smflimsup - Mathbox for Glauco Siliprandi

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Theorem smflimsup 47366

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 6863	. . . . . . . . 9 ⊢ (𝑛 = 𝑗 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑗))
7	6	iineq1d 45632	. . . . . . . 8 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑚))
8		nfcv 2923	. . . . . . . . . 10 ⊢ Ⅎ𝑞dom (𝐹‘𝑚)
9		smflimsup.n	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝐹
10		nfcv 2923	. . . . . . . . . . . 12 ⊢ Ⅎ𝑚𝑞
11	9, 10	nffv 6873	. . . . . . . . . . 11 ⊢ Ⅎ𝑚(𝐹‘𝑞)
12	11	nfdm 5925	. . . . . . . . . 10 ⊢ Ⅎ𝑚dom (𝐹‘𝑞)
13		fveq2 6863	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑞 → (𝐹‘𝑚) = (𝐹‘𝑞))
14	13	dmeqd 5879	. . . . . . . . . 10 ⊢ (𝑚 = 𝑞 → dom (𝐹‘𝑚) = dom (𝐹‘𝑞))
15	8, 12, 14	cbviin 4992	. . . . . . . . 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 2796	. . . . . . 7 ⊢ (𝑛 = 𝑗 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞))
18	17	cbviunv 4995	. . . . . 6 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
19	18	eleq2i 2853	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞))
20		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑞((𝐹‘𝑚)‘𝑥)
21		nfcv 2923	. . . . . . . . 9 ⊢ Ⅎ𝑚𝑥
22	11, 21	nffv 6873	. . . . . . . 8 ⊢ Ⅎ𝑚((𝐹‘𝑞)‘𝑥)
23	13	fveq1d 6865	. . . . . . . 8 ⊢ (𝑚 = 𝑞 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
24	20, 22, 23	cbvmpt 5201	. . . . . . 7 ⊢ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))
25	24	fveq2i 6866	. . . . . 6 ⊢ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
26	25	eleq1i 2852	. . . . 5 ⊢ ((lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ ↔ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ)
27	19, 26	anbi12i 637	. . . 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 3416	. . 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 2923	. . . . 5 ⊢ Ⅎ𝑥𝑍
30		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥(ℤ_≥‘𝑗)
31		smflimsup.x	. . . . . . . 8 ⊢ Ⅎ𝑥𝐹
32		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥𝑞
33	31, 32	nffv 6873	. . . . . . 7 ⊢ Ⅎ𝑥(𝐹‘𝑞)
34	33	nfdm 5925	. . . . . 6 ⊢ Ⅎ𝑥dom (𝐹‘𝑞)
35	30, 34	nfiin 4981	. . . . 5 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
36	29, 35	nfiun 4980	. . . 4 ⊢ Ⅎ𝑥∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
37		nfcv 2923	. . . 4 ⊢ Ⅎ𝑤∪ 𝑗 ∈ 𝑍 ∩ 𝑞 ∈ (ℤ_≥‘𝑗)dom (𝐹‘𝑞)
38		nfv 1933	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) ∈ ℝ
39		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥lim sup
40		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥𝑤
41	33, 40	nffv 6873	. . . . . . 7 ⊢ Ⅎ𝑥((𝐹‘𝑞)‘𝑤)
42	29, 41	nfmpt 5197	. . . . . 6 ⊢ Ⅎ𝑥(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))
43	39, 42	nffv 6873	. . . . 5 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
44		nfcv 2923	. . . . 5 ⊢ Ⅎ𝑥ℝ
45	43, 44	nfel 2937	. . . 4 ⊢ Ⅎ𝑥(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))) ∈ ℝ
46		fveq2 6863	. . . . . . 7 ⊢ (𝑥 = 𝑤 → ((𝐹‘𝑞)‘𝑥) = ((𝐹‘𝑞)‘𝑤))
47	46	mpteq2dv 5193	. . . . . 6 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤)))
48	47	fveq2d 6867	. . . . 5 ⊢ (𝑥 = 𝑤 → (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))) = (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
49	48	eleq1d 2846	. . . 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 3448	. . 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 2788	. 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 5195	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)))) = (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥))))
54		nfrab1 3433	. . . . 5 ⊢ Ⅎ𝑥{𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (lim sup‘(𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥))) ∈ ℝ}
55	5, 54	nfcxfr 2921	. . . 4 ⊢ Ⅎ𝑥𝐷
56		nfcv 2923	. . . 4 ⊢ Ⅎ𝑤𝐷
57		nfcv 2923	. . . 4 ⊢ Ⅎ𝑤(lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))
58	55, 56, 57, 43, 48	cbvmptf 5199	. . 3 ⊢ (𝑥 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑥)))) = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
59	52, 53, 58	3eqtri 2788	. 2 ⊢ 𝐺 = (𝑤 ∈ 𝐷 ↦ (lim sup‘(𝑞 ∈ 𝑍 ↦ ((𝐹‘𝑞)‘𝑤))))
60		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑥(ℤ_≥‘𝑖)
61	60, 34	nfiin 4981	. . . . . 6 ⊢ Ⅎ𝑥∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞)
62		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑤∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞)
63		nfv 1933	. . . . . 6 ⊢ Ⅎ𝑤sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ^*, < ) ∈ ℝ
64	60, 41	nfmpt 5197	. . . . . . . . 9 ⊢ Ⅎ𝑥(𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
65	64	nfrn 5926	. . . . . . . 8 ⊢ Ⅎ𝑥ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤))
66		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥ℝ^*
67		nfcv 2923	. . . . . . . 8 ⊢ Ⅎ𝑥 <
68	65, 66, 67	nfsup 9394	. . . . . . 7 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^*, < )
69	68, 44	nfel 2937	. . . . . 6 ⊢ Ⅎ𝑥sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^*, < ) ∈ ℝ
70	46	mpteq2dv 5193	. . . . . . . . 9 ⊢ (𝑥 = 𝑤 → (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
71	70	rneqd 5912	. . . . . . . 8 ⊢ (𝑥 = 𝑤 → ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)) = ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)))
