fnlimfvre - Mathbox for Glauco Siliprandi

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Theorem fnlimfvre 45679

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 2892	. . . . . . . 8 ⊢ Ⅎ𝑥𝑍
4		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎ𝑥(ℤ_≥‘𝑛)
5		fnlimfvre.n	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
6		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑚
7	5, 6	nffv 6871	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑚)
8	7	nfdm 5918	. . . . . . . . 9 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
9	4, 8	nfiin 4991	. . . . . . . 8 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
10	3, 9	nfiun 4990	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
11	10	ssrab2f 45118	. . . . . 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 3996	. . . . 5 ⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
13	12	sseli 3945	. . . 4 ⊢ (𝑋 ∈ 𝐷 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
14		eliun 4962	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
15	13, 14	sylib 218	. . 3 ⊢ (𝑋 ∈ 𝐷 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
16	1, 15	syl 17	. 2 ⊢ (𝜑 → ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
17		nfv 1914	. . 3 ⊢ Ⅎ𝑛𝜑
18		nfv 1914	. . 3 ⊢ Ⅎ𝑛( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ
19		fnlimfvre.p	. . . . . . 7 ⊢ Ⅎ𝑚𝜑
20		nfv 1914	. . . . . . 7 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
21		nfcv 2892	. . . . . . . 8 ⊢ Ⅎ𝑚𝑋
22		nfii1 4996	. . . . . . . 8 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
23	21, 22	nfel 2907	. . . . . . 7 ⊢ Ⅎ𝑚 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
24	19, 20, 23	nf3an 1901	. . . . . 6 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
25		uzssz 12821	. . . . . . . 8 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
26		fnlimfvre.z	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ_≥‘𝑀)
27	26	eleq2i 2821	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑀))
28	27	biimpi 216	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑀))
29	25, 28	sselid 3947	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
30	29	3ad2ant2 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑛 ∈ ℤ)
31		eqid 2730	. . . . . 6 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
32	26	fvexi 6875	. . . . . . 7 ⊢ 𝑍 ∈ V
33	32	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑍 ∈ V)
34	26	uztrn2 12819	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ 𝑍)
35	34	ssd 45081	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
36	35	3ad2ant2 1134	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ⊆ 𝑍)
37		fvexd 6876	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑋) ∈ V)
38		fvexd 6876	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ∈ V)
39		ssidd 3973	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑛))
40		fvexd 6876	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
41		eqidd 2731	. . . . . 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 45676	. . . . 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 2821	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐷 ↔ 𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ })
44	43	biimpi 216	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐷 → 𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ })
45		nfcv 2892	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑋
46	7, 45	nffv 6871	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑋)
47	3, 46	nfmpt 5208	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))
48		nfcv 2892	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥dom ⇝
49	47, 48	nfel 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝
50		fveq2 6861	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑋))
51	50	mpteq2dv 5204	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))
52	51	eleq1d 2814	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ))
53	45, 10, 49, 52	elrabf 3658	. . . . . . . . . . . . . 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 216	. . . . . . . . . . . . 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 495	. . . . . . . . . . . 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 17	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐷 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
57	56	adantr 480	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
58		nfmpt1 5209	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
59		nfcv 2892	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚dom ⇝
60	58, 59	nfel 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
61		nfv 1914	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚 𝑗 ∈ 𝑍
62	61	nfci 2880	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚𝑍
63	62, 22	nfiun 4990	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
64	60, 63	nfrabw 3446	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
65	2, 64	nfcxfr 2890	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚𝐷
66	21, 65	nfel 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝑋 ∈ 𝐷
67	66, 20	nfan 1899	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍)
68	29	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
69	32	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑍 ∈ V)
70	35	adantl 481	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
71		fvexd 6876	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑋) ∈ V)
72		fvexd 6876	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ∈ V)
73		ssidd 3973	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑛))
74		fvexd 6876	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
75		eqidd 2731	. . . . . . . . . . 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 45673	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ↔ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ))
77	57, 76	mpbid 232	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
78		climdm 15527	. . . . . . . . 9 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ↔ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
79	77, 78	sylib 218	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
80	1, 79	sylan 580	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
81	80	3adant3 1132	. . . . . 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 1192	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝜑)
83		simpl2 1193	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ 𝑍)
84		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗dom (𝐹‘𝑚)
85		fnlimfvre.m	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝐹
86		nfcv 2892	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝑗
87	85, 86	nffv 6871	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(𝐹‘𝑗)
88	87	nfdm 5918	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚dom (𝐹‘𝑗)
89		fveq2 6861	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
90	89	dmeqd 5872	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
91	84, 88, 90	cbviin 5004	. . . . . . . . . . . 12 ⊢ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗)
92	91	eleq2i 2821	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
93	92	biimpi 216	. . . . . . . . . 10 ⊢ (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
94	93	adantr 480	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
95		simpr 484	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ (ℤ_≥‘𝑛))
96		eliinid 45112	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
97	94, 95, 96	syl2anc 584	. . . . . . . 8 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
98	97	3ad2antl3 1188	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
99		simpr 484	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ (ℤ_≥‘𝑛))
100		id 22	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ_≥‘𝑛) → 𝑗 ∈ (ℤ_≥‘𝑛))
101		fvexd 6876	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ_≥‘𝑛) → ((𝐹‘𝑗)‘𝑋) ∈ V)
102	87, 21	nffv 6871	. . . . . . . . . . 11 ⊢ Ⅎ𝑚((𝐹‘𝑗)‘𝑋)
103	89	fveq1d 6863	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑋) = ((𝐹‘𝑗)‘𝑋))
104		eqid 2730	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
105	86, 102, 103, 104	fvmptf 6992	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℤ_≥‘𝑛) ∧ ((𝐹‘𝑗)‘𝑋) ∈ V) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
106	100, 101, 105	syl2anc 584	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ_≥‘𝑛) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
107	106	adantl 481	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
108		simpll 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝜑)
109	34	adantll 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ 𝑍)
110	19, 61	nfan 1899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑗 ∈ 𝑍)
111		nfcv 2892	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚ℝ
112	87, 88, 111	nff 6687	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚(𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ
113	110, 112	nfim 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
114		eleq1w 2812	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → (𝑚 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
115	114	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → ((𝜑 ∧ 𝑚 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
116	89, 90	feq12d 6679	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ ↔ (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ))
117	115, 116	imbi12d 344	. . . . . . . . . . . 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 2241	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
120	108, 109, 119	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
121	120	3adantl3 1169	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
122		simpl3 1194	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
123	121, 122	ffvelcdmd 7060	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑗)‘𝑋) ∈ ℝ)
124	107, 123	eqeltrd 2829	. . . . . . 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 1375	. . . . . 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 15556	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
127	42, 126	eqeltrd 2829	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
128	127	3exp 1119	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)))
129	17, 18, 128	rexlimd 3245	. 2 ⊢ (𝜑 → (∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ))
130	16, 129	mpd 15	1 ⊢ (𝜑 → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 Ⅎwnfc 2877 ∃wrex 3054 {crab 3408 Vcvv 3450 ⊆ wss 3917 ∪ ciun 4958 ∩ ciin 4959 class class class wbr 5110 ↦ cmpt 5191 dom cdm 5641 ⟶wf 6510 ‘cfv 6514 ℝcr 11074 ℤcz 12536 ℤ_≥cuz 12800 ⇝ cli 15457

