smflimsuplem4 - Mathbox for Glauco Siliprandi

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Theorem smflimsuplem4 47278

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 1922	. . . 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 45868	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
6		eqid 2741	. . . 4 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
7		fvexd 6845	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑥) ∈ V)
8		fvexd 6845	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑥) ∈ V)
9	1, 2, 5, 3, 6, 7, 8	limsupequzmpt 46184	. . 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 482	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑆 ∈ SAlg)
12	3, 4	uzssd2 45872	. . . . . . . . 9 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
13	12	sselda 3916	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ 𝑍)
14		smflimsuplem4.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
15	14	ffvelcdmda 7028	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
16	13, 15	syldan 598	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
17		eqid 2741	. . . . . . 7 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
18	11, 16, 17	smff 47187	. . . . . 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 47275	. . . . . . 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 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)dom (𝐻‘𝑛))
24		simpr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ (ℤ_≥‘𝑁))
25		fveq2 6830	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐻‘𝑛) = (𝐻‘𝑚))
26	25	dmeqd 5853	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → dom (𝐻‘𝑛) = dom (𝐻‘𝑚))
27	26	eleq2d 2827	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ dom (𝐻‘𝑛) ↔ 𝑥 ∈ dom (𝐻‘𝑚)))
28	23, 24, 27	eliind 45532	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ dom (𝐻‘𝑚))
29	21, 28	sseldd 3917	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ dom (𝐹‘𝑚))
30	18, 29	ffvelcdmd 7029	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
31	30	rexrd 11191	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ^*)
32	1, 5, 6, 31	limsupvaluzmpt 46172	. . 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 2776	. 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 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (ℤ_≥‘𝑁) ⊆ 𝑍)
36		simpr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ (ℤ_≥‘𝑁))
37	35, 36	sseldd 3917	. . . . . . . . . . . 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 6843	. . . . . . . . . . . . . . 15 ⊢ (𝐸‘𝑛) ∈ V
40	39	mptex 7170	. . . . . . . . . . . . . 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 6952	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
43	37, 42	syldan 598	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
44	43	dmeqd 5853	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → dom (𝐻‘𝑛) = dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
45		xrltso 13087	. . . . . . . . . . . . 13 ⊢ < Or ℝ^*
46	45	supex 9371	. . . . . . . . . . . 12 ⊢ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ V
47		eqid 2741	. . . . . . . . . . . 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 6632	. . . . . . . . . . 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 2776	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → dom (𝐻‘𝑛) = (𝐸‘𝑛))
51	34, 50	iineq2d 4947	. . . . . . . 8 ⊢ (𝜑 → ∩ 𝑛 ∈ (ℤ_≥‘𝑁)dom (𝐻‘𝑛) = ∩ 𝑛 ∈ (ℤ_≥‘𝑁)(𝐸‘𝑛))
52	22, 51	eleqtrd 2843	. . . . . . 7 ⊢ (𝜑 → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)(𝐸‘𝑛))
53	52	adantr 482	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)(𝐸‘𝑛))
54		eliinid 45570	. . . . . 6 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)(𝐸‘𝑛) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ (𝐸‘𝑛))
55	53, 36, 54	syl2anc 591	. . . . 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 6952	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑥 ∈ (𝐸‘𝑛)) → ((𝐻‘𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))
58	55, 57	mpdan 694	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))
59		eqid 2741	. . . . . . . . . 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 45858	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
61		eqid 2741	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
62	60, 61	uzn0d 45880	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
63		fvex 6843	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑚) ∈ V
64	63	dmex 7853	. . . . . . . . . . . . . 14 ⊢ dom (𝐹‘𝑚) ∈ V
65	64	rgenw 3059	. . . . . . . . . . . . 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 45592	. . . . . . . . . . 11 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
68	67	adantl 483	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
69	59, 68	rabexd 5270	. . . . . . . . 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 598	. . . . . . . 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 6950	. . . . . . . 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 591	. . . . . . 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 2843	. . . . . 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 3414	. . . . . 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 220	. . . . 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 497	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ)
77	58, 76	eqeltrd 2841	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) ∈ ℝ)
78	34, 58	mpteq2da 5166	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑥)) = (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
79		nfv 1922	. . . . 5 ⊢ Ⅎ𝑘𝜑
80		fveq2 6830	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑘))
81	80	mpteq1d 5164	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)))
82	81	rneqd 5886	. . . . . 6 ⊢ (𝑛 = 𝑘 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)))
83	82	supeq1d 9353	. . . . 5 ⊢ (𝑛 = 𝑘 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
84		nfv 1922	. . . . . . . 8 ⊢ Ⅎ𝑚(𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1))
85		eluzelz 12793	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ ℤ)
86	85	adantr 482	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℤ)
87		simpr 486	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 = (𝑛 + 1))
88	86	peano2zd 12631	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) ∈ ℤ)
89	87, 88	eqeltrd 2841	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℤ)
90	86	zred 12628	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℝ)
91	89	zred 12628	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℝ)
92	90	ltp1d 12081	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < (𝑛 + 1))
93	87	eqcomd 2747	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) = 𝑘)
94	92, 93	breqtrd 5100	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < 𝑘)
95	90, 91, 94	ltled 11290	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ≤ 𝑘)
96	61, 86, 89, 95	eluzd 45864	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ (ℤ_≥‘𝑛))
97		uzss 12806	. . . . . . . . 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 6845	. . . . . . . 8 ⊢ (((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → ((𝐹‘𝑚)‘𝑥) ∈ V)
100	84, 98, 99	rnmptss2 45713	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
101	100	3adant1 1137	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
102		nfv 1922	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁))
103		eqid 2741	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥))
104		simpll 773	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
105	37, 104	syldanl 609	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
106	6	uztrn2 12802	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
107	106	adantll 721	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
108	105, 107, 30	syl2anc 591	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
109	102, 103, 108	rnmptssd 7068	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ℝ)
110		ressxr 11185	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
111	110	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ℝ ⊆ ℝ^*)
112	109, 111	sstrd 3926	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ℝ^*)
113	112	3adant3 1139	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ℝ^*)
114		supxrss 13279	. . . . . 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 591	. . . . 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 6844	. . . . . . . . 9 ⊢ 𝑍 ∈ V
118	117	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ V)
119		fvexd 6845	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑥) ∈ V)
120		fvexd 6845	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘𝑁) ∈ V)
121	34, 36	ssdf 45536	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑁))
122		fvexd 6845	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) ∈ V)
123		eqidd 2742	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) = ((𝐻‘𝑛)‘𝑥))
124	34, 5, 6, 118, 12, 119, 120, 121, 122, 123	climeldmeqmpt 46123	. . . . . . 7 ⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ ↔ (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ ))
125	116, 124	mpbid 234	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ )
126	78, 125	eqeltrrd 2842	. . . . 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 46169	. . . 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 5096	. . 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 46139	. 2 ⊢ (𝜑 → inf(ran (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )), ℝ^, < ) ∈ ℝ)
130	33, 129	eqeltrd 2841	1 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 397 ∧ w3a 1093 = wceq 1548 Ⅎwnf 1791 ∈ wcel 2121 ∀wral 3055 {crab 3393 Vcvv 3433 ⊆ wss 3884 ∩ ciin 4924 class class class wbr 5074 ↦ cmpt 5155 dom cdm 5620 ran crn 5621 ⟶wf 6484 ‘cfv 6488 (class class class)co 7359 supcsup 9347 infcinf 9348 ℝcr 11033 1c1 11035 + caddc 11037 ℝ^*cxr 11174 < clt 11175 ≤ cle 11176 ℤcz 12519 ℤ_≥cuz 12783 lim supclsp 15427 ⇝ cli 15441 SAlgcsalg 46763 SMblFncsmblfn 47150