72	71	supeq1d 9389	. . . . . . 7 ⊢ (𝑥 = 𝑤 → sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < ))
73	72	eleq1d 2846	. . . . . 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 3448	. . . . 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 6863	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → (ℤ_≥‘𝑖) = (ℤ_≥‘𝑘))
77	76	iineq1d 45632	. . . . . . 7 ⊢ (𝑖 = 𝑘 → ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞) = ∩ 𝑞 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑞))
78	77	eleq2d 2847	. . . . . 6 ⊢ (𝑖 = 𝑘 → (𝑤 ∈ ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞) ↔ 𝑤 ∈ ∩ 𝑞 ∈ (ℤ_≥‘𝑘)dom (𝐹‘𝑞)))
79	76	mpteq1d 5189	. . . . . . . . 9 ⊢ (𝑖 = 𝑘 → (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
80	79	rneqd 5912	. . . . . . . 8 ⊢ (𝑖 = 𝑘 → ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
81	80	supeq1d 9389	. . . . . . 7 ⊢ (𝑖 = 𝑘 → sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < ))
82	81	eleq1d 2846	. . . . . 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 641	. . . . 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 3415	. . . 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 2796	. . 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 5203	. 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 6863	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → ((𝐹‘𝑝)‘𝑦) = ((𝐹‘𝑝)‘𝑤))
88	87	mpteq2dv 5193	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
89	88	rneqd 5912	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)) = ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)))
90	89	supeq1d 9389	. . . . . . 7 ⊢ (𝑦 = 𝑤 → sup(ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑦)), ℝ^, < ) = sup(ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ^, < ))
91	90	cbvmptv 5203	. . . . . 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 6863	. . . . . . . . . . . . . 14 ⊢ (𝑝 = 𝑞 → (𝐹‘𝑝) = (𝐹‘𝑞))
93	92	dmeqd 5879	. . . . . . . . . . . . 13 ⊢ (𝑝 = 𝑞 → dom (𝐹‘𝑝) = dom (𝐹‘𝑞))
94	93	cbviinv 4996	. . . . . . . . . . . 12 ⊢ ∩ 𝑝 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑝) = ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞)
95	94	eleq2i 2853	. . . . . . . . . . 11 ⊢ (𝑥 ∈ ∩ 𝑝 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑝) ↔ 𝑥 ∈ ∩ 𝑞 ∈ (ℤ_≥‘𝑖)dom (𝐹‘𝑞))
96		nfcv 2923	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑞((𝐹‘𝑝)‘𝑥)
97		nfcv 2923	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝(𝐹‘𝑞)
98		nfcv 2923	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑝𝑥
99	97, 98	nffv 6873	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑝((𝐹‘𝑞)‘𝑥)
100	92	fveq1d 6865	. . . . . . . . . . . . . . 15 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑥) = ((𝐹‘𝑞)‘𝑥))
101	96, 99, 100	cbvmpt 5201	. . . . . . . . . . . . . 14 ⊢ (𝑝 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
102	101	rneqi 5911	. . . . . . . . . . . . 13 ⊢ ran (𝑝 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)) = ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥))
103	102	supeq1i 9390	. . . . . . . . . . . 12 ⊢ sup(ran (𝑝 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑝)‘𝑥)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑖) ↦ ((𝐹‘𝑞)‘𝑥)), ℝ^, < )
104	103	eleq1i 2852	. . . . . . . . . . 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 637	. . . . . . . . . 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 3416	. . . . . . . . 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 5195	. . . . . . . 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 6864	. . . . . . 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 6865	. . . . . . . . . 10 ⊢ (𝑝 = 𝑞 → ((𝐹‘𝑝)‘𝑤) = ((𝐹‘𝑞)‘𝑤))
110	109	cbvmptv 5203	. . . . . . . . 9 ⊢ (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
111	110	rneqi 5911	. . . . . . . 8 ⊢ ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)) = ran (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤))
112	111	supeq1i 9390	. . . . . . 7 ⊢ sup(ran (𝑝 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑝)‘𝑤)), ℝ^, < ) = sup(ran (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)), ℝ^, < )
113	108, 112	mpteq12i 5196	. . . . . 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 2784	. . . . 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 6863	. . . . 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 6863	. . . . . . . 8 ⊢ (𝑙 = 𝑘 → (ℤ_≥‘𝑙) = (ℤ_≥‘𝑘))
118	117	mpteq1d 5189	. . . . . . 7 ⊢ (𝑙 = 𝑘 → (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
119	118	rneqd 5912	. . . . . 6 ⊢ (𝑙 = 𝑘 → ran (𝑞 ∈ (ℤ_≥‘𝑙) ↦ ((𝐹‘𝑞)‘𝑤)) = ran (𝑞 ∈ (ℤ_≥‘𝑘) ↦ ((𝐹‘𝑞)‘𝑤)))
120	119	supeq1d 9389	. . . . 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 5186	. . . 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 2796	. . 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 5203	. 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 47365	1 ⊢ (𝜑 → 𝐺 ∈ (SMblFn‘𝑆))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1559 ∈ wcel 2141 Ⅎwnfc 2908 {crab 3413 ∪ ciun 4948 ∩ ciin 4949 ↦ cmpt 5180 dom cdm 5645 ran crn 5646 ⟶wf 6513 ‘cfv 6517 supcsup 9383 ℝcr 11069 ℝ^*cxr 11212 < clt 11213 ℤcz 12565 ℤ_≥cuz 12836 lim supclsp 15480 SAlgcsalg 46846 SMblFncsmblfn 47233

