fnlimfvre - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > fnlimfvre		Structured version Visualization version GIF version

Theorem fnlimfvre 41953

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 2977	. . . . . . . 8 ⊢ Ⅎ𝑥𝑍
4		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑥(ℤ_≥‘𝑛)
5		fnlimfvre.n	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
6		nfcv 2977	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑚
7	5, 6	nffv 6679	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑚)
8	7	nfdm 5822	. . . . . . . . 9 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
9	4, 8	nfiin 4949	. . . . . . . 8 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
10	3, 9	nfiun 4948	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
11	10	ssrab2f 41381	. . . . . 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 4000	. . . . 5 ⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
13	12	sseli 3962	. . . 4 ⊢ (𝑋 ∈ 𝐷 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
14		eliun 4922	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
15	13, 14	sylib 220	. . 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 1911	. . 3 ⊢ Ⅎ𝑛𝜑
18		nfv 1911	. . 3 ⊢ Ⅎ𝑛( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ
19		fnlimfvre.p	. . . . . . 7 ⊢ Ⅎ𝑚𝜑
20		nfv 1911	. . . . . . 7 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
21		nfcv 2977	. . . . . . . 8 ⊢ Ⅎ𝑚𝑋
22		nfii1 4953	. . . . . . . 8 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
23	21, 22	nfel 2992	. . . . . . 7 ⊢ Ⅎ𝑚 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
24	19, 20, 23	nf3an 1898	. . . . . 6 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
25		uzssz 12263	. . . . . . . 8 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
26		fnlimfvre.z	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ_≥‘𝑀)
27	26	eleq2i 2904	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑀))
28	27	biimpi 218	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑀))
29	25, 28	sseldi 3964	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
30	29	3ad2ant2 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑛 ∈ ℤ)
31		eqid 2821	. . . . . 6 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
32	26	fvexi 6683	. . . . . . 7 ⊢ 𝑍 ∈ V
33	32	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑍 ∈ V)
34	26	uztrn2 12261	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ 𝑍)
35	34	ssd 41342	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
36	35	3ad2ant2 1130	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ⊆ 𝑍)
37		fvexd 6684	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑋) ∈ V)
38		fvexd 6684	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ∈ V)
39		ssidd 3989	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑛))
40		fvexd 6684	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
41		eqidd 2822	. . . . . 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 41950	. . . . 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 2904	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐷 ↔ 𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ })
44	43	biimpi 218	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐷 → 𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ })
45		nfcv 2977	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑋
46	7, 45	nffv 6679	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑋)
47	3, 46	nfmpt 5162	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))
48		nfcv 2977	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥dom ⇝
49	47, 48	nfel 2992	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝
50		fveq2 6669	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑋))
51	50	mpteq2dv 5161	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))
52	51	eleq1d 2897	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ))
53	45, 10, 49, 52	elrabf 3675	. . . . . . . . . . . . . 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 218	. . . . . . . . . . . . 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 498	. . . . . . . . . . . 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 483	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
58		nfmpt1 5163	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
59		nfcv 2977	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚dom ⇝
60	58, 59	nfel 2992	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
61		nfv 1911	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚 𝑗 ∈ 𝑍
62	61	nfci 2964	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚𝑍
63	62, 22	nfiun 4948	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
64	60, 63	nfrabw 3385	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
65	2, 64	nfcxfr 2975	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚𝐷
66	21, 65	nfel 2992	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝑋 ∈ 𝐷
67	66, 20	nfan 1896	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍)
68	29	adantl 484	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
69	32	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑍 ∈ V)
70	35	adantl 484	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
71		fvexd 6684	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑋) ∈ V)
72		fvexd 6684	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ∈ V)
73		ssidd 3989	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑛))
74		fvexd 6684	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
75		eqidd 2822	. . . . . . . . . . 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 41947	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ↔ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ))
77	57, 76	mpbid 234	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
78		climdm 14910	. . . . . . . . 9 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ↔ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
79	77, 78	sylib 220	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
80	1, 79	sylan 582	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
81	80	3adant3 1128	. . . . . 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 1187	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝜑)
83		simpl2 1188	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑛 ∈ 𝑍)
84		nfcv 2977	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗dom (𝐹‘𝑚)
85		fnlimfvre.m	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝐹
86		nfcv 2977	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝑗
87	85, 86	nffv 6679	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(𝐹‘𝑗)
88	87	nfdm 5822	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚dom (𝐹‘𝑗)
89		fveq2 6669	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
90	89	dmeqd 5773	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
91	84, 88, 90	cbviin 4961	. . . . . . . . . . . 12 ⊢ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗)
92	91	eleq2i 2904	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
93	92	biimpi 218	. . . . . . . . . 10 ⊢ (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
94	93	adantr 483	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
95		simpr 487	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ (ℤ_≥‘𝑛))
96		eliinid 41375	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
97	94, 95, 96	syl2anc 586	. . . . . . . 8 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
98	97	3ad2antl3 1183	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
99		simpr 487	. . . . . . 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 6684	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ_≥‘𝑛) → ((𝐹‘𝑗)‘𝑋) ∈ V)
102	87, 21	nffv 6679	. . . . . . . . . . 11 ⊢ Ⅎ𝑚((𝐹‘𝑗)‘𝑋)
103	89	fveq1d 6671	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑋) = ((𝐹‘𝑗)‘𝑋))
104		eqid 2821	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
105	86, 102, 103, 104	fvmptf 6788	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℤ_≥‘𝑛) ∧ ((𝐹‘𝑗)‘𝑋) ∈ V) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
106	100, 101, 105	syl2anc 586	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ_≥‘𝑛) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
107	106	adantl 484	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
108		simpll 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝜑)
109	34	adantll 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ 𝑍)
110	19, 61	nfan 1896	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑗 ∈ 𝑍)
111		nfcv 2977	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚ℝ
112	87, 88, 111	nff 6509	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚(𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ
113	110, 112	nfim 1893	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
114		eleq1w 2895	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → (𝑚 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
115	114	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → ((𝜑 ∧ 𝑚 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
116	89, 90	feq12d 6501	. . . . . . . . . . . . 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 2238	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
120	108, 109, 119	syl2anc 586	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
121	120	3adantl3 1164	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
122		simpl3 1189	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
123	121, 122	ffvelrnd 6851	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑗)‘𝑋) ∈ ℝ)
124	107, 123	eqeltrd 2913	. . . . . . 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 1369	. . . . . 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 14939	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
127	42, 126	eqeltrd 2913	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
128	127	3exp 1115	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)))
129	17, 18, 128	rexlimd 3317	. 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 398 ∧ w3a 1083 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 Ⅎwnfc 2961 ∃wrex 3139 {crab 3142 Vcvv 3494 ⊆ wss 3935 ∪ ciun 4918 ∩ ciin 4919 class class class wbr 5065 ↦ cmpt 5145 dom cdm 5554 ⟶wf 6350 ‘cfv 6354 ℝcr 10535 ℤcz 11980 ℤ_≥cuz 12242 ⇝ cli 14840

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613 ax-pre-sup 10614

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-iin 4921 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-pm 8408 df-en 8509 df-dom 8510 df-sdom 8511 df-sup 8905 df-inf 8906 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-2 11699 df-3 11700 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-fl 13161 df-seq 13369 df-exp 13429 df-cj 14457 df-re 14458 df-im 14459 df-sqrt 14593 df-abs 14594 df-clim 14844 df-rlim 14845

This theorem is referenced by: fnlimfvre2 41956 fnlimf 41957 smflimlem4 43049 smflim 43052

W3C validator