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 47362

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 2853	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ_≥‘𝑀))
5	4	biimpi 218	. . . . . . . . . 10 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ_≥‘𝑀))
6		uzss 12859	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
7	5, 6	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑍 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
8	7, 3	sseqtrrdi 3977	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 → (ℤ_≥‘𝑁) ⊆ 𝑍)
9	2, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
10	9	sselda 3936	. . . . . 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 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑍
13		nfrab1 3433	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ}
14	12, 13	nfmpt 5197	. . . . . . . . . 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 2921	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐸
16		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑛
17	15, 16	nffv 6873	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐸‘𝑛)
18		fvex 6876	. . . . . . . 8 ⊢ (𝐸‘𝑛) ∈ V
19	17, 18	mptexf 45776	. . . . . . 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 6983	. . . . . 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 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )))
24	23	fveq1d 6865	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) = ((𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))‘𝑋))
25		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑦(𝐸‘𝑛)
26		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑦sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )
27		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^*, < )
28		fveq2 6863	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
29	28	mpteq2dv 5193	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
30	29	rneqd 5912	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
31	30	supeq1d 9389	. . . . . 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 5199	. . . . 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 486	. . . . . . . . 9 ⊢ ((𝑦 = 𝑋 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑦 = 𝑋)
34	33	fveq2d 6867	. . . . . . . 8 ⊢ ((𝑦 = 𝑋 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
35	34	mpteq2dva 5192	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
36	35	rneqd 5912	. . . . . 6 ⊢ (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
37	36	supeq1d 9389	. . . . 5 ⊢ (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^, < ))
38	37	eleq1d 2846	. . . . . . . 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 12859	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑁))
40		iinss1 4964	. . . . . . . . . . 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 485	. . . . . . . . 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 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚))
45	42, 44	sseldd 3937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
46		smflimsuplem5.b	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝜑
47		nfv 1933	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑛 ∈ (ℤ_≥‘𝑁)
48	46, 47	nfan 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁))
49		eqid 2761	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
50		simpll 776	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
51	39	sselda 3936	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
52	51	adantll 724	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
53		smflimsuplem5.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ SAlg)
54	53	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑆 ∈ SAlg)
55		simpl 486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝜑)
56	9	sselda 3936	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ 𝑍)
57		smflimsuplem5.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
58	57	ffvelcdmda 7061	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
59	55, 56, 58	syl2anc 593	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
60		eqid 2761	. . . . . . . . . . . . 13 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
61	54, 59, 60	smff 47270	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
62		eliin 4953	. . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚))
65	64	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚))
66		simpr 488	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ (ℤ_≥‘𝑁))
67		rspa 3250	. . . . . . . . . . . . 13 ⊢ ((∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚) ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ dom (𝐹‘𝑚))
68	65, 66, 67	syl2anc 593	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ dom (𝐹‘𝑚))
69	61, 68	ffvelcdmd 7062	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
70	50, 52, 69	syl2anc 593	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
71		eluzelz 12846	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ ℤ)
72	71	adantl 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ ℤ)
73		smflimsuplem5.m	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℤ)
74	73	adantr 484	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑀 ∈ ℤ)
75		fvex 6876	. . . . . . . . . . . . . 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 46267	. . . . . . . . . . . 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 484	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
80	77, 79	eqeltrd 2861	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
81	80	renepnfd 11230	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
82	48, 49, 70, 81	limsupubuzmpt 46257	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦)
83		uzid2 45943	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ (ℤ_≥‘𝑛))
84	83	ne0d 4294	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑛) ≠ ∅)
85	84	adantl 485	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (ℤ_≥‘𝑛) ≠ ∅)
86	48, 85, 70	supxrre3rnmpt 45967	. . . . . . . . 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 259	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ) ∈ ℝ)
88	38, 45, 87	elrabd 3652	. . . . . . 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 486	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑥 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑦 = 𝑥)
90	89	fveq2d 6867	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑥 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑥))
91	90	mpteq2dva 5192	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
92	91	rneqd 5912	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
93	92	supeq1d 9389	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
94	93	eleq1d 2846	. . . . . . . 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 3423	. . . . . . 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 2871	. . . . . 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 2761	. . . . . . . 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 6876	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑚) ∈ V
99	98	dmex 7886	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) ∈ V
100	99	rgenw 3079	. . . . . . . . . . 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 45675	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
103	102	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
104	97, 103	rabexd 5295	. . . . . . 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 6983	. . . . . . 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 593	. . . . . 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 2864	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ (𝐸‘𝑛))
108	32, 37, 107, 87	fvmptd3 6995	. . . 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 2796	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ))
110	1, 109	mpteq2da 5191	. 2 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) = (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < )))
111	3	eluzelz2 45941	. . . 4 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)
112	2, 111	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
113		eqid 2761	. . 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 46267	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
117	116, 78	eqeltrd 2861	. . 3 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
118	46, 112, 113, 69, 117	supcnvlimsupmpt 46279	. 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 5121	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 208 ∧ wa 399 = wceq 1559 Ⅎwnf 1802 ∈ wcel 2141 ≠ wne 2956 ∀wral 3075 ∃wrex 3085 {crab 3413 Vcvv 3453 ⊆ wss 3904 ∅c0 4285 ∩ ciin 4949 class class class wbr 5099 ↦ cmpt 5180 dom cdm 5645 ran crn 5646 ⟶wf 6513 ‘cfv 6517 supcsup 9383 ℝcr 11069 ℝ^*cxr 11212 < clt 11213 ≤ cle 11214 ℤcz 12565 ℤ_≥cuz 12836 lim supclsp 15480 ⇝ cli 15494 SAlgcsalg 46846 SMblFncsmblfn 47233

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-er 8673 df-pm 8806 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-sup 9385 df-inf 9386 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-n0 12479 df-z 12566 df-uz 12837 df-q 12947 df-rp 12991 df-ioo 13350 df-ico 13352 df-fz 13510 df-fl 13799 df-ceil 13800 df-seq 14012 df-exp 14072 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-limsup 15481 df-clim 15498 df-smblfn 47234

This theorem is referenced by: smflimsuplem6 47363 smflimsuplem8 47365

W3C validator