Theorem smflimsup 47400

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 𝑗 𝑘 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 6871	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (ℤ≥‘𝑛) = (ℤ≥‘𝑗))
7	6	iineq1d 45666	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑚))
8		nfcv 2927	. . . . . . . . . 10 ⊢ Ⅎ𝑞dom (𝐹‘𝑚)
9		smflimsup.n	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝐹
10		nfcv 2927	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝑞
11	9, 10	nffv 6881	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝐹‘𝑞)
12	11	nfdm 5932	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑞)
13		fveq2 6871	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑞 → (𝐹‘𝑚) = (𝐹‘𝑞))
14	13	dmeqd 5886	. . . . . . . . . 10 ⊢ (𝑚 = 𝑞 → dom (𝐹‘𝑚) = dom (𝐹‘𝑞))
15	8, 12, 14	cbviin 4996	. . . . . . . . 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 2800	. . . . . . 7 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
18	17	cbviunv 4999	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
19	18	eleq2i 2857	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
20		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑞((𝐹‘𝑚)‘𝑥)
21		nfcv 2927	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑥
22	11, 21	nffv 6881	. . . . . . . 8 ⊢ Ⅎ𝑚((𝐹‘𝑞)‘𝑥)
23	13	fveq1d 6873	. . . . . . . 8 ⊢ (𝑚 = 𝑞 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
24	20, 22, 23	cbvmpt 5207	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))
25	24	fveq2i 6874	. . . . . 6 ⊢ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
26	25	eleq1i 2856	. . . . 5 ⊢ ((lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ)
27	19, 26	anbi12i 639	. . . 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 3420	. . 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 2927	. . . . 5 ⊢ Ⅎ𝑥𝑍
30		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑥(ℤ≥‘𝑗)
31		smflimsup.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
32		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑥𝑞
33	31, 32	nffv 6881	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑞)
34	33	nfdm 5932	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑞)
35	30, 34	nfiin 4985	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
36	29, 35	nfiun 4984	. . . 4 ⊢ Ⅎ𝑥∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
37		nfcv 2927	. . . 4 ⊢ Ⅎ𝑤∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
38		nfv 1937	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ
39		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑥lim sup
40		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
41	33, 40	nffv 6881	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑞)‘𝑤)
42	29, 41	nfmpt 5203	. . . . . 6 ⊢ Ⅎ𝑥(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))
43	39, 42	nffv 6881	. . . . 5 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
44		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑥ℝ
45	43, 44	nfel 2941	. . . 4 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ
46		fveq2 6871	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑞)‘𝑥) = ((𝐹‘𝑞)‘𝑤))
47	46	mpteq2dv 5199	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
48	47	fveq2d 6875	. . . . 5 ⊢ (𝑥 = 𝑤 → (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
49	48	eleq1d 2850	. . . 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	cbvrabw 3452	. . 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 2792	. 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 5201	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))))
54		nfrab1 3437	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
55	5, 54	nfcxfr 2925	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2927	. . . 4 ⊢ Ⅎ𝑤𝐷
57		nfcv 2927	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
58	55, 56, 57, 43, 48	cbvmptf 5205	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))) = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
59	52, 53, 58	3eqtri 2792	. 2 ⊢ 𝐺 = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
60		nfcv 2927	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑖)
61	60, 34	nfiin 4985	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
62		nfcv 2927	. . . . . 6 ⊢ Ⅎ𝑤∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
63		nfv 1937	. . . . . 6 ⊢ Ⅎ𝑤sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ
64	60, 41	nfmpt 5203	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
65	64	nfrn 5933	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
66		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑥ℝ*
67		nfcv 2927	. . . . . . . 8 ⊢ Ⅎ𝑥 <
68	65, 66, 67	nfsup 9399	. . . . . . 7 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )
69	68, 44	nfel 2941	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ
70	46	mpteq2dv 5199	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
71	70	rneqd 5919	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
72	71	supeq1d 9394	. . . . . . 7 ⊢ (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
73	72	eleq1d 2850	. . . . . 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	cbvrabw 3452	. . . . 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 6871	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘))
77	76	iineq1d 45666	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) = ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞))
78	77	eleq2d 2851	. . . . . 6 ⊢ (𝑖 = 𝑘 → (𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ↔ 𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞)))
79	76	mpteq1d 5195	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
80	79	rneqd 5919	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
81	80	supeq1d 9394	. . . . . . 7 ⊢ (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
82	81	eleq1d 2850	. . . . . 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 643	. . . . 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 3419	. . . 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 2800	. . 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 5209	. 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 6871	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑝)‘𝑦) = ((𝐹‘𝑝)‘𝑤))
88	87	mpteq2dv 5199	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
89	88	rneqd 5919	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
90	89	supeq1d 9394	. . . . . . 7 ⊢ (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ*, < ) = sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ))
91	90	cbvmptv 5209	. . . . . 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 6871	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
93	92	dmeqd 5886	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → dom (𝐹‘𝑝) = dom (𝐹‘𝑞))
94	93	cbviinv 5000	. . . . . . . . . . . 12 ⊢ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) = ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
95	94	eleq2i 2857	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ↔ 𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞))
96		nfcv 2927	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑞((𝐹‘𝑝)‘𝑥)
97		nfcv 2927	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝐹‘𝑞)
98		nfcv 2927	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝𝑥
99	97, 98	nffv 6881	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝((𝐹‘𝑞)‘𝑥)
100	92	fveq1d 6873	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
101	96, 99, 100	cbvmpt 5207	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
102	101	rneqi 5918	. . . . . . . . . . . . 13 ⊢ ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
103	102	supeq1i 9395	. . . . . . . . . . . 12 ⊢ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < )
104	103	eleq1i 2856	. . . . . . . . . . 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 639	. . . . . . . . . 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 3420	. . . . . . . . 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 5201	. . . . . . . 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 6872	. . . . . . 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 6873	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑤) = ((𝐹‘𝑞)‘𝑤))
110	109	cbvmptv 5209	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
111	110	rneqi 5918	. . . . . . . 8 ⊢ ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
112	111	supeq1i 9395	. . . . . . 7 ⊢ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )
113	108, 112	mpteq12i 5202	. . . . . 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 2788	. . . . 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 6871	. . . . 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 6871	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (ℤ≥‘𝑙) = (ℤ≥‘𝑘))
118	117	mpteq1d 5195	. . . . . . 7 ⊢ (𝑙 = 𝑘 → (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
119	118	rneqd 5919	. . . . . 6 ⊢ (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
120	119	supeq1d 9394	. . . . 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 5192	. . . 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 2800	. . 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 5209	. 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 47399	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  Ⅎwnfc 2912  {crab 3417  ∪ ciun 4952  ∩ ciin 4953   ↦ cmpt 5186  dom cdm 5652  ran crn 5653  ⟶wf 6521  ‘cfv 6525  supcsup 9388  ℝcr 11087  ℝ^*cxr 11230   < clt 11231  ℤcz 12582  ℤ_≥cuz 12853  lim supclsp 15511  SAlgcsalg 46880  SMblFncsmblfn 47267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-rep 5232  ax-sep 5251  ax-nul 5261  ax-pow 5327  ax-pr 5395  ax-un 7722  ax-inf2 9598  ax-cc 10407  ax-ac2 10435  ax-cnex 11144  ax-resscn 11145  ax-1cn 11146  ax-icn 11147  ax-addcl 11148  ax-addrcl 11149  ax-mulcl 11150  ax-mulrcl 11151  ax-mulcom 11152  ax-addass 11153  ax-mulass 11154  ax-distr 11155  ax-i2m1 11156  ax-1ne0 11157  ax-1rid 11158  ax-rnegex 11159  ax-rrecex 11160  ax-cnre 11161  ax-pre-lttri 11162  ax-pre-lttrn 11163  ax-pre-ltadd 11164  ax-pre-mulgt0 11165  ax-pre-sup 11166

