Theorem smflimsuplem2 48098

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 2763	. . . . . 6 ⊢ (ℤ≥‘𝑛) = (ℤ≥‘𝑛)
4		smflimsuplem2.n	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑛 ∈ 𝑍)
5		smflimsuplem2.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ≥‘𝑀)
6	4, 5	eleqtrdi 2874	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑛 ∈ (ℤ≥‘𝑀))
7		uzss 13026	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
8	6, 7	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
9	8, 5	sseqtrrdi 3954	. . . . . . . . . 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 3914	. . . . . . . 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 7114	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
17		eqid 2763	. . . . . . . . 9 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
18	14, 16, 17	smff 48009	. . . . . . . 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 5019	. . . . . . . . 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 3914	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑚))
24	19, 23	ffvelcdmd 7115	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
25		nfmpt1 5208	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
26		nfmpt1 5208	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))
27		eluzelz 13013	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
28	6, 27	syl 18	. . . . . . . . 9 ⊢ (𝜑 → 𝑛 ∈ ℤ)
29		eqid 2763	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
30	2, 24, 29	fmptdf 7147	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)):(ℤ≥‘𝑛)⟶ℝ)
31	30	ffnd 6738	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) Fn (ℤ≥‘𝑛))
32		smflimsuplem2.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
33		nfcv 2926	. . . . . . . . . 10 ⊢ Ⅎ𝑚(ℤ≥‘𝑀)
34		fvexd 6929	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
35	33, 2, 34	mptfnd 46521	. . . . . . . . 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 6929	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
38	36, 37	fvmpt2d 7036	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑚) = ((𝐹‘𝑚)‘𝑋))
39	12, 5	eleqtrdi 2874	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ (ℤ≥‘𝑀))
40		eqid 2763	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))
41	40	fvmpt2 7034	. . . . . . . . . . 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 2801	. . . . . . . . 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 47000	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))))
45	5	eqcomi 2772	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) = 𝑍
46	45	mpteq1i 5200	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))
47	46	fveq2i 6917	. . . . . . . . 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 2798	. . . . . . 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 11397	. . . . . . 7 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
52	49, 51	eqnetrd 3026	. . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
53	2, 3, 24, 52	limsupubuzmpt 46996	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦)
54		uzid 13018	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
55		ne0i 4290	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘𝑛) → (ℤ≥‘𝑛) ≠ ∅)
56	28, 54, 55	3syl 19	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑛) ≠ ∅)
57	2, 56, 24	supxrre3rnmpt 46706	. . . . 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 6914	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
61	60	mpteq2dv 5203	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
62	61	rneqd 5930	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
63	62	supeq1d 9473	. . . . . . 7 ⊢ (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ))
64	63	eleq1d 2849	. . . . . 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 3426	. . . . 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 2856	. . . 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 6914	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
68	67	mpteq2dv 5203	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
69	68	rneqd 5930	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
70	69	supeq1d 9473	. . . . . 6 ⊢ (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ))
71	70	eleq1d 2849	. . . . 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 3648	. . . 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 2926	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑍
80		nfrab1 3435	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
81	79, 80	nfmpt 5207	. . . . . . . . . 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 2924	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐸
83		nfcv 2926	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑛
84	82, 83	nffv 6924	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐸‘𝑛)
85		fvex 6927	. . . . . . . 8 ⊢ (𝐸‘𝑛) ∈ V
86	84, 85	mptexf 46516	. . . . . . 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 7036	. . . . 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 5897	. . 3 ⊢ (𝜑 → dom (𝐻‘𝑛) = dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
91		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑦(𝐸‘𝑛)
92		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑦sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )
93		nfcv 2926	. . . . 5 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < )
94	84, 91, 92, 93, 63	cbvmptf 5209	. . . 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 13308	. . . . . 6 ⊢ < Or ℝ*
96	95	supex 9491	. . . . 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 6712	. . 3 ⊢ (𝜑 → dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝐸‘𝑛))
99		eqid 2763	. . . . 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 6927	. . . . . . . . 9 ⊢ (𝐹‘𝑚) ∈ V
101	100	dmex 7953	. . . . . . . 8 ⊢ dom (𝐹‘𝑚) ∈ V
102	101	rgenw 3084	. . . . . . 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 46415	. . . . 5 ⊢ (𝜑 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
105	99, 104	rabexd 5306	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
106	78	fvmpt2 7034	. . . 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 2803	. 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 2866	1 ⊢ (𝜑 → 𝑋 ∈ dom (𝐻‘𝑛))

This proof depends on syntax axioms:   → wi 4   ∧ 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 5190  dom cdm 5656  ran crn 5657  ⟶wf 6560  ‘cfv 6564  supcsup 9467  ℝcr 11236  +∞cpnf 11377  ℝ^*cxr 11379   < clt 11380   ≤ cle 11381  ℤcz 12731  ℤ_≥cuz 13003  lim supclsp 15678  SAlgcsalg 47585  SMblFncsmblfn 47972

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 5236  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7781  ax-cnex 11293  ax-resscn 11294  ax-1cn 11295  ax-icn 11296  ax-addcl 11297  ax-addrcl 11298  ax-mulcl 11299  ax-mulrcl 11300  ax-mulcom 11301  ax-addass 11302  ax-mulass 11303  ax-distr 11304  ax-i2m1 11305  ax-1ne0 11306  ax-1rid 11307  ax-rnegex 11308  ax-rrecex 11309  ax-cnre 11310  ax-pre-lttri 11311  ax-pre-lttrn 11312  ax-pre-ltadd 11313  ax-pre-mulgt0 11314  ax-pre-sup 11315

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 5172  df-mpt 5191  df-tr 5217  df-id 5551  df-eprel 5556  df-po 5564  df-so 5565  df-fr 5609  df-we 5611  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-pred 6328  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6518  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-fv 6572  df-riota 7406  df-ov 7452  df-oprab 7453  df-mpo 7454  df-om 7910  df-1st 8033  df-2nd 8034  df-frecs 8335  df-wrecs 8366  df-recs 8415  df-rdg 8454  df-1o 8512  df-2o 8513  df-er 8753  df-pm 8886  df-en 9010  df-dom 9011  df-sdom 9012  df-fin 9013  df-sup 9469  df-inf 9470  df-pnf 11382  df-mnf 11383  df-xr 11384  df-ltxr 11385  df-le 11386  df-sub 11580  df-neg 11581  df-div 12012  df-nn 12374  df-n0 12645  df-z 12732  df-uz 13004  df-q 13114  df-ioo 13518  df-ico 13520  df-fz 13678  df-fl 13970  df-ceil 13971  df-limsup 15679  df-smblfn 47973

This theorem is used by:  smflimsuplem7  48103