hasheuni - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > hasheuni		Structured version Visualization version GIF version

Theorem hasheuni 34453

Description: The cardinality of a disjoint union, not necessarily finite. cf. hashuni 15880. (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 1944	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ∈ Fin
3		nfv 1944	. . . . . . . 8 ⊢ Ⅎ𝑥 𝐴 ⊆ Fin
4	1, 2, 3	nf3an 1931	. . . . . . 7 ⊢ Ⅎ𝑥(Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin)
5		simp2 1155	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
6		simp3 1156	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ⊆ Fin)
7		simp1 1154	. . . . . . 7 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Disj 𝑥 ∈ 𝐴 𝑥)
8	4, 5, 6, 7	hashunif 33129	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
9		simpl 487	. . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → 𝐴 ∈ Fin)
10		dfss3 3927	. . . . . . . . . . 11 ⊢ (𝐴 ⊆ Fin ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
11		hashcl 14394	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ ℕ₀)
12		nn0re 12514	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ ℕ₀ → (♯‘𝑥) ∈ ℝ)
13		nn0ge0 12530	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ ℕ₀ → 0 ≤ (♯‘𝑥))
14		elrege0 13482	. . . . . . . . . . . . . 14 ⊢ ((♯‘𝑥) ∈ (0[,)+∞) ↔ ((♯‘𝑥) ∈ ℝ ∧ 0 ≤ (♯‘𝑥)))
15	12, 13, 14	sylanbrc 594	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) ∈ ℕ₀ → (♯‘𝑥) ∈ (0[,)+∞))
16	11, 15	syl 18	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ Fin → (♯‘𝑥) ∈ (0[,)+∞))
17	16	ralimi 3102	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ Fin → ∀𝑥 ∈ 𝐴 (♯‘𝑥) ∈ (0[,)+∞))
18	10, 17	sylbi 220	. . . . . . . . . 10 ⊢ (𝐴 ⊆ Fin → ∀𝑥 ∈ 𝐴 (♯‘𝑥) ∈ (0[,)+∞))
19	18	r19.21bi 3257	. . . . . . . . 9 ⊢ ((𝐴 ⊆ Fin ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,)+∞))
20	19	adantll 726	. . . . . . . 8 ⊢ (((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) ∧ 𝑥 ∈ 𝐴) → (♯‘𝑥) ∈ (0[,)+∞))
21	9, 20	esumpfinval 34443	. . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
22	21	3adant1 1148	. . . . . 6 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = Σ𝑥 ∈ 𝐴 (♯‘𝑥))
23	8, 22	eqtr4d 2801	. . . . 5 ⊢ ((Disj 𝑥 ∈ 𝐴 𝑥 ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
24	23	3adant1l 1195	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
25	24	3expa 1136	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) ∧ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
26		uniexg 7740	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 ∈ V)
27	10	notbii 323	. . . . . . . . . 10 ⊢ (¬ 𝐴 ⊆ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
28		rexnal 3117	. . . . . . . . . 10 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin ↔ ¬ ∀𝑥 ∈ 𝐴 𝑥 ∈ Fin)
29	27, 28	bitr4i 281	. . . . . . . . 9 ⊢ (¬ 𝐴 ⊆ Fin ↔ ∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin)
30		elssuni 4905	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
31		ssfi 9158	. . . . . . . . . . . . 13 ⊢ ((∪ 𝐴 ∈ Fin ∧ 𝑥 ⊆ ∪ 𝐴) → 𝑥 ∈ Fin)
32	31	expcom 418	. . . . . . . . . . . 12 ⊢ (𝑥 ⊆ ∪ 𝐴 → (∪ 𝐴 ∈ Fin → 𝑥 ∈ Fin))
33	32	con3d 153	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ ∪ 𝐴 → (¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin))
34	30, 33	syl 18	. . . . . . . . . 10 ⊢ (𝑥 ∈ 𝐴 → (¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin))