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-rep 5237 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152 ax-pre-sup 11153

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-nel 3031 df-ral 3046 df-rex 3055 df-rmo 3356 df-reu 3357 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-pss 3937 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-iun 4960 df-iin 4961 df-br 5111 df-opab 5173 df-mpt 5192 df-tr 5218 df-id 5536 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5594 df-we 5596 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-pred 6277 df-ord 6338 df-on 6339 df-lim 6340 df-suc 6341 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-riota 7347 df-ov 7393 df-oprab 7394 df-mpo 7395 df-om 7846 df-2nd 7972 df-frecs 8263 df-wrecs 8294 df-recs 8343 df-rdg 8381 df-er 8674 df-pm 8805 df-en 8922 df-dom 8923 df-sdom 8924 df-sup 9400 df-inf 9401 df-pnf 11217 df-mnf 11218 df-xr 11219 df-ltxr 11220 df-le 11221 df-sub 11414 df-neg 11415 df-div 11843 df-nn 12194 df-2 12256 df-3 12257 df-n0 12450 df-z 12537 df-uz 12801 df-rp 12959 df-fl 13761 df-seq 13974 df-exp 14034 df-cj 15072 df-re 15073 df-im 15074 df-sqrt 15208 df-abs 15209 df-clim 15461 df-rlim 15462

This theorem is referenced by: fnlimfvre2 45682 fnlimf 45683 smflimlem4 46779 smflim 46782

W3C validator