fnlimfvre - Mathbox for Glauco Siliprandi

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Theorem fnlimfvre 46429

Description: The limit function of real functions, applied to elements in its domain, evaluates to Real values. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
fnlimfvre.p	⊢ Ⅎ𝑚𝜑
fnlimfvre.m	⊢ Ⅎ𝑚𝐹
fnlimfvre.n	⊢ Ⅎ𝑥𝐹
fnlimfvre.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
fnlimfvre.f	⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
fnlimfvre.d	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
fnlimfvre.x	⊢ (𝜑 → 𝑋 ∈ 𝐷)

**Assertion**
Ref	Expression
fnlimfvre	⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)

Distinct variable groups: 𝑛,𝐹 𝑚,𝑋,𝑛,𝑥 𝑚,𝑍,𝑛,𝑥 𝜑,𝑛 Allowed substitution hints: 𝜑(𝑥, 𝑚) 𝐷(𝑥, 𝑚, 𝑛) 𝐹(𝑥, 𝑚) 𝑀(𝑥, 𝑚, 𝑛)

Proof of Theorem fnlimfvre Dummy variable 𝑗 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fnlimfvre.x	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
2		fnlimfvre.d	. . . . . 6 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
3		nfcv 2928	. . . . . . . 8 ⊢ Ⅎ𝑥𝑍
4		nfcv 2928	. . . . . . . . 9 ⊢ Ⅎ𝑥(ℤ_≥‘𝑛)
5		fnlimfvre.n	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
6		nfcv 2928	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑚
7	5, 6	nffv 6898	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑚)
8	7	nfdm 5946	. . . . . . . . 9 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
9	4, 8	nfiin 4994	. . . . . . . 8 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
10	3, 9	nfiun 4993	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
11	10	ssrab2f 45876	. . . . . 6 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
12	2, 11	eqsstri 3986	. . . . 5 ⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
13	12	sseli 3936	. . . 4 ⊢ (𝑋 ∈ 𝐷 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
14		eliun 4965	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
15	13, 14	sylib 221	. . 3 ⊢ (𝑋 ∈ 𝐷 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
16	1, 15	syl 18	. 2 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
17		nfv 1947	. . 3 ⊢ Ⅎ𝑛𝜑
18		nfv 1947	. . 3 ⊢ Ⅎ𝑛( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ
19		fnlimfvre.p	. . . . . . 7 ⊢ Ⅎ𝑚𝜑
20		nfv 1947	. . . . . . 7 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
21		nfcv 2928	. . . . . . . 8 ⊢ Ⅎ𝑚𝑋
22		nfii1 4998	. . . . . . . 8 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
23	21, 22	nfel 2942	. . . . . . 7 ⊢ Ⅎ𝑚 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
24	19, 20, 23	nf3an 1934	. . . . . 6 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
25		uzssz 12901	. . . . . . . 8 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
26		fnlimfvre.z	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ_≥‘𝑀)
27	26	eleq2i 2858	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑀))
28	27	biimpi 219	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑀))
29	25, 28	sselid 3938	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
30	29	3ad2ant2 1152	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑛 ∈ ℤ)
31		eqid 2766	. . . . . 6 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
32	26	fvexi 6902	. . . . . . 7 ⊢ 𝑍 ∈ V
33	32	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑍 ∈ V)
34	26	uztrn2 12899	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ 𝑍)
35	34	ssd 45841	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
36	35	3ad2ant2 1152	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ⊆ 𝑍)
37		fvexd 6903	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑋) ∈ V)
38		fvexd 6903	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ∈ V)
39		ssidd 3963	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑛))
40		fvexd 6903	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
41		eqidd 2767	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) = ((𝐹‘𝑚)‘𝑋))
42	24, 30, 31, 33, 36, 37, 38, 39, 40, 41	climfveqmpt 46426	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) = ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
43	2	eleq2i 2858	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐷 ↔ 𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ })
44	43	biimpi 219	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐷 → 𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ })
45		nfcv 2928	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑋
46	7, 45	nffv 6898	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑋)
47	3, 46	nfmpt 5214	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))
48		nfcv 2928	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥dom ⇝
49	47, 48	nfel 2942	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝
50		fveq2 6888	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑋))
51	50	mpteq2dv 5210	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))
52	51	eleq1d 2851	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ))
53	45, 10, 49, 52	elrabf 3650	. . . . . . . . . . . . . 14 ⊢ (𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } ↔ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ))
54	53	biimpi 219	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } → (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ))
55	54	simprd 501	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
56	44, 55	syl 18	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐷 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
57	56	adantr 486	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
58		nfmpt1 5215	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
59		nfcv 2928	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚dom ⇝
60	58, 59	nfel 2942	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
61		nfv 1947	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚 𝑗 ∈ 𝑍
62	61	nfci 2916	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚𝑍
63	62, 22	nfiun 4993	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
64	60, 63	nfrabw 3455	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