35	34	rexlimiv 3159	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
36	29, 35	sylbi 220	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ¬ ∪ 𝐴 ∈ Fin)
37		hashinf 14373	. . . . . . . 8 ⊢ ((∪ 𝐴 ∈ V ∧ ¬ ∪ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = +∞)
38	26, 36, 37	syl2an 607	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = +∞)
39		vex 3459	. . . . . . . . . . 11 ⊢ 𝑥 ∈ V
40		hashinf 14373	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ V ∧ ¬ 𝑥 ∈ Fin) → (♯‘𝑥) = +∞)
41	39, 40	mpan 702	. . . . . . . . . 10 ⊢ (¬ 𝑥 ∈ Fin → (♯‘𝑥) = +∞)
42	41	reximi 3103	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 ¬ 𝑥 ∈ Fin → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
43	29, 42	sylbi 220	. . . . . . . 8 ⊢ (¬ 𝐴 ⊆ Fin → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
44		nfv 1944	. . . . . . . . . 10 ⊢ Ⅎ𝑥 𝐴 ∈ 𝑉
45		nfre1 3290	. . . . . . . . . 10 ⊢ Ⅎ𝑥∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞
46	44, 45	nfan 1929	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
47		simpl 487	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → 𝐴 ∈ 𝑉)
48		hashf2 34452	. . . . . . . . . . 11 ⊢ ♯:V⟶(0[,]+∞)
49		ffvelcdm 7078	. . . . . . . . . . 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 489	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞)
53	46, 47, 51, 52	esumpinfval 34441	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ∃𝑥 ∈ 𝐴 (♯‘𝑥) = +∞) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = +∞)
54	43, 53	sylan2 604	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = +∞)
55	38, 54	eqtr4d 2801	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
56	55	3adant2 1149	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐴 ∈ Fin ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
57	56	3adant1r 1196	. . . 4 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
58	57	3expa 1136	. . 3 ⊢ ((((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) ∧ ¬ 𝐴 ⊆ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
59	25, 58	pm2.61dan 824	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
60		pwfi 9279	. . . . . . 7 ⊢ (∪ 𝐴 ∈ Fin ↔ 𝒫 ∪ 𝐴 ∈ Fin)
61		pwuni 4912	. . . . . . . 8 ⊢ 𝐴 ⊆ 𝒫 ∪ 𝐴
62		ssfi 9158	. . . . . . . 8 ⊢ ((𝒫 ∪ 𝐴 ∈ Fin ∧ 𝐴 ⊆ 𝒫 ∪ 𝐴) → 𝐴 ∈ Fin)
63	61, 62	mpan2 703	. . . . . . 7 ⊢ (𝒫 ∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin)
64	60, 63	sylbi 220	. . . . . 6 ⊢ (∪ 𝐴 ∈ Fin → 𝐴 ∈ Fin)
65	64	con3i 155	. . . . 5 ⊢ (¬ 𝐴 ∈ Fin → ¬ ∪ 𝐴 ∈ Fin)
66	26, 65, 37	syl2an 607	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = +∞)
67		nftru 1834	. . . . . . . . 9 ⊢ Ⅎ𝑥⊤
68		unrab 4269	. . . . . . . . . . 11 ⊢ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)}
69		exmid 907	. . . . . . . . . . . . 13 ⊢ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)
70	69	rgenw 3083	. . . . . . . . . . . 12 ⊢ ∀𝑥 ∈ 𝐴 ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)
71		rabid2 3449	. . . . . . . . . . . 12 ⊢ (𝐴 = {𝑥 ∈ 𝐴 ∣ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)} ↔ ∀𝑥 ∈ 𝐴 ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0))
72	70, 71	mpbir 234	. . . . . . . . . . 11 ⊢ 𝐴 = {𝑥 ∈ 𝐴 ∣ ((♯‘𝑥) = 0 ∨ ¬ (♯‘𝑥) = 0)}
73	68, 72	eqtr4i 2789	. . . . . . . . . 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 34403	. . . . . . . 8 ⊢ (⊤ → Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})(♯‘𝑥) = Σ^𝑥 ∈ 𝐴(♯‘𝑥))
