hasheuni - Mathbox for Thierry Arnoux

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Theorem hasheuni 34082

Description: The cardinality of a disjoint union, not necessarily finite. cf. hashuni 15799. (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 5091	. . . . . . . 8 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝑥
2		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ Fin
3		nfv 1914	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ⊆ Fin
4	1, 2, 3	nf3an 1901	. . . . . . 7 ⊢ Ⅎ𝑥(Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin)
5		simp2 1137	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		simp3 1138	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
7		simp1 1136	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Disj 𝑥 ∈ 𝐴 𝑥)
8	4, 5, 6, 7	hashunif 32738	. . . . . 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 3938	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
11		hashcl 14328	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
12		nn0re 12458	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ ℕ₀ → (♯‘𝑥) ∈ ℝ)
13		nn0ge0 12474	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ ℕ₀ → 0 ≤ (♯‘𝑥))
14		elrege0 13422	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ (0[,)+∞) ↔ ((♯‘𝑥) ∈ ℝ ∧ 0 ≤ (♯‘𝑥)))
15	12, 13, 14	sylanbrc 583	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) ∈ ℕ₀ → (♯‘𝑥) ∈ (0[,)+∞))
16	11, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ (0[,)+∞))
17	16	ralimi 3067	. . . . . . . . . . 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 714	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,)+∞))
21	9, 20	esumpfinval 34072	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
22	21	3adant1 1130	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
23	8, 22	eqtr4d 2768	. . . . 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 1118	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
26		uniexg 7719	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
27	10	notbii 320	. . . . . . . . . 10 ⊢ (¬ 𝐴 ⊆ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
28		rexnal 3083	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
29	27, 28	bitr4i 278	. . . . . . . . 9 ⊢ (¬ 𝐴 ⊆ Fin ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin)
30		elssuni 4904	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
31		ssfi 9143	. . . . . . . . . . . . 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 3128	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
36	29, 35	sylbi 217	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ¬ ∪ 𝐴 ∈ Fin)
37		hashinf 14307	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ V ∧ ¬ ∪ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = +∞)
38	26, 36, 37	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = +∞)
39		vex 3454	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
40		hashinf 14307	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ Fin) → (♯‘𝑥) = +∞)
41	39, 40	mpan 690	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ Fin → (♯‘𝑥) = +∞)
42	41	reximi 3068	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
43	29, 42	sylbi 217	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
44		nfv 1914	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
45		nfre1 3263	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞
46	44, 45	nfan 1899	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
47		simpl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → 𝐴 ∈ 𝑉)
48		hashf2 34081	. . . . . . . . . . 11 ⊢ ♯:V⟶(0[,]+∞)
49		ffvelcdm 7056	. . . . . . . . . . 11 ⊢ ((♯:V⟶(0[,]+∞) ∧ 𝑥 ∈ V) → (♯‘𝑥) ∈ (0[,]+∞))
50	48, 39, 49	mp2an 692	. . . . . . . . . 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 34070	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = +∞)
54	43, 53	sylan2 593	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = +∞)
55	38, 54	eqtr4d 2768	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
56	55	3adant2 1131	. . . . 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 1118	. . 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 9275	. . . . . . 7 ⊢ (∪ 𝐴 ∈ Fin ↔ 𝒫 ∪ 𝐴 ∈ Fin)
61		pwuni 4912	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
62		ssfi 9143	. . . . . . . 8 ⊢ ((𝒫 ∪ 𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → 𝐴 ∈ Fin)
63	61, 62	mpan2 691	. . . . . . 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 596	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = +∞)
67		nftru 1804	. . . . . . . . 9 ⊢ Ⅎ𝑥⊤
68		unrab 4281	. . . . . . . . . . 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 3049	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ 𝐴 ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)
