hasheuni - Mathbox for Thierry Arnoux

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Theorem hasheuni 30147

Description: The cardinality of a disjoint union, not necessarily finite. cf. hashuni 14558. (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 4633	. . . . . . . 8 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝑥
2		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ Fin
3		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ⊆ Fin
4	1, 2, 3	nf3an 1831	. . . . . . 7 ⊢ Ⅎ𝑥(Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin)
5		simp2 1062	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		simp3 1063	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
7		simp1 1061	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Disj 𝑥 ∈ 𝐴 𝑥)
8	4, 5, 6, 7	hashunif 29562	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (#‘𝑥))
9		simpl 473	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
10		dfss3 3592	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
11		hashcl 13147	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → (#‘𝑥) ∈ ℕ₀)
12		nn0re 11301	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑥) ∈ ℕ₀ → (#‘𝑥) ∈ ℝ)
13		nn0ge0 11318	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑥) ∈ ℕ₀ → 0 ≤ (#‘𝑥))
14		elrege0 12278	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑥) ∈ (0[,)+∞) ↔ ((#‘𝑥) ∈ ℝ ∧ 0 ≤ (#‘𝑥)))
15	12, 13, 14	sylanbrc 698	. . . . . . . . . . . . 13 ⊢ ((#‘𝑥) ∈ ℕ₀ → (#‘𝑥) ∈ (0[,)+∞))
16	11, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Fin → (#‘𝑥) ∈ (0[,)+∞))
17	16	ralimi 2952	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ Fin → ∀𝑥 ∈ 𝐴 (#‘𝑥) ∈ (0[,)+∞))
18	10, 17	sylbi 207	. . . . . . . . . 10 ⊢ (𝐴 ⊆ Fin → ∀𝑥 ∈ 𝐴 (#‘𝑥) ∈ (0[,)+∞))
19	18	r19.21bi 2932	. . . . . . . . 9 ⊢ ((𝐴 ⊆ Fin ∧ 𝑥 ∈ 𝐴) → (#‘𝑥) ∈ (0[,)+∞))
20	19	adantll 750	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑥 ∈ 𝐴) → (#‘𝑥) ∈ (0[,)+∞))
21	9, 20	esumpfinval 30137	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(#‘𝑥) = Σ𝑥 ∈ 𝐴 (#‘𝑥))
22	21	3adant1 1079	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(#‘𝑥) = Σ𝑥 ∈ 𝐴 (#‘𝑥))
23	8, 22	eqtr4d 2659	. . . . 5 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
24	23	3adant1l 1318	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
25	24	3expa 1265	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
26		uniexg 6955	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
27	10	notbii 310	. . . . . . . . . 10 ⊢ (¬ 𝐴 ⊆ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
28		rexnal 2995	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
29	27, 28	bitr4i 267	. . . . . . . . 9 ⊢ (¬ 𝐴 ⊆ Fin ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin)
30		elssuni 4467	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
31		ssfi 8180	. . . . . . . . . . . . 13 ⊢ ((∪ 𝐴 ∈ Fin ∧ 𝑥 ⊆ ∪ 𝐴) → 𝑥 ∈ Fin)
32	31	expcom 451	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ ∪ 𝐴 → (∪ 𝐴 ∈ Fin → 𝑥 ∈ Fin))
33	32	con3d 148	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ ∪ 𝐴 → (¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin))
34	30, 33	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin))
35	34	rexlimiv 3027	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
36	29, 35	sylbi 207	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ¬ ∪ 𝐴 ∈ Fin)
37		hashinf 13122	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ V ∧ ¬ ∪ 𝐴 ∈ Fin) → (#‘∪ 𝐴) = +∞)
38	26, 36, 37	syl2an 494	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = +∞)
39		vex 3203	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
40		hashinf 13122	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ Fin) → (#‘𝑥) = +∞)
41	39, 40	mpan 706	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ Fin → (#‘𝑥) = +∞)
42	41	reximi 3011	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞)
43	29, 42	sylbi 207	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞)
44		nfv 1843	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
45		nfre1 3005	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞
46	44, 45	nfan 1828	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞)
47		simpl 473	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞) → 𝐴 ∈ 𝑉)
48		hashf2 30146	. . . . . . . . . . 11 ⊢ #:V⟶(0[,]+∞)
49		ffvelrn 6357	. . . . . . . . . . 11 ⊢ ((#:V⟶(0[,]+∞) ∧ 𝑥 ∈ V) → (#‘𝑥) ∈ (0[,]+∞))
50	48, 39, 49	mp2an 708	. . . . . . . . . 10 ⊢ (#‘𝑥) ∈ (0[,]+∞)
51	50	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞) ∧ 𝑥 ∈ 𝐴) → (#‘𝑥) ∈ (0[,]+∞))
52		simpr 477	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞) → ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞)
53	46, 47, 51, 52	esumpinfval 30135	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞) → Σ^*𝑥 ∈ 𝐴(#‘𝑥) = +∞)
54	43, 53	sylan2 491	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(#‘𝑥) = +∞)
55	38, 54	eqtr4d 2659	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
56	55	3adant2 1080	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ Fin ∧ ¬ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
57	56	3adant1r 1319	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin ∧ ¬ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
58	57	3expa 1265	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) ∧ ¬ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
59	25, 58	pm2.61dan 832	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
60		pwfi 8261	. . . . . . 7 ⊢ (∪ 𝐴 ∈ Fin ↔ 𝒫 ∪ 𝐴 ∈ Fin)
61		pwuni 4474	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
62		ssfi 8180	. . . . . . . 8 ⊢ ((𝒫 ∪ 𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → 𝐴 ∈ Fin)
63	61, 62	mpan2 707	. . . . . . 7 ⊢ (𝒫 ∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin)
64	60, 63	sylbi 207	. . . . . 6 ⊢ (∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin)
65	64	con3i 150	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
66	26, 65, 37	syl2an 494	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘∪ 𝐴) = +∞)
67		nftru 1730	. . . . . . . . 9 ⊢ Ⅎ𝑥⊤
68		unrab 3898	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0)}
69		exmid 431	. . . . . . . . . . . . 13 ⊢ ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0)
70	69	rgenw 2924	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ 𝐴 ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0)
71		rabid2 3118	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0)} ↔ ∀𝑥 ∈ 𝐴 ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0))
