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 44390

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 2904	. . . . . . . 8 ⊢ Ⅎ𝑥𝑍
4		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑥(ℤ_≥‘𝑛)
5		fnlimfvre.n	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝐹
6		nfcv 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑚
7	5, 6	nffv 6902	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹‘𝑚)
8	7	nfdm 5951	. . . . . . . . 9 ⊢ Ⅎ𝑥dom (𝐹‘𝑚)
9	4, 8	nfiin 5029	. . . . . . . 8 ⊢ Ⅎ𝑥∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
10	3, 9	nfiun 5028	. . . . . . 7 ⊢ Ⅎ𝑥∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
11	10	ssrab2f 43806	. . . . . 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 4017	. . . . 5 ⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
13	12	sseli 3979	. . . 4 ⊢ (𝑋 ∈ 𝐷 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
14		eliun 5002	. . . 4 ⊢ (𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ ∃𝑛 ∈ 𝑍 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
15	13, 14	sylib 217	. . 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 1918	. . 3 ⊢ Ⅎ𝑛𝜑
18		nfv 1918	. . 3 ⊢ Ⅎ𝑛( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ
19		fnlimfvre.p	. . . . . . 7 ⊢ Ⅎ𝑚𝜑
20		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑚 𝑛 ∈ 𝑍
21		nfcv 2904	. . . . . . . 8 ⊢ Ⅎ𝑚𝑋
22		nfii1 5033	. . . . . . . 8 ⊢ Ⅎ𝑚∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
23	21, 22	nfel 2918	. . . . . . 7 ⊢ Ⅎ𝑚 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
24	19, 20, 23	nf3an 1905	. . . . . 6 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
25		uzssz 12843	. . . . . . . 8 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
26		fnlimfvre.z	. . . . . . . . . 10 ⊢ 𝑍 = (ℤ_≥‘𝑀)
27	26	eleq2i 2826	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑀))
28	27	biimpi 215	. . . . . . . 8 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑀))
29	25, 28	sselid 3981	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
30	29	3ad2ant2 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑛 ∈ ℤ)
31		eqid 2733	. . . . . 6 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
32	26	fvexi 6906	. . . . . . 7 ⊢ 𝑍 ∈ V
33	32	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → 𝑍 ∈ V)
34	26	uztrn2 12841	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ 𝑍)
35	34	ssd 43769	. . . . . . 7 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
36	35	3ad2ant2 1135	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ⊆ 𝑍)
37		fvexd 6907	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑋) ∈ V)
38		fvexd 6907	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ∈ V)
39		ssidd 4006	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑛))
40		fvexd 6907	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
41		eqidd 2734	. . . . . 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 44387	. . . . 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 2826	. . . . . . . . . . . . 13 ⊢ (𝑋 ∈ 𝐷 ↔ 𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ })
44	43	biimpi 215	. . . . . . . . . . . 12 ⊢ (𝑋 ∈ 𝐷 → 𝑋 ∈ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ })
45		nfcv 2904	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥𝑋
46	7, 45	nffv 6902	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑥((𝐹‘𝑚)‘𝑋)
47	3, 46	nfmpt 5256	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))
48		nfcv 2904	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑥dom ⇝
49	47, 48	nfel 2918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑥(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝
50		fveq2 6892	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑋))
51	50	mpteq2dv 5251	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑋 → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))
52	51	eleq1d 2819	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑋 → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ ↔ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ))
53	45, 10, 49, 52	elrabf 3680	. . . . . . . . . . . . . 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 215	. . . . . . . . . . . . 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 497	. . . . . . . . . . . 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 482	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
58		nfmpt1 5257	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))
59		nfcv 2904	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚dom ⇝
60	58, 59	nfel 2918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝
61		nfv 1918	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑚 𝑗 ∈ 𝑍
62	61	nfci 2887	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑚𝑍
63	62, 22	nfiun 5028	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
64	60, 63	nfrabw 3469	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
65	2, 64	nfcxfr 2902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚𝐷
66	21, 65	nfel 2918	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚 𝑋 ∈ 𝐷
67	66, 20	nfan 1903	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍)
68	29	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
69	32	a1i 11	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → 𝑍 ∈ V)
70	35	adantl 483	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
71		fvexd 6907	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑋) ∈ V)
72		fvexd 6907	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ∈ V)
73		ssidd 4006	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑛))
74		fvexd 6907	. . . . . . . . . . 11 ⊢ (((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