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3370  df-reu 3371  df-rab 3418  df-v 3459  df-sbc 3748  df-csb 3856  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-pss 3927  df-nul 4289  df-if 4484  df-pw 4560  df-sn 4586  df-pr 4588  df-tp 4590  df-op 4592  df-uni 4869  df-int 4909  df-iun 4954  df-iin 4955  df-br 5106  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5547  df-eprel 5552  df-po 5560  df-so 5561  df-fr 5605  df-se 5606  df-we 5607  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-pred 6292  df-ord 6353  df-on 6354  df-lim 6355  df-suc 6356  df-iota 6481  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532  df-fv 6533  df-isom 6534  df-riota 7357  df-ov 7403  df-oprab 7404  df-mpo 7405  df-om 7851  df-1st 7974  df-2nd 7975  df-frecs 8266  df-wrecs 8297  df-recs 8346  df-rdg 8385  df-1o 8441  df-2o 8442  df-oadd 8445  df-omul 8446  df-er 8682  df-map 8814  df-pm 8815  df-en 8932  df-dom 8933  df-sdom 8934  df-fin 8935  df-sup 9390  df-inf 9391  df-oi 9460  df-card 9913  df-acn 9916  df-ac 10088  df-pnf 11233  df-mnf 11234  df-xr 11235  df-ltxr 11236  df-le 11237  df-sub 11431  df-neg 11432  df-div 11860  df-nn 12225  df-2 12294  df-3 12295  df-n0 12496  df-z 12583  df-uz 12854  df-q 12964  df-rp 13008  df-ioo 13367  df-ioc 13368  df-ico 13369  df-fz 13527  df-fl 13816  df-ceil 13817  df-seq 14029  df-exp 14089  df-cj 15140  df-re 15141  df-im 15142  df-sqrt 15276  df-abs 15277  df-limsup 15512  df-clim 15529  df-rlim 15530  df-rest 17465  df-topgen 17486  df-top 23012  df-bases 23064  df-salg 46881  df-salgen 46885  df-smblfn 47268

This theorem is referenced by:  smflimsupmpt  47401