smflimsuplem5 - Mathbox for Glauco Siliprandi

	Mathbox for Glauco Siliprandi		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > smflimsuplem5		Structured version Visualization version GIF version

Theorem smflimsuplem5 48039

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 2856	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ_≥‘𝑀))
5	4	biimpi 219	. . . . . . . . . 10 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ_≥‘𝑀))
6		uzss 13015	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
7	5, 6	syl 18	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑍 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
8	7, 3	sseqtrrdi 3954	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 → (ℤ_≥‘𝑁) ⊆ 𝑍)
9	2, 8	syl 18	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
10	9	sselda 3913	. . . . . 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 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑍
13		nfrab1 3435	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ}
14	12, 13	nfmpt 5206	. . . . . . . . . 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 2924	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐸
16		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑛
17	15, 16	nffv 6918	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐸‘𝑛)
18		fvex 6921	. . . . . . . 8 ⊢ (𝐸‘𝑛) ∈ V
19	17, 18	mptexf 46454	. . . . . . 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 7028	. . . . . 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 596	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
24	23	fveq1d 6910	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) = ((𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))‘𝑋))
25		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑦(𝐸‘𝑛)
26		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑦sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )
27		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^*, < )
28		fveq2 6908	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
29	28	mpteq2dv 5202	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
30	29	rneqd 5927	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
31	30	supeq1d 9465	. . . . . 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 5208	. . . . 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 488	. . . . . . . . 9 ⊢ ((𝑦 = 𝑋 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑦 = 𝑋)
34	33	fveq2d 6912	. . . . . . . 8 ⊢ ((𝑦 = 𝑋 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
35	34	mpteq2dva 5201	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
36	35	rneqd 5927	. . . . . 6 ⊢ (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
37	36	supeq1d 9465	. . . . 5 ⊢ (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^, < ))
38	37	eleq1d 2849	. . . . . . . 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 13015	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑁))
40		iinss1 4970	. . . . . . . . . . 11 ⊢ ((ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑁) → ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
41	39, 40	syl 18	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚) ⊆ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
42	41	adantl 487	. . . . . . . . 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 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚))
45	42, 44	sseldd 3914	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
46		smflimsuplem5.b	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝜑
47		nfv 1949	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑛 ∈ (ℤ_≥‘𝑁)
48	46, 47	nfan 1934	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁))
49		eqid 2763	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
50		simpll 779	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
51	39	sselda 3913	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
52	51	adantll 727	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
53		smflimsuplem5.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ SAlg)
54	53	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑆 ∈ SAlg)
55		simpl 488	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝜑)
56	9	sselda 3913	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ 𝑍)
57		smflimsuplem5.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
58	57	ffvelcdmda 7108	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
59	55, 56, 58	syl2anc 596	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
60		eqid 2763	. . . . . . . . . . . . 13 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
61	54, 59, 60	smff 47947	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
62		eliin 4959	. . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚)))
64	43, 63	mpbid 235	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚))
65	64	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚))
66		simpr 490	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ (ℤ_≥‘𝑁))
67		rspa 3255	. . . . . . . . . . . . 13 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚) ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ dom (𝐹‘𝑚))
68	65, 66, 67	syl2anc 596	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ dom (𝐹‘𝑚))
69	61, 68	ffvelcdmd 7109	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
70	50, 52, 69	syl2anc 596	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
71		eluzelz 13002	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ ℤ)
72	71	adantl 487	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ ℤ)
73		smflimsuplem5.m	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℤ)
74	73	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑀 ∈ ℤ)
75		fvex 6921	. . . . . . . . . . . . . 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 46944	. . . . . . . . . . . 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 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
80	77, 79	eqeltrd 2864	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
81	80	renepnfd 11387	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
82	48, 49, 70, 81	limsupubuzmpt 46934	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦)
83		uzid2 46620	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ (ℤ_≥‘𝑛))
84	83	ne0d 4291	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑛) ≠ ∅)
85	84	adantl 487	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (ℤ_≥‘𝑛) ≠ ∅)
86	48, 85, 70	supxrre3rnmpt 46644	. . . . . . . . 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 260	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ) ∈ ℝ)
88	38, 45, 87	elrabd 3650	. . . . . . 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 488	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑥 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑦 = 𝑥)
90	89	fveq2d 6912	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑥 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑥))
91	90	mpteq2dva 5201	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
92	91	rneqd 5927	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
93	92	supeq1d 9465	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
94	93	eleq1d 2849	. . . . . . . 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 3426	. . . . . . 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 2874	. . . . . 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 2763	. . . . . . . 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 6921	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑚) ∈ V
99	98	dmex 7945	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) ∈ V
100	99	rgenw 3084	. . . . . . . . . . 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 46353	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
103	102	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
104	97, 103	rabexd 5305	. . . . . . 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 7028	. . . . . . 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 596	. . . . . 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 2867	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ (𝐸‘𝑛))
108	32, 37, 107, 87	fvmptd3 7040	. . . 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 2798	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ))
110	1, 109	mpteq2da 5200	. 2 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) = (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < )))
111	3	eluzelz2 46618	. . . 4 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)
112	2, 111	syl 18	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
113		eqid 2763	. . 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 46944	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
117	116, 78	eqeltrd 2864	. . 3 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
118	46, 112, 113, 69, 117	supcnvlimsupmpt 46956	. 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 5130	1 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) ⇝ (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 = wceq 1570 Ⅎwnf 1816 ∈ wcel 2147 ≠ wne 2959 ∀wral 3080 ∃wrex 3090 {crab 3416 Vcvv 3454 ⊆ wss 3900 ∅c0 4281 ∩ ciin 4955 class class class wbr 5106 ↦ cmpt 5189 dom cdm 5655 ran crn 5656 ⟶wf 6554 ‘cfv 6558 supcsup 9459 ℝcr 11226 ℝ^*cxr 11369 < clt 11370 ≤ cle 11371 ℤcz 12720 ℤ_≥cuz 12992 lim supclsp 15666 ⇝ cli 15680 SAlgcsalg 47523 SMblFncsmblfn 47910

