smflimlem3 - Mathbox for Glauco Siliprandi

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Theorem smflimlem3 47727

Description: The limit of sigma-measurable functions is sigma-measurable. Proposition 121F (a) of [Fremlin1] p. 38 . (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
smflimlem3.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
smflimlem3.s	⊢ (𝜑 → 𝑆 ∈ SAlg)
smflimlem3.m	⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
smflimlem3.d	⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
smflimlem3.a	⊢ (𝜑 → 𝐴 ∈ ℝ)
smflimlem3.p	⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
smflimlem3.h	⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
smflimlem3.i	⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘)
smflimlem3.c	⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑃) → (𝐶‘𝑦) ∈ 𝑦)
smflimlem3.x	⊢ (𝜑 → 𝑋 ∈ (𝐷 ∩ 𝐼))
smflimlem3.k	⊢ (𝜑 → 𝐾 ∈ ℕ)
smflimlem3.y	⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
smflimlem3.l	⊢ (𝜑 → (1 / 𝐾) < 𝑌)

**Assertion**
Ref	Expression
smflimlem3	⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + 𝑌)))

Distinct variable groups: 𝐴,𝑘,𝑚,𝑠,𝑥 𝐶,𝑘,𝑚,𝑠 𝑦,𝐶 𝑖,𝐹,𝑘,𝑚,𝑛,𝑥 𝐹,𝑠,𝑖 𝑖,𝐻,𝑘,𝑚,𝑛 𝑖,𝐾,𝑘,𝑚,𝑠,𝑥 𝑦,𝐾,𝑖 𝑚,𝑀 𝑃,𝑘,𝑚,𝑠 𝑦,𝑃 𝑆,𝑘,𝑚,𝑠 𝑖,𝑋,𝑘,𝑚,𝑥 𝑖,𝑍,𝑘,𝑚,𝑛,𝑥 𝜑,𝑖,𝑘,𝑚 𝜑,𝑦 Allowed substitution hints: 𝜑(𝑥, 𝑛, 𝑠) 𝐴(𝑦, 𝑖, 𝑛) 𝐶(𝑥, 𝑖, 𝑛) 𝐷(𝑥, 𝑦, 𝑖, 𝑘, 𝑚, 𝑛, 𝑠) 𝑃(𝑥, 𝑖, 𝑛) 𝑆(𝑥, 𝑦, 𝑖, 𝑛) 𝐹(𝑦) 𝐻(𝑥, 𝑦, 𝑠) 𝐼(𝑥, 𝑦, 𝑖, 𝑘, 𝑚, 𝑛, 𝑠) 𝐾(𝑛) 𝑀(𝑥, 𝑦, 𝑖, 𝑘, 𝑛, 𝑠) 𝑋(𝑦, 𝑛, 𝑠) 𝑌(𝑥, 𝑦, 𝑖, 𝑘, 𝑚, 𝑛, 𝑠) 𝑍(𝑦, 𝑠)

**Proof of Theorem smflimlem3**
Step	Hyp	Ref	Expression
1		smflimlem3.d	. . . . . . . . 9 ⊢ 𝐷 = {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ }
2		ssrab2 4028	. . . . . . . . 9 ⊢ {𝑥 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) ∣ (𝑚 ∈ 𝑍 ↦ ((𝐹‘𝑚)‘𝑥)) ∈ dom ⇝ } ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
3	1, 2	eqsstri 3977	. . . . . . . 8 ⊢ 𝐷 ⊆ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚)
4		inss1 4182	. . . . . . . . 9 ⊢ (𝐷 ∩ 𝐼) ⊆ 𝐷
5		smflimlem3.x	. . . . . . . . 9 ⊢ (𝜑 → 𝑋 ∈ (𝐷 ∩ 𝐼))
6	4, 5	sselid 3929	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐷)
7	3, 6	sselid 3929	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚))
8		fveq2 6877	. . . . . . . . . . . . 13 ⊢ (𝑖 = 𝑚 → (𝐹‘𝑖) = (𝐹‘𝑚))
9	8	dmeqd 5887	. . . . . . . . . . . 12 ⊢ (𝑖 = 𝑚 → dom (𝐹‘𝑖) = dom (𝐹‘𝑚))
10		eqcom 2768	. . . . . . . . . . . . . 14 ⊢ (𝑖 = 𝑚 ↔ 𝑚 = 𝑖)
11	10	imbi1i 352	. . . . . . . . . . . . 13 ⊢ ((𝑖 = 𝑚 → dom (𝐹‘𝑖) = dom (𝐹‘𝑚)) ↔ (𝑚 = 𝑖 → dom (𝐹‘𝑖) = dom (𝐹‘𝑚)))
12		eqcom 2768	. . . . . . . . . . . . . 14 ⊢ (dom (𝐹‘𝑖) = dom (𝐹‘𝑚) ↔ dom (𝐹‘𝑚) = dom (𝐹‘𝑖))
13	12	imbi2i 339	. . . . . . . . . . . . 13 ⊢ ((𝑚 = 𝑖 → dom (𝐹‘𝑖) = dom (𝐹‘𝑚)) ↔ (𝑚 = 𝑖 → dom (𝐹‘𝑚) = dom (𝐹‘𝑖)))
14	11, 13	bitri 278	. . . . . . . . . . . 12 ⊢ ((𝑖 = 𝑚 → dom (𝐹‘𝑖) = dom (𝐹‘𝑚)) ↔ (𝑚 = 𝑖 → dom (𝐹‘𝑚) = dom (𝐹‘𝑖)))
15	9, 14	mpbi 233	. . . . . . . . . . 11 ⊢ (𝑚 = 𝑖 → dom (𝐹‘𝑚) = dom (𝐹‘𝑖))
16	15	cbviinv 4998	. . . . . . . . . 10 ⊢ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑖 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑖)
17	16	a1i 11	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∩ 𝑖 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑖))
18	17	iuneq2i 4973	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑖)
19		fveq2 6877	. . . . . . . . . 10 ⊢ (𝑛 = 𝑚 → (ℤ_≥‘𝑛) = (ℤ_≥‘𝑚))
20	19	iineq1d 46048	. . . . . . . . 9 ⊢ (𝑛 = 𝑚 → ∩ 𝑖 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑖) = ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖))