76	75	mptru 1577	. . . . . . 7 ⊢ Σ^𝑥 ∈ ({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∪ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0})(♯‘𝑥) = Σ^𝑥 ∈ 𝐴(♯‘𝑥)
77		nfrab1 3436	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}
78		nfrab1 3436	. . . . . . . 8 ⊢ Ⅎ𝑥{𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}
79		rabexg 5309	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ V)
80		rabexg 5309	. . . . . . . 8 ⊢ (𝐴 ∈ 𝑉 → {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ V)
81		rabnc 4349	. . . . . . . . 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 34421	. . . . . . 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 2812	. . . . . 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 485	. . . . 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 1944	. . . . . . 7 ⊢ Ⅎ𝑥(𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin)
89	80	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ V)
90		simpr 489	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ 𝐴 ∈ Fin)
91		dfrab3 4273	. . . . . . . . . . . 12 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} = (𝐴 ∩ {𝑥 ∣ (♯‘𝑥) = 0})
92		hasheq0 14401	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ V → ((♯‘𝑥) = 0 ↔ 𝑥 = ∅))
93	39, 92	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ ((♯‘𝑥) = 0 ↔ 𝑥 = ∅)
94	93	abbii 2830	. . . . . . . . . . . . . 14 ⊢ {𝑥 ∣ (♯‘𝑥) = 0} = {𝑥 ∣ 𝑥 = ∅}
95		df-sn 4591	. . . . . . . . . . . . . 14 ⊢ {∅} = {𝑥 ∣ 𝑥 = ∅}
96	94, 95	eqtr4i 2789	. . . . . . . . . . . . 13 ⊢ {𝑥 ∣ (♯‘𝑥) = 0} = {∅}
97	96	ineq2i 4171	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {𝑥 ∣ (♯‘𝑥) = 0}) = (𝐴 ∩ {∅})
98	91, 97	eqtri 2786	. . . . . . . . . . 11 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} = (𝐴 ∩ {∅})
99		snfi 9041	. . . . . . . . . . . 12 ⊢ {∅} ∈ Fin
100		inss2 4191	. . . . . . . . . . . 12 ⊢ (𝐴 ∩ {∅}) ⊆ {∅}
101		ssfi 9158	. . . . . . . . . . . 12 ⊢ (({∅} ∈ Fin ∧ (𝐴 ∩ {∅}) ⊆ {∅}) → (𝐴 ∩ {∅}) ∈ Fin)
102	99, 100, 101	mp2an 704	. . . . . . . . . . 11 ⊢ (𝐴 ∩ {∅}) ∈ Fin
103	98, 102	eqeltri 2859	. . . . . . . . . 10 ⊢ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin
104	103	a1i 11	. . . . . . . . 9 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin)
105		difinf 9272	. . . . . . . . 9 ⊢ ((¬ 𝐴 ∈ Fin ∧ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ Fin) → ¬ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) ∈ Fin)
106	90, 104, 105	syl2anc 595	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ¬ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) ∈ Fin)
107		notrab 4276	. . . . . . . . 9 ⊢ (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) = {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}
108	107	eleq1i 2854	. . . . . . . 8 ⊢ ((𝐴 ∖ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) ∈ Fin ↔ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ∈ Fin)
109	106, 108	sylnib 331	. . . . . . 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		rabid 3437	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} ↔ (𝑥 ∈ 𝐴 ∧ ¬ (♯‘𝑥) = 0))
113	112	bilani 509	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → (𝑥 ∈ 𝐴 ∧ ¬ (♯‘𝑥) = 0))
114	113	simprd 500	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → ¬ (♯‘𝑥) = 0)
115	93	biimpri 231	. . . . . . . . . 10 ⊢ (𝑥 = ∅ → (♯‘𝑥) = 0)
116	115	necon3bi 2984	. . . . . . . . 9 ⊢ (¬ (♯‘𝑥) = 0 → 𝑥 ≠ ∅)
117	114, 116	syl 18	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 𝑥 ≠ ∅)