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1945 ax-6 2002 ax-7 2043 ax-8 2149 ax-9 2157 ax-10 2180 ax-11 2197 ax-12 2216 ax-ext 2735 ax-rep 5235 ax-sep 5252 ax-nul 5263 ax-pow 5331 ax-pr 5395 ax-un 7775 ax-cnex 11283 ax-resscn 11284 ax-1cn 11285 ax-icn 11286 ax-addcl 11287 ax-addrcl 11288 ax-mulcl 11289 ax-mulrcl 11290 ax-mulcom 11291 ax-addass 11292 ax-mulass 11293 ax-distr 11294 ax-i2m1 11295 ax-1ne0 11296 ax-1rid 11297 ax-rnegex 11298 ax-rrecex 11299 ax-cnre 11300 ax-pre-lttri 11301 ax-pre-lttrn 11302 ax-pre-ltadd 11303 ax-pre-mulgt0 11304 ax-pre-sup 11305

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2102 df-mo 2567 df-eu 2597 df-clab 2742 df-cleq 2755 df-clel 2839 df-nfc 2913 df-ne 2960 df-nel 3066 df-ral 3081 df-rex 3091 df-rmo 3369 df-reu 3370 df-rab 3417 df-v 3456 df-sbc 3743 df-csb 3852 df-ss 3901 df-pss 4017 df-dif 4050 df-in 4094 df-un 4154 df-nul 4282 df-if 4486 df-pw 4562 df-sn 4589 df-pr 4591 df-tp 4593 df-op 4839 df-uni 4871 df-int 4911 df-iun 4956 df-iin 4957 df-br 5107 df-opab 5171 df-mpt 5190 df-tr 5216 df-id 5550 df-eprel 5555 df-po 5563 df-so 5564 df-fr 5608 df-we 5610 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-rn 5666 df-res 5667 df-ima 5668 df-pred 6322 df-ord 6383 df-on 6384 df-lim 6385 df-suc 6386 df-iota 6512 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-riota 7400 df-ov 7446 df-oprab 7447 df-mpo 7448 df-om 7902 df-1st 8025 df-2nd 8026 df-frecs 8327 df-wrecs 8358 df-recs 8407 df-rdg 8446 df-1o 8504 df-2o 8505 df-er 8745 df-pm 8878 df-en 9002 df-dom 9003 df-sdom 9004 df-fin 9005 df-sup 9461 df-inf 9462 df-pnf 11372 df-mnf 11373 df-xr 11374 df-ltxr 11375 df-le 11376 df-sub 11570 df-neg 11571 df-div 12001 df-nn 12363 df-2 12432 df-3 12433 df-n0 12634 df-z 12721 df-uz 12993 df-q 13103 df-rp 13148 df-ioo 13507 df-ico 13509 df-fz 13667 df-fl 13959 df-ceil 13960 df-seq 14172 df-exp 14232 df-cj 15295 df-re 15296 df-im 15297 df-sqrt 15431 df-abs 15432 df-limsup 15667 df-clim 15684 df-smblfn 47911

This theorem is used by: smflimsuplem6 48040 smflimsuplem8 48042

W3C validator