21	20	cbviunv 4997	. . . . . . . 8 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑖) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖)
22	18, 21	eqtri 2784	. . . . . . 7 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)dom (𝐹‘𝑚) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖)
23	7, 22	eleqtrdi 2871	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖))
24		smflimlem3.z	. . . . . . . 8 ⊢ 𝑍 = (ℤ_≥‘𝑀)
25		eqid 2761	. . . . . . . 8 ⊢ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖)
26	24, 25	allbutfi 46348	. . . . . . 7 ⊢ (𝑋 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖) ↔ ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)𝑋 ∈ dom (𝐹‘𝑖))
27	26	biimpi 219	. . . . . 6 ⊢ (𝑋 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)dom (𝐹‘𝑖) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)𝑋 ∈ dom (𝐹‘𝑖))
28	23, 27	syl 18	. . . . 5 ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)𝑋 ∈ dom (𝐹‘𝑖))
29	5	elin2d 4151	. . . . . . . 8 ⊢ (𝜑 → 𝑋 ∈ 𝐼)
30		smflimlem3.i	. . . . . . . . 9 ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘)
31		oveq1 7419	. . . . . . . . . . . . . . 15 ⊢ (𝑚 = 𝑖 → (𝑚𝐻𝑘) = (𝑖𝐻𝑘))
32	31	cbviinv 4998	. . . . . . . . . . . . . 14 ⊢ ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) = ∩ 𝑖 ∈ (ℤ_≥‘𝑛)(𝑖𝐻𝑘)
33	32	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑛 ∈ 𝑍 → ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) = ∩ 𝑖 ∈ (ℤ_≥‘𝑛)(𝑖𝐻𝑘))
34	33	iuneq2i 4973	. . . . . . . . . . . 12 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) = ∪ 𝑛 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑛)(𝑖𝐻𝑘)
35	19	iineq1d 46048	. . . . . . . . . . . . 13 ⊢ (𝑛 = 𝑚 → ∩ 𝑖 ∈ (ℤ_≥‘𝑛)(𝑖𝐻𝑘) = ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝑘))
36	35	cbviunv 4997	. . . . . . . . . . . 12 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑛)(𝑖𝐻𝑘) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝑘)
37	34, 36	eqtri 2784	. . . . . . . . . . 11 ⊢ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝑘)
38	37	a1i 11	. . . . . . . . . 10 ⊢ (𝑘 ∈ ℕ → ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝑘))
39	38	iineq2i 4974	. . . . . . . . 9 ⊢ ∩ 𝑘 ∈ ℕ ∪ 𝑛 ∈ 𝑍 ∩ 𝑚 ∈ (ℤ_≥‘𝑛)(𝑚𝐻𝑘) = ∩ 𝑘 ∈ ℕ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝑘)
40	30, 39	eqtri 2784	. . . . . . . 8 ⊢ 𝐼 = ∩ 𝑘 ∈ ℕ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝑘)
41	29, 40	eleqtrdi 2871	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ ∩ 𝑘 ∈ ℕ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝑘))
42		smflimlem3.k	. . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℕ)
43		oveq2 7420	. . . . . . . . . . 11 ⊢ (𝑘 = 𝐾 → (𝑖𝐻𝑘) = (𝑖𝐻𝐾))
44	43	adantr 486	. . . . . . . . . 10 ⊢ ((𝑘 = 𝐾 ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → (𝑖𝐻𝑘) = (𝑖𝐻𝐾))
45	44	iineq2dv 4977	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝑘) = ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝐾))
46	45	iuneq2d 4981	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝑘) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝐾))
47	46	eleq2d 2847	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑋 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝑘) ↔ 𝑋 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝐾)))
48	41, 42, 47	eliind 46031	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝐾))
