Theorem smflimsup 45059

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 6842	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (ℤ≥‘𝑛) = (ℤ≥‘𝑗))
7	6	iineq1d 43290	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑚))
8		nfcv 2907	. . . . . . . . . 10 ⊢ Ⅎ𝑞dom (𝐹‘𝑚)
9		smflimsup.n	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝐹
10		nfcv 2907	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝑞
11	9, 10	nffv 6852	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝐹‘𝑞)
12	11	nfdm 5906	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑞)
13		fveq2 6842	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑞 → (𝐹‘𝑚) = (𝐹‘𝑞))
14	13	dmeqd 5861	. . . . . . . . . 10 ⊢ (𝑚 = 𝑞 → dom (𝐹‘𝑚) = dom (𝐹‘𝑞))
15	8, 12, 14	cbviin 4997	. . . . . . . . 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 2776	. . . . . . 7 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
18	17	cbviunv 5000	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
19	18	eleq2i 2829	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞))
20		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑞((𝐹‘𝑚)‘𝑥)
21		nfcv 2907	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑥
22	11, 21	nffv 6852	. . . . . . . 8 ⊢ Ⅎ𝑚((𝐹‘𝑞)‘𝑥)
23	13	fveq1d 6844	. . . . . . . 8 ⊢ (𝑚 = 𝑞 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
24	20, 22, 23	cbvmpt 5216	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))
25	24	fveq2i 6845	. . . . . 6 ⊢ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
26	25	eleq1i 2828	. . . . 5 ⊢ ((lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ)
27	19, 26	anbi12i 627	. . . 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 3410	. . 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 2907	. . . . 5 ⊢ Ⅎ𝑥𝑍
30		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥(ℤ≥‘𝑗)
31		smflimsup.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
32		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥𝑞
33	31, 32	nffv 6852	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑞)
34	33	nfdm 5906	. . . . . 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 2907	. . . 4 ⊢ Ⅎ𝑤∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ≥‘𝑗)dom (𝐹‘𝑞)
38		nfv 1917	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ
39		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑥lim sup
40		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
41	33, 40	nffv 6852	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑞)‘𝑤)
42	29, 41	nfmpt 5212	. . . . . 6 ⊢ Ⅎ𝑥(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))
43	39, 42	nffv 6852	. . . . 5 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
44		nfcv 2907	. . . . 5 ⊢ Ⅎ𝑥ℝ
45	43, 44	nfel 2921	. . . 4 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ
46		fveq2 6842	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑞)‘𝑥) = ((𝐹‘𝑞)‘𝑤))
47	46	mpteq2dv 5207	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
48	47	fveq2d 6846	. . . . 5 ⊢ (𝑥 = 𝑤 → (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
49	48	eleq1d 2822	. . . 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 3439	. . 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 2768	. 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 5210	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))))
54		nfrab1 3426	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
55	5, 54	nfcxfr 2905	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2907	. . . 4 ⊢ Ⅎ𝑤𝐷
57		nfcv 2907	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
58	55, 56, 57, 43, 48	cbvmptf 5214	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))) = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
59	52, 53, 58	3eqtri 2768	. 2 ⊢ 𝐺 = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
60		nfcv 2907	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ≥‘𝑖)
61	60, 34	nfiin 4985	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
62		nfcv 2907	. . . . . 6 ⊢ Ⅎ𝑤∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
63		nfv 1917	. . . . . 6 ⊢ Ⅎ𝑤sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) ∈ ℝ
64	60, 41	nfmpt 5212	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
65	64	nfrn 5907	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
66		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥ℝ*
67		nfcv 2907	. . . . . . . 8 ⊢ Ⅎ𝑥 <
68	65, 66, 67	nfsup 9387	. . . . . . 7 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )
69	68, 44	nfel 2921	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) ∈ ℝ
70	46	mpteq2dv 5207	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
71	70	rneqd 5893	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
72	71	supeq1d 9382	. . . . . . 7 ⊢ (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
73	72	eleq1d 2822	. . . . . 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 3439	. . . . 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 6842	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (ℤ≥‘𝑖) = (ℤ≥‘𝑘))
77	76	iineq1d 43290	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) = ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞))
78	77	eleq2d 2823	. . . . . 6 ⊢ (𝑖 = 𝑘 → (𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞) ↔ 𝑤 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑘)dom (𝐹‘𝑞)))
79	76	mpteq1d 5200	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
80	79	rneqd 5893	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
81	80	supeq1d 9382	. . . . . . 7 ⊢ (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < ))
82	81	eleq1d 2822	. . . . . 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 631	. . . . 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 3409	. . . 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 2776	. . 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 5218	. 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 6842	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑝)‘𝑦) = ((𝐹‘𝑝)‘𝑤))
88	87	mpteq2dv 5207	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
89	88	rneqd 5893	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
90	89	supeq1d 9382	. . . . . . 7 ⊢ (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ*, < ) = sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ))
91	90	cbvmptv 5218	. . . . . 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 6842	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
93	92	dmeqd 5861	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → dom (𝐹‘𝑝) = dom (𝐹‘𝑞))
94	93	cbviinv 5001	. . . . . . . . . . . 12 ⊢ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) = ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞)
95	94	eleq2i 2829	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑝 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑝) ↔ 𝑥 ∈ ∩ 𝑞 ∈ (ℤ≥‘𝑖)dom (𝐹‘𝑞))
96		nfcv 2907	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑞((𝐹‘𝑝)‘𝑥)
97		nfcv 2907	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝐹‘𝑞)
98		nfcv 2907	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝𝑥
99	97, 98	nffv 6852	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝((𝐹‘𝑞)‘𝑥)
100	92	fveq1d 6844	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
101	96, 99, 100	cbvmpt 5216	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
102	101	rneqi 5892	. . . . . . . . . . . . 13 ⊢ ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
103	102	supeq1i 9383	. . . . . . . . . . . 12 ⊢ sup(ran (𝑝 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ*, < )
104	103	eleq1i 2828	. . . . . . . . . . 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 627	. . . . . . . . . 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 3410	. . . . . . . . 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 5210	. . . . . . . 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 6843	. . . . . . 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 6844	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑤) = ((𝐹‘𝑞)‘𝑤))
110	109	cbvmptv 5218	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
111	110	rneqi 5892	. . . . . . . 8 ⊢ ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
112	111	supeq1i 9383	. . . . . . 7 ⊢ sup(ran (𝑝 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ*, < ) = sup(ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ*, < )
113	108, 112	mpteq12i 5211	. . . . . 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 2764	. . . . 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 6842	. . . . 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 6842	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (ℤ≥‘𝑙) = (ℤ≥‘𝑘))
118	117	mpteq1d 5200	. . . . . . 7 ⊢ (𝑙 = 𝑘 → (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
119	118	rneqd 5893	. . . . . 6 ⊢ (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
120	119	supeq1d 9382	. . . . 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 5196	. . . 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 2776	. . 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 5218	. 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 45058	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2106  Ⅎwnfc 2887  {crab 3407  ∪ ciun 4954  ∩ ciin 4955   ↦ cmpt 5188  dom cdm 5633  ran crn 5634  ⟶wf 6492  ‘cfv 6496  supcsup 9376  ℝcr 11050  ℝ^*cxr 11188   < clt 11189  ℤcz 12499  ℤ_≥cuz 12763  lim supclsp 15352  SAlgcsalg 44539  SMblFncsmblfn 44926