72	70, 71	mpbir 221	. . . . . . . . . . 11 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0)}
73	68, 72	eqtr4i 2647	. . . . . . . . . 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 30097	. . . . . . . 8 ⊢ (⊤ → Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0})(#‘𝑥) = Σ^𝑥 ∈ 𝐴(#‘𝑥))
76	75	trud 1493	. . . . . . 7 ⊢ Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0})(#‘𝑥) = Σ^𝑥 ∈ 𝐴(#‘𝑥)
77		nfrab1 3122	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}
78		nfrab1 3122	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}
79		rabexg 4812	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∈ V)
80		rabexg 4812	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0} ∈ V)
81		rabnc 3962	. . . . . . . . 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 30115	. . . . . . 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	syl5eqr 2670	. . . . . 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 1843	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin)
89	80	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0} ∈ V)
90		simpr 477	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
91		dfrab3 3902	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} = (𝐴 ∩ {𝑥 ∣ (#‘𝑥) = 0})
92		hasheq0 13154	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
93	39, 92	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑥) = 0 ↔ 𝑥 = ∅)
94	93	abbii 2739	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∣ (#‘𝑥) = 0} = {𝑥 ∣ 𝑥 = ∅}
95		df-sn 4178	. . . . . . . . . . . . . 14 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
96	94, 95	eqtr4i 2647	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ (#‘𝑥) = 0} = {∅}
97	96	ineq2i 3811	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑥 ∣ (#‘𝑥) = 0}) = (𝐴 ∩ {∅})
98	91, 97	eqtri 2644	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} = (𝐴 ∩ {∅})
99		snfi 8038	. . . . . . . . . . . 12 ⊢ {∅} ∈ Fin
100		inss2 3834	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {∅}) ⊆ {∅}
101		ssfi 8180	. . . . . . . . . . . 12 ⊢ (({∅} ∈ Fin ∧ (𝐴 ∩ {∅}) ⊆ {∅}) → (𝐴 ∩ {∅}) ∈ Fin)
102	99, 100, 101	mp2an 708	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {∅}) ∈ Fin
103	98, 102	eqeltri 2697	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∈ Fin
104	103	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∈ Fin)
105		difinf 8230	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∈ Fin) → ¬ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}) ∈ Fin)
106	90, 104, 105	syl2anc 693	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}) ∈ Fin)
107		notrab 3904	. . . . . . . . 9 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}
108	107	eleq1i 2692	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}) ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0} ∈ Fin)
109	106, 108	sylnib 318	. . . . . . 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 477	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0})
113		rabid 3116	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0} ↔ (𝑥 ∈ 𝐴 ∧ ¬ (#‘𝑥) = 0))
114	112, 113	sylib 208	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) → (𝑥 ∈ 𝐴 ∧ ¬ (#‘𝑥) = 0))
115	114	simprd 479	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) → ¬ (#‘𝑥) = 0)
116	93	biimpri 218	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (#‘𝑥) = 0)
117	116	necon3bi 2820	. . . . . . . . 9 ⊢ (¬ (#‘𝑥) = 0 → 𝑥 ≠ ∅)
118	115, 117	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) → 𝑥 ≠ ∅)
119		hashge1 13178	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → 1 ≤ (#‘𝑥))
120	111, 118, 119	syl2anc 693	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) → 1 ≤ (#‘𝑥))
121		1re 10039	. . . . . . . . 9 ⊢ 1 ∈ ℝ
122	121	rexri 10097	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
123	122	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 1 ∈ ℝ^*)
124		0lt1 10550	. . . . . . . 8 ⊢ 0 < 1
125	124	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 0 < 1)
126	88, 78, 89, 109, 110, 120, 123, 125	esumpinfsum 30139	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0} (#‘𝑥) = +∞)
127	126	oveq2d 6666	. . . . 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 +∞))
128		iccssxr 12256	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ^*
129	79	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∈ V)
130	50	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}) → (#‘𝑥) ∈ (0[,]+∞))
131	130	ralrimiva 2966	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ∈ (0[,]+∞))
132	77	esumcl 30092	. . . . . . . 8 ⊢ (({𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∈ V ∧ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ∈ (0[,]+∞)) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ∈ (0[,]+∞))
133	129, 131, 132	syl2anc 693	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ∈ (0[,]+∞))
134	128, 133	sseldi 3601	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ∈ ℝ^)
135		xrge0neqmnf 12276	. . . . . . 7 ⊢ (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ∈ (0[,]+∞) → Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ≠ -∞)
136	133, 135	syl 17	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ≠ -∞)
137		xaddpnf1 12057	. . . . . 6 ⊢ ((Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ∈ ℝ^ ∧ Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ≠ -∞) → (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) +_𝑒 +∞) = +∞)
138	134, 136, 137	syl2anc 693	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) +_𝑒 +∞) = +∞)
139	87, 127, 138	3eqtrd 2660	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ 𝐴(#‘𝑥) = +∞)
140	66, 139	eqtr4d 2659	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
141	140	adantlr 751	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ ¬ 𝐴 ∈ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
142	59, 141	pm2.61dan 832	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 196 ∨ wo 383 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ⊤wtru 1484 ∈ wcel 1990 {cab 2608 ≠ wne 2794 ∀wral 2912 ∃wrex 2913 {crab 2916 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ∩ cin 3573 ⊆ wss 3574 ∅c0 3915 𝒫 cpw 4158 {csn 4177 ∪ cuni 4436 Disj wdisj 4620 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 Fincfn 7955 ℝcr 9935 0cc0 9936 1c1 9937 +∞cpnf 10071 -∞cmnf 10072 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℕ₀cn0 11292 +_𝑒 cxad 11944 [,)cico 12177 [,]cicc 12178 #chash 13117 Σcsu 14416 Σ^*cesum 30089

