Theorem smflimsuplem3 46944

Description: The limit of the (𝐻‘𝑛) functions is sigma-measurable. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Hypotheses

Ref	Expression
smflimsuplem3.m	⊢ (𝜑 → 𝑀 ∈ ℤ)
smflimsuplem3.z	⊢ 𝑍 = (ℤ≥‘𝑀)
smflimsuplem3.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimsuplem3.f	⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
smflimsuplem3.e	⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
smflimsuplem3.h	⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))

Assertion

Ref	Expression
smflimsuplem3	⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) ∈ (SMblFn‘𝑆))

Distinct variable groups:   𝑥,𝐸   𝑚,𝐹,𝑥   𝑘,𝐻,𝑥   𝑆,𝑘,𝑛   𝑘,𝑍,𝑛,𝑥   𝑚,𝑍,𝑛   𝜑,𝑘,𝑛,𝑥   𝜑,𝑚

Allowed substitution hints:   𝑆(𝑥,𝑚)   𝐸(𝑘,𝑚,𝑛)   𝐹(𝑘,𝑛)   𝐻(𝑚,𝑛)   𝑀(𝑥,𝑘,𝑚,𝑛)

Proof of Theorem smflimsuplem3

Step	Hyp	Ref	Expression
1		nfv 1915	. 2 ⊢ Ⅎ𝑛𝜑
2		nfv 1915	. 2 ⊢ Ⅎ𝑥𝜑
3		nfv 1915	. 2 ⊢ Ⅎ𝑘𝜑
4		smflimsuplem3.m	. 2 ⊢ (𝜑 → 𝑀 ∈ ℤ)
5		smflimsuplem3.z	. 2 ⊢ 𝑍 = (ℤ≥‘𝑀)
6		fvex 6841	. . . 4 ⊢ (𝐻‘𝑛) ∈ V
7	6	dmex 7845	. . 3 ⊢ dom (𝐻‘𝑛) ∈ V
8	7	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐻‘𝑛) ∈ V)
9		fvexd 6843	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐻‘𝑛)) → ((𝐻‘𝑛)‘𝑥) ∈ V)
10		smflimsuplem3.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11	10	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑆 ∈ SAlg)
12		smflimsuplem3.e	. . . . . . . . . . . . 13 ⊢ 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
13	12	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐸 = (𝑛 ∈ 𝑍 ↦ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}))
14		eqid 2733	. . . . . . . . . . . . 13 ⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
15	5	eluzelz2 45525	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
16		eqid 2733	. . . . . . . . . . . . . . . 16 ⊢ (ℤ≥‘𝑛) = (ℤ≥‘𝑛)
17	15, 16	uzn0d 45547	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ≠ ∅)
18		fvex 6841	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹‘𝑚) ∈ V
19	18	dmex 7845	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝐹‘𝑚) ∈ V
20	19	rgenw 3052	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V
21	20	a1i 11	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → ∀𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
22	17, 21	iinexd 45254	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
23	22	adantl 481	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
24	14, 23	rabexd 5280	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ∈ V)
25	13, 24	fvmpt2d 6948	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
26		fvres 6847	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) → ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) = (𝐹‘𝑚))
27	26	eqcomd 2739	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚))
28	27	adantl 481	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚))
29	28	dmeqd 5849	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ≥‘𝑛)) → dom (𝐹‘𝑚) = dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚))
30	29	iineq2dv 4967	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚))
31	30	eleq2d 2819	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)))
32	27	fveq1d 6830	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) → ((𝐹‘𝑚)‘𝑥) = (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥))
33	32	mpteq2ia 5188	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥))
34	33	rneqi 5881	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥))
35	34	supeq1i 9338	. . . . . . . . . . . . . . 15 ⊢ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < )
36	35	a1i 11	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) = sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ))
37	36	eleq1d 2818	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ ↔ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ))
38	31, 37	anbi12d 632	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ((𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ) ↔ (𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∧ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ)))
39	38	rabbidva2 3398	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom (𝐹‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
40	25, 39	eqtrd 2768	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐸‘𝑛) = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ})
41	40, 36	mpteq12dv 5180	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < )))
42		nfcv 2895	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝐹 ↾ (ℤ≥‘𝑛))
43		nfcv 2895	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹 ↾ (ℤ≥‘𝑛))
44	15	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
45		smflimsuplem3.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
46	45	adantr 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶(SMblFn‘𝑆))
47	5	eleq2i 2825	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ≥‘𝑀))
48	47	biimpi 216	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ≥‘𝑀))
49		uzss 12761	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ≥‘𝑀) → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
50	48, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ (ℤ≥‘𝑀))
51	50, 5	sseqtrrdi 3972	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (ℤ≥‘𝑛) ⊆ 𝑍)
52	51	adantl 481	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ≥‘𝑛) ⊆ 𝑍)
53	46, 52	fssresd 6695	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ≥‘𝑛)):(ℤ≥‘𝑛)⟶(SMblFn‘𝑆))
54		eqid 2733	. . . . . . . . . 10 ⊢ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} = {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ}
55		eqid 2733	. . . . . . . . . 10 ⊢ (𝑥 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < )) = (𝑥 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ))
56	42, 43, 44, 16, 11, 53, 54, 55	smfsupxr 46938	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ {𝑥 ∈ ∩ 𝑚 ∈ (ℤ≥‘𝑛)dom ((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚) ∣ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < ) ∈ ℝ} ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ (((𝐹 ↾ (ℤ≥‘𝑛))‘𝑚)‘𝑥)), ℝ*, < )) ∈ (SMblFn‘𝑆))
57	41, 56	eqeltrd 2833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )) ∈ (SMblFn‘𝑆))
58		smflimsuplem3.h	. . . . . . . 8 ⊢ 𝐻 = (𝑛 ∈ 𝑍 ↦ (𝑥 ∈ (𝐸‘𝑛) ↦ sup(ran (𝑚 ∈ (ℤ≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)), ℝ*, < )))
59	57, 58	fmptd 7053	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝑍⟶(SMblFn‘𝑆))
60	59	ffvelcdmda 7023	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ (SMblFn‘𝑆))
61		eqid 2733	. . . . . 6 ⊢ dom (𝐻‘𝑛) = dom (𝐻‘𝑛)
62	11, 60, 61	smff 46854	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):dom (𝐻‘𝑛)⟶ℝ)
63	62	feqmptd 6896	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)))
64	63	eqcomd 2739	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)) = (𝐻‘𝑛))
65	64, 60	eqeltrd 2833	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
66		eqid 2733	. 2 ⊢ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } = {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ }
67		eqid 2733	. 2 ⊢ (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) = (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥))))
68	1, 2, 3, 4, 5, 8, 9, 10, 65, 66, 67	smflimmpt 46932	1 ⊢ (𝜑 → (𝑥 ∈ {𝑥 ∈ ∪ 𝑘 ∈ 𝑍 ∩ 𝑛 ∈ (ℤ≥‘𝑘)dom (𝐻‘𝑛) ∣ (𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)) ∈ dom ⇝ } ↦ ( ⇝ ‘(𝑛 ∈ 𝑍 ↦ ((𝐻‘𝑛)‘𝑥)))) ∈ (SMblFn‘𝑆))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  ∀wral 3048  {crab 3396  Vcvv 3437   ⊆ wss 3898  ∪ ciun 4941  ∩ ciin 4942   ↦ cmpt 5174  dom cdm 5619  ran crn 5620   ↾ cres 5621  ⟶wf 6482  ‘cfv 6486  supcsup 9331  ℝcr 11012  ℝ^*cxr 11152   < clt 11153  ℤcz 12475  ℤ_≥cuz 12738   ⇝ cli 15393  SAlgcsalg 46430  SMblFncsmblfn 46817

