smflimsuplem5 - Mathbox for Glauco Siliprandi

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Theorem smflimsuplem5 42530

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 2857	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ_≥‘𝑀))
5	4	biimpi 208	. . . . . . . . . 10 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ_≥‘𝑀))
6		uzss 12082	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
7	5, 6	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑍 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
8	7, 3	syl6sseqr 3910	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 → (ℤ_≥‘𝑁) ⊆ 𝑍)
9	2, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
10	9	sselda 3860	. . . . . 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 2932	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑍
13		nfrab1 3324	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ}
14	12, 13	nfmpt 5025	. . . . . . . . . 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 2930	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐸
16		nfcv 2932	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑛
17	15, 16	nffv 6511	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐸‘𝑛)
18		fvex 6514	. . . . . . . 8 ⊢ (𝐸‘𝑛) ∈ V
19	17, 18	mptexf 40935	. . . . . . 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 6607	. . . . . 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 576	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
24	23	fveq1d 6503	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) = ((𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))‘𝑋))
25		nfcv 2932	. . . . . 6 ⊢ Ⅎ𝑦(𝐸‘𝑛)
26		nfcv 2932	. . . . . 6 ⊢ Ⅎ𝑦sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )
27		nfcv 2932	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^*, < )
28		fveq2 6501	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
29	28	mpteq2dv 5024	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
30	29	rneqd 5652	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
31	30	supeq1d 8707	. . . . . 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 5027	. . . . 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 475	. . . . . . . . 9 ⊢ ((𝑦 = 𝑋 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑦 = 𝑋)
34	33	fveq2d 6505	. . . . . . . 8 ⊢ ((𝑦 = 𝑋 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
35	34	mpteq2dva 5023	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
36	35	rneqd 5652	. . . . . 6 ⊢ (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
37	36	supeq1d 8707	. . . . 5 ⊢ (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^, < ))
38	37	eleq1d 2850	. . . . . . . 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 12082	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑁))
40		iinss1 4807	. . . . . . . . . . 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 474	. . . . . . . . 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 473	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚))
45	42, 44	sseldd 3861	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
46		smflimsuplem5.b	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝜑
47		nfv 1873	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑛 ∈ (ℤ_≥‘𝑁)
48	46, 47	nfan 1862	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁))
49		eqid 2778	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
50		simpll 754	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
51	39	sselda 3860	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
52	51	adantll 701	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
53		smflimsuplem5.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ SAlg)
54	53	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑆 ∈ SAlg)
55		simpl 475	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝜑)
56	9	sselda 3860	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ 𝑍)
57		smflimsuplem5.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
58	57	ffvelrnda 6678	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
59	55, 56, 58	syl2anc 576	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
60		eqid 2778	. . . . . . . . . . . . 13 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
61	54, 59, 60	smff 42441	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
62		eliin 4798	. . . . . . . . . . . . . . . 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 224	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚))
65	64	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚))
66		simpr 477	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ (ℤ_≥‘𝑁))
67		rspa 3156	. . . . . . . . . . . . 13 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚) ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ dom (𝐹‘𝑚))
68	65, 66, 67	syl2anc 576	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ dom (𝐹‘𝑚))
69	61, 68	ffvelrnd 6679	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
70	50, 52, 69	syl2anc 576	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
71		eluzelz 12071	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ ℤ)
72	71	adantl 474	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ ℤ)
73		smflimsuplem5.m	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℤ)
74	73	adantr 473	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑀 ∈ ℤ)
75		fvex 6514	. . . . . . . . . . . . . 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 41442	. . . . . . . . . . . 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 473	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
80	77, 79	eqeltrd 2866	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
81	80	renepnfd 10493	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
82	48, 49, 70, 81	limsupubuzmpt 41432	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦)
83		uzid2 41109	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ (ℤ_≥‘𝑛))
84	83	ne0d 4189	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑛) ≠ ∅)
85	84	adantl 474	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (ℤ_≥‘𝑛) ≠ ∅)
86	48, 85, 70	supxrre3rnmpt 41135	. . . . . . . . 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 249	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ) ∈ ℝ)
88	38, 45, 87	elrabd 3598	. . . . . . 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 475	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑥 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑦 = 𝑥)
90	89	fveq2d 6505	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑥 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑥))
91	90	mpteq2dva 5023	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
92	91	rneqd 5652	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
93	92	supeq1d 8707	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
94	93	eleq1d 2850	. . . . . . . 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 3412	. . . . . . 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	syl6eleq 2876	. . . . . 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 2778	. . . . . . . 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 6514	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑚) ∈ V
99	98	dmex 7433	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) ∈ V
100	99	rgenw 3100	. . . . . . . . . . 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 40823	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
103	102	adantl 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
104	97, 103	rabexd 5093	. . . . . . 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 6607	. . . . . . 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 576	. . . . . 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 2869	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ (𝐸‘𝑛))
108	32, 37, 107, 87	fvmptd3 6619	. . . 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 2814	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ))
110	1, 109	mpteq2da 5022	. 2 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) = (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < )))
111	3	eluzelz2 41107	. . . 4 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)
112	2, 111	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
113		eqid 2778	. . 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 41442	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
117	116, 78	eqeltrd 2866	. . 3 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
118	46, 112, 113, 69, 117	supcnvlimsupmpt 41454	. 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 4952	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 198 ∧ wa 387 = wceq 1507 Ⅎwnf 1746 ∈ wcel 2050 ≠ wne 2967 ∀wral 3088 ∃wrex 3089 {crab 3092 Vcvv 3415 ⊆ wss 3831 ∅c0 4180 ∩ ciin 4794 class class class wbr 4930 ↦ cmpt 5009 dom cdm 5408 ran crn 5409 ⟶wf 6186 ‘cfv 6190 supcsup 8701 ℝcr 10336 ℝ^*cxr 10475 < clt 10476 ≤ cle 10477 ℤcz 11796 ℤ_≥cuz 12061 lim supclsp 14691 ⇝ cli 14705 SAlgcsalg 42025 SMblFncsmblfn 42409

