Theorem smflimsuplem2 44696

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 2736	. . . . . 6 ⊢ (ℤ≥‘𝑛) = (ℤ≥‘𝑛)
4		smflimsuplem2.n	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑛 ∈ 𝑍)
5		smflimsuplem2.z	. . . . . . . . . . . . 13 ⊢ 𝑍 = (ℤ≥‘𝑀)
6	4, 5	eleqtrdi 2847	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝑛 ∈ (ℤ≥‘𝑀))
7		uzss 12706	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
8	6, 7	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
9	8, 5	sseqtrrdi 3983	. . . . . . . . . 10 ⊢ (𝜑 → (ℤ≥‘𝑛) ⊆ 𝑍)
10	9	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (ℤ≥‘𝑛) ⊆ 𝑍)
11		simpr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ (ℤ≥‘𝑛))
12	10, 11	sseldd 3933	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ 𝑍)
13		smflimsuplem2.s	. . . . . . . . . 10 ⊢ (𝜑 → 𝑆 ∈ SAlg)
14	13	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
15		smflimsuplem2.f	. . . . . . . . . 10 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
16	15	ffvelcdmda 7017	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
17		eqid 2736	. . . . . . . . 9 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
18	14, 16, 17	smff 44607	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
19	12, 18	syldan 591	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
20		iinss2 5004	. . . . . . . . 9 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ dom (𝐹‘𝑚))
21	20	adantl 482	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ⊆ dom (𝐹‘𝑚))
22	1	adantr 481	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚))
23	21, 22	sseldd 3933	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑋 ∈ dom (𝐹‘𝑚))
24	19, 23	ffvelcdmd 7018	. . . . . 6 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ ℝ)
25		nfmpt1 5200	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
26		nfmpt1 5200	. . . . . . . . 9 ⊢ Ⅎ𝑚(𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))
27		eluzelz 12693	. . . . . . . . . 10 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → 𝑛 ∈ ℤ)
28	6, 27	syl 17	. . . . . . . . 9 ⊢ (𝜑 → 𝑛 ∈ ℤ)
29		eqid 2736	. . . . . . . . . . 11 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))
30	2, 24, 29	fmptdf 7047	. . . . . . . . . 10 ⊢ (𝜑 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)):(ℤ≥‘𝑛)⟶ℝ)
31	30	ffnd 6652	. . . . . . . . 9 ⊢ (𝜑 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)) Fn (ℤ≥‘𝑛))
32		smflimsuplem2.m	. . . . . . . . 9 ⊢ (𝜑 → 𝑀 ∈ ℤ)
33		nfcv 2904	. . . . . . . . . 10 ⊢ Ⅎ𝑚(ℤ≥‘𝑀)
34		fvexd 6840	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑀)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
35	33, 2, 34	mptfnd 43114	. . . . . . . . 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 6840	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝐹‘𝑚)‘𝑋) ∈ V)
38	36, 37	fvmpt2d 6944	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑚) = ((𝐹‘𝑚)‘𝑋))
39	12, 5	eleqtrdi 2847	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → 𝑚 ∈ (ℤ≥‘𝑀))
40		eqid 2736	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))
41	40	fvmpt2 6942	. . . . . . . . . . 11 ⊢ ((𝑚 ∈ (ℤ≥‘𝑀) ∧ ((𝐹‘𝑚)‘𝑋) ∈ V) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑚) = ((𝐹‘𝑚)‘𝑋))
42	39, 37, 41	syl2anc 584	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → ((𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))‘𝑚) = ((𝐹‘𝑚)‘𝑋))
43	38, 42	eqtr4d 2779	. . . . . . . . 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 43600	. . . . . . . 8 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) = (lim sup‘(𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋))))
45	5	eqcomi 2745	. . . . . . . . . . 11 ⊢ (ℤ≥‘𝑀) = 𝑍
46	45	mpteq1i 5188	. . . . . . . . . 10 ⊢ (𝑚 ∈ (ℤ≥‘𝑀) ↦ ((𝐹‘𝑚)‘𝑋)) = (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))
47	46	fveq2i 6828	. . . . . . . . 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 2776	. . . . . . 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 11127	. . . . . . 7 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
52	49, 51	eqnetrd 3008	. . . . . 6 ⊢ (𝜑 → (lim sup‘(𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋))) ≠ +∞)
53	2, 3, 24, 52	limsupubuzmpt 43596	. . . . 5 ⊢ (𝜑 → ∃𝑦 ∈ ℝ ∀𝑚 ∈ (ℤ≥‘𝑛)((𝐹‘𝑚)‘𝑋) ≤ 𝑦)
54		uzid 12698	. . . . . . 7 ⊢ (𝑛 ∈ ℤ → 𝑛 ∈ (ℤ≥‘𝑛))
55		ne0i 4281	. . . . . . 7 ⊢ (𝑛 ∈ (ℤ≥‘𝑛) → (ℤ≥‘𝑛) ≠ ∅)
56	28, 54, 55	3syl 18	. . . . . 6 ⊢ (𝜑 → (ℤ≥‘𝑛) ≠ ∅)
57	2, 56, 24	supxrre3rnmpt 43304	. . . . 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 256	. . . 4 ⊢ (𝜑 → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ)
59	1, 58	jca 512	. . 3 ⊢ (𝜑 → (𝑋 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ) ∈ ℝ))
60		fveq2 6825	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑚)‘𝑦))
61	60	mpteq2dv 5194	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
62	61	rneqd 5879	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)))
63	62	supeq1d 9303	. . . . . . 7 ⊢ (𝑥 = 𝑦 → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ))
64	63	eleq1d 2821	. . . . . 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 3413	. . . . 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 2828	. . . 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 6825	. . . . . . . . 9 ⊢ (𝑦 = 𝑋 → ((𝐹‘𝑚)‘𝑦) = ((𝐹‘𝑚)‘𝑋))
68	67	mpteq2dv 5194	. . . . . . . 8 ⊢ (𝑦 = 𝑋 → (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
69	68	rneqd 5879	. . . . . . 7 ⊢ (𝑦 = 𝑋 → ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)))
70	69	supeq1d 9303	. . . . . 6 ⊢ (𝑦 = 𝑋 → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑋)), ℝ*, < ))
71	70	eleq1d 2821	. . . . 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 3634	. . . 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 274	. . 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 233	. 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 2904	. . . . . . . . . . 11 ⊢ Ⅎ𝑥𝑍
80		nfrab1 3422	. . . . . . . . . . 11 ⊢ Ⅎ𝑥{𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
81	79, 80	nfmpt 5199	. . . . . . . . . 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 2902	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐸
83		nfcv 2904	. . . . . . . . 9 ⊢ Ⅎ𝑥𝑛
84	82, 83	nffv 6835	. . . . . . . 8 ⊢ Ⅎ𝑥(𝐸‘𝑛)
85		fvex 6838	. . . . . . . 8 ⊢ (𝐸‘𝑛) ∈ V
86	84, 85	mptexf 43109	. . . . . . 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 6944	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
89	75, 4, 88	syl2anc 584	. . . 4 ⊢ (𝜑 → (𝐻‘𝑛) = (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
90	89	dmeqd 5847	. . 3 ⊢ (𝜑 → dom (𝐻‘𝑛) = dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
91		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑦(𝐸‘𝑛)
92		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑦sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )
93		nfcv 2904	. . . . 5 ⊢ Ⅎ𝑥sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑦)), ℝ*, < )
94	84, 91, 92, 93, 63	cbvmptf 5201	. . . 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 12976	. . . . . 6 ⊢ < Or ℝ*
96	95	supex 9320	. . . . 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 6629	. . 3 ⊢ (𝜑 → dom (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝐸‘𝑛))
99		eqid 2736	. . . . 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 6838	. . . . . . . . 9 ⊢ (𝐹‘𝑚) ∈ V
101	100	dmex 7826	. . . . . . . 8 ⊢ dom (𝐹‘𝑚) ∈ V
102	101	rgenw 3065	. . . . . . 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 43003	. . . . 5 ⊢ (𝜑 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
105	99, 104	rabexd 5277	. . . 4 ⊢ (𝜑 → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
106	78	fvmpt2 6942	. . . 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 584	. . 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 2781	. 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 2839	1 ⊢ (𝜑 → 𝑋 ∈ dom (𝐻‘𝑛))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540  Ⅎwnf 1784   ∈ wcel 2105   ≠ wne 2940  ∀wral 3061  ∃wrex 3070  {crab 3403  Vcvv 3441   ⊆ wss 3898  ∅c0 4269  ∩ ciin 4942   class class class wbr 5092   ↦ cmpt 5175  dom cdm 5620  ran crn 5621  ⟶wf 6475  ‘cfv 6479  supcsup 9297  ℝcr 10971  +∞cpnf 11107  ℝ^*cxr 11109   < clt 11110   ≤ cle 11111  ℤcz 12420  ℤ_≥cuz 12683  lim supclsp 15278  SAlgcsalg 44185  SMblFncsmblfn 44570