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5219  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-inf2 9538  ax-cc 10333  ax-ac2 10361  ax-cnex 11069  ax-resscn 11070  ax-1cn 11071  ax-icn 11072  ax-addcl 11073  ax-addrcl 11074  ax-mulcl 11075  ax-mulrcl 11076  ax-mulcom 11077  ax-addass 11078  ax-mulass 11079  ax-distr 11080  ax-i2m1 11081  ax-1ne0 11082  ax-1rid 11083  ax-rnegex 11084  ax-rrecex 11085  ax-cnre 11086  ax-pre-lttri 11087  ax-pre-lttrn 11088  ax-pre-ltadd 11089  ax-pre-mulgt0 11090  ax-pre-sup 11091

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-int 4898  df-iun 4943  df-iin 4944  df-br 5094  df-opab 5156  df-mpt 5175  df-tr 5201  df-id 5514  df-eprel 5519  df-po 5527  df-so 5528  df-fr 5572  df-se 5573  df-we 5574  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-pred 6253  df-ord 6314  df-on 6315  df-lim 6316  df-suc 6317  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-isom 6495  df-riota 7309  df-ov 7355  df-oprab 7356  df-mpo 7357  df-om 7803  df-1st 7927  df-2nd 7928  df-frecs 8217  df-wrecs 8248  df-recs 8297  df-rdg 8335  df-1o 8391  df-2o 8392  df-oadd 8395  df-omul 8396  df-er 8628  df-map 8758  df-pm 8759  df-en 8876  df-dom 8877  df-sdom 8878  df-fin 8879  df-sup 9333  df-inf 9334  df-oi 9403  df-card 9839  df-acn 9842  df-ac 10014  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158  df-le 11159  df-sub 11353  df-neg 11354  df-div 11782  df-nn 12133  df-2 12195  df-3 12196  df-n0 12389  df-z 12476  df-uz 12739  df-q 12849  df-rp 12893  df-ioo 13251  df-ioc 13252  df-ico 13253  df-fl 13698  df-seq 13911  df-exp 13971  df-cj 15008  df-re 15009  df-im 15010  df-sqrt 15144  df-abs 15145  df-clim 15397  df-rlim 15398  df-rest 17328  df-topgen 17349  df-top 22810  df-bases 22862  df-salg 46431  df-salgen 46435  df-smblfn 46818

This theorem is referenced by:  smflimsuplem8  46949