smflimsup - Mathbox for Glauco Siliprandi

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Theorem smflimsup 46139

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 6891	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑗))
7	6	iineq1d 44379	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑚))
8		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑞dom (𝐹‘𝑚)
9		smflimsup.n	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝐹
10		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝑞
11	9, 10	nffv 6901	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝐹‘𝑞)
12	11	nfdm 5947	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑞)
13		fveq2 6891	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑞 → (𝐹‘𝑚) = (𝐹‘𝑞))
14	13	dmeqd 5902	. . . . . . . . . 10 ⊢ (𝑚 = 𝑞 → dom (𝐹‘𝑚) = dom (𝐹‘𝑞))
15	8, 12, 14	cbviin 5034	. . . . . . . . 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 2767	. . . . . . 7 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞))
18	17	cbviunv 5037	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
19	18	eleq2i 2820	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞))
20		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑞((𝐹‘𝑚)‘𝑥)
21		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑥
22	11, 21	nffv 6901	. . . . . . . 8 ⊢ Ⅎ𝑚((𝐹‘𝑞)‘𝑥)
23	13	fveq1d 6893	. . . . . . . 8 ⊢ (𝑚 = 𝑞 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
24	20, 22, 23	cbvmpt 5253	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))
25	24	fveq2i 6894	. . . . . 6 ⊢ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
26	25	eleq1i 2819	. . . . 5 ⊢ ((lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ)
27	19, 26	anbi12i 626	. . . 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 3430	. . 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 2898	. . . . 5 ⊢ Ⅎ𝑥𝑍
30		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑥(ℤ_≥‘𝑗)
31		smflimsup.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
32		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑥𝑞
33	31, 32	nffv 6901	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑞)
34	33	nfdm 5947	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑞)
35	30, 34	nfiin 5022	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
36	29, 35	nfiun 5021	. . . 4 ⊢ Ⅎ𝑥∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
37		nfcv 2898	. . . 4 ⊢ Ⅎ𝑤∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
38		nfv 1910	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ
39		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑥lim sup
40		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
41	33, 40	nffv 6901	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑞)‘𝑤)
42	29, 41	nfmpt 5249	. . . . . 6 ⊢ Ⅎ𝑥(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))
43	39, 42	nffv 6901	. . . . 5 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
44		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑥ℝ
45	43, 44	nfel 2912	. . . 4 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ
46		fveq2 6891	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑞)‘𝑥) = ((𝐹‘𝑞)‘𝑤))
47	46	mpteq2dv 5244	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
48	47	fveq2d 6895	. . . . 5 ⊢ (𝑥 = 𝑤 → (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
49	48	eleq1d 2813	. . . 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 3462	. . 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 2759	. 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 5247	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))))
54		nfrab1 3446	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
55	5, 54	nfcxfr 2896	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2898	. . . 4 ⊢ Ⅎ𝑤𝐷
57		nfcv 2898	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
58	55, 56, 57, 43, 48	cbvmptf 5251	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))) = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
59	52, 53, 58	3eqtri 2759	. 2 ⊢ 𝐺 = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
60		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ_≥‘𝑖)
61	60, 34	nfiin 5022	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞)
62		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑤∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞)
63		nfv 1910	. . . . . 6 ⊢ Ⅎ𝑤sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ^*, < ) ∈ ℝ
64	60, 41	nfmpt 5249	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
65	64	nfrn 5948	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
66		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑥ℝ^*
67		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑥 <
68	65, 66, 67	nfsup 9466	. . . . . . 7 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^*, < )
69	68, 44	nfel 2912	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^*, < ) ∈ ℝ
70	46	mpteq2dv 5244	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
71	70	rneqd 5934	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
72	71	supeq1d 9461	. . . . . . 7 ⊢ (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < ))
73	72	eleq1d 2813	. . . . . 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 3462	. . . . 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 6891	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (ℤ_≥‘𝑖) = (ℤ_≥‘𝑘))
77	76	iineq1d 44379	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞) = ∩ 𝑞 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑞))
78	77	eleq2d 2814	. . . . . 6 ⊢ (𝑖 = 𝑘 → (𝑤 ∈ ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞) ↔ 𝑤 ∈ ∩ 𝑞 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑞)))
79	76	mpteq1d 5237	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
80	79	rneqd 5934	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
81	80	supeq1d 9461	. . . . . . 7 ⊢ (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < ))
82	81	eleq1d 2813	. . . . . 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 630	. . . . 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 3429	. . . 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 2767	. . 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 5255	. 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 6891	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑝)‘𝑦) = ((𝐹‘𝑝)‘𝑤))
88	87	mpteq2dv 5244	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
89	88	rneqd 5934	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
90	89	supeq1d 9461	. . . . . . 7 ⊢ (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ^, < ) = sup(ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ^, < ))
91	90	cbvmptv 5255	. . . . . 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 6891	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
93	92	dmeqd 5902	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → dom (𝐹‘𝑝) = dom (𝐹‘𝑞))
94	93	cbviinv 5038	. . . . . . . . . . . 12 ⊢ ∩ 𝑝 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑝) = ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞)
95	94	eleq2i 2820	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑝 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑝) ↔ 𝑥 ∈ ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞))
96		nfcv 2898	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑞((𝐹‘𝑝)‘𝑥)
97		nfcv 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝐹‘𝑞)
98		nfcv 2898	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝𝑥
99	97, 98	nffv 6901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝((𝐹‘𝑞)‘𝑥)
100	92	fveq1d 6893	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
101	96, 99, 100	cbvmpt 5253	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
102	101	rneqi 5933	. . . . . . . . . . . . 13 ⊢ ran (𝑝 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
103	102	supeq1i 9462	. . . . . . . . . . . 12 ⊢ sup(ran (𝑝 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ^, < )
104	103	eleq1i 2819	. . . . . . . . . . 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 626	. . . . . . . . . 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 3430	. . . . . . . . 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 5247	. . . . . . . 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 6892	. . . . . . 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 6893	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑤) = ((𝐹‘𝑞)‘𝑤))
110	109	cbvmptv 5255	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
111	110	rneqi 5933	. . . . . . . 8 ⊢ ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
112	111	supeq1i 9462	. . . . . . 7 ⊢ sup(ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < )
113	108, 112	mpteq12i 5248	. . . . . 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 2755	. . . . 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 6891	. . . . 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 6891	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (ℤ_≥‘𝑙) = (ℤ_≥‘𝑘))
118	117	mpteq1d 5237	. . . . . . 7 ⊢ (𝑙 = 𝑘 → (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
119	118	rneqd 5934	. . . . . 6 ⊢ (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
120	119	supeq1d 9461	. . . . 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 5233	. . . 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 2767	. . 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 5255	. 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 46138	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 Ⅎwnfc 2878 {crab 3427 ∪ ciun 4991 ∩ ciin 4992 ↦ cmpt 5225 dom cdm 5672 ran crn 5673 ⟶wf 6538 ‘cfv 6542 supcsup 9455 ℝcr 11129 ℝ^*cxr 11269 < clt 11270 ℤcz 12580 ℤ_≥cuz 12844 lim supclsp 15438 SAlgcsalg 45619 SMblFncsmblfn 46006