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pow 5308  ax-pr 5372  ax-un 7650  ax-cnex 11028  ax-resscn 11029  ax-1cn 11030  ax-icn 11031  ax-addcl 11032  ax-addrcl 11033  ax-mulcl 11034  ax-mulrcl 11035  ax-mulcom 11036  ax-addass 11037  ax-mulass 11038  ax-distr 11039  ax-i2m1 11040  ax-1ne0 11041  ax-1rid 11042  ax-rnegex 11043  ax-rrecex 11044  ax-cnre 11045  ax-pre-lttri 11046  ax-pre-lttrn 11047  ax-pre-ltadd 11048  ax-pre-mulgt0 11049  ax-pre-sup 11050

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3349  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3917  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-tp 4578  df-op 4580  df-uni 4853  df-int 4895  df-iun 4943  df-iin 4944  df-br 5093  df-opab 5155  df-mpt 5176  df-tr 5210  df-id 5518  df-eprel 5524  df-po 5532  df-so 5533  df-fr 5575  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-ord 6305  df-on 6306  df-lim 6307  df-suc 6308  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-riota 7293  df-ov 7340  df-oprab 7341  df-mpo 7342  df-om 7781  df-1st 7899  df-2nd 7900  df-frecs 8167  df-wrecs 8198  df-recs 8272  df-rdg 8311  df-1o 8367  df-er 8569  df-pm 8689  df-en 8805  df-dom 8806  df-sdom 8807  df-fin 8808  df-sup 9299  df-inf 9300  df-pnf 11112  df-mnf 11113  df-xr 11114  df-ltxr 11115  df-le 11116  df-sub 11308  df-neg 11309  df-div 11734  df-nn 12075  df-n0 12335  df-z 12421  df-uz 12684  df-q 12790  df-ioo 13184  df-ico 13186  df-fz 13341  df-fl 13613  df-ceil 13614  df-limsup 15279  df-smblfn 44571

This theorem is referenced by:  smflimsuplem7  44701