49		eqid 2761	. . . . . . 7 ⊢ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝐾) = ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝐾)
50	24, 49	allbutfi 46348	. . . . . 6 ⊢ (𝑋 ∈ ∪ 𝑚 ∈ 𝑍 ∩ 𝑖 ∈ (ℤ_≥‘𝑚)(𝑖𝐻𝐾) ↔ ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)𝑋 ∈ (𝑖𝐻𝐾))
51	48, 50	sylib 221	. . . . 5 ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)𝑋 ∈ (𝑖𝐻𝐾))
52	28, 51	jca 521	. . . 4 ⊢ (𝜑 → (∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)𝑋 ∈ dom (𝐹‘𝑖) ∧ ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)𝑋 ∈ (𝑖𝐻𝐾)))
53	24	rexanuz2 15497	. . . 4 ⊢ (∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) ↔ (∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)𝑋 ∈ dom (𝐹‘𝑖) ∧ ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)𝑋 ∈ (𝑖𝐻𝐾)))
54	52, 53	sylibr 237	. . 3 ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)))
55		simpll 779	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → 𝜑)
56		simpr 490	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑚 ∈ 𝑍)
57	24	uztrn2 12965	. . . . . . 7 ⊢ ((𝑚 ∈ 𝑍 ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → 𝑖 ∈ 𝑍)
58	56, 57	sylan 592	. . . . . 6 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → 𝑖 ∈ 𝑍)
59		simprl 783	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ dom (𝐹‘𝑖))
60		simp3 1156	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → 𝑋 ∈ (𝑖𝐻𝐾))
61		smflimlem3.h	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘)))
62	61	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐻 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ (𝐶‘(𝑚𝑃𝑘))))
63		oveq12 7421	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 = 𝑖 ∧ 𝑘 = 𝐾) → (𝑚𝑃𝑘) = (𝑖𝑃𝐾))
64	63	fveq2d 6881	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 = 𝑖 ∧ 𝑘 = 𝐾) → (𝐶‘(𝑚𝑃𝑘)) = (𝐶‘(𝑖𝑃𝐾)))
65	64	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑚 = 𝑖 ∧ 𝑘 = 𝐾)) → (𝐶‘(𝑚𝑃𝑘)) = (𝐶‘(𝑖𝑃𝐾)))
66		simpr 490	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑖 ∈ 𝑍)
67	42	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝐾 ∈ ℕ)
68		fvexd 6892	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐶‘(𝑖𝑃𝐾)) ∈ V)
69	62, 65, 66, 67, 68	ovmpod 7564	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑖𝐻𝐾) = (𝐶‘(𝑖𝑃𝐾)))
70	69	3adant3 1150	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → (𝑖𝐻𝐾) = (𝐶‘(𝑖𝑃𝐾)))
71	60, 70	eleqtrd 2863	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍 ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → 𝑋 ∈ (𝐶‘(𝑖𝑃𝐾)))
72	71	3expa 1136	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → 𝑋 ∈ (𝐶‘(𝑖𝑃𝐾)))
73	72	adantrl 729	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ (𝐶‘(𝑖𝑃𝐾)))
74	73, 59	elind 4146	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹‘𝑖)))
75		eqid 2761	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}
76		smflimlem3.s	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ (𝜑 → 𝑆 ∈ SAlg)
77	75, 76	rabexd 5301	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝜑 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
78	77	ralrimivw 3159	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝜑 → ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
79	78	a1d 26	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜑 → (𝑚 ∈ 𝑍 → ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V))
80	79	imp 412	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