71		rabid2 3442	. . . . . . . . . . . 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 2756	. . . . . . . . . 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 34032	. . . . . . . 8 ⊢ (⊤ → Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})(♯‘𝑥) = Σ^𝑥 ∈ 𝐴(♯‘𝑥))
76	75	mptru 1547	. . . . . . 7 ⊢ Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})(♯‘𝑥) = Σ^𝑥 ∈ 𝐴(♯‘𝑥)
77		nfrab1 3429	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}
78		nfrab1 3429	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}
79		rabexg 5295	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ V)
80		rabexg 5295	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ V)
81		rabnc 4357	. . . . . . . . 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 34050	. . . . . . 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 2779	. . . . . 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 1914	. . . . . . 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 4285	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} = (𝐴 ∩ {𝑥 ∣ (♯‘𝑥) = 0})
92		hasheq0 14335	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
93	39, 92	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
94	93	abbii 2797	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∣ (♯‘𝑥) = 0} = {𝑥 ∣ 𝑥 = ∅}
95		df-sn 4593	. . . . . . . . . . . . . 14 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
96	94, 95	eqtr4i 2756	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ (♯‘𝑥) = 0} = {∅}
97	96	ineq2i 4183	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑥 ∣ (♯‘𝑥) = 0}) = (𝐴 ∩ {∅})
98	91, 97	eqtri 2753	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} = (𝐴 ∩ {∅})
99		snfi 9017	. . . . . . . . . . . 12 ⊢ {∅} ∈ Fin
100		inss2 4204	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {∅}) ⊆ {∅}
101		ssfi 9143	. . . . . . . . . . . 12 ⊢ (({∅} ∈ Fin ∧ (𝐴 ∩ {∅}) ⊆ {∅}) → (𝐴 ∩ {∅}) ∈ Fin)
102	99, 100, 101	mp2an 692	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {∅}) ∈ Fin
103	98, 102	eqeltri 2825	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin
104	103	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin)
105		difinf 9267	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin) → ¬ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) ∈ Fin)
106	90, 104, 105	syl2anc 584	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) ∈ Fin)
107		notrab 4288	. . . . . . . . 9 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}
108	107	eleq1i 2820	. . . . . . . 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 3430	. . . . . . . . . . 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 2952	. . . . . . . . 9 ⊢ (¬ (♯‘𝑥) = 0 → 𝑥 ≠ ∅)
118	115, 117	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 𝑥 ≠ ∅)
119		hashge1 14361	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → 1 ≤ (♯‘𝑥))
120	111, 118, 119	syl2anc 584	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 1 ≤ (♯‘𝑥))
121		1xr 11240	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
122	121	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 1 ∈ ℝ^*)
123		0lt1 11707	. . . . . . . 8 ⊢ 0 < 1
124	123	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 0 < 1)
125	88, 78, 89, 109, 110, 120, 122, 124	esumpinfsum 34074	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} (♯‘𝑥) = +∞)
126	125	oveq2d 7406	. . . . 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 13398	. . . . . . 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 3126	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞))
131	77	esumcl 34027	. . . . . . . 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 584	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞))
133	127, 132	sselid 3947	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ ℝ^)
134		xrge0neqmnf 13420	. . . . . . 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 13193	. . . . . 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 584	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) +_𝑒 +∞) = +∞)
138	87, 126, 137	3eqtrd 2769	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = +∞)
139	66, 138	eqtr4d 2768	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
140	139	adantlr 715	. 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 206 ∧ wa 395 ∨ wo 847 ∧ w3a 1086 = wceq 1540 ⊤wtru 1541 ∈ wcel 2109 {cab 2708 ≠ wne 2926 ∀wral 3045 ∃wrex 3054 {crab 3408 Vcvv 3450 ∖ cdif 3914 ∪ cun 3915 ∩ cin 3916 ⊆ wss 3917 ∅c0 4299 𝒫 cpw 4566 {csn 4592 ∪ cuni 4874 Disj wdisj 5077 class class class wbr 5110 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 Fincfn 8921 ℝcr 11074 0cc0 11075 1c1 11076 +∞cpnf 11212 -∞cmnf 11213 ℝ^*cxr 11214 < clt 11215 ≤ cle 11216 ℕ₀cn0 12449 +_𝑒 cxad 13077 [,)cico 13315 [,]cicc 13316 ♯chash 14302 Σcsu 15659 Σ^*cesum 34024

