Theorem smflimsup 41034

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
smflimsup.n	⊢ Ⅎ𝑚𝐹
smflimsup.x	⊢ Ⅎ𝑥𝐹
smflimsup.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflimsup.z	⊢ 𝑍 = (ℤ≥‘𝑀)
smflimsup.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimsup.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsup.d	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
smflimsup.g	⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))

Assertion

Ref	Expression
smflimsup	⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

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

Allowed substitution hints:   𝜑(𝑥,𝑚,𝑛)   𝐷(𝑥,𝑚,𝑛)   𝑆(𝑥,𝑚,𝑛)   𝐹(𝑥,𝑚)   𝐺(𝑥,𝑚,𝑛)   𝑀(𝑥,𝑚,𝑛)

Proof of Theorem smflimsup

Dummy variables 𝑘 𝑗 𝑞 𝑤 𝑖 𝑙 𝑝 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		smflimsup.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
2		smflimsup.z	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
3		smflimsup.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
4		smflimsup.f	. 2 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
5		smflimsup.d	. . 3 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
6		fveq2 6191	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (ℤ≥‘𝑛) = (ℤ≥‘𝑗))
7	6	iineq1d 39267	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑚))
8		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑞dom (𝐹‘𝑚)
9		smflimsup.n	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝐹
10		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝑞
11	9, 10	nffv 6198	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝐹‘𝑞)
12	11	nfdm 5367	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑞)
13		fveq2 6191	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑞 → (𝐹‘𝑚) = (𝐹‘𝑞))
14	13	dmeqd 5326	. . . . . . . . . 10 ⊢ (𝑚 = 𝑞 → dom (𝐹‘𝑚) = dom (𝐹‘𝑞))
15	8, 12, 14	cbviin 4558	. . . . . . . . 9 ⊢ ∩ 𝑚 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑚) = ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
16	15	a1i 11	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑚) = ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
17	7, 16	eqtrd 2656	. . . . . . 7 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
18	17	cbviunv 4559	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
19	18	eleq2i 2693	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
20		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑞((𝐹‘𝑚)‘𝑥)
21		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑥
22	11, 21	nffv 6198	. . . . . . . 8 ⊢ Ⅎ𝑚((𝐹‘𝑞)‘𝑥)
23	13	fveq1d 6193	. . . . . . . 8 ⊢ (𝑚 = 𝑞 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
24	20, 22, 23	cbvmpt 4749	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))
25	24	fveq2i 6194	. . . . . 6 ⊢ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
26	25	eleq1i 2692	. . . . 5 ⊢ ((lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ)
27	19, 26	anbi12i 733	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ) ↔ (𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞) ∧ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ))
28	27	rabbia2 3187	. . 3 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ} = {𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞) ∣ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ}
29		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑥𝑍
30		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥(ℤ≥‘𝑗)
31		smflimsup.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
32		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥𝑞
33	31, 32	nffv 6198	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑞)
34	33	nfdm 5367	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑞)
35	30, 34	nfiin 4549	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
36	29, 35	nfiun 4548	. . . 4 ⊢ Ⅎ𝑥∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
37		nfcv 2764	. . . 4 ⊢ Ⅎ𝑤∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
38		nfv 1843	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ
39		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑥lim sup
40		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
41	33, 40	nffv 6198	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑞)‘𝑤)
42	29, 41	nfmpt 4746	. . . . . 6 ⊢ Ⅎ𝑥(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))
43	39, 42	nffv 6198	. . . . 5 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
44		nfcv 2764	. . . . 5 ⊢ Ⅎ𝑥ℝ
45	43, 44	nfel 2777	. . . 4 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ
46		fveq2 6191	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑞)‘𝑥) = ((𝐹‘𝑞)‘𝑤))
47	46	mpteq2dv 4745	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
48	47	fveq2d 6195	. . . . 5 ⊢ (𝑥 = 𝑤 → (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
49	48	eleq1d 2686	. . . 4 ⊢ (𝑥 = 𝑤 → ((lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ))