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-rep 5050 ax-sep 5061 ax-nul 5068 ax-pow 5120 ax-pr 5187 ax-un 7281 ax-cnex 10393 ax-resscn 10394 ax-1cn 10395 ax-icn 10396 ax-addcl 10397 ax-addrcl 10398 ax-mulcl 10399 ax-mulrcl 10400 ax-mulcom 10401 ax-addass 10402 ax-mulass 10403 ax-distr 10404 ax-i2m1 10405 ax-1ne0 10406 ax-1rid 10407 ax-rnegex 10408 ax-rrecex 10409 ax-cnre 10410 ax-pre-lttri 10411 ax-pre-lttrn 10412 ax-pre-ltadd 10413 ax-pre-mulgt0 10414 ax-pre-sup 10415

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2583 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-reu 3095 df-rmo 3096 df-rab 3097 df-v 3417 df-sbc 3684 df-csb 3789 df-dif 3834 df-un 3836 df-in 3838 df-ss 3845 df-pss 3847 df-nul 4181 df-if 4352 df-pw 4425 df-sn 4443 df-pr 4445 df-tp 4447 df-op 4449 df-uni 4714 df-int 4751 df-iun 4795 df-iin 4796 df-br 4931 df-opab 4993 df-mpt 5010 df-tr 5032 df-id 5313 df-eprel 5318 df-po 5327 df-so 5328 df-fr 5367 df-we 5369 df-xp 5414 df-rel 5415 df-cnv 5416 df-co 5417 df-dm 5418 df-rn 5419 df-res 5420 df-ima 5421 df-pred 5988 df-ord 6034 df-on 6035 df-lim 6036 df-suc 6037 df-iota 6154 df-fun 6192 df-fn 6193 df-f 6194 df-f1 6195 df-fo 6196 df-f1o 6197 df-fv 6198 df-riota 6939 df-ov 6981 df-oprab 6982 df-mpo 6983 df-om 7399 df-1st 7503 df-2nd 7504 df-wrecs 7752 df-recs 7814 df-rdg 7852 df-1o 7907 df-oadd 7911 df-er 8091 df-pm 8211 df-en 8309 df-dom 8310 df-sdom 8311 df-fin 8312 df-sup 8703 df-inf 8704 df-pnf 10478 df-mnf 10479 df-xr 10480 df-ltxr 10481 df-le 10482 df-sub 10674 df-neg 10675 df-div 11101 df-nn 11442 df-2 11506 df-3 11507 df-n0 11711 df-z 11797 df-uz 12062 df-q 12166 df-rp 12208 df-ioo 12561 df-ico 12563 df-fz 12712 df-fl 12980 df-ceil 12981 df-seq 13188 df-exp 13248 df-cj 14322 df-re 14323 df-im 14324 df-sqrt 14458 df-abs 14459 df-limsup 14692 df-clim 14709 df-smblfn 42410

This theorem is referenced by: smflimsuplem6 42531 smflimsuplem8 42533

W3C validator