smflimsuplem5 - Mathbox for Glauco Siliprandi

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Theorem smflimsuplem5 46861

Description: 𝐻 converges to the superior limit of 𝐹. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Hypotheses**
Ref	Expression
smflimsuplem5.a	⊢ Ⅎ𝑛𝜑
smflimsuplem5.b	⊢ Ⅎ𝑚𝜑
smflimsuplem5.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflimsuplem5.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
smflimsuplem5.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimsuplem5.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsuplem5.e	⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
smflimsuplem5.h	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
smflimsuplem5.r	⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
smflimsuplem5.n	⊢ (𝜑 → 𝑁 ∈ 𝑍)
smflimsuplem5.x	⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚))

**Assertion**
Ref	Expression
smflimsuplem5	⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))))

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

Proof of Theorem smflimsuplem5 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smflimsuplem5.a	. . 3 ⊢ Ⅎ𝑛𝜑
2		smflimsuplem5.n	. . . . . . . 8 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
3		smflimsuplem5.z	. . . . . . . . . . . 12 ⊢ 𝑍 = (ℤ_≥‘𝑀)
4	3	eleq2i 2823	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ_≥‘𝑀))
5	4	biimpi 216	. . . . . . . . . 10 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ_≥‘𝑀))
6		uzss 12752	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
7	5, 6	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑍 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
8	7, 3	sseqtrrdi 3976	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 → (ℤ_≥‘𝑁) ⊆ 𝑍)
9	2, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
10	9	sselda 3934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ 𝑍)
11		smflimsuplem5.e	. . . . . . . . . 10 ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
12		nfcv 2894	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑍
13		nfrab1 3415	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ}
14	12, 13	nfmpt 5189	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
15	11, 14	nfcxfr 2892	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐸
16		nfcv 2894	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑛
17	15, 16	nffv 6832	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐸‘𝑛)
18		fvex 6835	. . . . . . . 8 ⊢ (𝐸‘𝑛) ∈ V
19	17, 18	mptexf 45273	. . . . . . 7 ⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) ∈ V
20	19	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )) ∈ V)
21		smflimsuplem5.h	. . . . . . 7 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
22	21	fvmpt2 6940	. . . . . 6 ⊢ ((𝑛 ∈ 𝑍 ∧ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )) ∈ V) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )))
23	10, 20, 22	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
24	23	fveq1d 6824	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) = ((𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))‘𝑋))
25		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑦(𝐸‘𝑛)
26		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑦sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )
27		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^*, < )
28		fveq2 6822	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
29	28	mpteq2dv 5185	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
30	29	rneqd 5878	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
31	30	supeq1d 9330	. . . . . 6 ⊢ (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ))
32	17, 25, 26, 27, 31	cbvmptf 5191	. . . . 5 ⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < )) = (𝑦 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ))
33		simpl 482	. . . . . . . . 9 ⊢ ((𝑦 = 𝑋 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑦 = 𝑋)
34	33	fveq2d 6826	. . . . . . . 8 ⊢ ((𝑦 = 𝑋 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
35	34	mpteq2dva 5184	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
36	35	rneqd 5878	. . . . . 6 ⊢ (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
37	36	supeq1d 9330	. . . . 5 ⊢ (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^, < ))
38	37	eleq1d 2816	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^, < ) ∈ ℝ))
39		uzss 12752	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑁))
40		iinss1 4957	. . . . . . . . . . 11 ⊢ ((ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑁) → ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
41	39, 40	syl 17	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
42	41	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
43		smflimsuplem5.x	. . . . . . . . . 10 ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚))
44	43	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚))
45	42, 44	sseldd 3935	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
46		smflimsuplem5.b	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝜑
47		nfv 1915	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑛 ∈ (ℤ_≥‘𝑁)
48	46, 47	nfan 1900	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁))
49		eqid 2731	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
50		simpll 766	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
51	39	sselda 3934	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
52	51	adantll 714	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
53		smflimsuplem5.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ SAlg)
54	53	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑆 ∈ SAlg)
55		simpl 482	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝜑)
56	9	sselda 3934	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ 𝑍)
57		smflimsuplem5.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
58	57	ffvelcdmda 7017	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
