smflimsuplem4 - Mathbox for Glauco Siliprandi

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Theorem smflimsuplem4 41029

Description: If 𝐻 converges, the lim sup of 𝐹 is real. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
smflimsuplem4.1	⊢ Ⅎ𝑛𝜑
smflimsuplem4.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflimsuplem4.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
smflimsuplem4.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimsuplem4.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsuplem4.e	⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
smflimsuplem4.h	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
smflimsuplem4.n	⊢ (𝜑 → 𝑁 ∈ 𝑍)
smflimsuplem4.i	⊢ (𝜑 → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)dom (𝐻‘𝑛))
smflimsuplem4.c	⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ )

**Assertion**
Ref	Expression
smflimsuplem4	⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)

Distinct variable groups: 𝑛,𝐸,𝑥 𝑚,𝐹,𝑛,𝑥 𝑛,𝐻 𝑚,𝑀 𝑚,𝑁,𝑛 𝑚,𝑍,𝑛 𝜑,𝑚 Allowed substitution hints: 𝜑(𝑥,𝑛) 𝑆(𝑥,𝑚,𝑛) 𝐸(𝑚) 𝐻(𝑥,𝑚) 𝑀(𝑥,𝑛) 𝑁(𝑥) 𝑍(𝑥)

Proof of Theorem smflimsuplem4 Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1843	. . . 4 ⊢ Ⅎ𝑚𝜑
2		smflimsuplem4.m	. . . 4 ⊢ (𝜑 → 𝑀 ∈ ℤ)
3		smflimsuplem4.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4		smflimsuplem4.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
5	3, 4	eluzelz2d 39640	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
6		eqid 2622	. . . 4 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
7		fvexd 6203	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑥) ∈ V)
8		fvexd 6203	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑥) ∈ V)
9	1, 2, 5, 3, 6, 7, 8	limsupequzmpt 39961	. . 3 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑥))))
10		smflimsuplem4.s	. . . . . . . 8 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11	10	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑆 ∈ SAlg)
12	3, 4	uzssd2 39644	. . . . . . . . 9 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
13	12	sselda 3603	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ 𝑍)
14		smflimsuplem4.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
15	14	ffvelrnda 6359	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
16	13, 15	syldan 487	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
17		eqid 2622	. . . . . . 7 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
18	11, 16, 17	smff 40941	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
19		smflimsuplem4.e	. . . . . . . 8 ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
20		smflimsuplem4.h	. . . . . . . 8 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
21	3, 19, 20, 13	smflimsuplem1 41026	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → dom (𝐻‘𝑚) ⊆ dom (𝐹‘𝑚))
22		smflimsuplem4.i	. . . . . . . . 9 ⊢ (𝜑 → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)dom (𝐻‘𝑛))
23	22	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)dom (𝐻‘𝑛))
24		simpr 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ (ℤ_≥‘𝑁))
25		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐻‘𝑛) = (𝐻‘𝑚))
26	25	dmeqd 5326	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → dom (𝐻‘𝑛) = dom (𝐻‘𝑚))
27	26	eleq2d 2687	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ dom (𝐻‘𝑛) ↔ 𝑥 ∈ dom (𝐻‘𝑚)))
28	23, 24, 27	eliind 39240	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ dom (𝐻‘𝑚))
29	21, 28	sseldd 3604	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ dom (𝐹‘𝑚))
30	18, 29	ffvelrnd 6360	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
31	30	rexrd 10089	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ^*)
32	1, 5, 6, 31	limsupvaluzmpt 39949	. . 3 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑥))) = inf(ran (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )), ℝ^, < ))
33	9, 32	eqtrd 2656	. 2 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = inf(ran (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )), ℝ^, < ))
34		smflimsuplem4.1	. . 3 ⊢ Ⅎ𝑛𝜑
35	12	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (ℤ_≥‘𝑁) ⊆ 𝑍)
36		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ (ℤ_≥‘𝑁))
37	35, 36	sseldd 3604	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ 𝑍)
38	20	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))))
39		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐸‘𝑛) ∈ V
40	39	mptex 6486	. . . . . . . . . . . . . 14 ⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) ∈ V
41	40	a1i 11	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) ∈ V)
42	38, 41	fvmpt2d 6293	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
43	37, 42	syldan 487	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
44	43	dmeqd 5326	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → dom (𝐻‘𝑛) = dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
45		xrltso 11974	. . . . . . . . . . . . 13 ⊢ < Or ℝ^*
46	45	supex 8369	. . . . . . . . . . . 12 ⊢ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ V
47		eqid 2622	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
48	46, 47	dmmpti 6023	. . . . . . . . . . 11 ⊢ dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) = (𝐸‘𝑛)
49	48	a1i 11	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) = (𝐸‘𝑛))
50	44, 49	eqtrd 2656	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → dom (𝐻‘𝑛) = (𝐸‘𝑛))
51	34, 50	iineq2d 4541	. . . . . . . 8 ⊢ (𝜑 → ∩ 𝑛 ∈ (ℤ_≥‘𝑁)dom (𝐻‘𝑛) = ∩ 𝑛 ∈ (ℤ_≥‘𝑁)(𝐸‘𝑛))
52	22, 51	eleqtrd 2703	. . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)(𝐸‘𝑛))
53	52	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)(𝐸‘𝑛))
54		eliinid 39294	. . . . . 6 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)(𝐸‘𝑛) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ (𝐸‘𝑛))
55	53, 36, 54	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ (𝐸‘𝑛))
56	46	a1i 11	. . . . . 6 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑥 ∈ (𝐸‘𝑛)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ V)
57	43, 56	fvmpt2d 6293	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑥 ∈ (𝐸‘𝑛)) → ((𝐻‘𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))
58	55, 57	mpdan 702	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))
59		eqid 2622	. . . . . . . . . 10 ⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ}
60	3	eluzelz2 39627	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
61		eqid 2622	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
62	60, 61	uzn0d 39652	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
63		fvex 6201	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑚) ∈ V
64	63	dmex 7099	. . . . . . . . . . . . . 14 ⊢ dom (𝐹‘𝑚) ∈ V
65	64	rgenw 2924	. . . . . . . . . . . . 13 ⊢ ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V