81	80	ralrimiva 3155	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝜑 → ∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V)
82		smflimlem3.p	. . . . . . . . . . . . . . . . . . . . 21 ⊢ 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))})
83	82	fnmpo 8069	. . . . . . . . . . . . . . . . . . . 20 ⊢ (∀𝑚 ∈ 𝑍 ∀𝑘 ∈ ℕ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} ∈ V → 𝑃 Fn (𝑍 × ℕ))
84	81, 83	syl 18	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝜑 → 𝑃 Fn (𝑍 × ℕ))
85	84	adantr 486	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑃 Fn (𝑍 × ℕ))
86		fnovrn 7588	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑃 Fn (𝑍 × ℕ) ∧ 𝑖 ∈ 𝑍 ∧ 𝐾 ∈ ℕ) → (𝑖𝑃𝐾) ∈ ran 𝑃)
87	85, 66, 67, 86	syl3anc 1398	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑖𝑃𝐾) ∈ ran 𝑃)
88		ovex 7445	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑖𝑃𝐾) ∈ V
89		eleq1 2849	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑖𝑃𝐾) → (𝑦 ∈ ran 𝑃 ↔ (𝑖𝑃𝐾) ∈ ran 𝑃))
90	89	anbi2d 642	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑖𝑃𝐾) → ((𝜑 ∧ 𝑦 ∈ ran 𝑃) ↔ (𝜑 ∧ (𝑖𝑃𝐾) ∈ ran 𝑃)))
91		fveq2 6877	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑖𝑃𝐾) → (𝐶‘𝑦) = (𝐶‘(𝑖𝑃𝐾)))
92		id 23	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = (𝑖𝑃𝐾) → 𝑦 = (𝑖𝑃𝐾))
93	91, 92	eleq12d 2855	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = (𝑖𝑃𝐾) → ((𝐶‘𝑦) ∈ 𝑦 ↔ (𝐶‘(𝑖𝑃𝐾)) ∈ (𝑖𝑃𝐾)))
94	90, 93	imbi12d 347	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = (𝑖𝑃𝐾) → (((𝜑 ∧ 𝑦 ∈ ran 𝑃) → (𝐶‘𝑦) ∈ 𝑦) ↔ ((𝜑 ∧ (𝑖𝑃𝐾) ∈ ran 𝑃) → (𝐶‘(𝑖𝑃𝐾)) ∈ (𝑖𝑃𝐾))))
95		smflimlem3.c	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝜑 ∧ 𝑦 ∈ ran 𝑃) → (𝐶‘𝑦) ∈ 𝑦)
96	88, 94, 95	vtocl 3521	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ (𝑖𝑃𝐾) ∈ ran 𝑃) → (𝐶‘(𝑖𝑃𝐾)) ∈ (𝑖𝑃𝐾))
97	87, 96	syldan 603	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐶‘(𝑖𝑃𝐾)) ∈ (𝑖𝑃𝐾))
98	82	a1i 11	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → 𝑃 = (𝑚 ∈ 𝑍, 𝑘 ∈ ℕ ↦ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))}))
99	15	adantr 486	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 = 𝑖 ∧ 𝑘 = 𝐾) → dom (𝐹‘𝑚) = dom (𝐹‘𝑖))
100	8	fveq1d 6879	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑖 = 𝑚 → ((𝐹‘𝑖)‘𝑥) = ((𝐹‘𝑚)‘𝑥))
101	10	imbi1i 352	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑖 = 𝑚 → ((𝐹‘𝑖)‘𝑥) = ((𝐹‘𝑚)‘𝑥)) ↔ (𝑚 = 𝑖 → ((𝐹‘𝑖)‘𝑥) = ((𝐹‘𝑚)‘𝑥)))
102		eqcom 2768	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝐹‘𝑖)‘𝑥) = ((𝐹‘𝑚)‘𝑥) ↔ ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑖)‘𝑥))
103	102	imbi2i 339	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑚 = 𝑖 → ((𝐹‘𝑖)‘𝑥) = ((𝐹‘𝑚)‘𝑥)) ↔ (𝑚 = 𝑖 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑖)‘𝑥)))
104	101, 103	bitri 278	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑖 = 𝑚 → ((𝐹‘𝑖)‘𝑥) = ((𝐹‘𝑚)‘𝑥)) ↔ (𝑚 = 𝑖 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑖)‘𝑥)))
105	100, 104	mpbi 233	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑚 = 𝑖 → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑖)‘𝑥))
106	105	adantr 486	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 = 𝑖 ∧ 𝑘 = 𝐾) → ((𝐹‘𝑚)‘𝑥) = ((𝐹‘𝑖)‘𝑥))
107		oveq2 7420	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑘 = 𝐾 → (1 / 𝑘) = (1 / 𝐾))
108	107	oveq2d 7428	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑘 = 𝐾 → (𝐴 + (1 / 𝑘)) = (𝐴 + (1 / 𝐾)))
109	108	adantl 487	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑚 = 𝑖 ∧ 𝑘 = 𝐾) → (𝐴 + (1 / 𝑘)) = (𝐴 + (1 / 𝐾)))
