-er 8703 df-map 8822 df-pm 8823 df-ixp 8892 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-fsupp 9362 df-fi 9406 df-sup 9437 df-inf 9438 df-oi 9505 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-4 12277 df-5 12278 df-6 12279 df-7 12280 df-8 12281 df-9 12282 df-n0 12473 df-xnn0 12545 df-z 12559 df-dec 12678 df-uz 12823 df-q 12933 df-rp 12975 df-xneg 13092 df-xadd 13093 df-xmul 13094 df-ioo 13328 df-ioc 13329 df-ico 13330 df-icc 13331 df-fz 13485 df-fzo 13628 df-fl 13757 df-mod 13835 df-seq 13967 df-exp 14028 df-fac 14234 df-bc 14263 df-hash 14291 df-shft 15014 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-limsup 15415 df-clim 15432 df-rlim 15433 df-sum 15633 df-ef 16011 df-sin 16013 df-cos 16014 df-pi 16016 df-struct 17080 df-sets 17097 df-slot 17115 df-ndx 17127 df-base 17145 df-ress 17174 df-plusg 17210 df-mulr 17211 df-starv 17212 df-sca 17213 df-vsca 17214 df-ip 17215 df-tset 17216 df-ple 17217 df-ds 17219 df-unif 17220 df-hom 17221 df-cco 17222 df-rest 17368 df-topn 17369 df-0g 17387 df-gsum 17388 df-topgen 17389 df-pt 17390 df-prds 17393 df-ordt 17447 df-xrs 17448 df-qtop 17453 df-imas 17454 df-xps 17456 df-mre 17530 df-mrc 17531 df-acs 17533 df-ps 18519 df-tsr 18520 df-plusf 18560 df-mgm 18561 df-sgrp 18610 df-mnd 18626 df-mhm 18671 df-submnd 18672 df-grp 18822 df-minusg 18823 df-sbg 18824 df-mulg 18951 df-subg 19003 df-cntz 19181 df-cmn 19650 df-abl 19651 df-mgp 19988 df-ur 20005 df-ring 20058 df-cring 20059 df-subrg 20317 df-abv 20425 df-lmod 20473 df-scaf 20474 df-sra 20785 df-rgmod 20786 df-psmet 20936 df-xmet 20937 df-met 20938 df-bl 20939 df-mopn 20940 df-fbas 20941 df-fg 20942 df-cnfld 20945 df-top 22396 df-topon 22413 df-topsp 22435 df-bases 22449 df-cld 22523 df-ntr 22524 df-cls 22525 df-nei 22602 df-lp 22640 df-perf 22641 df-cn 22731 df-cnp 22732 df-haus 22819 df-tx 23066 df-hmeo 23259 df-fil 23350 df-fm 23442 df-flim 23443 df-flf 23444 df-tmd 23576 df-tgp 23577 df-tsms 23631 df-trg 23664 df-xms 23826 df-ms 23827 df-tms 23828 df-nm 24091 df-ngp 24092 df-nrg 24094 df-nlm 24095 df-ii 24393 df-cncf 24394 df-limc 25383 df-dv 25384 df-log 26065 df-esum 33026

This theorem is referenced by: cntmeas 33224

W3C validator