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-inf2 9656 ax-cc 10450 ax-ac2 10478 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 ax-pre-sup 11208

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rmo 3371 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-tp 4629 df-op 4631 df-uni 4904 df-int 4945 df-iun 4993 df-iin 4994 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-se 5628 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 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 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-1st 7987 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-1o 8480 df-oadd 8484 df-omul 8485 df-er 8718 df-map 8838 df-pm 8839 df-en 8956 df-dom 8957 df-sdom 8958 df-fin 8959 df-sup 9457 df-inf 9458 df-oi 9525 df-card 9954 df-acn 9957 df-ac 10131 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-div 11894 df-nn 12235 df-2 12297 df-3 12298 df-n0 12495 df-z 12581 df-uz 12845 df-q 12955 df-rp 12999 df-ioo 13352 df-ioc 13353 df-ico 13354 df-fz 13509 df-fl 13781 df-ceil 13782 df-seq 13991 df-exp 14051 df-cj 15070 df-re 15071 df-im 15072 df-sqrt 15206 df-abs 15207 df-limsup 15439 df-clim 15456 df-rlim 15457 df-rest 17395 df-topgen 17416 df-top 22783 df-bases 22836 df-salg 45620 df-salgen 45624 df-smblfn 46007

This theorem is referenced by: smflimsupmpt 46140

W3C validator