smflimsuplem5 - Mathbox for Glauco Siliprandi

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Theorem smflimsuplem5 45055

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 2829	. . . . . . . . . . 11 ⊢ (𝑁 ∈ 𝑍 ↔ 𝑁 ∈ (ℤ_≥‘𝑀))
5	4	biimpi 215	. . . . . . . . . 10 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ (ℤ_≥‘𝑀))
6		uzss 12786	. . . . . . . . . 10 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
7	5, 6	syl 17	. . . . . . . . 9 ⊢ (𝑁 ∈ 𝑍 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
8	7, 3	sseqtrrdi 3995	. . . . . . . 8 ⊢ (𝑁 ∈ 𝑍 → (ℤ_≥‘𝑁) ⊆ 𝑍)
9	2, 8	syl 17	. . . . . . 7 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
10	9	sselda 3944	. . . . . 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 2907	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑍
13		nfrab1 3426	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ) ∈ ℝ}
14	12, 13	nfmpt 5212	. . . . . . . . . 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 2905	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐸
16		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑛
17	15, 16	nffv 6852	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐸‘𝑛)
18		fvex 6855	. . . . . . . 8 ⊢ (𝐸‘𝑛) ∈ V
19	17, 18	mptexf 43453	. . . . . . 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 6959	. . . . . 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 6844	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) = ((𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < ))‘𝑋))
25		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑦(𝐸‘𝑛)
26		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑦sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^*, < )
27		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^*, < )
28		fveq2 6842	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
29	28	mpteq2dv 5207	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
30	29	rneqd 5893	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
31	30	supeq1d 9382	. . . . . 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 5214	. . . . 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 483	. . . . . . . . 9 ⊢ ((𝑦 = 𝑋 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑦 = 𝑋)
34	33	fveq2d 6846	. . . . . . . 8 ⊢ ((𝑦 = 𝑋 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
35	34	mpteq2dva 5205	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
36	35	rneqd 5893	. . . . . 6 ⊢ (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
37	36	supeq1d 9382	. . . . 5 ⊢ (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^, < ))
38	37	eleq1d 2822	. . . . . . . 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 12786	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑁))
40		iinss1 4969	. . . . . . . . . . 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 482	. . . . . . . . 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 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚))
45	42, 44	sseldd 3945	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
46		smflimsuplem5.b	. . . . . . . . . . 11 ⊢ Ⅎ𝑚𝜑
47		nfv 1917	. . . . . . . . . . 11 ⊢ Ⅎ𝑚 𝑛 ∈ (ℤ_≥‘𝑁)
48	46, 47	nfan 1902	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁))
49		eqid 2736	. . . . . . . . . 10 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
50		simpll 765	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝜑)
51	39	sselda 3944	. . . . . . . . . . . 12 ⊢ ((𝑛 ∈ (ℤ_≥‘𝑁) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
52	51	adantll 712	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑚 ∈ (ℤ_≥‘𝑁))
53		smflimsuplem5.s	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑆 ∈ SAlg)
54	53	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑆 ∈ SAlg)
55		simpl 483	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝜑)
56	9	sselda 3944	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ 𝑍)
57		smflimsuplem5.f	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
58	57	ffvelcdmda 7035	. . . . . . . . . . . . . 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 2736	. . . . . . . . . . . . 13 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
61	54, 59, 60	smff 44963	. . . . . . . . . . . 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 17	. . . . . . . . . . . . . . 15 ⊢ (𝜑 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑁)dom (𝐹‘𝑚) ↔ ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚)))
64	43, 63	mpbid 231	. . . . . . . . . . . . . 14 ⊢ (𝜑 → ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚))
65	64	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → ∀𝑚 ∈ (ℤ_≥‘𝑁)𝑋 ∈ dom (𝐹‘𝑚))
66		simpr 485	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ_≥‘𝑁)) → 𝑚 ∈ (ℤ_≥‘𝑁))
