smflimsuplem3 - Mathbox for Glauco Siliprandi

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Theorem smflimsuplem3 45836

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 6903	. . . 4 ⊢ (𝐻‘𝑛) ∈ V
7	6	dmex 7904	. . 3 ⊢ dom (𝐻‘𝑛) ∈ V
8	7	a1i 11	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → dom (𝐻‘𝑛) ∈ V)
9		fvexd 6905	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍 ∧ 𝑥 ∈ dom (𝐻‘𝑛)) → ((𝐻‘𝑛)‘𝑥) ∈ V)
10		smflimsuplem3.s	. 2 ⊢ (𝜑 → 𝑆 ∈ SAlg)
11	10	adantr 479	. . . . . 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 2730	. . . . . . . . . . . . 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 44411	. . . . . . . . . . . . . . . 16 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ ℤ)
16		eqid 2730	. . . . . . . . . . . . . . . 16 ⊢ (ℤ_≥‘𝑛) = (ℤ_≥‘𝑛)
17	15, 16	uzn0d 44433	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ≠ ∅)
18		fvex 6903	. . . . . . . . . . . . . . . . . 18 ⊢ (𝐹‘𝑚) ∈ V
19	18	dmex 7904	. . . . . . . . . . . . . . . . 17 ⊢ dom (𝐹‘𝑚) ∈ V
20	19	rgenw 3063	. . . . . . . . . . . . . . . 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 44123	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
23	22	adantl 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∈ V)
24	14, 23	rabexd 5332	. . . . . . . . . . . 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 7010	. . . . . . . . . . 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 6909	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) → ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚) = (𝐹‘𝑚))
27	26	eqcomd 2736	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚))
28	27	adantl 480	. . . . . . . . . . . . . . . 16 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → (𝐹‘𝑚) = ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚))
29	28	dmeqd 5904	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑛 ∈ 𝑍) ∧ 𝑚 ∈ (ℤ_≥‘𝑛)) → dom (𝐹‘𝑚) = dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚))
30	29	iineq2dv 5021	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚))
31	30	eleq2d 2817	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ↔ 𝑥 ∈ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom ((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)))
32	27	fveq1d 6892	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) → ((𝐹‘𝑚)‘𝑥) = (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥))
33	32	mpteq2ia 5250	. . . . . . . . . . . . . . . . 17 ⊢ (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥))
34	33	rneqi 5935	. . . . . . . . . . . . . . . 16 ⊢ ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ ((𝐹‘𝑚)‘𝑥)) = ran (𝑚 ∈ (ℤ_≥‘𝑛) ↦ (((𝐹 ↾ (ℤ_≥‘𝑛))‘𝑚)‘𝑥))
35	34	supeq1i 9444	. . . . . . . . . . . . . . 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 2816	. . . . . . . . . . . . 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 629	. . . . . . . . . . . 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 3432	. . . . . . . . . . 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 2770	. . . . . . . . . 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 5238	. . . . . . . . 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 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑚(𝐹 ↾ (ℤ_≥‘𝑛))
43		nfcv 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑥(𝐹 ↾ (ℤ_≥‘𝑛))
44	15	adantl 480	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝑛 ∈ ℤ)
45		smflimsuplem3.f	. . . . . . . . . . . 12 ⊢ (𝜑 → 𝐹:𝑍⟶(SMblFn‘𝑆))
46	45	adantr 479	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → 𝐹:𝑍⟶(SMblFn‘𝑆))
47	5	eleq2i 2823	. . . . . . . . . . . . . . 15 ⊢ (𝑛 ∈ 𝑍 ↔ 𝑛 ∈ (ℤ_≥‘𝑀))
48	47	biimpi 215	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ 𝑍 → 𝑛 ∈ (ℤ_≥‘𝑀))
49		uzss 12849	. . . . . . . . . . . . . 14 ⊢ (𝑛 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑀))
50	48, 49	syl 17	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ (ℤ_≥‘𝑀))
51	50, 5	sseqtrrdi 4032	. . . . . . . . . . . 12 ⊢ (𝑛 ∈ 𝑍 → (ℤ_≥‘𝑛) ⊆ 𝑍)
52	51	adantl 480	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (ℤ_≥‘𝑛) ⊆ 𝑍)
53	46, 52	fssresd 6757	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐹 ↾ (ℤ_≥‘𝑛)):(ℤ_≥‘𝑛)⟶(SMblFn‘𝑆))
54		eqid 2730	. . . . . . . . . 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 2730	. . . . . . . . . 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 45830	. . . . . . . . 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 2831	. . . . . . . 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 7114	. . . . . . 7 ⊢ (𝜑 → 𝐻:𝑍⟶(SMblFn‘𝑆))
60	59	ffvelcdmda 7085	. . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) ∈ (SMblFn‘𝑆))
61		eqid 2730	. . . . . 6 ⊢ dom (𝐻‘𝑛) = dom (𝐻‘𝑛)
62	11, 60, 61	smff 45746	. . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛):dom (𝐻‘𝑛)⟶ℝ)
63	62	feqmptd 6959	. . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝐻‘𝑛) = (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)))
64	63	eqcomd 2736	. . 3 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)) = (𝐻‘𝑛))
65	64, 60	eqeltrd 2831	. 2 ⊢ ((𝜑 ∧ 𝑛 ∈ 𝑍) → (𝑥 ∈ dom (𝐻‘𝑛) ↦ ((𝐻‘𝑛)‘𝑥)) ∈ (SMblFn‘𝑆))
66		eqid 2730	. 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 2730	. 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 45824	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))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 394 ∧ w3a 1085 = wceq 1539 ∈ wcel 2104 ∀wral 3059 {crab 3430 Vcvv 3472 ⊆ wss 3947 ∪ ciun 4996 ∩ ciin 4997 ↦ cmpt 5230 dom cdm 5675 ran crn 5676 ↾ cres 5677 ⟶wf 6538 ‘cfv 6542 supcsup 9437 ℝcr 11111 ℝ^*cxr 11251 < clt 11252 ℤcz 12562 ℤ_≥cuz 12826 ⇝ cli 15432 SAlgcsalg 45322 SMblFncsmblfn 45709