110	106, 109	breq12d 5116	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑚 = 𝑖 ∧ 𝑘 = 𝐾) → (((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘)) ↔ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))))
111	99, 110	rabeqbidv 3430	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = 𝑖 ∧ 𝑘 = 𝐾) → {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
112	15	ineq2d 4166	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑚 = 𝑖 → (𝑠 ∩ dom (𝐹‘𝑚)) = (𝑠 ∩ dom (𝐹‘𝑖)))
113	112	adantr 486	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑚 = 𝑖 ∧ 𝑘 = 𝐾) → (𝑠 ∩ dom (𝐹‘𝑚)) = (𝑠 ∩ dom (𝐹‘𝑖)))
114	111, 113	eqeq12d 2777	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑚 = 𝑖 ∧ 𝑘 = 𝐾) → ({𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚)) ↔ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖))))
115	114	rabbidv 3420	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑚 = 𝑖 ∧ 𝑘 = 𝐾) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖))})
116	115	adantl 487	. . . . . . . . . . . . . . . . 17 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑚 = 𝑖 ∧ 𝑘 = 𝐾)) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑚) ∣ ((𝐹‘𝑚)‘𝑥) < (𝐴 + (1 / 𝑘))} = (𝑠 ∩ dom (𝐹‘𝑚))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖))})
117		eqid 2761	. . . . . . . . . . . . . . . . . . 19 ⊢ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖))} = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖))}
118	117, 76	rabexd 5301	. . . . . . . . . . . . . . . . . 18 ⊢ (𝜑 → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖))} ∈ V)
119	118	adantr 486	. . . . . . . . . . . . . . . . 17 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖))} ∈ V)
120	98, 116, 66, 67, 119	ovmpod 7564	. . . . . . . . . . . . . . . 16 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝑖𝑃𝐾) = {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖))})
121	97, 120	eleqtrd 2863	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐶‘(𝑖𝑃𝐾)) ∈ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖))})
122		ineq1 4159	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 = (𝐶‘(𝑖𝑃𝐾)) → (𝑠 ∩ dom (𝐹‘𝑖)) = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹‘𝑖)))
123	122	eqeq2d 2772	. . . . . . . . . . . . . . . 16 ⊢ (𝑠 = (𝐶‘(𝑖𝑃𝐾)) → ({𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖)) ↔ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹‘𝑖))))
124	123	elrab 3645	. . . . . . . . . . . . . . 15 ⊢ ((𝐶‘(𝑖𝑃𝐾)) ∈ {𝑠 ∈ 𝑆 ∣ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = (𝑠 ∩ dom (𝐹‘𝑖))} ↔ ((𝐶‘(𝑖𝑃𝐾)) ∈ 𝑆 ∧ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹‘𝑖))))
125	121, 124	sylib 221	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝐶‘(𝑖𝑃𝐾)) ∈ 𝑆 ∧ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹‘𝑖))))
126	125	simprd 501	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} = ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹‘𝑖)))
127	126	eqcomd 2767	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹‘𝑖)) = {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
128	127	adantr 486	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → ((𝐶‘(𝑖𝑃𝐾)) ∩ dom (𝐹‘𝑖)) = {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
129	74, 128	eleqtrd 2863	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → 𝑋 ∈ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))})
