hasheuni - Mathbox for Thierry Arnoux

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Theorem hasheuni 29474

Description: The cardinality of a disjoint union, not necessarily finite. cf. hashuni 14397. (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 4566	. . . . . . . 8 ⊢ Ⅎ𝑥Disj 𝑥 ∈ 𝐴 𝑥
2		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ Fin
3		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ⊆ Fin
4	1, 2, 3	nf3an 1819	. . . . . . 7 ⊢ Ⅎ𝑥(Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin)
5		simp2 1055	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		simp3 1056	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
7		simp1 1054	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Disj 𝑥 ∈ 𝐴 𝑥)
8	4, 5, 6, 7	hashunif 28949	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (#‘𝑥))
9		simpl 472	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
10		dfss3 3558	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
11		hashcl 13009	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → (#‘𝑥) ∈ ℕ₀)
12		nn0re 11178	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑥) ∈ ℕ₀ → (#‘𝑥) ∈ ℝ)
13		nn0ge0 11195	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑥) ∈ ℕ₀ → 0 ≤ (#‘𝑥))
14		elrege0 12149	. . . . . . . . . . . . . 14 ⊢ ((#‘𝑥) ∈ (0[,)+∞) ↔ ((#‘𝑥) ∈ ℝ ∧ 0 ≤ (#‘𝑥)))
15	12, 13, 14	sylanbrc 695	. . . . . . . . . . . . 13 ⊢ ((#‘𝑥) ∈ ℕ₀ → (#‘𝑥) ∈ (0[,)+∞))
16	11, 15	syl 17	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Fin → (#‘𝑥) ∈ (0[,)+∞))
17	16	ralimi 2936	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ Fin → ∀𝑥 ∈ 𝐴 (#‘𝑥) ∈ (0[,)+∞))
18	10, 17	sylbi 206	. . . . . . . . . 10 ⊢ (𝐴 ⊆ Fin → ∀𝑥 ∈ 𝐴 (#‘𝑥) ∈ (0[,)+∞))
19	18	r19.21bi 2916	. . . . . . . . 9 ⊢ ((𝐴 ⊆ Fin ∧ 𝑥 ∈ 𝐴) → (#‘𝑥) ∈ (0[,)+∞))
20	19	adantll 746	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑥 ∈ 𝐴) → (#‘𝑥) ∈ (0[,)+∞))
21	9, 20	esumpfinval 29464	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(#‘𝑥) = Σ𝑥 ∈ 𝐴 (#‘𝑥))
22	21	3adant1 1072	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(#‘𝑥) = Σ𝑥 ∈ 𝐴 (#‘𝑥))
23	8, 22	eqtr4d 2647	. . . . 5 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
24	23	3adant1l 1310	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
25	24	3expa 1257	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) ∧ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
26		uniexg 6853	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
27	10	notbii 309	. . . . . . . . . 10 ⊢ (¬ 𝐴 ⊆ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
28		rexnal 2978	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
29	27, 28	bitr4i 266	. . . . . . . . 9 ⊢ (¬ 𝐴 ⊆ Fin ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin)
30		elssuni 4403	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
31		ssfi 8065	. . . . . . . . . . . . 13 ⊢ ((∪ 𝐴 ∈ Fin ∧ 𝑥 ⊆ ∪ 𝐴) → 𝑥 ∈ Fin)
32	31	expcom 450	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ ∪ 𝐴 → (∪ 𝐴 ∈ Fin → 𝑥 ∈ Fin))
33	32	con3d 147	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ ∪ 𝐴 → (¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin))
34	30, 33	syl 17	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin))
35	34	rexlimiv 3009	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
36	29, 35	sylbi 206	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ¬ ∪ 𝐴 ∈ Fin)
37		hashinf 12984	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ V ∧ ¬ ∪ 𝐴 ∈ Fin) → (#‘∪ 𝐴) = +∞)
38	26, 36, 37	syl2an 493	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = +∞)
39		vex 3176	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
40		hashinf 12984	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ Fin) → (#‘𝑥) = +∞)
41	39, 40	mpan 702	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ Fin → (#‘𝑥) = +∞)
42	41	reximi 2994	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞)
43	29, 42	sylbi 206	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞)
44		nfv 1830	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
45		nfre1 2988	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞
46	44, 45	nfan 1816	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞)
47		simpl 472	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞) → 𝐴 ∈ 𝑉)
48		hashf2 29473	. . . . . . . . . . 11 ⊢ #:V⟶(0[,]+∞)
49		ffvelrn 6265	. . . . . . . . . . 11 ⊢ ((#:V⟶(0[,]+∞) ∧ 𝑥 ∈ V) → (#‘𝑥) ∈ (0[,]+∞))
50	48, 39, 49	mp2an 704	. . . . . . . . . 10 ⊢ (#‘𝑥) ∈ (0[,]+∞)
51	50	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞) ∧ 𝑥 ∈ 𝐴) → (#‘𝑥) ∈ (0[,]+∞))
52		simpr 476	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞) → ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞)
53	46, 47, 51, 52	esumpinfval 29462	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (#‘𝑥) = +∞) → Σ^*𝑥 ∈ 𝐴(#‘𝑥) = +∞)
54	43, 53	sylan2 490	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(#‘𝑥) = +∞)
55	38, 54	eqtr4d 2647	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
56	55	3adant2 1073	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ Fin ∧ ¬ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
57	56	3adant1r 1311	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin ∧ ¬ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
58	57	3expa 1257	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) ∧ ¬ 𝐴 ⊆ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
59	25, 58	pm2.61dan 828	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
60		pwfi 8144	. . . . . . 7 ⊢ (∪ 𝐴 ∈ Fin ↔ 𝒫 ∪ 𝐴 ∈ Fin)
61		pwuni 4825	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
62		ssfi 8065	. . . . . . . 8 ⊢ ((𝒫 ∪ 𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → 𝐴 ∈ Fin)
63	61, 62	mpan2 703	. . . . . . 7 ⊢ (𝒫 ∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin)
64	60, 63	sylbi 206	. . . . . 6 ⊢ (∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin)
65	64	con3i 149	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
66	26, 65, 37	syl2an 493	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘∪ 𝐴) = +∞)
67		nftru 1721	. . . . . . . . 9 ⊢ Ⅎ𝑥⊤
68		unrab 3857	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0)}
69		exmid 430	. . . . . . . . . . . . 13 ⊢ ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0)