50	36, 37, 38, 45, 49	cbvrab 3198	. . 3 ⊢ {𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞) ∣ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ} = {𝑤 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞) ∣ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ}
51	5, 28, 50	3eqtri 2648	. 2 ⊢ 𝐷 = {𝑤 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞) ∣ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ}
52		smflimsup.g	. . 3 ⊢ 𝐺 = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))))
53	25	mpteq2i 4741	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))))
54		nfrab1 3122	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
55	5, 54	nfcxfr 2762	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2764	. . . 4 ⊢ Ⅎ𝑤𝐷
57		nfcv 2764	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
58	55, 56, 57, 43, 48	cbvmptf 4748	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))) = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
59	52, 53, 58	3eqtri 2648	. 2 ⊢ 𝐺 = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
60		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑖)
61	60, 34	nfiin 4549	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
62		nfcv 2764	. . . . . 6 ⊢ Ⅎ𝑤∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
63		nfv 1843	. . . . . 6 ⊢ Ⅎ𝑤sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ
64	60, 41	nfmpt 4746	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
65	64	nfrn 5368	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
66		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥ℝ*
67		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑥 <
68	65, 66, 67	nfsup 8357	. . . . . . 7 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )
69	68, 44	nfel 2777	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ
70	46	mpteq2dv 4745	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
71	70	rneqd 5353	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
72	71	supeq1d 8352	. . . . . . 7 ⊢ (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
73	72	eleq1d 2686	. . . . . 6 ⊢ (𝑥 = 𝑤 → (sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
74	61, 62, 63, 69, 73	cbvrab 3198	. . . . 5 ⊢ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ}
75	74	a1i 11	. . . 4 ⊢ (𝑖 = 𝑘 → {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ})
76		fveq2 6191	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘))
77	76	iineq1d 39267	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) = ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞))
78	77	eleq2d 2687	. . . . . 6 ⊢ (𝑖 = 𝑘 → (𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ↔ 𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞)))
79	76	mpteq1d 4738	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
80	79	rneqd 5353	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
81	80	supeq1d 8352	. . . . . . 7 ⊢ (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
82	81	eleq1d 2686	. . . . . 6 ⊢ (𝑖 = 𝑘 → (sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ))
83	78, 82	anbi12d 747	. . . . 5 ⊢ (𝑖 = 𝑘 → ((𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∧ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ) ↔ (𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞) ∧ sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ)))
84	83	rabbidva2 3186	. . . 4 ⊢ (𝑖 = 𝑘 → {𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ} = {𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ})
85	75, 84	eqtrd 2656	. . 3 ⊢ (𝑖 = 𝑘 → {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ})
86	85	cbvmptv 4750	. 2 ⊢ (𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ}) = (𝑘 ∈ 𝑍 ↦ {𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ})
87		fveq2 6191	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑝)‘𝑦) = ((𝐹‘𝑝)‘𝑤))
88	87	mpteq2dv 4745	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
89	88	rneqd 5353	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
90	89	supeq1d 8352	. . . . . . 7 ⊢ (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ*, < ) = sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ))
91	90	cbvmptv 4750	. . . . . 6 ⊢ (𝑦 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∣ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ*, < )) = (𝑤 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∣ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ))
92		fveq2 6191	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
93	92	dmeqd 5326	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → dom (𝐹‘𝑝) = dom (𝐹‘𝑞))
94	93	cbviinv 4560	. . . . . . . . . . . 12 ⊢ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) = ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
95	94	eleq2i 2693	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ↔ 𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞))
96		nfcv 2764	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑞((𝐹‘𝑝)‘𝑥)
97		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝐹‘𝑞)
98		nfcv 2764	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝𝑥
