stoweidlem45 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem45 44846

Description: This lemma proves that, given an appropriate 𝐾 (in another theorem we prove such a 𝐾 exists), there exists a function q_n as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 91 ( at the top of page 91): 0 <= q_n <= 1 , q_n < ε on T \ U, and q_n > 1 - ε on 𝑉. We use y to represent the final q_n in the paper (the one with n large enough), 𝑁 to represent 𝑛 in the paper, 𝐾 to represent 𝑘, 𝐷 to represent δ, 𝐸 to represent ε, and 𝑃 to represent 𝑝. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem45.1	⊢ Ⅎ𝑡𝑃
stoweidlem45.2	⊢ Ⅎ𝑡𝜑
stoweidlem45.3	⊢ 𝑉 = {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)}
stoweidlem45.4	⊢ 𝑄 = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
stoweidlem45.5	⊢ (𝜑 → 𝑁 ∈ ℕ)
stoweidlem45.6	⊢ (𝜑 → 𝐾 ∈ ℕ)
stoweidlem45.7	⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
stoweidlem45.8	⊢ (𝜑 → 𝐷 < 1)
stoweidlem45.9	⊢ (𝜑 → 𝑃 ∈ 𝐴)
stoweidlem45.10	⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
stoweidlem45.11	⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
stoweidlem45.12	⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡))
stoweidlem45.13	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem45.14	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem45.15	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem45.16	⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
stoweidlem45.17	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
stoweidlem45.18	⊢ (𝜑 → (1 − 𝐸) < (1 − (((𝐾 · 𝐷) / 2)↑𝑁)))
stoweidlem45.19	⊢ (𝜑 → (1 / ((𝐾 · 𝐷)↑𝑁)) < 𝐸)

**Assertion**
Ref	Expression
stoweidlem45	⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸))

Distinct variable groups: 𝑓,𝑔,𝑡,𝐴 𝑓,𝑁,𝑔,𝑡 𝑃,𝑓,𝑔 𝑇,𝑓,𝑔,𝑡 𝜑,𝑓,𝑔 𝑥,𝑡,𝐴 𝑦,𝑡,𝐴 𝑡,𝐾 𝑥,𝑇 𝜑,𝑥 𝑦,𝐸 𝑦,𝑄 𝑦,𝑇 𝑦,𝑈 𝑦,𝑉 Allowed substitution hints: 𝜑(𝑦,𝑡) 𝐷(𝑥,𝑦,𝑡,𝑓,𝑔) 𝑃(𝑥,𝑦,𝑡) 𝑄(𝑥,𝑡,𝑓,𝑔) 𝑈(𝑥,𝑡,𝑓,𝑔) 𝐸(𝑥,𝑡,𝑓,𝑔) 𝐾(𝑥,𝑦,𝑓,𝑔) 𝑁(𝑥,𝑦) 𝑉(𝑥,𝑡,𝑓,𝑔)

