Theorem stoweidlem45 45966

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 2740	. . 3 ⊢ (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁))) = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
5		eqid 2740	. . 3 ⊢ (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ 1)
6		eqid 2740	. . 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 12613	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
16	14, 15	nnexpcld 14294	. . 3 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℕ)
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16	stoweidlem40 45961	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
18		1red 11291	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
19	8	ffvelcdmda 7118	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ∈ ℝ)
20	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℕ0)
21	19, 20	reexpcld 14213	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ∈ ℝ)
22	18, 21	resubcld 11718	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ)
23	14	nnnn0d 12613	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
24	23, 15	nn0expcld 14295	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℕ0)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐾↑𝑁) ∈ ℕ0)
26		1m1e0 12365	. . . . . . . 8 ⊢ (1 − 1) = 0
27		stoweidlem45.11	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
28	27	r19.21bi 3257	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
29	28	simpld 494	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑃‘𝑡))
30	28	simprd 495	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ≤ 1)
31		exple1 14226	. . . . . . . . . 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 11907	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − 1) ≤ (1 − ((𝑃‘𝑡)↑𝑁)))
34	26, 33	eqbrtrrid 5202	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (1 − ((𝑃‘𝑡)↑𝑁)))
35	22, 25, 34	expge0d 14214	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
36	3, 8, 15, 23	stoweidlem12 45933	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑄‘𝑡) = ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
37	35, 36	breqtrrd 5194	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑄‘𝑡))
38		0red 11293	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ)
39	19, 20, 29	expge0d 14214	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑃‘𝑡)↑𝑁))
40	38, 21, 18, 39	lesub2dd 11907	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ≤ (1 − 0))
41		1m0e1 12414	. . . . . . . 8 ⊢ (1 − 0) = 1
42	40, 41	breqtrdi 5207	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ≤ 1)
43		exple1 14226	. . . . . . 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 5188	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑄‘𝑡) ≤ 1)
46	37, 45	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1))
47	46	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)))
48	2, 47	ralrimi 3263	. 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 45945	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − 𝐸) < (𝑄‘𝑡))
54	53	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑉 → (1 − 𝐸) < (𝑄‘𝑡)))
55	2, 54	ralrimi 3263	. 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 45946	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑄‘𝑡) < 𝐸)
59	58	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → (𝑄‘𝑡) < 𝐸))
60	2, 59	ralrimi 3263	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸)
61		nfmpt1 5274	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
62	3, 61	nfcxfr 2906	. . . . . 6 ⊢ Ⅎ𝑡𝑄
63	62	nfeq2 2926	. . . . 5 ⊢ Ⅎ𝑡 𝑦 = 𝑄
64		fveq1 6919	. . . . . . 7 ⊢ (𝑦 = 𝑄 → (𝑦‘𝑡) = (𝑄‘𝑡))
65	64	breq2d 5178	. . . . . 6 ⊢ (𝑦 = 𝑄 → (0 ≤ (𝑦‘𝑡) ↔ 0 ≤ (𝑄‘𝑡)))
66	64	breq1d 5176	. . . . . 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 3279	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)))
69	64	breq2d 5178	. . . . 5 ⊢ (𝑦 = 𝑄 → ((1 − 𝐸) < (𝑦‘𝑡) ↔ (1 − 𝐸) < (𝑄‘𝑡)))
70	63, 69	ralbid 3279	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ↔ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡)))
71	64	breq1d 5176	. . . . 5 ⊢ (𝑦 = 𝑄 → ((𝑦‘𝑡) < 𝐸 ↔ (𝑄‘𝑡) < 𝐸))
72	63, 71	ralbid 3279	. . . 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 3635	. 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))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087   = wceq 1537  Ⅎwnf 1781   ∈ wcel 2108  Ⅎwnfc 2893  ∀wral 3067  ∃wrex 3076  {crab 3443   ∖ cdif 3973   class class class wbr 5166   ↦ cmpt 5249  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  ℝcr 11183  0cc0 11184  1c1 11185   + caddc 11187   · cmul 11189   < clt 11324   ≤ cle 11325   − cmin 11520   / cdiv 11947  ℕcn 12293  2c2 12348  ℕ₀cn0 12553  ℝ⁺crp 13057  ↑cexp 14112

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241  ax-1cn 11242  ax-icn 11243  ax-addcl 11244  ax-addrcl 11245  ax-mulcl 11246  ax-mulrcl 11247  ax-mulcom 11248  ax-addass 11249  ax-mulass 11250  ax-distr 11251  ax-i2m1 11252  ax-1ne0 11253  ax-1rid 11254  ax-rnegex 11255  ax-rrecex 11256  ax-cnre 11257  ax-pre-lttri 11258  ax-pre-lttrn 11259  ax-pre-ltadd 11260  ax-pre-mulgt0 11261

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-nel 3053  df-ral 3068  df-rex 3077  df-rmo 3388  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-2nd 8031  df-frecs 8322  df-wrecs 8353  df-recs 8427  df-rdg 8466  df-er 8763  df-en 9004  df-dom 9005  df-sdom 9006  df-pnf 11326  df-mnf 11327  df-xr 11328  df-ltxr 11329  df-le 11330  df-sub 11522  df-neg 11523  df-div 11948  df-nn 12294  df-2 12356  df-n0 12554  df-z 12640  df-uz 12904  df-rp 13058  df-seq 14053  df-exp 14113

This theorem is referenced by:  stoweidlem49  45970