Theorem smflimsuplem2 47359

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 12859	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
8	6, 7	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
9	8, 5	sseqtrrdi 3977	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ≥‘𝑛) ⊆ 𝑍)
10	9	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (ℤ≥‘𝑛) ⊆ 𝑍)
11		simpr 488	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ (ℤ≥‘𝑛))
12	10, 11	sseldd 3937	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
13		smflimsuplem2.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ SAlg)
14	13	adantr 484	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
15		smflimsuplem2.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
16	15	ffvelcdmda 7061	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
17		eqid 2761	. . . . . . . . 9 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
18	14, 16, 17	smff 47270	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
19	12, 18	syldan 600	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
20		iinss2 5014	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ dom (𝐹‘𝑚))
21	20	adantl 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ dom (𝐹‘𝑚))
22	1	adantr 484	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚))
23	21, 22	sseldd 3937	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑚))
24	19, 23	ffvelcdmd 7062	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
25		nfmpt1 5198	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
26		nfmpt1 5198	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))
27		eluzelz 12846	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
28	6, 27	syl 17	. . . . . . . . 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 7094	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)):(ℤ≥‘𝑛)⟶ℝ)
31	30	ffnd 6688	. . . . . . . . 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 6878	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
35	33, 2, 34	mptfnd 45781	. . . . . . . . 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 6878	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
38	36, 37	fvmpt2d 6985	. . . . . . . . . 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 6983	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (ℤ≥‘𝑀) ∧ ((𝐹‘𝑚)‘𝑋) ∈ V) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑚) = ((𝐹‘𝑚)‘𝑋))
42	39, 37, 41	syl2anc 593	. . . . . . . . . 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 46261	. . . . . . . 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 5190	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))
47	46	fveq2i 6866	. . . . . . . . 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 11230	. . . . . . 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 46257	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦)
54		uzid 12851	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
55		ne0i 4293	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘𝑛) → (ℤ≥‘𝑛) ≠ ∅)
56	28, 54, 55	3syl 18	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑛) ≠ ∅)
57	2, 56, 24	supxrre3rnmpt 45967	. . . . 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 259	. . . 4 ⊢ (𝜑 → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ)
59	1, 58	jca 519	. . 3 ⊢ (𝜑 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
60		fveq2 6863	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
61	60	mpteq2dv 5193	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
62	61	rneqd 5912	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
63	62	supeq1d 9389	. . . . . . 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 6863	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
68	67	mpteq2dv 5193	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
69	68	rneqd 5912	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
70	69	supeq1d 9389	. . . . . 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 3650	. . . 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 277	. . 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 236	. 2 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
75		id 22	. . . . 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 3433	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
81	79, 80	nfmpt 5197	. . . . . . . . . 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 6873	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐸‘𝑛)
85		fvex 6876	. . . . . . . 8 ⊢ (𝐸‘𝑛) ∈ V
86	84, 85	mptexf 45776	. . . . . . 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 6985	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
89	75, 4, 88	syl2anc 593	. . . 4 ⊢ (𝜑 → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
90	89	dmeqd 5879	. . 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 5199	. . . 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 13140	. . . . . 6 ⊢ < Or ℝ*
96	95	supex 9407	. . . . 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 6662	. . 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 6876	. . . . . . . . 9 ⊢ (𝐹‘𝑚) ∈ V
101	100	dmex 7886	. . . . . . . 8 ⊢ dom (𝐹‘𝑚) ∈ V
102	101	rgenw 3079	. . . . . . 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 45675	. . . . 5 ⊢ (𝜑 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
105	99, 104	rabexd 5295	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
106	78	fvmpt2 6983	. . . 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 593	. . 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))

Syntax hints:   → wi 4   ∧ 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  +∞cpnf 11210  ℝ^*cxr 11212   < clt 11213   ≤ cle 11214  ℤcz 12565  ℤ_≥cuz 12836  lim supclsp 15480  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-n0 12479  df-z 12566  df-uz 12837  df-q 12947  df-ioo 13350  df-ico 13352  df-fz 13510  df-fl 13799  df-ceil 13800  df-limsup 15481  df-smblfn 47234

This theorem is referenced by:  smflimsuplem7  47364