smflimsuplem4 - Mathbox for Glauco Siliprandi

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Theorem smflimsuplem4 42659

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 1892	. . . 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 41248	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
6		eqid 2795	. . . 4 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
7		fvexd 6553	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑥) ∈ V)
8		fvexd 6553	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑥) ∈ V)
9	1, 2, 5, 3, 6, 7, 8	limsupequzmpt 41571	. . 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 41252	. . . . . . . . 9 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
13	12	sselda 3889	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ 𝑍)
14		smflimsuplem4.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
15	14	ffvelrnda 6716	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
16	13, 15	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
17		eqid 2795	. . . . . . 7 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
18	11, 16, 17	smff 42571	. . . . . 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 42656	. . . . . . 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 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ (ℤ_≥‘𝑁))
25		fveq2 6538	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (𝐻‘𝑛) = (𝐻‘𝑚))
26	25	dmeqd 5660	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → dom (𝐻‘𝑛) = dom (𝐻‘𝑚))
27	26	eleq2d 2868	. . . . . . . 8 ⊢ (𝑛 = 𝑚 → (𝑥 ∈ dom (𝐻‘𝑛) ↔ 𝑥 ∈ dom (𝐻‘𝑚)))
28	23, 24, 27	eliind 40891	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ dom (𝐻‘𝑚))
29	21, 28	sseldd 3890	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ dom (𝐹‘𝑚))
30	18, 29	ffvelrnd 6717	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
31	30	rexrd 10537	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ^*)
32	1, 5, 6, 31	limsupvaluzmpt 41559	. . 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 2831	. 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 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ (ℤ_≥‘𝑁))
37	35, 36	sseldd 3890	. . . . . . . . . . . 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 6551	. . . . . . . . . . . . . . 15 ⊢ (𝐸‘𝑛) ∈ V
40	39	mptex 6852	. . . . . . . . . . . . . 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 6647	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
43	37, 42	syldan 591	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
44	43	dmeqd 5660	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → dom (𝐻‘𝑛) = dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
45		xrltso 12384	. . . . . . . . . . . . 13 ⊢ < Or ℝ^*
46	45	supex 8773	. . . . . . . . . . . 12 ⊢ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ V
47		eqid 2795	. . . . . . . . . . . 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 6360	. . . . . . . . . . 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 2831	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → dom (𝐻‘𝑛) = (𝐸‘𝑛))
51	34, 50	iineq2d 4847	. . . . . . . 8 ⊢ (𝜑 → ∩ 𝑛 ∈ (ℤ_≥‘𝑁)dom (𝐻‘𝑛) = ∩ 𝑛 ∈ (ℤ_≥‘𝑁)(𝐸‘𝑛))
52	22, 51	eleqtrd 2885	. . . . . . 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 40936	. . . . . 6 ⊢ ((𝑥 ∈ ∩ 𝑛 ∈ (ℤ_≥‘𝑁)(𝐸‘𝑛) ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑥 ∈ (𝐸‘𝑛))
55	53, 36, 54	syl2anc 584	. . . . 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 6647	. . . . 5 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑥 ∈ (𝐸‘𝑛)) → ((𝐻‘𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))
58	55, 57	mpdan 683	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))
59		eqid 2795	. . . . . . . . . 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 41236	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
61		eqid 2795	. . . . . . . . . . . . 13 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
62	60, 61	uzn0d 41260	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
63		fvex 6551	. . . . . . . . . . . . . . 15 ⊢ (𝐹‘𝑚) ∈ V
64	63	dmex 7472	. . . . . . . . . . . . . 14 ⊢ dom (𝐹‘𝑚) ∈ V
65	64	rgenw 3117	. . . . . . . . . . . . 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 40958	. . . . . . . . . . 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 5127	. . . . . . . . 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 591	. . . . . . . 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 6645	. . . . . . . 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 584	. . . . . . 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 2885	. . . . . 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 3337	. . . . . 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 219	. . . . 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 496	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ)
77	58, 76	eqeltrd 2883	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) ∈ ℝ)
78	34, 58	mpteq2da 5054	. . . 4 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑥)) = (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
79		nfv 1892	. . . . 5 ⊢ Ⅎ𝑘𝜑
80		fveq2 6538	. . . . . . . 8 ⊢ (𝑛 = 𝑘 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑘))
