Theorem smflimsuplem2 47800

Description: The superior limit of a sequence of sigma-measurable functions is sigma-measurable. Proposition 121F (d) of [Fremlin1] p. 39 . (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smflimsuplem2.p	⊢ Ⅎ𝑚𝜑
smflimsuplem2.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflimsuplem2.z	⊢ 𝑍 = (ℤ≥‘𝑀)
smflimsuplem2.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimsuplem2.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsuplem2.e	⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
smflimsuplem2.h	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
smflimsuplem2.n	⊢ (𝜑 → 𝑛 ∈ 𝑍)
smflimsuplem2.r	⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
smflimsuplem2.x	⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚))

Assertion

Ref	Expression
smflimsuplem2	⊢ (𝜑 → 𝑋 ∈ dom (𝐻‘𝑛))

Distinct variable groups:   𝑥,𝐹   𝑚,𝑀   𝑚,𝑋   𝑚,𝑍,𝑛,𝑥

Allowed substitution hints:   𝜑(𝑥, 𝑚, 𝑛)   𝑆(𝑥, 𝑚, 𝑛)   𝐸(𝑥, 𝑚, 𝑛)   𝐹(𝑚, 𝑛)   𝐻(𝑥, 𝑚, 𝑛)   𝑀(𝑥, 𝑛)   𝑋(𝑥, 𝑛)

Proof of Theorem smflimsuplem2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		smflimsuplem2.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚))
2		smflimsuplem2.p	. . . . . 6 ⊢ Ⅎ𝑚𝜑
3		eqid 2761	. . . . . 6 ⊢ (ℤ≥‘𝑛) = (ℤ≥‘𝑛)
4		smflimsuplem2.n	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑛 ∈ 𝑍)
5		smflimsuplem2.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ≥‘𝑀)
6	4, 5	eleqtrdi 2871	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑛 ∈ (ℤ≥‘𝑀))
7		uzss 12981	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
8	6, 7	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
9	8, 5	sseqtrrdi 3972	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ≥‘𝑛) ⊆ 𝑍)
10	9	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (ℤ≥‘𝑛) ⊆ 𝑍)
11		simpr 490	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ (ℤ≥‘𝑛))
12	10, 11	sseldd 3932	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
13		smflimsuplem2.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ SAlg)
14	13	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
15		smflimsuplem2.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
16	15	ffvelcdmda 7082	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
17		eqid 2761	. . . . . . . . 9 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
18	14, 16, 17	smff 47711	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
19	12, 18	syldan 603	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
20		iinss2 5016	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ dom (𝐹‘𝑚))
21	20	adantl 487	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ dom (𝐹‘𝑚))
22	1	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚))
23	21, 22	sseldd 3932	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑚))
24	19, 23	ffvelcdmd 7083	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
25		nfmpt1 5204	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
26		nfmpt1 5204	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))
27		eluzelz 12968	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
28	6, 27	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑛 ∈ ℤ)
29		eqid 2761	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
30	2, 24, 29	fmptdf 7115	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)):(ℤ≥‘𝑛)⟶ℝ)
31	30	ffnd 6708	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) Fn (ℤ≥‘𝑛))
32		smflimsuplem2.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
33		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑚(ℤ≥‘𝑀)
34		fvexd 6898	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
35	33, 2, 34	mptfnd 46223	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋)) Fn (ℤ≥‘𝑀))
36	29	a1i 11	. . . . . . . . . . 11 ⊢ (𝜑 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
37		fvexd 6898	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
38	36, 37	fvmpt2d 7005	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑚) = ((𝐹‘𝑚)‘𝑋))
39	12, 5	eleqtrdi 2871	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ (ℤ≥‘𝑀))
40		eqid 2761	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))
41	40	fvmpt2 7003	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (ℤ≥‘𝑀) ∧ ((𝐹‘𝑚)‘𝑋) ∈ V) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑚) = ((𝐹‘𝑚)‘𝑋))
42	39, 37, 41	syl2anc 596	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑚) = ((𝐹‘𝑚)‘𝑋))
43	38, 42	eqtr4d 2799	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑚) = ((𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑚))
44	2, 25, 26, 28, 31, 32, 35, 28, 43	limsupequz 46702	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))))
45	5	eqcomi 2770	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) = 𝑍
46	45	mpteq1i 5196	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))
47	46	fveq2i 6886	. . . . . . . . 9 ⊢ (lim sup‘(𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋)))
48	47	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
49	44, 48	eqtrd 2796	. . . . . . 7 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))))
50		smflimsuplem2.r	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ∈ ℝ)
51	50	renepnfd 11353	. . . . . . 7 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
52	49, 51	eqnetrd 3023	. . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
53	2, 3, 24, 52	limsupubuzmpt 46698	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦)
54		uzid 12973	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
55		ne0i 4287	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘𝑛) → (ℤ≥‘𝑛) ≠ ∅)
56	28, 54, 55	3syl 19	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑛) ≠ ∅)
57	2, 56, 24	supxrre3rnmpt 46408	. . . . 5 ⊢ (𝜑 → (sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ ↔ ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦))
58	53, 57	mpbird 260	. . . 4 ⊢ (𝜑 → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ)
59	1, 58	jca 521	. . 3 ⊢ (𝜑 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
60		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
61	60	mpteq2dv 5199	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
62	61	rneqd 5920	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
63	62	supeq1d 9431	. . . . . . 7 ⊢ (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ))
64	63	eleq1d 2846	. . . . . 6 ⊢ (𝑥 = 𝑦 → (sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ))
65	64	cbvrabv 3423	. . . . 5 ⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑦 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ}
66	65	eleq2i 2853	. . . 4 ⊢ (𝑋 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ↔ 𝑋 ∈ {𝑦 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ})
67		fveq2 6883	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
68	67	mpteq2dv 5199	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
69	68	rneqd 5920	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
70	69	supeq1d 9431	. . . . . 6 ⊢ (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ))
71	70	eleq1d 2846	. . . . 5 ⊢ (𝑦 = 𝑋 → (sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
72	71	elrab 3645	. . . 4 ⊢ (𝑋 ∈ {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))
73	66, 72	bitri 278	. . 3 ⊢ (𝑋 ∈ {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))
74	59, 73	sylibr 237	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
75		id 23	. . . . 5 ⊢ (𝜑 → 𝜑)
76		smflimsuplem2.h	. . . . . . 7 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
77	76	a1i 11	. . . . . 6 ⊢ (𝜑 → 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ))))
78		smflimsuplem2.e	. . . . . . . . . 10 ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
79		nfcv 2923	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑍
80		nfrab1 3432	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
81	79, 80	nfmpt 5203	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
82	78, 81	nfcxfr 2921	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐸
83		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑛
84	82, 83	nffv 6893	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐸‘𝑛)
85		fvex 6896	. . . . . . . 8 ⊢ (𝐸‘𝑛) ∈ V
86	84, 85	mptexf 46218	. . . . . . 7 ⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) ∈ V
87	86	a1i 11	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) ∈ V)
88	77, 87	fvmpt2d 7005	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
89	75, 4, 88	syl2anc 596	. . . 4 ⊢ (𝜑 → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
90	89	dmeqd 5887	. . 3 ⊢ (𝜑 → dom (𝐻‘𝑛) = dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
91		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑦(𝐸‘𝑛)
92		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑦sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )
93		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < )
94	84, 91, 92, 93, 63	cbvmptf 5205	. . . 4 ⊢ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝑦 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ))
95		xrltso 13263	. . . . . 6 ⊢ < Or ℝ*
96	95	supex 9449	. . . . 5 ⊢ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) ∈ V
97	96	a1i 11	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ (𝐸‘𝑛)) → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) ∈ V)
98	94, 97	dmmptd 6682	. . 3 ⊢ (𝜑 → dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝐸‘𝑛))
99		eqid 2761	. . . . 5 ⊢ {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}
100		fvex 6896	. . . . . . . . 9 ⊢ (𝐹‘𝑚) ∈ V
101	100	dmex 7919	. . . . . . . 8 ⊢ dom (𝐹‘𝑚) ∈ V
102	101	rgenw 3081	. . . . . . 7 ⊢ ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
103	102	a1i 11	. . . . . 6 ⊢ (𝜑 → ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
104	56, 103	iinexd 46117	. . . . 5 ⊢ (𝜑 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
105	99, 104	rabexd 5301	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
106	78	fvmpt2 7003	. . . 4 ⊢ ((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})
107	4, 105, 106	syl2anc 596	. . 3 ⊢ (𝜑 → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
108	90, 98, 107	3eqtrrd 2801	. 2 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = dom (𝐻‘𝑛))
109	74, 108	eleqtrd 2863	1 ⊢ (𝜑 → 𝑋 ∈ dom (𝐻‘𝑛))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2145   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  {crab 3413  Vcvv 3451   ⊆ wss 3899  ∅c0 4279  ∩ ciin 4952   class class class wbr 5103   ↦ cmpt 5186  dom cdm 5651  ran crn 5652  ⟶wf 6533  ‘cfv 6537  supcsup 9425  ℝcr 11192  +∞cpnf 11333  ℝ^*cxr 11335   < clt 11336   ≤ cle 11337  ℤcz 12686  ℤ_≥cuz 12958  lim supclsp 15630  SAlgcsalg 47287  SMblFncsmblfn 47674