118		hashge1 14427	. . . . . . . 8 ⊢ ((𝑥 ∈ V ∧ 𝑥 ≠ ∅) → 1 ≤ (♯‘𝑥))
119	111, 117, 118	syl2anc 595	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0}) → 1 ≤ (♯‘𝑥))
120		1xr 11269	. . . . . . . 8 ⊢ 1 ∈ ℝ^*
121	120	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 1 ∈ ℝ^*)
122		0lt1 11737	. . . . . . . 8 ⊢ 0 < 1
123	122	a1i 11	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → 0 < 1)
124	88, 78, 89, 109, 110, 119, 121, 123	esumpinfsum 34445	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ ¬ (♯‘𝑥) = 0} (♯‘𝑥) = +∞)
125	124	oveq2d 7428	. . . . 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 +∞))
126		iccssxr 13458	. . . . . . 7 ⊢ (0[,]+∞) ⊆ ℝ^*
127	79	adantr 485	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ V)
128	50	a1i 11	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) ∧ 𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0}) → (♯‘𝑥) ∈ (0[,]+∞))
129	128	ralrimiva 3157	. . . . . . . 8 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞))
130	77	esumcl 34398	. . . . . . . 8 ⊢ (({𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} ∈ V ∧ ∀𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞)) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞))
131	127, 129, 130	syl2anc 595	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞))
132	126, 131	sselid 3936	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ ℝ^)
133		xrge0neqmnf 13480	. . . . . . 7 ⊢ (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ (0[,]+∞) → Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ≠ -∞)
134	131, 133	syl 18	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ≠ -∞)
135		xaddpnf1 13253	. . . . . 6 ⊢ ((Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ∈ ℝ^ ∧ Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) ≠ -∞) → (Σ^𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) +_𝑒 +∞) = +∞)
136	132, 134, 135	syl2anc 595	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (Σ^*𝑥 ∈ {𝑥 ∈ 𝐴 ∣ (♯‘𝑥) = 0} (♯‘𝑥) +_𝑒 +∞) = +∞)
137	87, 125, 136	3eqtrd 2802	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → Σ^*𝑥 ∈ 𝐴(♯‘𝑥) = +∞)
138	66, 137	eqtr4d 2801	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
139	138	adantlr 727	. 2 ⊢ (((𝐴 ∈ 𝑉 ∧ Disj 𝑥 ∈ 𝐴 𝑥) ∧ ¬ 𝐴 ∈ Fin) → (♯‘∪ 𝐴) = Σ^*𝑥 ∈ 𝐴(♯‘𝑥))
140	59, 139	pm2.61dan 824	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 209 ∧ wa 400 ∨ wo 860 ∧ w3a 1103 = wceq 1570 ⊤wtru 1571 ∈ wcel 2143 {cab 2741 ≠ wne 2958 ∀wral 3079 ∃wrex 3089 {crab 3416 Vcvv 3455 ∖ cdif 3903 ∪ cun 3904 ∩ cin 3905 ⊆ wss 3906 ∅c0 4287 𝒫 cpw 4563 {csn 4590 ∪ cuni 4873 Disj wdisj 5077 class class class wbr 5110 ⟶wf 6534 ‘cfv 6538 (class class class)co 7412 Fincfn 8944 ℝcr 11100 0cc0 11101 1c1 11102 +∞cpnf 11241 -∞cmnf 11242 ℝ^*cxr 11243 < clt 11244 ≤ cle 11245 ℕ₀cn0 12505 +_𝑒 cxad 13136 [,)cico 13375 [,]cicc 13376 ♯chash 14368 Σcsu 15739 Σ^*cesum 34395

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1825 ax-4 1839 ax-5 1940 ax-6 1997 ax-7 2038 ax-8 2145 ax-9 2153 ax-10 2176 ax-11 2192 ax-12 2213 ax-ext 2735 ax-rep 5239 ax-sep 5258 ax-nul 5270 ax-pow 5338 ax-pr 5406 ax-un 7734 ax-inf2 9611 ax-cnex 11157 ax-resscn 11158 ax-1cn 11159 ax-icn 11160 ax-addcl 11161 ax-addrcl 11162 ax-mulcl 11163 ax-mulrcl 11164 ax-mulcom 11165 ax-addass 11166 ax-mulass 11167 ax-distr 11168 ax-i2m1 11169 ax-1ne0 11170 ax-1rid 11171 ax-rnegex 11172 ax-rrecex 11173 ax-cnre 11174 ax-pre-lttri 11175 ax-pre-lttrn 11176 ax-pre-ltadd 11177 ax-pre-mulgt0 11178 ax-pre-sup 11179 ax-addf 11180 ax-mulf 11181