65	2, 64	nfcxfr 2926	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚𝐷
66	21, 65	nfel 2942	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝑋 ∈ 𝐷
67	66, 20	nfan 1932	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍)
68	29	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
69	32	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑍 ∈ V)
70	35	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
71		fvexd 6903	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑋) ∈ V)
72		fvexd 6903	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ∈ V)
73		ssidd 3963	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑛))
74		fvexd 6903	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
75		eqidd 2767	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) = ((𝐹‘𝑚)‘𝑋))
76	67, 68, 31, 69, 70, 71, 72, 73, 74, 75	climeldmeqmpt 46423	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ↔ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ))
77	57, 76	mpbid 235	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
78		climdm 15631	. . . . . . . . 9 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ↔ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
79	77, 78	sylib 221	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
80	1, 79	sylan 592	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
81	80	3adant3 1150	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
82		simpl1 1210	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝜑)
83		simpl2 1211	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ 𝑍)
84		nfcv 2928	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗dom (𝐹‘𝑚)
85		fnlimfvre.m	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝐹
86		nfcv 2928	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝑗
87	85, 86	nffv 6898	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(𝐹‘𝑗)
88	87	nfdm 5946	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚dom (𝐹‘𝑗)
89		fveq2 6888	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
90	89	dmeqd 5900	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
91	84, 88, 90	cbviin 5005	. . . . . . . . . . . 12 ⊢ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗)
92	91	eleq2i 2858	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
93	92	biimpi 219	. . . . . . . . . 10 ⊢ (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
94	93	adantr 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
95		simpr 490	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ (ℤ_≥‘𝑛))
96		eliinid 45870	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
97	94, 95, 96	syl2anc 596	. . . . . . . 8 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
98	97	3ad2antl3 1206	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
99		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ (ℤ_≥‘𝑛))
100		id 23	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ_≥‘𝑛) → 𝑗 ∈ (ℤ_≥‘𝑛))
101		fvexd 6903	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ_≥‘𝑛) → ((𝐹‘𝑗)‘𝑋) ∈ V)
102	87, 21	nffv 6898	. . . . . . . . . . 11 ⊢ Ⅎ𝑚((𝐹‘𝑗)‘𝑋)
103	89	fveq1d 6890	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑋) = ((𝐹‘𝑗)‘𝑋))
104		eqid 2766	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
105	86, 102, 103, 104	fvmptf 7018	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℤ_≥‘𝑛) ∧ ((𝐹‘𝑗)‘𝑋) ∈ V) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
106	100, 101, 105	syl2anc 596	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ_≥‘𝑛) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
107	106	adantl 487	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
108		simpll 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝜑)
109	34	adantll 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ 𝑍)
110	19, 61	nfan 1932	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑗 ∈ 𝑍)
111		nfcv 2928	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚ℝ
112	87, 88, 111	nff 6708	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚(𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ
113	110, 112	nfim 1929	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
114		eleq1w 2849	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → (𝑚 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
115	114	anbi2d 642	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → ((𝜑 ∧ 𝑚 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
116	89, 90	feq12d 6700	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ ↔ (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ))
117	115, 116	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑗 → (((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) ↔ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)))
118		fnlimfvre.f	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
119	113, 117, 118	chvarfv 2279	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
120	108, 109, 119	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
121	120	3adantl3 1187	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
122		simpl3 1212	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
123	121, 122	ffvelcdmd 7087	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑗)‘𝑋) ∈ ℝ)
124	107, 123	eqeltrd 2866	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) ∈ ℝ)
125	82, 83, 98, 99, 124	syl31anc 1400	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) ∈ ℝ)
126	31, 30, 81, 125	climrecl 15660	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
127	42, 126	eqeltrd 2866	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
128	127	3exp 1137	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)))
129	17, 18, 128	rexlimd 3275	. 2 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ))
130	16, 129	mpd 16	1 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2146 Ⅎwnfc 2913 ∃wrex 3092 {crab 3419 Vcvv 3458 ⊆ wss 3908 ∪ ciun 4961 ∩ ciin 4962 class class class wbr 5114 ↦ cmpt 5197 dom cdm 5666 ⟶wf 6539 ‘cfv 6543 ℝcr 11117 ℤcz 12609 ℤ_≥cuz 12880 ⇝ cli 15561