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-rep 5232  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7749  ax-cnex 11249  ax-resscn 11250  ax-1cn 11251  ax-icn 11252  ax-addcl 11253  ax-addrcl 11254  ax-mulcl 11255  ax-mulrcl 11256  ax-mulcom 11257  ax-addass 11258  ax-mulass 11259  ax-distr 11260  ax-i2m1 11261  ax-1ne0 11262  ax-1rid 11263  ax-rnegex 11264  ax-rrecex 11265  ax-cnre 11266  ax-pre-lttri 11267  ax-pre-lttrn 11268  ax-pre-ltadd 11269  ax-pre-mulgt0 11270  ax-pre-sup 11271

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-nel 3063  df-ral 3078  df-rex 3088  df-rmo 3366  df-reu 3367  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-tp 4589  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-iin 4954  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-riota 7375  df-ov 7421  df-oprab 7422  df-mpo 7423  df-om 7876  df-1st 7999  df-2nd 8000  df-frecs 8292  df-wrecs 8323  df-recs 8372  df-rdg 8411  df-1o 8469  df-2o 8470  df-er 8710  df-pm 8843  df-en 8967  df-dom 8968  df-sdom 8969  df-fin 8970  df-sup 9427  df-inf 9428  df-pnf 11338  df-mnf 11339  df-xr 11340  df-ltxr 11341  df-le 11342  df-sub 11536  df-neg 11537  df-div 11967  df-nn 12329  df-n0 12600  df-z 12687  df-uz 12959  df-q 13069  df-ioo 13473  df-ico 13475  df-fz 13633  df-fl 13925  df-ceil 13926  df-limsup 15631  df-smblfn 47675

This theorem is used by:  smflimsuplem7  47805