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5226 ax-sep 5245 ax-nul 5255 ax-pow 5321 ax-pr 5389 ax-un 7714 ax-inf2 9593 ax-cc 10389 ax-ac2 10417 ax-cnex 11126 ax-resscn 11127 ax-1cn 11128 ax-icn 11129 ax-addcl 11130 ax-addrcl 11131 ax-mulcl 11132 ax-mulrcl 11133 ax-mulcom 11134 ax-addass 11135 ax-mulass 11136 ax-distr 11137 ax-i2m1 11138 ax-1ne0 11139 ax-1rid 11140 ax-rnegex 11141 ax-rrecex 11142 ax-cnre 11143 ax-pre-lttri 11144 ax-pre-lttrn 11145 ax-pre-ltadd 11146 ax-pre-mulgt0 11147 ax-pre-sup 11148

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3745 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-pss 3924 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-tp 4586 df-op 4588 df-uni 4865 df-int 4905 df-iun 4950 df-iin 4951 df-br 5100 df-opab 5162 df-mpt 5181 df-tr 5207 df-id 5540 df-eprel 5545 df-po 5553 df-so 5554 df-fr 5598 df-se 5599 df-we 5600 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-pred 6284 df-ord 6345 df-on 6346 df-lim 6347 df-suc 6348 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-isom 6526 df-riota 7349 df-ov 7395 df-oprab 7396 df-mpo 7397 df-om 7843 df-1st 7966 df-2nd 7967 df-frecs 8257 df-wrecs 8288 df-recs 8337 df-rdg 8376 df-1o 8432 df-2o 8433 df-oadd 8436 df-omul 8437 df-er 8673 df-map 8805 df-pm 8806 df-en 8924 df-dom 8925 df-sdom 8926 df-fin 8927 df-sup 9385 df-inf 9386 df-oi 9455 df-card 9894 df-acn 9897 df-ac 10069 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11413 df-neg 11414 df-div 11842 df-nn 12208 df-2 12277 df-3 12278 df-n0 12479 df-z 12566 df-uz 12837 df-q 12947 df-rp 12991 df-ioo 13350 df-ioc 13351 df-ico 13352 df-fz 13510 df-fl 13799 df-ceil 13800 df-seq 14012 df-exp 14072 df-cj 15109 df-re 15110 df-im 15111 df-sqrt 15245 df-abs 15246 df-limsup 15481 df-clim 15498 df-rlim 15499 df-rest 17434 df-topgen 17455 df-top 22934 df-bases 22986 df-salg 46847 df-salgen 46851 df-smblfn 47234

This theorem is referenced by: smflimsupmpt 47367

W3C validator