hasheuni - Mathbox for Thierry Arnoux

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Theorem hasheuni 33152

Description: The cardinality of a disjoint union, not necessarily finite. cf. hashuni 15774. (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 5127	. . . . . . . 8 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝑥
2		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ Fin
3		nfv 1917	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ⊆ Fin
4	1, 2, 3	nf3an 1904	. . . . . . 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 32056	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
9		simpl 483	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
10		dfss3 3970	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
11		hashcl 14318	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
12		nn0re 12483	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ ℕ₀ → (♯‘𝑥) ∈ ℝ)
13		nn0ge0 12499	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ ℕ₀ → 0 ≤ (♯‘𝑥))
14		elrege0 13433	. . . . . . . . . . . . . 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 3083	. . . . . . . . . . 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 3248	. . . . . . . . 9 ⊢ ((𝐴 ⊆ Fin ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,)+∞))
20	19	adantll 712	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,)+∞))
21	9, 20	esumpfinval 33142	. . . . . . 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 2775	. . . . 5 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
24	23	3adant1l 1176	. . . 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 7732	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
27	10	notbii 319	. . . . . . . . . 10 ⊢ (¬ 𝐴 ⊆ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
28		rexnal 3100	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
29	27, 28	bitr4i 277	. . . . . . . . 9 ⊢ (¬ 𝐴 ⊆ Fin ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin)
30		elssuni 4941	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
31		ssfi 9175	. . . . . . . . . . . . 13 ⊢ ((∪ 𝐴 ∈ Fin ∧ 𝑥 ⊆ ∪ 𝐴) → 𝑥 ∈ Fin)
32	31	expcom 414	. . . . . . . . . . . 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 3148	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
36	29, 35	sylbi 216	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ¬ ∪ 𝐴 ∈ Fin)
37		hashinf 14297	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ V ∧ ¬ ∪ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = +∞)
38	26, 36, 37	syl2an 596	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = +∞)
39		vex 3478	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
40		hashinf 14297	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ Fin) → (♯‘𝑥) = +∞)
41	39, 40	mpan 688	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ Fin → (♯‘𝑥) = +∞)
42	41	reximi 3084	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
43	29, 42	sylbi 216	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
44		nfv 1917	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
45		nfre1 3282	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞
46	44, 45	nfan 1902	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
47		simpl 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → 𝐴 ∈ 𝑉)
48		hashf2 33151	. . . . . . . . . . 11 ⊢ ♯:V⟶(0[,]+∞)
49		ffvelcdm 7083	. . . . . . . . . . 11 ⊢ ((♯:V⟶(0[,]+∞) ∧ 𝑥 ∈ V) → (♯‘𝑥) ∈ (0[,]+∞))
50	48, 39, 49	mp2an 690	. . . . . . . . . 10 ⊢ (♯‘𝑥) ∈ (0[,]+∞)
51	50	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,]+∞))
52		simpr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
53	46, 47, 51, 52	esumpinfval 33140	. . . . . . . 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 2775	. . . . . 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 1177	. . . 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 811	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
60		pwfi 9180	. . . . . . 7 ⊢ (∪ 𝐴 ∈ Fin ↔ 𝒫 ∪ 𝐴 ∈ Fin)
61		pwuni 4949	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
62		ssfi 9175	. . . . . . . 8 ⊢ ((𝒫 ∪ 𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → 𝐴 ∈ Fin)
63	61, 62	mpan2 689	. . . . . . 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 596	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = +∞)
67		nftru 1806	. . . . . . . . 9 ⊢ Ⅎ𝑥⊤
68		unrab 4305	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)}
69		exmid 893	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)
70	69	rgenw 3065	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ 𝐴 ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)
71		rabid2 3464	. . . . . . . . . . . 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 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 33102	. . . . . . . 8 ⊢ (⊤ → Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})(♯‘𝑥) = Σ^𝑥 ∈ 𝐴(♯‘𝑥))
76	75	mptru 1548	. . . . . . 7 ⊢ Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})(♯‘𝑥) = Σ^𝑥 ∈ 𝐴(♯‘𝑥)
77		nfrab1 3451	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}
78		nfrab1 3451	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}
79		rabexg 5331	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ V)
80		rabexg 5331	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ V)
81		rabnc 4387	. . . . . . . . 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 33120	. . . . . . 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 481	. . . . 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 1917	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin)
89	80	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ V)
90		simpr 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
91		dfrab3 4309	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} = (𝐴 ∩ {𝑥 ∣ (♯‘𝑥) = 0})
92		hasheq0 14325	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
93	39, 92	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
94	93	abbii 2802	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∣ (♯‘𝑥) = 0} = {𝑥 ∣ 𝑥 = ∅}
95		df-sn 4629	. . . . . . . . . . . . . 14 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
96	94, 95	eqtr4i 2763	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ (♯‘𝑥) = 0} = {∅}
97	96	ineq2i 4209	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑥 ∣ (♯‘𝑥) = 0}) = (𝐴 ∩ {∅})
98	91, 97	eqtri 2760	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} = (𝐴 ∩ {∅})
99		snfi 9046	. . . . . . . . . . . 12 ⊢ {∅} ∈ Fin
100		inss2 4229	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {∅}) ⊆ {∅}
101		ssfi 9175	. . . . . . . . . . . 12 ⊢ (({∅} ∈ Fin ∧ (𝐴 ∩ {∅}) ⊆ {∅}) → (𝐴 ∩ {∅}) ∈ Fin)
102	99, 100, 101	mp2an 690	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {∅}) ∈ Fin
103	98, 102	eqeltri 2829	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin
104	103	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin)
105		difinf 9318	. . . . . . . . 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 4311	. . . . . . . . 9 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}
108	107	eleq1i 2824	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ Fin)
109	106, 108	sylnib 327	. . . . . . 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 485	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})
113		rabid 3452	. . . . . . . . . . 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 496	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → ¬ (♯‘𝑥) = 0)
116	93	biimpri 227	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
117	116	necon3bi 2967	. . . . . . . . 9 ⊢ (¬ (♯‘𝑥) = 0 → 𝑥 ≠ ∅)
118	115, 117	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 𝑥 ≠ ∅)
119		hashge1 14351	. . . . . . . 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 11275	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
122	121	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 1 ∈ ℝ^*)
123		0lt1 11738	. . . . . . . 8 ⊢ 0 < 1
124	123	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 0 < 1)
125	88, 78, 89, 109, 110, 120, 122, 124	esumpinfsum 33144	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} (♯‘𝑥) = +∞)
126	125	oveq2d 7427	. . . . 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 13409	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ^*
128	79	adantr 481	. . . . . . . 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 3146	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞))
131	77	esumcl 33097	. . . . . . . 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 3980	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ ℝ^)
134		xrge0neqmnf 13431	. . . . . . 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 13207	. . . . . 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 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 713	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
141	59, 140	pm2.61dan 811	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 396 ∨ wo 845 ∧ w3a 1087 = wceq 1541 ⊤wtru 1542 ∈ wcel 2106 {cab 2709 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3432 Vcvv 3474 ∖ cdif 3945 ∪ cun 3946 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 𝒫 cpw 4602 {csn 4628 ∪ cuni 4908 Disj wdisj 5113 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 Fincfn 8941 ℝcr 11111 0cc0 11112 1c1 11113 +∞cpnf 11247 -∞cmnf 11248 ℝ^*cxr 11249 < clt 11250 ≤ cle 11251 ℕ₀cn0 12474 +_𝑒 cxad 13092 [,)cico 13328 [,]cicc 13329 ♯chash 14292 Σcsu 15634 Σ^*cesum 33094