81	80	mpteq1d 5049	. . . . . . 7 ⊢ (𝑛 = 𝑘 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)))
82	81	rneqd 5690	. . . . . 6 ⊢ (𝑛 = 𝑘 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)))
83	82	supeq1d 8756	. . . . 5 ⊢ (𝑛 = 𝑘 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
84		nfv 1892	. . . . . . . 8 ⊢ Ⅎ𝑚(𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1))
85		eluzelz 12103	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ ℤ)
86	85	adantr 481	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℤ)
87		simpr 485	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 = (𝑛 + 1))
88	86	peano2zd 11939	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) ∈ ℤ)
89	87, 88	eqeltrd 2883	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℤ)
90	86	zred 11936	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ∈ ℝ)
91	89	zred 11936	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ ℝ)
92	90	ltp1d 11418	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < (𝑛 + 1))
93	87	eqcomd 2801	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → (𝑛 + 1) = 𝑘)
94	92, 93	breqtrd 4988	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 < 𝑘)
95	90, 91, 94	ltled 10635	. . . . . . . . . 10 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑛 ≤ 𝑘)
96	61, 86, 89, 95	eluzd 41243	. . . . . . . . 9 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → 𝑘 ∈ (ℤ_≥‘𝑛))
97		uzss 12114	. . . . . . . . 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 6553	. . . . . . . 8 ⊢ (((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) ∧ 𝑚 ∈ (ℤ_≥‘𝑘)) → ((𝐹‘𝑚)‘𝑥) ∈ V)
100	84, 98, 99	rnmptss2 41089	. . . . . . 7 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
101	100	3adant1 1123	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
102		nfv 1892	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁))
103		eqid 2795	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥))
104		simpll 763	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
105	37, 104	syldanl 601	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
106	6	uztrn2 12111	. . . . . . . . . . 11 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
107	106	adantll 710	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
108	105, 107, 30	syl2anc 584	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑥) ∈ ℝ)
109	102, 103, 108	rnmptssd 41017	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ℝ)
110		ressxr 10531	. . . . . . . . 9 ⊢ ℝ ⊆ ℝ^*
111	110	a1i 11	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ℝ ⊆ ℝ^*)
112	109, 111	sstrd 3899	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ℝ^*)
113	112	3adant3 1125	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑘 = (𝑛 + 1)) → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) ⊆ ℝ^*)
114		supxrss 12575	. . . . . 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 584	. . . . 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 6552	. . . . . . . . 9 ⊢ 𝑍 ∈ V
118	117	a1i 11	. . . . . . . 8 ⊢ (𝜑 → 𝑍 ∈ V)
119		fvexd 6553	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝐻‘𝑛)‘𝑥) ∈ V)
120		fvexd 6553	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘𝑁) ∈ V)
121	34, 36	ssdf 40897	. . . . . . . 8 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑁))
122		fvexd 6553	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) ∈ V)
123		eqidd 2796	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑥) = ((𝐻‘𝑛)‘𝑥))
124	34, 5, 6, 118, 12, 119, 120, 121, 122, 123	climeldmeqmpt 41510	. . . . . . 7 ⊢ (𝜑 → ((𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ ↔ (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ ))
125	116, 124	mpbid 233	. . . . . 6 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ )
126	78, 125	eqeltrrd 2884	. . . . 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 41556	. . . 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 4984	. . 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 41526	. 2 ⊢ (𝜑 → inf(ran (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )), ℝ^, < ) ∈ ℝ)
130	33, 129	eqeltrd 2883	1 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1080 = wceq 1522 Ⅎwnf 1765 ∈ wcel 2081 ∀wral 3105 {crab 3109 Vcvv 3437 ⊆ wss 3859 ∩ ciin 4826 class class class wbr 4962 ↦ cmpt 5041 dom cdm 5443 ran crn 5444 ⟶wf 6221 ‘cfv 6225 (class class class)co 7016 supcsup 8750 infcinf 8751 ℝcr 10382 1c1 10384 + caddc 10386 ℝ^*cxr 10520 < clt 10521 ≤ cle 10522 ℤcz 11829 ℤ_≥cuz 12093 lim supclsp 14661 ⇝ cli 14675 SAlgcsalg 42155 SMblFncsmblfn 42539

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 ax-pre-sup 10461

This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-int 4783 df-iun 4827 df-iin 4828 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-1o 7953 df-oadd 7957 df-er 8139 df-pm 8259 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-n0 11746 df-z 11830 df-uz 12094 df-q 12198 df-rp 12240 df-ioo 12592 df-ico 12594 df-fz 12743 df-fl 13012 df-seq 13220 df-exp 13280 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-smblfn 42540

This theorem is referenced by: smflimsuplem7 42662

W3C validator