Theorem smflimsup 47525

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 6883	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (ℤ≥‘𝑛) = (ℤ≥‘𝑗))
7	6	iineq1d 45791	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑚))
8		nfcv 2925	. . . . . . . . . 10 ⊢ Ⅎ𝑞dom (𝐹‘𝑚)
9		smflimsup.n	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝐹
10		nfcv 2925	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝑞
11	9, 10	nffv 6893	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝐹‘𝑞)
12	11	nfdm 5943	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑞)
13		fveq2 6883	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑞 → (𝐹‘𝑚) = (𝐹‘𝑞))
14	13	dmeqd 5897	. . . . . . . . . 10 ⊢ (𝑚 = 𝑞 → dom (𝐹‘𝑚) = dom (𝐹‘𝑞))
15	8, 12, 14	cbviin 5001	. . . . . . . . 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 2798	. . . . . . 7 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
18	17	cbviunv 5004	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
19	18	eleq2i 2855	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
20		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑞((𝐹‘𝑚)‘𝑥)
21		nfcv 2925	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑥
22	11, 21	nffv 6893	. . . . . . . 8 ⊢ Ⅎ𝑚((𝐹‘𝑞)‘𝑥)
23	13	fveq1d 6885	. . . . . . . 8 ⊢ (𝑚 = 𝑞 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
24	20, 22, 23	cbvmpt 5214	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))
25	24	fveq2i 6886	. . . . . 6 ⊢ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
26	25	eleq1i 2854	. . . . 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 3419	. . 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 2925	. . . . 5 ⊢ Ⅎ𝑥𝑍
30		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑥(ℤ≥‘𝑗)
31		smflimsup.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
32		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥𝑞
33	31, 32	nffv 6893	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑞)
34	33	nfdm 5943	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑞)
35	30, 34	nfiin 4990	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
36	29, 35	nfiun 4989	. . . 4 ⊢ Ⅎ𝑥∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
37		nfcv 2925	. . . 4 ⊢ Ⅎ𝑤∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
38		nfv 1944	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ
39		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑥lim sup
40		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
41	33, 40	nffv 6893	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑞)‘𝑤)
42	29, 41	nfmpt 5210	. . . . . 6 ⊢ Ⅎ𝑥(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))
43	39, 42	nffv 6893	. . . . 5 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
44		nfcv 2925	. . . . 5 ⊢ Ⅎ𝑥ℝ
45	43, 44	nfel 2939	. . . 4 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ
46		fveq2 6883	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑞)‘𝑥) = ((𝐹‘𝑞)‘𝑤))
47	46	mpteq2dv 5206	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
48	47	fveq2d 6887	. . . . 5 ⊢ (𝑥 = 𝑤 → (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
49	48	eleq1d 2848	. . . 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 3451	. . 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 2790	. 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 5208	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))))
54		nfrab1 3436	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
55	5, 54	nfcxfr 2923	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2925	. . . 4 ⊢ Ⅎ𝑤𝐷
57		nfcv 2925	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
58	55, 56, 57, 43, 48	cbvmptf 5212	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))) = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
59	52, 53, 58	3eqtri 2790	. 2 ⊢ 𝐺 = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
60		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑖)
61	60, 34	nfiin 4990	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
62		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑤∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
63		nfv 1944	. . . . . 6 ⊢ Ⅎ𝑤sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ
64	60, 41	nfmpt 5210	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
65	64	nfrn 5944	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
66		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥ℝ*
67		nfcv 2925	. . . . . . . 8 ⊢ Ⅎ𝑥 <
68	65, 66, 67	nfsup 9412	. . . . . . 7 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )
69	68, 44	nfel 2939	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ
70	46	mpteq2dv 5206	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
71	70	rneqd 5930	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
72	71	supeq1d 9407	. . . . . . 7 ⊢ (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
73	72	eleq1d 2848	. . . . . 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 3451	. . . . 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 6883	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘))
77	76	iineq1d 45791	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) = ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞))
78	77	eleq2d 2849	. . . . . 6 ⊢ (𝑖 = 𝑘 → (𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ↔ 𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞)))
79	76	mpteq1d 5202	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
80	79	rneqd 5930	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
81	80	supeq1d 9407	. . . . . . 7 ⊢ (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
82	81	eleq1d 2848	. . . . . 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 3418	. . . 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 2798	. . 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 5216	. 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 6883	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑝)‘𝑦) = ((𝐹‘𝑝)‘𝑤))
88	87	mpteq2dv 5206	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
89	88	rneqd 5930	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
90	89	supeq1d 9407	. . . . . . 7 ⊢ (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ*, < ) = sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ))
91	90	cbvmptv 5216	. . . . . 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 6883	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
93	92	dmeqd 5897	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → dom (𝐹‘𝑝) = dom (𝐹‘𝑞))
94	93	cbviinv 5005	. . . . . . . . . . . 12 ⊢ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) = ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
95	94	eleq2i 2855	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ↔ 𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞))
96		nfcv 2925	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑞((𝐹‘𝑝)‘𝑥)
97		nfcv 2925	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝐹‘𝑞)
98		nfcv 2925	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝𝑥
99	97, 98	nffv 6893	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝((𝐹‘𝑞)‘𝑥)
100	92	fveq1d 6885	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
101	96, 99, 100	cbvmpt 5214	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
102	101	rneqi 5929	. . . . . . . . . . . . 13 ⊢ ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
103	102	supeq1i 9408	. . . . . . . . . . . 12 ⊢ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < )
104	103	eleq1i 2854	. . . . . . . . . . 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 3419	. . . . . . . . 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 5208	. . . . . . . 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 6884	. . . . . . 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 6885	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑤) = ((𝐹‘𝑞)‘𝑤))
110	109	cbvmptv 5216	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
111	110	rneqi 5929	. . . . . . . 8 ⊢ ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
112	111	supeq1i 9408	. . . . . . 7 ⊢ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )
113	108, 112	mpteq12i 5209	. . . . . 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 2786	. . . . 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 6883	. . . . 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 6883	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (ℤ≥‘𝑙) = (ℤ≥‘𝑘))
118	117	mpteq1d 5202	. . . . . . 7 ⊢ (𝑙 = 𝑘 → (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
119	118	rneqd 5930	. . . . . 6 ⊢ (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
120	119	supeq1d 9407	. . . . 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 5199	. . . 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 2798	. . 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 5216	. 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 47524	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  Ⅎwnfc 2910  {crab 3416  ∪ ciun 4957  ∩ ciin 4958   ↦ cmpt 5193  dom cdm 5663  ran crn 5664  ⟶wf 6534  ‘cfv 6538  supcsup 9401  ℝcr 11100  ℝ^*cxr 11243   < clt 11244  ℤcz 12592  ℤ_≥cuz 12863  lim supclsp 15523  SAlgcsalg 47005  SMblFncsmblfn 47392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734  ax-inf2 9611  ax-cc 10420  ax-ac2 10448  ax-cnex 11157  ax-resscn 11158  ax-1cn 11159  ax-icn 11160  ax-addcl 11161  ax-addrcl 11162  ax-mulcl 11163  ax-mulrcl 11164  ax-mulcom 11165  ax-addass 11166  ax-mulass 11167  ax-distr 11168  ax-i2m1 11169  ax-1ne0 11170  ax-1rid 11171  ax-rnegex 11172  ax-rrecex 11173  ax-cnre 11174  ax-pre-lttri 11175  ax-pre-lttrn 11176  ax-pre-ltadd 11177  ax-pre-mulgt0 11178  ax-pre-sup 11179