70	69	rgenw 2908	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ 𝐴 ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0)
71		rabid2 3096	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0)} ↔ ∀𝑥 ∈ 𝐴 ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0))
72	70, 71	mpbir 220	. . . . . . . . . . 11 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ ((#‘𝑥) = 0 ∨ ¬ (#‘𝑥) = 0)}
73	68, 72	eqtr4i 2635	. . . . . . . . . 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 29424	. . . . . . . 8 ⊢ (⊤ → Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0})(#‘𝑥) = Σ^𝑥 ∈ 𝐴(#‘𝑥))
76	75	trud 1484	. . . . . . 7 ⊢ Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0})(#‘𝑥) = Σ^𝑥 ∈ 𝐴(#‘𝑥)
77		nfrab1 3099	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}
78		nfrab1 3099	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}
79		rabexg 4739	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∈ V)
80		rabexg 4739	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0} ∈ V)
81		rabnc 3916	. . . . . . . . 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 29442	. . . . . . 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 2658	. . . . . 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 1830	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin)
89	80	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0} ∈ V)
90		simpr 476	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
91		dfrab3 3861	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} = (𝐴 ∩ {𝑥 ∣ (#‘𝑥) = 0})
92		hasheq0 13015	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((#‘𝑥) = 0 ↔ 𝑥 = ∅))
93	39, 92	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((#‘𝑥) = 0 ↔ 𝑥 = ∅)
94	93	abbii 2726	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∣ (#‘𝑥) = 0} = {𝑥 ∣ 𝑥 = ∅}
95		df-sn 4126	. . . . . . . . . . . . . 14 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
96	94, 95	eqtr4i 2635	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ (#‘𝑥) = 0} = {∅}
97	96	ineq2i 3773	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑥 ∣ (#‘𝑥) = 0}) = (𝐴 ∩ {∅})
98	91, 97	eqtri 2632	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} = (𝐴 ∩ {∅})
99		snfi 7923	. . . . . . . . . . . 12 ⊢ {∅} ∈ Fin
100		inss2 3796	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {∅}) ⊆ {∅}
101		ssfi 8065	. . . . . . . . . . . 12 ⊢ (({∅} ∈ Fin ∧ (𝐴 ∩ {∅}) ⊆ {∅}) → (𝐴 ∩ {∅}) ∈ Fin)
102	99, 100, 101	mp2an 704	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {∅}) ∈ Fin
103	98, 102	eqeltri 2684	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∈ Fin
104	103	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∈ Fin)
105		difinf 8115	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} ∈ Fin) → ¬ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}) ∈ Fin)
106	90, 104, 105	syl2anc 691	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}) ∈ Fin)
107		notrab 3863	. . . . . . . . 9 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}
108	107	eleq1i 2679	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0}) ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0} ∈ Fin)
109	106, 108	sylnib 317	. . . . . . 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 476	. . . . . . . . . . 11 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) → 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0})
113		rabid 3095	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0} ↔ (𝑥 ∈ 𝐴 ∧ ¬ (#‘𝑥) = 0))
114	112, 113	sylib 207	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) → (𝑥 ∈ 𝐴 ∧ ¬ (#‘𝑥) = 0))
115	114	simprd 478	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) → ¬ (#‘𝑥) = 0)
116	93	biimpri 217	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (#‘𝑥) = 0)
117	116	necon3bi 2808	. . . . . . . . 9 ⊢ (¬ (#‘𝑥) = 0 → 𝑥 ≠ ∅)
118	115, 117	syl 17	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) → 𝑥 ≠ ∅)
119		hashge1 13039	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → 1 ≤ (#‘𝑥))
120	111, 118, 119	syl2anc 691	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0}) → 1 ≤ (#‘𝑥))
121		1re 9918	. . . . . . . . 9 ⊢ 1 ∈ ℝ
122	121	rexri 9976	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
123	122	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 1 ∈ ℝ^*)
124		0lt1 10429	. . . . . . . 8 ⊢ 0 < 1
125	124	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 0 < 1)
126	88, 78, 89, 109, 110, 120, 123, 125	esumpinfsum 29466	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (#‘𝑥) = 0} (#‘𝑥) = +∞)
127	126	oveq2d 6565	. . . . 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 12127	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ^*
129	79	adantr 480	. . . . . . . 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 2949	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ∈ (0[,]+∞))
132	77	esumcl 29419	. . . . . . . 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 691	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ∈ (0[,]+∞))
134	128, 133	sseldi 3566	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) ∈ ℝ^)
135		xrge0neqmnf 12147	. . . . . . 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 11931	. . . . . 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 691	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (#‘𝑥) = 0} (#‘𝑥) +_𝑒 +∞) = +∞)
139	87, 127, 138	3eqtrd 2648	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ 𝐴(#‘𝑥) = +∞)
140	66, 139	eqtr4d 2647	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
141	140	adantlr 747	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ ¬ 𝐴 ∈ Fin) → (#‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(#‘𝑥))
142	59, 141	pm2.61dan 828	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 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ⊤wtru 1476 ∈ wcel 1977 {cab 2596 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∖ cdif 3537 ∪ cun 3538 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 {csn 4125 ∪ cuni 4372 Disj wdisj 4553 class class class wbr 4583 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℝcr 9814 0cc0 9815 1c1 9816 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953 ≤ cle 9954 ℕ₀cn0 11169 +_𝑒 cxad 11820 [,)cico 12048 [,]cicc 12049 #chash 12979 Σcsu 14264 Σ^*cesum 29416

