Theorem smflimsup 45842

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 6890	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (ℤ≥‘𝑛) = (ℤ≥‘𝑗))
7	6	iineq1d 44080	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑚))
8		nfcv 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑞dom (𝐹‘𝑚)
9		smflimsup.n	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝐹
10		nfcv 2901	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝑞
11	9, 10	nffv 6900	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝐹‘𝑞)
12	11	nfdm 5949	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑞)
13		fveq2 6890	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑞 → (𝐹‘𝑚) = (𝐹‘𝑞))
14	13	dmeqd 5904	. . . . . . . . . 10 ⊢ (𝑚 = 𝑞 → dom (𝐹‘𝑚) = dom (𝐹‘𝑞))
15	8, 12, 14	cbviin 5039	. . . . . . . . 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 2770	. . . . . . 7 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
18	17	cbviunv 5042	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
19	18	eleq2i 2823	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
20		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑞((𝐹‘𝑚)‘𝑥)
21		nfcv 2901	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑥
22	11, 21	nffv 6900	. . . . . . . 8 ⊢ Ⅎ𝑚((𝐹‘𝑞)‘𝑥)
23	13	fveq1d 6892	. . . . . . . 8 ⊢ (𝑚 = 𝑞 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
24	20, 22, 23	cbvmpt 5258	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))
25	24	fveq2i 6893	. . . . . 6 ⊢ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
26	25	eleq1i 2822	. . . . 5 ⊢ ((lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ)
27	19, 26	anbi12i 625	. . . 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 3433	. . 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 2901	. . . . 5 ⊢ Ⅎ𝑥𝑍
30		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑥(ℤ≥‘𝑗)
31		smflimsup.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
32		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑥𝑞
33	31, 32	nffv 6900	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑞)
34	33	nfdm 5949	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑞)
35	30, 34	nfiin 5027	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
36	29, 35	nfiun 5026	. . . 4 ⊢ Ⅎ𝑥∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
37		nfcv 2901	. . . 4 ⊢ Ⅎ𝑤∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
38		nfv 1915	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ
39		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑥lim sup
40		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
41	33, 40	nffv 6900	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑞)‘𝑤)
42	29, 41	nfmpt 5254	. . . . . 6 ⊢ Ⅎ𝑥(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))
43	39, 42	nffv 6900	. . . . 5 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
44		nfcv 2901	. . . . 5 ⊢ Ⅎ𝑥ℝ
45	43, 44	nfel 2915	. . . 4 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ
46		fveq2 6890	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑞)‘𝑥) = ((𝐹‘𝑞)‘𝑤))
47	46	mpteq2dv 5249	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
48	47	fveq2d 6894	. . . . 5 ⊢ (𝑥 = 𝑤 → (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
49	48	eleq1d 2816	. . . 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 3465	. . 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 2762	. 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 5252	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))))
54		nfrab1 3449	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
55	5, 54	nfcxfr 2899	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2901	. . . 4 ⊢ Ⅎ𝑤𝐷
57		nfcv 2901	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
58	55, 56, 57, 43, 48	cbvmptf 5256	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))) = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
59	52, 53, 58	3eqtri 2762	. 2 ⊢ 𝐺 = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
60		nfcv 2901	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑖)
61	60, 34	nfiin 5027	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
62		nfcv 2901	. . . . . 6 ⊢ Ⅎ𝑤∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
63		nfv 1915	. . . . . 6 ⊢ Ⅎ𝑤sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ
64	60, 41	nfmpt 5254	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
65	64	nfrn 5950	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
66		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑥ℝ*
67		nfcv 2901	. . . . . . . 8 ⊢ Ⅎ𝑥 <
68	65, 66, 67	nfsup 9448	. . . . . . 7 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )
69	68, 44	nfel 2915	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ
70	46	mpteq2dv 5249	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
71	70	rneqd 5936	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
72	71	supeq1d 9443	. . . . . . 7 ⊢ (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
73	72	eleq1d 2816	. . . . . 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 3465	. . . . 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 6890	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘))
77	76	iineq1d 44080	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) = ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞))