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-tp 4595  df-op 4597  df-uni 4874  df-int 4914  df-iun 4959  df-iin 4960  df-br 5111  df-opab 5175  df-mpt 5194  df-tr 5220  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-se 5617  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6304  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-isom 6547  df-riota 7369  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7864  df-1st 7987  df-2nd 7988  df-frecs 8279  df-wrecs 8310  df-recs 8359  df-rdg 8398  df-1o 8454  df-2o 8455  df-oadd 8458  df-omul 8459  df-er 8695  df-map 8827  df-pm 8828  df-en 8945  df-dom 8946  df-sdom 8947  df-fin 8948  df-sup 9403  df-inf 9404  df-oi 9473  df-card 9926  df-acn 9929  df-ac 10101  df-pnf 11246  df-mnf 11247  df-xr 11248  df-ltxr 11249  df-le 11250  df-sub 11444  df-neg 11445  df-div 11873  df-nn 12235  df-2 12304  df-3 12305  df-n0 12506  df-z 12593  df-uz 12864  df-q 12974  df-rp 13018  df-ioo 13377  df-ioc 13378  df-ico 13379  df-fz 13537  df-fl 13827  df-ceil 13828  df-seq 14040  df-exp 14100  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-limsup 15524  df-clim 15541  df-rlim 15542  df-rest 17476  df-topgen 17497  df-top 23032  df-bases 23084  df-salg 47006  df-salgen 47010  df-smblfn 47393

This theorem is referenced by:  smflimsupmpt  47526