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1975 ax-7 2016 ax-8 2123 ax-9 2131 ax-10 2154 ax-11 2170 ax-12 2191 ax-ext 2713 ax-rep 5201 ax-sep 5220 ax-nul 5230 ax-pow 5296 ax-pr 5364 ax-un 7681 ax-cnex 11090 ax-resscn 11091 ax-1cn 11092 ax-icn 11093 ax-addcl 11094 ax-addrcl 11095 ax-mulcl 11096 ax-mulrcl 11097 ax-mulcom 11098 ax-addass 11099 ax-mulass 11100 ax-distr 11101 ax-i2m1 11102 ax-1ne0 11103 ax-1rid 11104 ax-rnegex 11105 ax-rrecex 11106 ax-cnre 11107 ax-pre-lttri 11108 ax-pre-lttrn 11109 ax-pre-ltadd 11110 ax-pre-mulgt0 11111 ax-pre-sup 11112

This theorem depends on definitions: df-bi 209 df-an 398 df-or 855 df-3or 1094 df-3an 1095 df-tru 1551 df-fal 1561 df-ex 1788 df-nf 1792 df-sb 2075 df-mo 2545 df-eu 2575 df-clab 2720 df-cleq 2733 df-clel 2816 df-nfc 2890 df-ne 2937 df-nel 3041 df-ral 3056 df-rex 3066 df-rmo 3346 df-reu 3347 df-rab 3394 df-v 3435 df-sbc 3725 df-csb 3833 df-dif 3887 df-un 3889 df-in 3891 df-ss 3901 df-pss 3904 df-nul 4264 df-if 4457 df-pw 4533 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4841 df-int 4880 df-iun 4925 df-iin 4926 df-br 5075 df-opab 5137 df-mpt 5156 df-tr 5182 df-id 5515 df-eprel 5520 df-po 5528 df-so 5529 df-fr 5573 df-we 5575 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6255 df-ord 6316 df-on 6317 df-lim 6318 df-suc 6319 df-iota 6444 df-fun 6490 df-fn 6491 df-f 6492 df-f1 6493 df-fo 6494 df-f1o 6495 df-fv 6496 df-riota 7316 df-ov 7362 df-oprab 7363 df-mpo 7364 df-om 7810 df-1st 7933 df-2nd 7934 df-frecs 8224 df-wrecs 8255 df-recs 8304 df-rdg 8343 df-1o 8399 df-2o 8400 df-er 8637 df-pm 8770 df-en 8888 df-dom 8889 df-sdom 8890 df-fin 8891 df-sup 9349 df-inf 9350 df-pnf 11177 df-mnf 11178 df-xr 11179 df-ltxr 11180 df-le 11181 df-sub 11375 df-neg 11376 df-div 11804 df-nn 12170 df-2 12239 df-3 12240 df-n0 12433 df-z 12520 df-uz 12784 df-q 12894 df-rp 12938 df-ioo 13297 df-ico 13299 df-fz 13457 df-fl 13746 df-seq 13959 df-exp 14019 df-cj 15056 df-re 15057 df-im 15058 df-sqrt 15192 df-abs 15193 df-limsup 15428 df-clim 15445 df-rlim 15446 df-smblfn 47151

This theorem is referenced by: smflimsuplem7 47281

W3C validator