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1810 df-nf 1814 df-sb 2097 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2838 df-nfc 2912 df-ne 2959 df-nel 3065 df-ral 3080 df-rex 3090 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3457 df-sbc 3746 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4288 df-if 4489 df-pw 4565 df-sn 4591 df-pr 4593 df-tp 4595 df-op 4597 df-uni 4874 df-int 4914 df-iun 4959 df-iin 4960 df-disj 5078 df-br 5111 df-opab 5175 df-mpt 5194 df-tr 5220 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-se 5617 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6304 df-ord 6365 df-on 6366 df-lim 6367 df-suc 6368 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-isom 6547 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-of 7676 df-om 7864 df-1st 7987 df-2nd 7988 df-supp 8158 df-frecs 8279 df-wrecs 8310 df-recs 8359 df-rdg 8398 df-1o 8454 df-2o 8455 df-oadd 8458 df-er 8695 df-map 8827 df-pm 8828 df-ixp 8897 df-en 8945 df-dom 8946 df-sdom 8947 df-fin 8948 df-fsupp 9323 df-fi 9372 df-sup 9403 df-inf 9404 df-oi 9473 df-card 9926 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11444 df-neg 11445 df-div 11873 df-nn 12235 df-2 12304 df-3 12305 df-4 12306 df-5 12307 df-6 12308 df-7 12309 df-8 12310 df-9 12311 df-n0 12506 df-xnn0 12579 df-z 12593 df-dec 12713 df-uz 12864 df-q 12974 df-rp 13018 df-xneg 13138 df-xadd 13139 df-xmul 13140 df-ioo 13377 df-ioc 13378 df-ico 13379 df-icc 13380 df-fz 13537 df-fzo 13685 df-fl 13827 df-mod 13905 df-seq 14040 df-exp 14100 df-fac 14312 df-bc 14341 df-hash 14369 df-shft 15106 df-cj 15152 df-re 15153 df-im 15154 df-sqrt 15288 df-abs 15289 df-limsup 15524 df-clim 15541 df-rlim 15542 df-sum 15740 df-ef 16122 df-sin 16124 df-cos 16125 df-pi 16127 df-struct 17208 df-sets 17225 df-slot 17243 df-ndx 17255 df-base 17271 df-ress 17292 df-plusg 17324 df-mulr 17325 df-starv 17326 df-sca 17327 df-vsca 17328 df-ip 17329 df-tset 17330 df-ple 17331 df-ds 17333 df-unif 17334 df-hom 17335 df-cco 17336 df-rest 17476 df-topn 17477 df-0g 17495 df-gsum 17496 df-topgen 17497 df-pt 17498 df-prds 17501 df-ordt 17556 df-xrs 17557 df-qtop 17562 df-imas 17563 df-xps 17565 df-mre 17639 df-mrc 17640 df-acs 17642 df-ps 18623 df-tsr 18624 df-plusf 18698 df-mgm 18699 df-sgrp 18778 df-mnd 18794 df-mhm 18842 df-submnd 18843 df-grp 19004 df-minusg 19005 df-sbg 19006 df-mulg 19135 df-subg 19190 df-cntz 19388 df-cmn 19853 df-abl 19854 df-mgp 20218 df-rng 20232 df-ur 20265 df-ring 20318 df-cring 20319 df-subrng 20632 df-subrg 20656 df-abv 20893 df-lmod 20964 df-scaf 20965 df-sra 21275 df-rgmod 21276 df-psmet 21495 df-xmet 21496 df-met 21497 df-bl 21498 df-mopn 21499 df-fbas 21500 df-fg 21501 df-cnfld 21504 df-top 23032 df-topon 23049 df-topsp 23071 df-bases 23084 df-cld 23157 df-ntr 23158 df-cls 23159 df-nei 23236 df-lp 23274 df-perf 23275 df-cn 23365 df-cnp 23366 df-haus 23453 df-tx 23700 df-hmeo 23893 df-fil 23984 df-fm 24076 df-flim 24077 df-flf 24078 df-tmd 24210 df-tgp 24211 df-tsms 24265 df-trg 24298 df-xms 24458 df-ms 24459 df-tms 24460 df-nm 24720 df-ngp 24721 df-nrg 24723 df-nlm 24724 df-ii 25017 df-cncf 25018 df-limc 26006 df-dv 26007 df-log 26699 df-esum 34396

This theorem is referenced by: cntmeas 34594

W3C validator