78	77	eleq2d 2817	. . . . . 6 ⊢ (𝑖 = 𝑘 → (𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ↔ 𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞)))
79	76	mpteq1d 5242	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
80	79	rneqd 5936	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
81	80	supeq1d 9443	. . . . . . 7 ⊢ (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
82	81	eleq1d 2816	. . . . . 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 629	. . . . 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 3432	. . . 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 2770	. . 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 5260	. 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 6890	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑝)‘𝑦) = ((𝐹‘𝑝)‘𝑤))
88	87	mpteq2dv 5249	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
89	88	rneqd 5936	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
90	89	supeq1d 9443	. . . . . . 7 ⊢ (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ*, < ) = sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ))
91	90	cbvmptv 5260	. . . . . 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 6890	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
93	92	dmeqd 5904	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → dom (𝐹‘𝑝) = dom (𝐹‘𝑞))
94	93	cbviinv 5043	. . . . . . . . . . . 12 ⊢ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) = ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
95	94	eleq2i 2823	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ↔ 𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞))
96		nfcv 2901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑞((𝐹‘𝑝)‘𝑥)
97		nfcv 2901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝐹‘𝑞)
98		nfcv 2901	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝𝑥
99	97, 98	nffv 6900	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝((𝐹‘𝑞)‘𝑥)
100	92	fveq1d 6892	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
101	96, 99, 100	cbvmpt 5258	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
102	101	rneqi 5935	. . . . . . . . . . . . 13 ⊢ ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
103	102	supeq1i 9444	. . . . . . . . . . . 12 ⊢ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < )
104	103	eleq1i 2822	. . . . . . . . . . 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 625	. . . . . . . . . 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 3433	. . . . . . . . 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 5252	. . . . . . . 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 6891	. . . . . . 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 6892	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑤) = ((𝐹‘𝑞)‘𝑤))
110	109	cbvmptv 5260	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
111	110	rneqi 5935	. . . . . . . 8 ⊢ ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
112	111	supeq1i 9444	. . . . . . 7 ⊢ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )
113	108, 112	mpteq12i 5253	. . . . . 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 2758	. . . . 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 6890	. . . . 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 6890	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (ℤ≥‘𝑙) = (ℤ≥‘𝑘))
118	117	mpteq1d 5242	. . . . . . 7 ⊢ (𝑙 = 𝑘 → (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
119	118	rneqd 5936	. . . . . 6 ⊢ (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
120	119	supeq1d 9443	. . . . 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 5238	. . . 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 2770	. . 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 5260	. 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 45841	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2104  Ⅎwnfc 2881  {crab 3430  ∪ ciun 4996  ∩ ciin 4997   ↦ cmpt 5230  dom cdm 5675  ran crn 5676  ⟶wf 6538  ‘cfv 6542  supcsup 9437  ℝcr 11111  ℝ^*cxr 11251   < clt 11252  ℤcz 12562  ℤ_≥cuz 12826  lim supclsp 15418  SAlgcsalg 45322  SMblFncsmblfn 45709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-inf2 9638  ax-cc 10432  ax-ac2 10460  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189  ax-pre-sup 11190

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-iin 4999  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-se 5631  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-isom 6551  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-omul 8473  df-er 8705  df-map 8824  df-pm 8825  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-sup 9439  df-inf 9440  df-oi 9507  df-card 9936  df-acn 9939  df-ac 10113  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-div 11876  df-nn 12217  df-2 12279  df-3 12280  df-n0 12477  df-z 12563  df-uz 12827  df-q 12937  df-rp 12979  df-ioo 13332  df-ioc 13333  df-ico 13334  df-fz 13489  df-fl 13761  df-ceil 13762  df-seq 13971  df-exp 14032  df-cj 15050  df-re 15051  df-im 15052  df-sqrt 15186  df-abs 15187  df-limsup 15419  df-clim 15436  df-rlim 15437  df-rest 17372  df-topgen 17393  df-top 22616  df-bases 22669  df-salg 45323  df-salgen 45327  df-smblfn 45710

This theorem is referenced by:  smflimsupmpt  45843