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2738 ax-rep 5243 ax-sep 5262 ax-nul 5274 ax-pow 5341 ax-pr 5409 ax-un 7745 ax-cnex 11174 ax-resscn 11175 ax-1cn 11176 ax-icn 11177 ax-addcl 11178 ax-addrcl 11179 ax-mulcl 11180 ax-mulrcl 11181 ax-mulcom 11182 ax-addass 11183 ax-mulass 11184 ax-distr 11185 ax-i2m1 11186 ax-1ne0 11187 ax-1rid 11188 ax-rnegex 11189 ax-rrecex 11190 ax-cnre 11191 ax-pre-lttri 11192 ax-pre-lttrn 11193 ax-pre-ltadd 11194 ax-pre-mulgt0 11195 ax-pre-sup 11196

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2570 df-eu 2600 df-clab 2745 df-cleq 2758 df-clel 2841 df-nfc 2915 df-ne 2962 df-nel 3068 df-ral 3083 df-rex 3093 df-rmo 3372 df-reu 3373 df-rab 3420 df-v 3460 df-sbc 3748 df-csb 3857 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3928 df-nul 4290 df-if 4493 df-pw 4569 df-sn 4595 df-pr 4597 df-op 4601 df-uni 4878 df-iun 4963 df-iin 4964 df-br 5115 df-opab 5179 df-mpt 5198 df-tr 5224 df-id 5561 df-eprel 5566 df-po 5574 df-so 5575 df-fr 5619 df-we 5621 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6309 df-ord 6370 df-on 6371 df-lim 6372 df-suc 6373 df-iota 6499 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7380 df-ov 7426 df-oprab 7427 df-mpo 7428 df-om 7872 df-2nd 7996 df-frecs 8287 df-wrecs 8318 df-recs 8367 df-rdg 8406 df-er 8703 df-pm 8836 df-en 8953 df-dom 8954 df-sdom 8955 df-sup 9412 df-inf 9413 df-pnf 11263 df-mnf 11264 df-xr 11265 df-ltxr 11266 df-le 11267 df-sub 11461 df-neg 11462 df-div 11890 df-nn 12252 df-2 12321 df-3 12322 df-n0 12523 df-z 12610 df-uz 12881 df-rp 13035 df-fl 13845 df-seq 14058 df-exp 14118 df-cj 15176 df-re 15177 df-im 15178 df-sqrt 15312 df-abs 15313 df-clim 15565 df-rlim 15566

This theorem is used by: fnlimfvre2 46432 fnlimf 46433 smflimlem4 47529 smflim 47532

W3C validator