130		fveq2 6877	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → ((𝐹‘𝑖)‘𝑥) = ((𝐹‘𝑖)‘𝑋))
131		eqidd 2762	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑋 → (𝐴 + (1 / 𝐾)) = (𝐴 + (1 / 𝐾)))
132	130, 131	breq12d 5116	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑋 → (((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾)) ↔ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
133	132	elrab 3645	. . . . . . . . . 10 ⊢ (𝑋 ∈ {𝑥 ∈ dom (𝐹‘𝑖) ∣ ((𝐹‘𝑖)‘𝑥) < (𝐴 + (1 / 𝐾))} ↔ (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
134	129, 133	sylib 221	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
135	134	simprd 501	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))
136	59, 135	jca 521	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾))) → (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
137	136	ex 418	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
138	55, 58, 137	syl2anc 596	. . . . 5 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → ((𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
139	138	ralimdva 3175	. . . 4 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
140	139	reximdva 3176	. . 3 ⊢ (𝜑 → (∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ 𝑋 ∈ (𝑖𝐻𝐾)) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))))
141	54, 140	mpd 16	. 2 ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))))
142		simprl 783	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → 𝑋 ∈ dom (𝐹‘𝑖))
143		eleq1 2849	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑖 → (𝑚 ∈ 𝑍 ↔ 𝑖 ∈ 𝑍))
144	143	anbi2d 642	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑖 → ((𝜑 ∧ 𝑚 ∈ 𝑍) ↔ (𝜑 ∧ 𝑖 ∈ 𝑍)))
145		fveq2 6877	. . . . . . . . . . . . . 14 ⊢ (𝑚 = 𝑖 → (𝐹‘𝑚) = (𝐹‘𝑖))
146	145, 15	feq12d 6689	. . . . . . . . . . . . 13 ⊢ (𝑚 = 𝑖 → ((𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ ↔ (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ))
147	144, 146	imbi12d 347	. . . . . . . . . . . 12 ⊢ (𝑚 = 𝑖 → (((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ) ↔ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ)))
148	76	adantr 486	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → 𝑆 ∈ SAlg)
149		smflimlem3.m	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚) ∈ (SMblFn‘𝑆))
150		eqid 2761	. . . . . . . . . . . . 13 ⊢ dom (𝐹‘𝑚) = dom (𝐹‘𝑚)
151	148, 149, 150	smff 47686	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (𝐹‘𝑚):dom (𝐹‘𝑚)⟶ℝ)
152	147, 151	chvarvv 2022	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ)
153	152	adantr 486	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑋 ∈ dom (𝐹‘𝑖)) → (𝐹‘𝑖):dom (𝐹‘𝑖)⟶ℝ)
154		simpr 490	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑋 ∈ dom (𝐹‘𝑖)) → 𝑋 ∈ dom (𝐹‘𝑖))
155	153, 154	ffvelcdmd 7077	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ 𝑋 ∈ dom (𝐹‘𝑖)) → ((𝐹‘𝑖)‘𝑋) ∈ ℝ)
156	155	adantrr 730	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → ((𝐹‘𝑖)‘𝑋) ∈ ℝ)
157		smflimlem3.a	. . . . . . . . . 10 ⊢ (𝜑 → 𝐴 ∈ ℝ)
158	42	nnrecred 12370	. . . . . . . . . 10 ⊢ (𝜑 → (1 / 𝐾) ∈ ℝ)
159	157, 158	readdcld 11319	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + (1 / 𝐾)) ∈ ℝ)
160	159	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝐴 + (1 / 𝐾)) ∈ ℝ)
161		smflimlem3.y	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑌 ∈ ℝ⁺)
162	161	rpred 13145	. . . . . . . . . 10 ⊢ (𝜑 → 𝑌 ∈ ℝ)
163	157, 162	readdcld 11319	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + 𝑌) ∈ ℝ)