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-iin 4458 df-disj 4554 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-of 6795 df-om 6958 df-1st 7059 df-2nd 7060 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-2o 7448 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-ixp 7795 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-fsupp 8159 df-fi 8200 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-cda 8873 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-xnn0 11241 df-z 11255 df-dec 11370 df-uz 11564 df-q 11665 df-rp 11709 df-xneg 11822 df-xadd 11823 df-xmul 11824 df-ioo 12050 df-ioc 12051 df-ico 12052 df-icc 12053 df-fz 12198 df-fzo 12335 df-fl 12455 df-mod 12531 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 df-sin 14639 df-cos 14640 df-pi 14642 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-mulr 15782 df-starv 15783 df-sca 15784 df-vsca 15785 df-ip 15786 df-tset 15787 df-ple 15788 df-ds 15791 df-unif 15792 df-hom 15793 df-cco 15794 df-rest 15906 df-topn 15907 df-0g 15925 df-gsum 15926 df-topgen 15927 df-pt 15928 df-prds 15931 df-ordt 15984 df-xrs 15985 df-qtop 15990 df-imas 15991 df-xps 15993 df-mre 16069 df-mrc 16070 df-acs 16072 df-ps 17023 df-tsr 17024 df-plusf 17064 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-mhm 17158 df-submnd 17159 df-grp 17248 df-minusg 17249 df-sbg 17250 df-mulg 17364 df-subg 17414 df-cntz 17573 df-cmn 18018 df-abl 18019 df-mgp 18313 df-ur 18325 df-ring 18372 df-cring 18373 df-subrg 18601 df-abv 18640 df-lmod 18688 df-scaf 18689 df-sra 18993 df-rgmod 18994 df-psmet 19559 df-xmet 19560 df-met 19561 df-bl 19562 df-mopn 19563 df-fbas 19564 df-fg 19565 df-cnfld 19568 df-top 20521 df-bases 20522 df-topon 20523 df-topsp 20524 df-cld 20633 df-ntr 20634 df-cls 20635 df-nei 20712 df-lp 20750 df-perf 20751 df-cn 20841 df-cnp 20842 df-haus 20929 df-tx 21175 df-hmeo 21368 df-fil 21460 df-fm 21552 df-flim 21553 df-flf 21554 df-tmd 21686 df-tgp 21687 df-tsms 21740 df-trg 21773 df-xms 21935 df-ms 21936 df-tms 21937 df-nm 22197 df-ngp 22198 df-nrg 22200 df-nlm 22201 df-ii 22488 df-cncf 22489 df-limc 23436 df-dv 23437 df-log 24107 df-esum 29417

This theorem is referenced by: cntmeas 29616

W3C validator