59	55, 56, 58	syl2anc 584	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
60		eqid 2731	. . . . . . . . . . . . 13 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
61	54, 59, 60	smff 46769	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
62		eliin 4946	. . . . . . . . . . . . . . . 16 ⊢ (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚) → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚)))
63	43, 62	syl 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚)))
64	43, 63	mpbid 232	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚))
65	64	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚))
66		simpr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ (ℤ_≥‘𝑁))
67		rspa 3221	. . . . . . . . . . . . 13 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚) ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ dom (𝐹‘𝑚))
68	65, 66, 67	syl2anc 584	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ dom (𝐹‘𝑚))
69	61, 68	ffvelcdmd 7018	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
70	50, 52, 69	syl2anc 584	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
71		eluzelz 12739	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ ℤ)
72	71	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ ℤ)
73		smflimsuplem5.m	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℤ)
74	73	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑀 ∈ ℤ)
75		fvex 6835	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑚)‘𝑋) ∈ V
76	75	a1i 11	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑋) ∈ V)
77	48, 72, 74, 49, 3, 70, 76	limsupequzmpt 45766	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
78		smflimsuplem5.r	. . . . . . . . . . . . 13 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
79	78	adantr 480	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
80	77, 79	eqeltrd 2831	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
81	80	renepnfd 11160	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
82	48, 49, 70, 81	limsupubuzmpt 45756	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦)
83		uzid2 45442	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ (ℤ_≥‘𝑛))
84	83	ne0d 4292	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑛) ≠ ∅)
85	84	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (ℤ_≥‘𝑛) ≠ ∅)
86	48, 85, 70	supxrre3rnmpt 45466	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦))
87	82, 86	mpbird 257	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ) ∈ ℝ)
88	38, 45, 87	elrabd 3649	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ {𝑦 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^*, < ) ∈ ℝ})
89		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑥 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑦 = 𝑥)
90	89	fveq2d 6826	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑥 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑥))
91	90	mpteq2dva 5184	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
92	91	rneqd 5878	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
93	92	supeq1d 9330	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
94	93	eleq1d 2816	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ))
95	94	cbvrabv 3405	. . . . . . 7 ⊢ {𝑦 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ) ∈ ℝ}
96	88, 95	eleqtrdi 2841	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
97		eqid 2731	. . . . . . . 8 ⊢ {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}
98		fvex 6835	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑚) ∈ V
99	98	dmex 7839	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) ∈ V
100	99	rgenw 3051	. . . . . . . . . . 11 ⊢ ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V
101	100	a1i 11	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → ∀𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
102	84, 101	iinexd 45169	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
103	102	adantl 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
104	97, 103	rabexd 5278	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ} ∈ V)
105	11	fvmpt2 6940	. . . . . . 7 ⊢ ((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})
106	10, 104, 105	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ})
107	96, 106	eleqtrrd 2834	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ (𝐸‘𝑛))
108	32, 37, 107, 87	fvmptd3 6952	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))‘𝑋) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^, < ))
109	24, 108	eqtrd 2766	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ))
110	1, 109	mpteq2da 5183	. 2 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) = (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < )))
111	3	eluzelz2 45440	. . . 4 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)
112	2, 111	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
113		eqid 2731	. . 3 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
114	75	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
115	75	a1i 11	. . . . 5 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ((𝐹‘𝑚)‘𝑋) ∈ V)
116	46, 112, 73, 113, 3, 114, 115	limsupequzmpt 45766	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
117	116, 78	eqeltrd 2831	. . 3 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
118	46, 112, 113, 69, 117	supcnvlimsupmpt 45778	. 2 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < )) ⇝ (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))))
119	110, 118	eqbrtrd 5113	1 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1541 Ⅎwnf 1784 ∈ wcel 2111 ≠ wne 2928 ∀wral 3047 ∃wrex 3056 {crab 3395 Vcvv 3436 ⊆ wss 3902 ∅c0 4283 ∩ ciin 4942 class class class wbr 5091 ↦ cmpt 5172 dom cdm 5616 ran crn 5617 ⟶wf 6477 ‘cfv 6481 supcsup 9324 ℝcr 11002 ℝ^*cxr 11142 < clt 11143 ≤ cle 11144 ℤcz 12465 ℤ_≥cuz 12729 lim supclsp 15374 ⇝ cli 15388 SAlgcsalg 46345 SMblFncsmblfn 46732