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7727 ax-inf2 9638 ax-cc 10432 ax-ac2 10460 ax-cnex 11168 ax-resscn 11169 ax-1cn 11170 ax-icn 11171 ax-addcl 11172 ax-addrcl 11173 ax-mulcl 11174 ax-mulrcl 11175 ax-mulcom 11176 ax-addass 11177 ax-mulass 11178 ax-distr 11179 ax-i2m1 11180 ax-1ne0 11181 ax-1rid 11182 ax-rnegex 11183 ax-rrecex 11184 ax-cnre 11185 ax-pre-lttri 11186 ax-pre-lttrn 11187 ax-pre-ltadd 11188 ax-pre-mulgt0 11189 ax-pre-sup 11190

This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 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 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-1st 7977 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-oadd 8472 df-omul 8473 df-er 8705 df-map 8824 df-pm 8825 df-en 8942 df-dom 8943 df-sdom 8944 df-fin 8945 df-sup 9439 df-inf 9440 df-oi 9507 df-card 9936 df-acn 9939 df-ac 10113 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-sub 11450 df-neg 11451 df-div 11876 df-nn 12217 df-2 12279 df-3 12280 df-n0 12477 df-z 12563 df-uz 12827 df-q 12937 df-rp 12979 df-ioo 13332 df-ioc 13333 df-ico 13334 df-fl 13761 df-seq 13971 df-exp 14032 df-cj 15050 df-re 15051 df-im 15052 df-sqrt 15186 df-abs 15187 df-clim 15436 df-rlim 15437 df-rest 17372 df-topgen 17393 df-top 22616 df-bases 22669 df-salg 45323 df-salgen 45327 df-smblfn 45710

This theorem is referenced by: smflimsuplem8 45841

W3C validator