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-disj 4621 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-of 6897 df-om 7066 df-1st 7168 df-2nd 7169 df-supp 7296 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-2o 7561 df-oadd 7564 df-er 7742 df-map 7859 df-pm 7860 df-ixp 7909 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-fsupp 8276 df-fi 8317 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 df-cda 8990 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-xnn0 11364 df-z 11378 df-dec 11494 df-uz 11688 df-q 11789 df-rp 11833 df-xneg 11946 df-xadd 11947 df-xmul 11948 df-ioo 12179 df-ioc 12180 df-ico 12181 df-icc 12182 df-fz 12327 df-fzo 12466 df-fl 12593 df-mod 12669 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 df-sin 14800 df-cos 14801 df-pi 14803 df-struct 15859 df-ndx 15860 df-slot 15861 df-base 15863 df-sets 15864 df-ress 15865 df-plusg 15954 df-mulr 15955 df-starv 15956 df-sca 15957 df-vsca 15958 df-ip 15959 df-tset 15960 df-ple 15961 df-ds 15964 df-unif 15965 df-hom 15966 df-cco 15967 df-rest 16083 df-topn 16084 df-0g 16102 df-gsum 16103 df-topgen 16104 df-pt 16105 df-prds 16108 df-ordt 16161 df-xrs 16162 df-qtop 16167 df-imas 16168 df-xps 16170 df-mre 16246 df-mrc 16247 df-acs 16249 df-ps 17200 df-tsr 17201 df-plusf 17241 df-mgm 17242 df-sgrp 17284 df-mnd 17295 df-mhm 17335 df-submnd 17336 df-grp 17425 df-minusg 17426 df-sbg 17427 df-mulg 17541 df-subg 17591 df-cntz 17750 df-cmn 18195 df-abl 18196 df-mgp 18490 df-ur 18502 df-ring 18549 df-cring 18550 df-subrg 18778 df-abv 18817 df-lmod 18865 df-scaf 18866 df-sra 19172 df-rgmod 19173 df-psmet 19738 df-xmet 19739 df-met 19740 df-bl 19741 df-mopn 19742 df-fbas 19743 df-fg 19744 df-cnfld 19747 df-top 20699 df-topon 20716 df-topsp 20737 df-bases 20750 df-cld 20823 df-ntr 20824 df-cls 20825 df-nei 20902 df-lp 20940 df-perf 20941 df-cn 21031 df-cnp 21032 df-haus 21119 df-tx 21365 df-hmeo 21558 df-fil 21650 df-fm 21742 df-flim 21743 df-flf 21744 df-tmd 21876 df-tgp 21877 df-tsms 21930 df-trg 21963 df-xms 22125 df-ms 22126 df-tms 22127 df-nm 22387 df-ngp 22388 df-nrg 22390 df-nlm 22391 df-ii 22680 df-cncf 22681 df-limc 23630 df-dv 23631 df-log 24303 df-esum 30090

This theorem is referenced by: cntmeas 30289

W3C validator