Theorem stoweidlem45 46473

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 2736	. . 3 ⊢ (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁))) = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
5		eqid 2736	. . 3 ⊢ (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ 1)
6		eqid 2736	. . 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 12498	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
16	14, 15	nnexpcld 14207	. . 3 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℕ)
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16	stoweidlem40 46468	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
18		1red 11145	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
19	8	ffvelcdmda 7036	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ∈ ℝ)
20	15	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℕ0)
21	19, 20	reexpcld 14125	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ∈ ℝ)
22	18, 21	resubcld 11578	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ)
23	14	nnnn0d 12498	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
24	23, 15	nn0expcld 14208	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℕ0)
25	24	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐾↑𝑁) ∈ ℕ0)
26		1m1e0 12253	. . . . . . . 8 ⊢ (1 − 1) = 0
27		stoweidlem45.11	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
28	27	r19.21bi 3229	. . . . . . . . . . 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 14139	. . . . . . . . . 10 ⊢ ((((𝑃‘𝑡) ∈ ℝ ∧ 0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ 𝑁 ∈ ℕ0) → ((𝑃‘𝑡)↑𝑁) ≤ 1)
32	19, 29, 30, 20, 31	syl31anc 1376	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ≤ 1)
33	21, 18, 18, 32	lesub2dd 11767	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − 1) ≤ (1 − ((𝑃‘𝑡)↑𝑁)))
34	26, 33	eqbrtrrid 5121	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (1 − ((𝑃‘𝑡)↑𝑁)))
35	22, 25, 34	expge0d 14126	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
36	3, 8, 15, 23	stoweidlem12 46440	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑄‘𝑡) = ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
37	35, 36	breqtrrd 5113	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑄‘𝑡))
38		0red 11147	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ)
39	19, 20, 29	expge0d 14126	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑃‘𝑡)↑𝑁))
40	38, 21, 18, 39	lesub2dd 11767	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ≤ (1 − 0))
41		1m0e1 12297	. . . . . . . 8 ⊢ (1 − 0) = 1
42	40, 41	breqtrdi 5126	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ≤ 1)
43		exple1 14139	. . . . . . 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 1376	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)) ≤ 1)
45	36, 44	eqbrtrd 5107	. . . . 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 3235	. 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 46452	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − 𝐸) < (𝑄‘𝑡))
54	53	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑉 → (1 − 𝐸) < (𝑄‘𝑡)))
55	2, 54	ralrimi 3235	. 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 46453	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑄‘𝑡) < 𝐸)
59	58	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → (𝑄‘𝑡) < 𝐸))
60	2, 59	ralrimi 3235	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸)
61		nfmpt1 5184	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
62	3, 61	nfcxfr 2896	. . . . . 6 ⊢ Ⅎ𝑡𝑄
63	62	nfeq2 2916	. . . . 5 ⊢ Ⅎ𝑡 𝑦 = 𝑄
64		fveq1 6839	. . . . . . 7 ⊢ (𝑦 = 𝑄 → (𝑦‘𝑡) = (𝑄‘𝑡))
65	64	breq2d 5097	. . . . . 6 ⊢ (𝑦 = 𝑄 → (0 ≤ (𝑦‘𝑡) ↔ 0 ≤ (𝑄‘𝑡)))
66	64	breq1d 5095	. . . . . 6 ⊢ (𝑦 = 𝑄 → ((𝑦‘𝑡) ≤ 1 ↔ (𝑄‘𝑡) ≤ 1))
67	65, 66	anbi12d 633	. . . . 5 ⊢ (𝑦 = 𝑄 → ((0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ↔ (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)))
68	63, 67	ralbid 3250	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)))
69	64	breq2d 5097	. . . . 5 ⊢ (𝑦 = 𝑄 → ((1 − 𝐸) < (𝑦‘𝑡) ↔ (1 − 𝐸) < (𝑄‘𝑡)))
70	63, 69	ralbid 3250	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ↔ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡)))
71	64	breq1d 5095	. . . . 5 ⊢ (𝑦 = 𝑄 → ((𝑦‘𝑡) < 𝐸 ↔ (𝑄‘𝑡) < 𝐸))
72	63, 71	ralbid 3250	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸))
73	68, 70, 72	3anbi123d 1439	. . 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 3564	. 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 1375	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 1542  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2883  ∀wral 3051  ∃wrex 3061  {crab 3389   ∖ cdif 3886   class class class wbr 5085   ↦ cmpt 5166  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ℝcr 11037  0cc0 11038  1c1 11039   + caddc 11041   · cmul 11043   < clt 11179   ≤ cle 11180   − cmin 11377   / cdiv 11807  ℕcn 12174  2c2 12236  ℕ₀cn0 12437  ℝ⁺crp 12942  ↑cexp 14023

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-om 7818  df-2nd 7943  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-er 8643  df-en 8894  df-dom 8895  df-sdom 8896  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-n0 12438  df-z 12525  df-uz 12789  df-rp 12943  df-seq 13964  df-exp 14024

This theorem is referenced by:  stoweidlem49  46477