99	97, 98	nffv 6198	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝((𝐹‘𝑞)‘𝑥)
100	92	fveq1d 6193	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
101	96, 99, 100	cbvmpt 4749	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
102	101	rneqi 5352	. . . . . . . . . . . . 13 ⊢ ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
103	102	supeq1i 8353	. . . . . . . . . . . 12 ⊢ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < )
104	103	eleq1i 2692	. . . . . . . . . . 11 ⊢ (sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ)
105	95, 104	anbi12i 733	. . . . . . . . . 10 ⊢ ((𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∧ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ) ↔ (𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∧ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ))
106	105	rabbia2 3187	. . . . . . . . 9 ⊢ {𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∣ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ}
107	106	mpteq2i 4741	. . . . . . . 8 ⊢ (𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∣ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ}) = (𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})
108	107	fveq1i 6192	. . . . . . 7 ⊢ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∣ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) = ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙)
109	92	fveq1d 6193	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑤) = ((𝐹‘𝑞)‘𝑤))
110	109	cbvmptv 4750	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
111	110	rneqi 5352	. . . . . . . 8 ⊢ ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
112	111	supeq1i 8353	. . . . . . 7 ⊢ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )
113	108, 112	mpteq12i 4742	. . . . . 6 ⊢ (𝑤 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∣ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < )) = (𝑤 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
114	91, 113	eqtri 2644	. . . . 5 ⊢ (𝑦 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∣ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ*, < )) = (𝑤 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
115	114	a1i 11	. . . 4 ⊢ (𝑙 = 𝑘 → (𝑦 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∣ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ*, < )) = (𝑤 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )))
116		fveq2 6191	. . . . 5 ⊢ (𝑙 = 𝑘 → ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) = ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑘))
117		fveq2 6191	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (ℤ≥‘𝑙) = (ℤ≥‘𝑘))
118	117	mpteq1d 4738	. . . . . . 7 ⊢ (𝑙 = 𝑘 → (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
119	118	rneqd 5353	. . . . . 6 ⊢ (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
120	119	supeq1d 8352	. . . . 5 ⊢ (𝑙 = 𝑘 → sup(ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
121	116, 120	mpteq12dv 4733	. . . 4 ⊢ (𝑙 = 𝑘 → (𝑤 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )) = (𝑤 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑘) ↦ sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )))
122	115, 121	eqtrd 2656	. . 3 ⊢ (𝑙 = 𝑘 → (𝑦 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∣ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ*, < )) = (𝑤 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑘) ↦ sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )))
123	122	cbvmptv 4750	. 2 ⊢ (𝑙 ∈ 𝑍 ↦ (𝑦 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ∣ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑙) ↦ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ*, < ))) = (𝑘 ∈ 𝑍 ↦ (𝑤 ∈ ((𝑖 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ∣ sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ})‘𝑘) ↦ sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )))
124	1, 2, 3, 4, 51, 59, 86, 123	smflimsuplem8 41033	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Ⅎwnfc 2751  {crab 2916  ∪ ciun 4520  ∩ ciin 4521   ↦ cmpt 4729  dom cdm 5114  ran crn 5115  ⟶wf 5884  ‘cfv 5888  supcsup 8346  ℝcr 9935  ℝ^*cxr 10073   < clt 10074  ℤcz 11377  ℤ_≥cuz 11687  lim supclsp 14201  SAlgcsalg 40528  SMblFncsmblfn 40909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cc 9257  ax-ac2 9285  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-oadd 7564  df-omul 7565  df-er 7742  df-map 7859  df-pm 7860  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-acn 8768  df-ac 8939  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-n0 11293  df-z 11378  df-uz 11688  df-q 11789  df-rp 11833  df-ioo 12179  df-ioc 12180  df-ico 12181  df-fz 12327  df-fl 12593  df-ceil 12594  df-seq 12802  df-exp 12861  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-limsup 14202  df-clim 14219  df-rlim 14220  df-rest 16083  df-topgen 16104  df-top 20699  df-bases 20750  df-salg 40529  df-salgen 40533  df-smblfn 40910

This theorem is referenced by:  smflimsupmpt  41035