**Proof of Theorem stoweidlem45**
Step	Hyp	Ref	Expression
1		stoweidlem45.1	. . 3 ⊢ Ⅎ𝑡𝑃
2		stoweidlem45.2	. . 3 ⊢ Ⅎ𝑡𝜑
3		stoweidlem45.4	. . 3 ⊢ 𝑄 = (𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
4		eqid 2732	. . 3 ⊢ (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁))) = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
5		eqid 2732	. . 3 ⊢ (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ 1)
6		eqid 2732	. . 3 ⊢ (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁)) = (𝑡 ∈ 𝑇 ↦ ((𝑃‘𝑡)↑𝑁))
7		stoweidlem45.9	. . 3 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
8		stoweidlem45.10	. . 3 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
9		stoweidlem45.13	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
10		stoweidlem45.14	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
11		stoweidlem45.15	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
12		stoweidlem45.16	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑥) ∈ 𝐴)
13		stoweidlem45.5	. . 3 ⊢ (𝜑 → 𝑁 ∈ ℕ)
14		stoweidlem45.6	. . . 4 ⊢ (𝜑 → 𝐾 ∈ ℕ)
15	13	nnnn0d 12534	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ₀)
16	14, 15	nnexpcld 14210	. . 3 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℕ)
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16	stoweidlem40 44841	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
18		1red 11217	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
19	8	ffvelcdmda 7086	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ∈ ℝ)
20	15	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℕ₀)
21	19, 20	reexpcld 14130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ∈ ℝ)
22	18, 21	resubcld 11644	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ)
23	14	nnnn0d 12534	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ₀)
24	23, 15	nn0expcld 14211	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℕ₀)
25	24	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐾↑𝑁) ∈ ℕ₀)
26		1m1e0 12286	. . . . . . . 8 ⊢ (1 − 1) = 0
27		stoweidlem45.11	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
28	27	r19.21bi 3248	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
29	28	simpld 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑃‘𝑡))
30	28	simprd 496	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ≤ 1)
31		exple1 14143	. . . . . . . . . 10 ⊢ ((((𝑃‘𝑡) ∈ ℝ ∧ 0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ 𝑁 ∈ ℕ₀) → ((𝑃‘𝑡)↑𝑁) ≤ 1)
32	19, 29, 30, 20, 31	syl31anc 1373	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ≤ 1)
33	21, 18, 18, 32	lesub2dd 11833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − 1) ≤ (1 − ((𝑃‘𝑡)↑𝑁)))
34	26, 33	eqbrtrrid 5184	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (1 − ((𝑃‘𝑡)↑𝑁)))
35	22, 25, 34	expge0d 14131	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
36	3, 8, 15, 23	stoweidlem12 44813	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑄‘𝑡) = ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
37	35, 36	breqtrrd 5176	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑄‘𝑡))
38		0red 11219	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ)
39	19, 20, 29	expge0d 14131	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑃‘𝑡)↑𝑁))
40	38, 21, 18, 39	lesub2dd 11833	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ≤ (1 − 0))
41		1m0e1 12335	. . . . . . . 8 ⊢ (1 − 0) = 1
42	40, 41	breqtrdi 5189	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ≤ 1)
43		exple1 14143	. . . . . . 7 ⊢ ((((1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ ∧ 0 ≤ (1 − ((𝑃‘𝑡)↑𝑁)) ∧ (1 − ((𝑃‘𝑡)↑𝑁)) ≤ 1) ∧ (𝐾↑𝑁) ∈ ℕ₀) → ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)) ≤ 1)
44	22, 34, 42, 25, 43	syl31anc 1373	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)) ≤ 1)
45	36, 44	eqbrtrd 5170	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑄‘𝑡) ≤ 1)
46	37, 45	jca 512	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1))
47	46	ex 413	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)))
48	2, 47	ralrimi 3254	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1))
49		stoweidlem45.3	. . . . 5 ⊢ 𝑉 = {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)}
50		stoweidlem45.7	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ⁺)
51		stoweidlem45.17	. . . . 5 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
52		stoweidlem45.18	. . . . 5 ⊢ (𝜑 → (1 − 𝐸) < (1 − (((𝐾 · 𝐷) / 2)↑𝑁)))
53	49, 3, 8, 15, 23, 50, 51, 52, 27	stoweidlem24 44825	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − 𝐸) < (𝑄‘𝑡))
54	53	ex 413	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑉 → (1 − 𝐸) < (𝑄‘𝑡)))
55	2, 54	ralrimi 3254	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡))
56		stoweidlem45.12	. . . . 5 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡))
57		stoweidlem45.19	. . . . 5 ⊢ (𝜑 → (1 / ((𝐾 · 𝐷)↑𝑁)) < 𝐸)
58	3, 13, 14, 50, 8, 27, 56, 51, 57	stoweidlem25 44826	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑄‘𝑡) < 𝐸)
59	58	ex 413	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → (𝑄‘𝑡) < 𝐸))
60	2, 59	ralrimi 3254	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸)
61		nfmpt1 5256	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
62	3, 61	nfcxfr 2901	. . . . . 6 ⊢ Ⅎ𝑡𝑄
63	62	nfeq2 2920	. . . . 5 ⊢ Ⅎ𝑡 𝑦 = 𝑄
64		fveq1 6890	. . . . . . 7 ⊢ (𝑦 = 𝑄 → (𝑦‘𝑡) = (𝑄‘𝑡))
65	64	breq2d 5160	. . . . . 6 ⊢ (𝑦 = 𝑄 → (0 ≤ (𝑦‘𝑡) ↔ 0 ≤ (𝑄‘𝑡)))
66	64	breq1d 5158	. . . . . 6 ⊢ (𝑦 = 𝑄 → ((𝑦‘𝑡) ≤ 1 ↔ (𝑄‘𝑡) ≤ 1))
67	65, 66	anbi12d 631	. . . . 5 ⊢ (𝑦 = 𝑄 → ((0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ↔ (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)))
68	63, 67	ralbid 3270	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)))
69	64	breq2d 5160	. . . . 5 ⊢ (𝑦 = 𝑄 → ((1 − 𝐸) < (𝑦‘𝑡) ↔ (1 − 𝐸) < (𝑄‘𝑡)))
70	63, 69	ralbid 3270	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ↔ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡)))
71	64	breq1d 5158	. . . . 5 ⊢ (𝑦 = 𝑄 → ((𝑦‘𝑡) < 𝐸 ↔ (𝑄‘𝑡) < 𝐸))
72	63, 71	ralbid 3270	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸))
73	68, 70, 72	3anbi123d 1436	. . 3 ⊢ (𝑦 = 𝑄 → ((∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸)))
74	73	rspcev 3612	. 2 ⊢ ((𝑄 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸)) → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸))
75	17, 48, 55, 60, 74	syl13anc 1372	1 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1087 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 ∀wral 3061 ∃wrex 3070 {crab 3432 ∖ cdif 3945 class class class wbr 5148 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7411 ℝcr 11111 0cc0 11112 1c1 11113 + caddc 11115 · cmul 11117 < clt 11250 ≤ cle 11251 − cmin 11446 / cdiv 11873 ℕcn 12214 2c2 12269 ℕ₀cn0 12474 ℝ⁺crp 12976 ↑cexp 14029

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7727 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

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-pss 3967 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5574 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-pred 6300 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7367 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11252 df-mnf 11253 df-xr 11254 df-ltxr 11255 df-le 11256 df-sub 11448 df-neg 11449 df-div 11874 df-nn 12215 df-2 12277 df-n0 12475 df-z 12561 df-uz 12825 df-rp 12977 df-seq 13969 df-exp 14030

This theorem is referenced by: stoweidlem49 44850

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