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-inf2 9577  ax-cc 10371  ax-ac2 10399  ax-cnex 11107  ax-resscn 11108  ax-1cn 11109  ax-icn 11110  ax-addcl 11111  ax-addrcl 11112  ax-mulcl 11113  ax-mulrcl 11114  ax-mulcom 11115  ax-addass 11116  ax-mulass 11117  ax-distr 11118  ax-i2m1 11119  ax-1ne0 11120  ax-1rid 11121  ax-rnegex 11122  ax-rrecex 11123  ax-cnre 11124  ax-pre-lttri 11125  ax-pre-lttrn 11126  ax-pre-ltadd 11127  ax-pre-mulgt0 11128  ax-pre-sup 11129

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3065  df-rex 3074  df-rmo 3353  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-tp 4591  df-op 4593  df-uni 4866  df-int 4908  df-iun 4956  df-iin 4957  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-se 5589  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-isom 6505  df-riota 7313  df-ov 7360  df-oprab 7361  df-mpo 7362  df-om 7803  df-1st 7921  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-1o 8412  df-oadd 8416  df-omul 8417  df-er 8648  df-map 8767  df-pm 8768  df-en 8884  df-dom 8885  df-sdom 8886  df-fin 8887  df-sup 9378  df-inf 9379  df-oi 9446  df-card 9875  df-acn 9878  df-ac 10052  df-pnf 11191  df-mnf 11192  df-xr 11193  df-ltxr 11194  df-le 11195  df-sub 11387  df-neg 11388  df-div 11813  df-nn 12154  df-2 12216  df-3 12217  df-n0 12414  df-z 12500  df-uz 12764  df-q 12874  df-rp 12916  df-ioo 13268  df-ioc 13269  df-ico 13270  df-fz 13425  df-fl 13697  df-ceil 13698  df-seq 13907  df-exp 13968  df-cj 14984  df-re 14985  df-im 14986  df-sqrt 15120  df-abs 15121  df-limsup 15353  df-clim 15370  df-rlim 15371  df-rest 17304  df-topgen 17325  df-top 22243  df-bases 22296  df-salg 44540  df-salgen 44544  df-smblfn 44927

This theorem is referenced by:  smflimsupmpt  45060