smflimsup - Mathbox for Glauco Siliprandi

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Theorem smflimsup 43109

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 6672	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑗))
7	6	iineq1d 41363	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑚))
8		nfcv 2979	. . . . . . . . . 10 ⊢ Ⅎ𝑞dom (𝐹‘𝑚)
9		smflimsup.n	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝐹
10		nfcv 2979	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝑞
11	9, 10	nffv 6682	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝐹‘𝑞)
12	11	nfdm 5825	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑞)
13		fveq2 6672	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑞 → (𝐹‘𝑚) = (𝐹‘𝑞))
14	13	dmeqd 5776	. . . . . . . . . 10 ⊢ (𝑚 = 𝑞 → dom (𝐹‘𝑚) = dom (𝐹‘𝑞))
15	8, 12, 14	cbviin 4964	. . . . . . . . 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 2858	. . . . . . 7 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞))
18	17	cbviunv 4967	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
19	18	eleq2i 2906	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞))
20		nfcv 2979	. . . . . . . 8 ⊢ Ⅎ𝑞((𝐹‘𝑚)‘𝑥)
21		nfcv 2979	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑥
22	11, 21	nffv 6682	. . . . . . . 8 ⊢ Ⅎ𝑚((𝐹‘𝑞)‘𝑥)
23	13	fveq1d 6674	. . . . . . . 8 ⊢ (𝑚 = 𝑞 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
24	20, 22, 23	cbvmpt 5169	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))
25	24	fveq2i 6675	. . . . . 6 ⊢ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
26	25	eleq1i 2905	. . . . 5 ⊢ ((lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ)
27	19, 26	anbi12i 628	. . . 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 3479	. . 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 2979	. . . . 5 ⊢ Ⅎ𝑥𝑍
30		nfcv 2979	. . . . . 6 ⊢ Ⅎ𝑥(ℤ_≥‘𝑗)
31		smflimsup.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
32		nfcv 2979	. . . . . . . 8 ⊢ Ⅎ𝑥𝑞
33	31, 32	nffv 6682	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑞)
34	33	nfdm 5825	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑞)
35	30, 34	nfiin 4952	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
36	29, 35	nfiun 4951	. . . 4 ⊢ Ⅎ𝑥∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
37		nfcv 2979	. . . 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 2979	. . . . . 6 ⊢ Ⅎ𝑥lim sup
40		nfcv 2979	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
41	33, 40	nffv 6682	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑞)‘𝑤)
42	29, 41	nfmpt 5165	. . . . . 6 ⊢ Ⅎ𝑥(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))
43	39, 42	nffv 6682	. . . . 5 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
44		nfcv 2979	. . . . 5 ⊢ Ⅎ𝑥ℝ
45	43, 44	nfel 2994	. . . 4 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ
46		fveq2 6672	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑞)‘𝑥) = ((𝐹‘𝑞)‘𝑤))
47	46	mpteq2dv 5164	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
48	47	fveq2d 6676	. . . . 5 ⊢ (𝑥 = 𝑤 → (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
49	48	eleq1d 2899	. . . 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 3491	. . 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 2850	. 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 5160	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))))
54		nfrab1 3386	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
55	5, 54	nfcxfr 2977	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2979	. . . 4 ⊢ Ⅎ𝑤𝐷
57		nfcv 2979	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
58	55, 56, 57, 43, 48	cbvmptf 5167	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))) = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
59	52, 53, 58	3eqtri 2850	. 2 ⊢ 𝐺 = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
60		nfcv 2979	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ_≥‘𝑖)
61	60, 34	nfiin 4952	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞)
62		nfcv 2979	. . . . . 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 5165	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
65	64	nfrn 5826	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
66		nfcv 2979	. . . . . . . 8 ⊢ Ⅎ𝑥ℝ^*
67		nfcv 2979	. . . . . . . 8 ⊢ Ⅎ𝑥 <
68	65, 66, 67	nfsup 8917	. . . . . . 7 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^*, < )
69	68, 44	nfel 2994	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^*, < ) ∈ ℝ
70	46	mpteq2dv 5164	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
71	70	rneqd 5810	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
72	71	supeq1d 8912	. . . . . . 7 ⊢ (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < ))
73	72	eleq1d 2899	. . . . . 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 3491	. . . . 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 6672	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (ℤ_≥‘𝑖) = (ℤ_≥‘𝑘))
77	76	iineq1d 41363	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞) = ∩ 𝑞 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑞))