67		rspa 3231	. . . . . . . . . . . . 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 7036	. . . . . . . . . . 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 12773	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ ℤ)
72	71	adantl 482	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑛 ∈ ℤ)
73		smflimsuplem5.m	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝑀 ∈ ℤ)
74	73	adantr 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑀 ∈ ℤ)
75		fvex 6855	. . . . . . . . . . . . . 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 43960	. . . . . . . . . . . 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 481	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
80	77, 79	eqeltrd 2838	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
81	80	renepnfd 11206	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
82	48, 49, 70, 81	limsupubuzmpt 43950	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ_≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦)
83		uzid2 43630	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → 𝑛 ∈ (ℤ_≥‘𝑛))
84	83	ne0d 4295	. . . . . . . . . . 11 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑛) ≠ ∅)
85	84	adantl 482	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → (ℤ_≥‘𝑛) ≠ ∅)
86	48, 85, 70	supxrre3rnmpt 43654	. . . . . . . . 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 256	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ) ∈ ℝ)
88	38, 45, 87	elrabd 3647	. . . . . . 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 483	. . . . . . . . . . . . 13 ⊢ ((𝑦 = 𝑥 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → 𝑦 = 𝑥)
90	89	fveq2d 6846	. . . . . . . . . . . 12 ⊢ ((𝑦 = 𝑥 ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑥))
91	90	mpteq2dva 5205	. . . . . . . . . . 11 ⊢ (𝑦 = 𝑥 → (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
92	91	rneqd 5893	. . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)))
93	92	supeq1d 9382	. . . . . . . . 9 ⊢ (𝑦 = 𝑥 → sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ^, < ) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ^, < ))
94	93	eleq1d 2822	. . . . . . . 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 3417	. . . . . . 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 2848	. . . . . 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 2736	. . . . . . . 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 6855	. . . . . . . . . . . . 13 ⊢ (𝐹‘𝑚) ∈ V
99	98	dmex 7848	. . . . . . . . . . . 12 ⊢ dom (𝐹‘𝑚) ∈ V
100	99	rgenw 3068	. . . . . . . . . . 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 43333	. . . . . . . . 9 ⊢ (𝑛 ∈ (ℤ_≥‘𝑁) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
103	102	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
104	97, 103	rabexd 5290	. . . . . . 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 6959	. . . . . . 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 2841	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → 𝑋 ∈ (𝐸‘𝑛))
108	32, 37, 107, 87	fvmptd3 6971	. . . 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 2776	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ (ℤ_≥‘𝑁)) → ((𝐻‘𝑛)‘𝑋) = sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < ))
110	1, 109	mpteq2da 5203	. 2 ⊢ (𝜑 → (𝑛 ∈ (ℤ_≥‘𝑁) ↦ ((𝐻‘𝑛)‘𝑋)) = (𝑛 ∈ (ℤ_≥‘𝑁) ↦ sup(ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ^*, < )))
111	3	eluzelz2 43628	. . . 4 ⊢ (𝑁 ∈ 𝑍 → 𝑁 ∈ ℤ)
112	2, 111	syl 17	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℤ)
113		eqid 2736	. . 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 43960	. . . 4 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
117	116, 78	eqeltrd 2838	. . 3 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ_≥‘𝑁) ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
118	46, 112, 113, 69, 117	supcnvlimsupmpt 43972	. 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 5127	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 205 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 ≠ wne 2943 ∀wral 3064 ∃wrex 3073 {crab 3407 Vcvv 3445 ⊆ wss 3910 ∅c0 4282 ∩ ciin 4955 class class class wbr 5105 ↦ cmpt 5188 dom cdm 5633 ran crn 5634 ⟶wf 6492 ‘cfv 6496 supcsup 9376 ℝcr 11050 ℝ^*cxr 11188 < clt 11189 ≤ cle 11190 ℤcz 12499 ℤ_≥cuz 12763 lim supclsp 15352 ⇝ cli 15366 SAlgcsalg 44539 SMblFncsmblfn 44926

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-er 8648 df-pm 8768 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-sup 9378 df-inf 9379 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-n0 12414 df-z 12500 df-uz 12764 df-q 12874 df-rp 12916 df-ioo 13268 df-ico 13270 df-fz 13425 df-fl 13697 df-ceil 13698 df-seq 13907 df-exp 13968 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-limsup 15353 df-clim 15370 df-smblfn 44927

This theorem is referenced by: smflimsuplem6 45056 smflimsuplem8 45058

W3C validator