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 ax-inf2 9638 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190 ax-addf 11191 ax-mulf 11192

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-disj 5114 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-se 5632 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-of 7672 df-om 7858 df-1st 7977 df-2nd 7978 df-supp 8149 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-er 8705 df-map 8824 df-pm 8825 df-ixp 8894 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-fsupp 9364 df-fi 9408 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-3 12278 df-4 12279 df-5 12280 df-6 12281 df-7 12282 df-8 12283 df-9 12284 df-n0 12475 df-xnn0 12547 df-z 12561 df-dec 12680 df-uz 12825 df-q 12935 df-rp 12977 df-xneg 13094 df-xadd 13095 df-xmul 13096 df-ioo 13330 df-ioc 13331 df-ico 13332 df-icc 13333 df-fz 13487 df-fzo 13630 df-fl 13759 df-mod 13837 df-seq 13969 df-exp 14030 df-fac 14236 df-bc 14265 df-hash 14293 df-shft 15016 df-cj 15048 df-re 15049 df-im 15050 df-sqrt 15184 df-abs 15185 df-limsup 15417 df-clim 15434 df-rlim 15435 df-sum 15635 df-ef 16013 df-sin 16015 df-cos 16016 df-pi 16018 df-struct 17082 df-sets 17099 df-slot 17117 df-ndx 17129 df-base 17147 df-ress 17176 df-plusg 17212 df-mulr 17213 df-starv 17214 df-sca 17215 df-vsca 17216 df-ip 17217 df-tset 17218 df-ple 17219 df-ds 17221 df-unif 17222 df-hom 17223 df-cco 17224 df-rest 17370 df-topn 17371 df-0g 17389 df-gsum 17390 df-topgen 17391 df-pt 17392 df-prds 17395 df-ordt 17449 df-xrs 17450 df-qtop 17455 df-imas 17456 df-xps 17458 df-mre 17532 df-mrc 17533 df-acs 17535 df-ps 18521 df-tsr 18522 df-plusf 18562 df-mgm 18563 df-sgrp 18612 df-mnd 18628 df-mhm 18673 df-submnd 18674 df-grp 18824 df-minusg 18825 df-sbg 18826 df-mulg 18953 df-subg 19005 df-cntz 19183 df-cmn 19652 df-abl 19653 df-mgp 19990 df-ur 20007 df-ring 20060 df-cring 20061 df-subrg 20321 df-abv 20429 df-lmod 20477 df-scaf 20478 df-sra 20791 df-rgmod 20792 df-psmet 20942 df-xmet 20943 df-met 20944 df-bl 20945 df-mopn 20946 df-fbas 20947 df-fg 20948 df-cnfld 20951 df-top 22403 df-topon 22420 df-topsp 22442 df-bases 22456 df-cld 22530 df-ntr 22531 df-cls 22532 df-nei 22609 df-lp 22647 df-perf 22648 df-cn 22738 df-cnp 22739 df-haus 22826 df-tx 23073 df-hmeo 23266 df-fil 23357 df-fm 23449 df-flim 23450 df-flf 23451 df-tmd 23583 df-tgp 23584 df-tsms 23638 df-trg 23671 df-xms 23833 df-ms 23834 df-tms 23835 df-nm 24098 df-ngp 24099 df-nrg 24101 df-nlm 24102 df-ii 24400 df-cncf 24401 df-limc 25390 df-dv 25391 df-log 26072 df-esum 33095

This theorem is referenced by: cntmeas 33293

W3C validator