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2113 ax-9 2121 ax-10 2144 ax-11 2160 ax-12 2180 ax-ext 2703 ax-rep 5217 ax-sep 5234 ax-nul 5244 ax-pow 5303 ax-pr 5370 ax-un 7668 ax-cnex 11059 ax-resscn 11060 ax-1cn 11061 ax-icn 11062 ax-addcl 11063 ax-addrcl 11064 ax-mulcl 11065 ax-mulrcl 11066 ax-mulcom 11067 ax-addass 11068 ax-mulass 11069 ax-distr 11070 ax-i2m1 11071 ax-1ne0 11072 ax-1rid 11073 ax-rnegex 11074 ax-rrecex 11075 ax-cnre 11076 ax-pre-lttri 11077 ax-pre-lttrn 11078 ax-pre-ltadd 11079 ax-pre-mulgt0 11080 ax-pre-sup 11081

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2535 df-eu 2564 df-clab 2710 df-cleq 2723 df-clel 2806 df-nfc 2881 df-ne 2929 df-nel 3033 df-ral 3048 df-rex 3057 df-rmo 3346 df-reu 3347 df-rab 3396 df-v 3438 df-sbc 3742 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4476 df-pw 4552 df-sn 4577 df-pr 4579 df-tp 4581 df-op 4583 df-uni 4860 df-int 4898 df-iun 4943 df-iin 4944 df-br 5092 df-opab 5154 df-mpt 5173 df-tr 5199 df-id 5511 df-eprel 5516 df-po 5524 df-so 5525 df-fr 5569 df-we 5571 df-xp 5622 df-rel 5623 df-cnv 5624 df-co 5625 df-dm 5626 df-rn 5627 df-res 5628 df-ima 5629 df-pred 6248 df-ord 6309 df-on 6310 df-lim 6311 df-suc 6312 df-iota 6437 df-fun 6483 df-fn 6484 df-f 6485 df-f1 6486 df-fo 6487 df-f1o 6488 df-fv 6489 df-riota 7303 df-ov 7349 df-oprab 7350 df-mpo 7351 df-om 7797 df-1st 7921 df-2nd 7922 df-frecs 8211 df-wrecs 8242 df-recs 8291 df-rdg 8329 df-1o 8385 df-2o 8386 df-er 8622 df-pm 8753 df-en 8870 df-dom 8871 df-sdom 8872 df-fin 8873 df-sup 9326 df-inf 9327 df-pnf 11145 df-mnf 11146 df-xr 11147 df-ltxr 11148 df-le 11149 df-sub 11343 df-neg 11344 df-div 11772 df-nn 12123 df-2 12185 df-3 12186 df-n0 12379 df-z 12466 df-uz 12730 df-q 12844 df-rp 12888 df-ioo 13246 df-ico 13248 df-fz 13405 df-fl 13693 df-ceil 13694 df-seq 13906 df-exp 13966 df-cj 15003 df-re 15004 df-im 15005 df-sqrt 15139 df-abs 15140 df-limsup 15375 df-clim 15392 df-smblfn 46733

This theorem is referenced by: smflimsuplem6 46862 smflimsuplem8 46864

W3C validator