75		eqidd 2734	. . . . . . . . . . 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 44384	. . . . . . . . . 10 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → ((𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ↔ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ))
77	57, 76	mpbid 231	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ )
78		climdm 15498	. . . . . . . . 9 ⊢ ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ∈ dom ⇝ ↔ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
79	77, 78	sylib 217	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝐷 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
80	1, 79	sylan 581	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) ⇝ ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))))
81	80	3adant3 1133	. . . . . 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 2904	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑗dom (𝐹‘𝑚)
85		fnlimfvre.m	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝐹
86		nfcv 2904	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑚𝑗
87	85, 86	nffv 6902	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚(𝐹‘𝑗)
88	87	nfdm 5951	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚dom (𝐹‘𝑗)
89		fveq2 6892	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → (𝐹‘𝑚) = (𝐹‘𝑗))
90	89	dmeqd 5906	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → dom (𝐹‘𝑚) = dom (𝐹‘𝑗))
91	84, 88, 90	cbviin 5041	. . . . . . . . . . . 12 ⊢ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗)
92	91	eleq2i 2826	. . . . . . . . . . 11 ⊢ (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
93	92	biimpi 215	. . . . . . . . . 10 ⊢ (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
94	93	adantr 482	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗))
95		simpr 486	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ (ℤ_≥‘𝑛))
96		eliinid 43800	. . . . . . . . 9 ⊢ ((𝑋 ∈ ∩ 𝑗 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑗) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑗))
97	94, 95, 96	syl2anc 585	. . . . . . . 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 486	. . . . . . 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 6907	. . . . . . . . . 10 ⊢ (𝑗 ∈ (ℤ_≥‘𝑛) → ((𝐹‘𝑗)‘𝑋) ∈ V)
102	87, 21	nffv 6902	. . . . . . . . . . 11 ⊢ Ⅎ𝑚((𝐹‘𝑗)‘𝑋)
103	89	fveq1d 6894	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚)‘𝑋) = ((𝐹‘𝑗)‘𝑋))
104		eqid 2733	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
105	86, 102, 103, 104	fvmptf 7020	. . . . . . . . . 10 ⊢ ((𝑗 ∈ (ℤ_≥‘𝑛) ∧ ((𝐹‘𝑗)‘𝑋) ∈ V) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
106	100, 101, 105	syl2anc 585	. . . . . . . . 9 ⊢ (𝑗 ∈ (ℤ_≥‘𝑛) → ((𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑗) = ((𝐹‘𝑗)‘𝑋))
107	106	adantl 483	. . . . . . . 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 713	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → 𝑗 ∈ 𝑍)
110	19, 61	nfan 1903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑗 ∈ 𝑍)
111		nfcv 2904	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑚ℝ
112	87, 88, 111	nff 6714	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑚(𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ
113	110, 112	nfim 1900	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
114		eleq1w 2817	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑗 → (𝑚 ∈ 𝑍 ↔ 𝑗 ∈ 𝑍))
115	114	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → ((𝜑 ∧ 𝑚 ∈ 𝑍) ↔ (𝜑 ∧ 𝑗 ∈ 𝑍)))
116	89, 90	feq12d 6706	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑗 → ((𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ ↔ (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ))
117	115, 116	imbi12d 345	. . . . . . . . . . . 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 2234	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑗 ∈ 𝑍) → (𝐹‘𝑗):dom (𝐹‘𝑗)⟶ℝ)
120	108, 109, 119	syl2anc 585	. . . . . . . . . 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 7088	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ dom (𝐹‘𝑗)) ∧ 𝑗 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑗)‘𝑋) ∈ ℝ)
124	107, 123	eqeltrd 2834	. . . . . . 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 1374	. . . . . 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 15527	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ( ⇝ ‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
127	42, 126	eqeltrd 2834	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
128	127	3exp 1120	. . 3 ⊢ (𝜑 → (𝑛 ∈ 𝑍 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) → ( ⇝ ‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)))
129	17, 18, 128	rexlimd 3264	. 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 397 ∧ w3a 1088 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 Ⅎwnfc 2884 ∃wrex 3071 {crab 3433 Vcvv 3475 ⊆ wss 3949 ∪ ciun 4998 ∩ ciin 4999 class class class wbr 5149 ↦ cmpt 5232 dom cdm 5677 ⟶wf 6540 ‘cfv 6544 ℝcr 11109 ℤcz 12558 ℤ_≥cuz 12822 ⇝ cli 15428

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 ax-pre-sup 11188

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3377 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-iin 5001 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-er 8703 df-pm 8823 df-en 8940 df-dom 8941 df-sdom 8942 df-sup 9437 df-inf 9438 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-div 11872 df-nn 12213 df-2 12275 df-3 12276 df-n0 12473 df-z 12559 df-uz 12823 df-rp 12975 df-fl 13757 df-seq 13967 df-exp 14028 df-cj 15046 df-re 15047 df-im 15048 df-sqrt 15182 df-abs 15183 df-clim 15432 df-rlim 15433

This theorem is referenced by: fnlimfvre2 44393 fnlimf 44394 smflimlem4 45490 smflim 45493

W3C validator