78	77	eleq2d 2900	. . . . . 6 ⊢ (𝑖 = 𝑘 → (𝑤 ∈ ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞) ↔ 𝑤 ∈ ∩ 𝑞 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑞)))
79	76	mpteq1d 5157	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
80	79	rneqd 5810	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
81	80	supeq1d 8912	. . . . . . 7 ⊢ (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < ))
82	81	eleq1d 2899	. . . . . 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 632	. . . . 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 3478	. . . 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 2858	. . 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 5171	. 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 6672	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑝)‘𝑦) = ((𝐹‘𝑝)‘𝑤))
88	87	mpteq2dv 5164	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
89	88	rneqd 5810	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
90	89	supeq1d 8912	. . . . . . 7 ⊢ (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ^, < ) = sup(ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ^, < ))
91	90	cbvmptv 5171	. . . . . 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 6672	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
93	92	dmeqd 5776	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → dom (𝐹‘𝑝) = dom (𝐹‘𝑞))
94	93	cbviinv 4968	. . . . . . . . . . . 12 ⊢ ∩ 𝑝 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑝) = ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞)
95	94	eleq2i 2906	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑝 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑝) ↔ 𝑥 ∈ ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞))
96		nfcv 2979	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑞((𝐹‘𝑝)‘𝑥)
97		nfcv 2979	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝐹‘𝑞)
98		nfcv 2979	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝𝑥
99	97, 98	nffv 6682	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝((𝐹‘𝑞)‘𝑥)
100	92	fveq1d 6674	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
101	96, 99, 100	cbvmpt 5169	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
102	101	rneqi 5809	. . . . . . . . . . . . 13 ⊢ ran (𝑝 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
103	102	supeq1i 8913	. . . . . . . . . . . 12 ⊢ sup(ran (𝑝 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ^, < )
104	103	eleq1i 2905	. . . . . . . . . . 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 628	. . . . . . . . . 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 3479	. . . . . . . . 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 5160	. . . . . . . 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 6673	. . . . . . 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 6674	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑤) = ((𝐹‘𝑞)‘𝑤))
110	109	cbvmptv 5171	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
111	110	rneqi 5809	. . . . . . . 8 ⊢ ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
112	111	supeq1i 8913	. . . . . . 7 ⊢ sup(ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < )
113	108, 112	mpteq12i 5161	. . . . . 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 2846	. . . . 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 6672	. . . . 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 6672	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (ℤ_≥‘𝑙) = (ℤ_≥‘𝑘))
118	117	mpteq1d 5157	. . . . . . 7 ⊢ (𝑙 = 𝑘 → (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
119	118	rneqd 5810	. . . . . 6 ⊢ (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
120	119	supeq1d 8912	. . . . 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 5153	. . . 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 2858	. . 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 5171	. 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 43108	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 Ⅎwnfc 2963 {crab 3144 ∪ ciun 4921 ∩ ciin 4922 ↦ cmpt 5148 dom cdm 5557 ran crn 5558 ⟶wf 6353 ‘cfv 6357 supcsup 8906 ℝcr 10538 ℝ^*cxr 10676 < clt 10677 ℤcz 11984 ℤ_≥cuz 12246 lim supclsp 14829 SAlgcsalg 42600 SMblFncsmblfn 42984

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cc 9859 ax-ac2 9887 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-iin 4924 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-omul 8109 df-er 8291 df-map 8410 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-acn 9373 df-ac 9544 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-n0 11901 df-z 11985 df-uz 12247 df-q 12352 df-rp 12393 df-ioo 12745 df-ioc 12746 df-ico 12747 df-fz 12896 df-fl 13165 df-ceil 13166 df-seq 13373 df-exp 13433 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-rest 16698 df-topgen 16719 df-top 21504 df-bases 21556 df-salg 42601 df-salgen 42605 df-smblfn 42985

This theorem is referenced by: smflimsupmpt 43110

W3C validator