66	65	a1i 11	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
67	62, 66	iinexd 39318	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
68	67	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
69	59, 68	rabexd 4814	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ} ∈ V)
70	37, 69	syldan 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ} ∈ V)
71	19	fvmpt2 6291	. . . . . . . 8 ⊢ ((𝑛 ∈ 𝑍 ∧ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} ∈ V) → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ})
72	37, 70, 71	syl2anc 693	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
73	55, 72	eleqtrd 2703	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
74		rabid 3116	. . . . . 6 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ} ↔ (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ))
75	73, 74	sylib 208	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ))
76	75	simprd 479	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ)
77	58, 76	eqeltrd 2701	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) ∈ ℝ)
78	34, 58	mpteq2da 4743	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑥)) = (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
79		nfv 1843	. . . . 5 ⊢ Ⅎ𝑘𝜑
80		fveq2 6191	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑘))
81	80	mpteq1d 4738	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)))
82	81	rneqd 5353	. . . . . 6 ⊢ (𝑛 = 𝑘 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)))
83	82	supeq1d 8352	. . . . 5 ⊢ (𝑛 = 𝑘 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
84		nfv 1843	. . . . . . . 8 ⊢ Ⅎ𝑚(𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1))
85		eluzelz 11697	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ ℤ)
86	85	adantr 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℤ)
87		simpr 477	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 = (𝑛 + 1))
88	86	peano2zd 11485	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) ∈ ℤ)
89	87, 88	eqeltrd 2701	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℤ)
90	86	zred 11482	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℝ)
91	89	zred 11482	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℝ)
92	90	ltp1d 10954	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < (𝑛 + 1))
93	87	eqcomd 2628	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) = 𝑘)
94	92, 93	breqtrd 4679	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < 𝑘)
95	90, 91, 94	ltled 10185	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ≤ 𝑘)
96	61, 86, 89, 95	eluzd 39635	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ (ℤ_≥‘𝑛))
97		uzss 11708	. . . . . . . . 9 ⊢ (𝑘 ∈ (ℤ_≥‘𝑛) → (ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑛))
98	96, 97	syl 17	. . . . . . . 8 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → (ℤ_≥‘𝑘) ⊆ (ℤ_≥‘𝑛))
99		fvexd 6203	. . . . . . . 8 ⊢ (((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → ((𝐹‘𝑚)‘𝑥) ∈ V)
100	84, 98, 99	rnmptss2 39472	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
101	100	3adant1 1079	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
102		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁))
103		eqid 2622	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥))
104		simpll 790	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
105	37, 104	syldanl 735	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
106	6	uztrn2 11705	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
107	106	adantll 750	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
108	105, 107, 30	syl2anc 693	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
109	102, 103, 108	rnmptssd 39385	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ℝ)
110		ressxr 10083	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
111	110	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ℝ ⊆ ℝ^*)
112	109, 111	sstrd 3613	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ℝ^*)
113	112	3adant3 1081	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ℝ^*)
114		supxrss 12162	. . . . . 6 ⊢ ((ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ∧ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ℝ^) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ≤ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))
115	101, 113, 114	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ≤ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
116		smflimsuplem4.c	. . . . . . 7 ⊢ (𝜑 → (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ )
117	3	fvexi 6202	. . . . . . . . 9 ⊢ 𝑍 ∈ V
118	117	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ V)
119		fvexd 6203	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑥) ∈ V)
120		fvexd 6203	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘𝑁) ∈ V)
121	34, 36	ssdf 39247	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑁))
122		fvexd 6203	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) ∈ V)
123		eqidd 2623	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) = ((𝐻‘𝑛)‘𝑥))
124	34, 5, 6, 118, 12, 119, 120, 121, 122, 123	climeldmeqmpt 39900	. . . . . . 7 ⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ ↔ (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ ))
125	116, 124	mpbid 222	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ )
126	78, 125	eqeltrrd 2702	. . . . 5 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) ∈ dom ⇝ )
127	34, 79, 5, 6, 76, 83, 115, 126	climinf2mpt 39946	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )) ⇝ inf(ran (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )), ℝ^*, < ))
128	78, 127	eqbrtrd 4675	. . 3 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑥)) ⇝ inf(ran (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )), ℝ^, < ))
129	34, 5, 6, 77, 128	climreclmpt 39916	. 2 ⊢ (𝜑 → inf(ran (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )), ℝ^, < ) ∈ ℝ)
130	33, 129	eqeltrd 2701	1 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 {crab 2916 Vcvv 3200 ⊆ wss 3574 ∩ ciin 4521 class class class wbr 4653 ↦ cmpt 4729 dom cdm 5114 ran crn 5115 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 supcsup 8346 infcinf 8347 ℝcr 9935 1c1 9937 + caddc 9939 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075 ℤcz 11377 ℤ_≥cuz 11687 lim supclsp 14201 ⇝ cli 14215 SAlgcsalg 40528 SMblFncsmblfn 40909

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-iin 4523 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-n0 11293 df-z 11378 df-uz 11688 df-q 11789 df-rp 11833 df-ioo 12179 df-ico 12181 df-fz 12327 df-fl 12593 df-seq 12802 df-exp 12861 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-smblfn 40910

This theorem is referenced by: smflimsuplem7 41032

W3C validator