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-inf2 9601 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153 ax-addf 11154 ax-mulf 11155

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-tp 4597 df-op 4599 df-uni 4875 df-int 4914 df-iun 4960 df-iin 4961 df-disj 5078 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-se 5595 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-isom 6523 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-of 7656 df-om 7846 df-1st 7971 df-2nd 7972 df-supp 8143 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-1o 8437 df-2o 8438 df-oadd 8441 df-er 8674 df-map 8804 df-pm 8805 df-ixp 8874 df-en 8922 df-dom 8923 df-sdom 8924 df-fin 8925 df-fsupp 9320 df-fi 9369 df-sup 9400 df-inf 9401 df-oi 9470 df-card 9899 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-4 12258 df-5 12259 df-6 12260 df-7 12261 df-8 12262 df-9 12263 df-n0 12450 df-xnn0 12523 df-z 12537 df-dec 12657 df-uz 12801 df-q 12915 df-rp 12959 df-xneg 13079 df-xadd 13080 df-xmul 13081 df-ioo 13317 df-ioc 13318 df-ico 13319 df-icc 13320 df-fz 13476 df-fzo 13623 df-fl 13761 df-mod 13839 df-seq 13974 df-exp 14034 df-fac 14246 df-bc 14275 df-hash 14303 df-shft 15040 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-limsup 15444 df-clim 15461 df-rlim 15462 df-sum 15660 df-ef 16040 df-sin 16042 df-cos 16043 df-pi 16045 df-struct 17124 df-sets 17141 df-slot 17159 df-ndx 17171 df-base 17187 df-ress 17208 df-plusg 17240 df-mulr 17241 df-starv 17242 df-sca 17243 df-vsca 17244 df-ip 17245 df-tset 17246 df-ple 17247 df-ds 17249 df-unif 17250 df-hom 17251 df-cco 17252 df-rest 17392 df-topn 17393 df-0g 17411 df-gsum 17412 df-topgen 17413 df-pt 17414 df-prds 17417 df-ordt 17471 df-xrs 17472 df-qtop 17477 df-imas 17478 df-xps 17480 df-mre 17554 df-mrc 17555 df-acs 17557 df-ps 18532 df-tsr 18533 df-plusf 18573 df-mgm 18574 df-sgrp 18653 df-mnd 18669 df-mhm 18717 df-submnd 18718 df-grp 18875 df-minusg 18876 df-sbg 18877 df-mulg 19007 df-subg 19062 df-cntz 19256 df-cmn 19719 df-abl 19720 df-mgp 20057 df-rng 20069 df-ur 20098 df-ring 20151 df-cring 20152 df-subrng 20462 df-subrg 20486 df-abv 20725 df-lmod 20775 df-scaf 20776 df-sra 21087 df-rgmod 21088 df-psmet 21263 df-xmet 21264 df-met 21265 df-bl 21266 df-mopn 21267 df-fbas 21268 df-fg 21269 df-cnfld 21272 df-top 22788 df-topon 22805 df-topsp 22827 df-bases 22840 df-cld 22913 df-ntr 22914 df-cls 22915 df-nei 22992 df-lp 23030 df-perf 23031 df-cn 23121 df-cnp 23122 df-haus 23209 df-tx 23456 df-hmeo 23649 df-fil 23740 df-fm 23832 df-flim 23833 df-flf 23834 df-tmd 23966 df-tgp 23967 df-tsms 24021 df-trg 24054 df-xms 24215 df-ms 24216 df-tms 24217 df-nm 24477 df-ngp 24478 df-nrg 24480 df-nlm 24481 df-ii 24777 df-cncf 24778 df-limc 25774 df-dv 25775 df-log 26472 df-esum 34025

This theorem is referenced by: cntmeas 34223

W3C validator