Theorem stoweidlem45 46836

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 2765	. . 3 ⊢ (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁))) = (𝑡 ∈ 𝑇 ↦ (1 − ((𝑃‘𝑡)↑𝑁)))
5		eqid 2765	. . 3 ⊢ (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ 1)
6		eqid 2765	. . 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 12584	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ0)
16	14, 15	nnexpcld 14303	. . 3 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℕ)
17	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 16	stoweidlem40 46831	. 2 ⊢ (𝜑 → 𝑄 ∈ 𝐴)
18		1red 11228	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
19	8	ffvelcdmda 7083	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ∈ ℝ)
20	15	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑁 ∈ ℕ0)
21	19, 20	reexpcld 14221	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ∈ ℝ)
22	18, 21	resubcld 11661	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ∈ ℝ)
23	14	nnnn0d 12584	. . . . . . . . 9 ⊢ (𝜑 → 𝐾 ∈ ℕ0)
24	23, 15	nn0expcld 14304	. . . . . . . 8 ⊢ (𝜑 → (𝐾↑𝑁) ∈ ℕ0)
25	24	adantr 486	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐾↑𝑁) ∈ ℕ0)
26		1m1e0 12332	. . . . . . . 8 ⊢ (1 − 1) = 0
27		stoweidlem45.11	. . . . . . . . . . . 12 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
28	27	r19.21bi 3259	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
29	28	simpld 500	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑃‘𝑡))
30	28	simprd 501	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑃‘𝑡) ≤ 1)
31		exple1 14235	. . . . . . . . . 10 ⊢ ((((𝑃‘𝑡) ∈ ℝ ∧ 0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1) ∧ 𝑁 ∈ ℕ0) → ((𝑃‘𝑡)↑𝑁) ≤ 1)
32	19, 29, 30, 20, 31	syl31anc 1400	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝑃‘𝑡)↑𝑁) ≤ 1)
33	21, 18, 18, 32	lesub2dd 11850	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − 1) ≤ (1 − ((𝑃‘𝑡)↑𝑁)))
34	26, 33	eqbrtrrid 5149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (1 − ((𝑃‘𝑡)↑𝑁)))
35	22, 25, 34	expge0d 14222	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
36	3, 8, 15, 23	stoweidlem12 46803	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑄‘𝑡) = ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
37	35, 36	breqtrrd 5141	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑄‘𝑡))
38		0red 11230	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ)
39	19, 20, 29	expge0d 14222	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ ((𝑃‘𝑡)↑𝑁))
40	38, 21, 18, 39	lesub2dd 11850	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ≤ (1 − 0))
41		1m0e1 12379	. . . . . . . 8 ⊢ (1 − 0) = 1
42	40, 41	breqtrdi 5154	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − ((𝑃‘𝑡)↑𝑁)) ≤ 1)
43		exple1 14235	. . . . . . 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 1400	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)) ≤ 1)
45	36, 44	eqbrtrd 5135	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑄‘𝑡) ≤ 1)
46	37, 45	jca 521	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1))
47	46	ex 418	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)))
48	2, 47	ralrimi 3265	. 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 46815	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − 𝐸) < (𝑄‘𝑡))
54	53	ex 418	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑉 → (1 − 𝐸) < (𝑄‘𝑡)))
55	2, 54	ralrimi 3265	. 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 46816	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑄‘𝑡) < 𝐸)
59	58	ex 418	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → (𝑄‘𝑡) < 𝐸))
60	2, 59	ralrimi 3265	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸)
61		nfmpt1 5212	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((1 − ((𝑃‘𝑡)↑𝑁))↑(𝐾↑𝑁)))
62	3, 61	nfcxfr 2925	. . . . . 6 ⊢ Ⅎ𝑡𝑄
63	62	nfeq2 2944	. . . . 5 ⊢ Ⅎ𝑡 𝑦 = 𝑄
64		fveq1 6884	. . . . . . 7 ⊢ (𝑦 = 𝑄 → (𝑦‘𝑡) = (𝑄‘𝑡))
65	64	breq2d 5123	. . . . . 6 ⊢ (𝑦 = 𝑄 → (0 ≤ (𝑦‘𝑡) ↔ 0 ≤ (𝑄‘𝑡)))
66	64	breq1d 5121	. . . . . 6 ⊢ (𝑦 = 𝑄 → ((𝑦‘𝑡) ≤ 1 ↔ (𝑄‘𝑡) ≤ 1))
67	65, 66	anbi12d 644	. . . . 5 ⊢ (𝑦 = 𝑄 → ((0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ↔ (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)))
68	63, 67	ralbid 3280	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑄‘𝑡) ∧ (𝑄‘𝑡) ≤ 1)))
69	64	breq2d 5123	. . . . 5 ⊢ (𝑦 = 𝑄 → ((1 − 𝐸) < (𝑦‘𝑡) ↔ (1 − 𝐸) < (𝑄‘𝑡)))
70	63, 69	ralbid 3280	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ↔ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑄‘𝑡)))
71	64	breq1d 5121	. . . . 5 ⊢ (𝑦 = 𝑄 → ((𝑦‘𝑡) < 𝐸 ↔ (𝑄‘𝑡) < 𝐸))
72	63, 71	ralbid 3280	. . . 4 ⊢ (𝑦 = 𝑄 → (∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑄‘𝑡) < 𝐸))
73	68, 70, 72	3anbi123d 1464	. . 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 3583	. 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 1399	1 ⊢ (𝜑 → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2146  Ⅎwnfc 2912  ∀wral 3081  ∃wrex 3091  {crab 3418   ∖ cdif 3903   class class class wbr 5111   ↦ cmpt 5194  ⟶wf 6536  ‘cfv 6540  (class class class)co 7419  ℝcr 11118  0cc0 11119  1c1 11120   + caddc 11122   · cmul 11124   < clt 11262   ≤ cle 11263   − cmin 11460   / cdiv 11890  ℕcn 12252  2c2 12314  ℕ₀cn0 12523  ℝ⁺crp 13036  ↑cexp 14119

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742  ax-cnex 11175  ax-resscn 11176  ax-1cn 11177  ax-icn 11178  ax-addcl 11179  ax-addrcl 11180  ax-mulcl 11181  ax-mulrcl 11182  ax-mulcom 11183  ax-addass 11184  ax-mulass 11185  ax-distr 11186  ax-i2m1 11187  ax-1ne0 11188  ax-1rid 11189  ax-rnegex 11190  ax-rrecex 11191  ax-cnre 11192  ax-pre-lttri 11193  ax-pre-lttrn 11194  ax-pre-ltadd 11195  ax-pre-mulgt0 11196

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-nel 3067  df-ral 3082  df-rex 3092  df-rmo 3371  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-riota 7376  df-ov 7422  df-oprab 7423  df-mpo 7424  df-om 7869  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-er 8700  df-en 8950  df-dom 8951  df-sdom 8952  df-pnf 11264  df-mnf 11265  df-xr 11266  df-ltxr 11267  df-le 11268  df-sub 11462  df-neg 11463  df-div 11891  df-nn 12253  df-2 12322  df-n0 12524  df-z 12611  df-uz 12883  df-rp 13037  df-seq 14060  df-exp 14120

This theorem is used by:  stoweidlem49  46840