164	163	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝐴 + 𝑌) ∈ ℝ)
165		simprr 785	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))
166		smflimlem3.l	. . . . . . . . . 10 ⊢ (𝜑 → (1 / 𝐾) < 𝑌)
167	158, 162, 157, 166	ltadd2dd 11450	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 + (1 / 𝐾)) < (𝐴 + 𝑌))
168	167	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝐴 + (1 / 𝐾)) < (𝐴 + 𝑌))
169	156, 160, 164, 165, 168	lttrd 11452	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → ((𝐹‘𝑖)‘𝑋) < (𝐴 + 𝑌))
170	142, 169	jca 521	. . . . . 6 ⊢ (((𝜑 ∧ 𝑖 ∈ 𝑍) ∧ (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾)))) → (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + 𝑌)))
171	170	ex 418	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ 𝑍) → ((𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + 𝑌))))
172	55, 58, 171	syl2anc 596	. . . 4 ⊢ (((𝜑 ∧ 𝑚 ∈ 𝑍) ∧ 𝑖 ∈ (ℤ_≥‘𝑚)) → ((𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → (𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + 𝑌))))
173	172	ralimdva 3175	. . 3 ⊢ ((𝜑 ∧ 𝑚 ∈ 𝑍) → (∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + 𝑌))))
174	173	reximdva 3176	. 2 ⊢ (𝜑 → (∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + (1 / 𝐾))) → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + 𝑌))))
175	141, 174	mpd 16	1 ⊢ (𝜑 → ∃𝑚 ∈ 𝑍 ∀𝑖 ∈ (ℤ_≥‘𝑚)(𝑋 ∈ dom (𝐹‘𝑖) ∧ ((𝐹‘𝑖)‘𝑋) < (𝐴 + 𝑌)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2145 ∀wral 3077 ∃wrex 3087 {crab 3413 Vcvv 3451 ∩ cin 3898 ∪ ciun 4951 ∩ ciin 4952 class class class wbr 5103 ↦ cmpt 5186 × cxp 5649 dom cdm 5651 ran crn 5652 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 (class class class)co 7412 ∈ cmpo 7414 ℝcr 11180 1c1 11182 + caddc 11184 < clt 11324 / cdiv 11954 ℕcn 12316 ℤ_≥cuz 12946 ℝ⁺crp 13101 ⇝ cli 15631 SAlgcsalg 47262 SMblFncsmblfn 47649

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2733 ax-sep 5249 ax-nul 5260 ax-pow 5327 ax-pr 5391 ax-un 7740 ax-cnex 11237 ax-resscn 11238 ax-1cn 11239 ax-icn 11240 ax-addcl 11241 ax-addrcl 11242 ax-mulcl 11243 ax-mulrcl 11244 ax-mulcom 11245 ax-addass 11246 ax-mulass 11247 ax-distr 11248 ax-i2m1 11249 ax-1ne0 11250 ax-1rid 11251 ax-rnegex 11252 ax-rrecex 11253 ax-cnre 11254 ax-pre-lttri 11255 ax-pre-lttrn 11256 ax-pre-ltadd 11257 ax-pre-mulgt0 11258

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3063 df-ral 3078 df-rex 3088 df-rmo 3366 df-reu 3367 df-rab 3414 df-v 3453 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-pss 3919 df-nul 4280 df-if 4483 df-pw 4559 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-iun 4953 df-iin 4954 df-br 5104 df-opab 5168 df-mpt 5187 df-tr 5213 df-id 5546 df-eprel 5551 df-po 5559 df-so 5560 df-fr 5604 df-we 5606 df-xp 5657 df-rel 5658 df-cnv 5659 df-co 5660 df-dm 5661 df-rn 5662 df-res 5663 df-ima 5664 df-pred 6297 df-ord 6358 df-on 6359 df-lim 6360 df-suc 6361 df-iota 6487 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-riota 7369 df-ov 7415 df-oprab 7416 df-mpo 7417 df-om 7867 df-1st 7990 df-2nd 7991 df-frecs 8283 df-wrecs 8314 df-recs 8363 df-rdg 8402 df-er 8701 df-pm 8834 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11326 df-mnf 11327 df-xr 11328 df-ltxr 11329 df-le 11330 df-sub 11524 df-neg 11525 df-div 11955 df-nn 12317 df-z 12675 df-uz 12947 df-rp 13102 df-ioo 13461 df-ico 13463 df-smblfn 47650

This theorem is used by: smflimlem4 47728

W3C validator