Theorem stoweidlem41 42683

Description: This lemma is used to prove that there exists x as in Lemma 1 of [BrosowskiDeutsh] p. 90: 0 <= x(t) <= 1 for all t in T, x(t) < epsilon for all t in V, x(t) > 1 - epsilon for all t in T \ U. Here we prove the very last step of the proof of Lemma 1: "The result follows from taking x = 1 - q_n";. Here 𝐸 is used to represent ε in the paper, and 𝑦 to represent q_n in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem41.1	⊢ Ⅎ𝑡𝜑
stoweidlem41.2	⊢ 𝑋 = (𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡)))
stoweidlem41.3	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ 1)
stoweidlem41.4	⊢ 𝑉 ⊆ 𝑇
stoweidlem41.5	⊢ (𝜑 → 𝑦 ∈ 𝐴)
stoweidlem41.6	⊢ (𝜑 → 𝑦:𝑇⟶ℝ)
stoweidlem41.7	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem41.8	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem41.9	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem41.10	⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑤) ∈ 𝐴)
stoweidlem41.11	⊢ (𝜑 → 𝐸 ∈ ℝ+)
stoweidlem41.12	⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1))
stoweidlem41.13	⊢ (𝜑 → ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡))
stoweidlem41.14	⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸)

Assertion

Ref	Expression
stoweidlem41	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑥‘𝑡)))

Distinct variable groups:   𝑓,𝑔,𝑡,𝑦   𝐴,𝑓,𝑔,𝑡   𝑓,𝐹,𝑔   𝑇,𝑓,𝑔,𝑡   𝜑,𝑓,𝑔   𝑤,𝑡,𝐴   𝑥,𝑡,𝐴   𝑤,𝑇   𝜑,𝑤   𝑥,𝐸   𝑥,𝑇   𝑥,𝑈   𝑥,𝑉   𝑥,𝑋

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑡)   𝐴(𝑦)   𝑇(𝑦)   𝑈(𝑦,𝑤,𝑡,𝑓,𝑔)   𝐸(𝑦,𝑤,𝑡,𝑓,𝑔)   𝐹(𝑥,𝑦,𝑤,𝑡)   𝑉(𝑦,𝑤,𝑡,𝑓,𝑔)   𝑋(𝑦,𝑤,𝑡,𝑓,𝑔)

Proof of Theorem stoweidlem41

Step	Hyp	Ref	Expression
1		stoweidlem41.1	. . . . 5 ⊢ Ⅎ𝑡𝜑
2		1re 10630	. . . . . . . 8 ⊢ 1 ∈ ℝ
3		stoweidlem41.3	. . . . . . . . 9 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ 1)
4	3	fvmpt2 6756	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐹‘𝑡) = 1)
5	2, 4	mpan2 690	. . . . . . 7 ⊢ (𝑡 ∈ 𝑇 → (𝐹‘𝑡) = 1)
6	5	adantl 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = 1)
7	6	oveq1d 7150	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) − (𝑦‘𝑡)) = (1 − (𝑦‘𝑡)))
8	1, 7	mpteq2da 5124	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝑦‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡))))
9		stoweidlem41.2	. . . 4 ⊢ 𝑋 = (𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡)))
10	8, 9	eqtr4di 2851	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝑦‘𝑡))) = 𝑋)
11		stoweidlem41.10	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑤) ∈ 𝐴)
12	11	stoweidlem4 42646	. . . . . 6 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
13	2, 12	mpan2 690	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
14	3, 13	eqeltrid 2894	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
15		stoweidlem41.5	. . . 4 ⊢ (𝜑 → 𝑦 ∈ 𝐴)
16		nfmpt1 5128	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
17	3, 16	nfcxfr 2953	. . . . 5 ⊢ Ⅎ𝑡𝐹
18		nfcv 2955	. . . . 5 ⊢ Ⅎ𝑡𝑦
19		stoweidlem41.7	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
20		stoweidlem41.8	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
21		stoweidlem41.9	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
22	17, 18, 1, 19, 20, 21, 11	stoweidlem33 42675	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝑦‘𝑡))) ∈ 𝐴)
23	14, 15, 22	mpd3an23 1460	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝑦‘𝑡))) ∈ 𝐴)
24	10, 23	eqeltrrd 2891	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
25		stoweidlem41.6	. . . . . . . 8 ⊢ (𝜑 → 𝑦:𝑇⟶ℝ)
26	25	ffvelrnda 6828	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦‘𝑡) ∈ ℝ)
27		1red 10631	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
28		0red 10633	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ)
29		stoweidlem41.12	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1))
30	29	r19.21bi 3173	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1))
31	30	simprd 499	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦‘𝑡) ≤ 1)
32		1m0e1 11746	. . . . . . . 8 ⊢ (1 − 0) = 1
33	31, 32	breqtrrdi 5072	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦‘𝑡) ≤ (1 − 0))
34	26, 27, 28, 33	lesubd 11233	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (1 − (𝑦‘𝑡)))
35		simpr 488	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
36	27, 26	resubcld 11057	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − (𝑦‘𝑡)) ∈ ℝ)
37	9	fvmpt2 6756	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ (1 − (𝑦‘𝑡)) ∈ ℝ) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
38	35, 36, 37	syl2anc 587	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
39	34, 38	breqtrrd 5058	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑋‘𝑡))
40	30	simpld 498	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑦‘𝑡))
41	28, 26, 27, 40	lesub2dd 11246	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − (𝑦‘𝑡)) ≤ (1 − 0))
42	41, 32	breqtrdi 5071	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − (𝑦‘𝑡)) ≤ 1)
43	38, 42	eqbrtrd 5052	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) ≤ 1)
44	39, 43	jca 515	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1))
45	44	ex 416	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
46	1, 45	ralrimi 3180	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1))
47		stoweidlem41.4	. . . . . . 7 ⊢ 𝑉 ⊆ 𝑇
48	47	sseli 3911	. . . . . 6 ⊢ (𝑡 ∈ 𝑉 → 𝑡 ∈ 𝑇)
49	48, 38	sylan2 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
50		1red 10631	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 1 ∈ ℝ)
51		stoweidlem41.11	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
52	51	rpred 12419	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
53	52	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝐸 ∈ ℝ)
54	48, 26	sylan2 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑦‘𝑡) ∈ ℝ)
55		stoweidlem41.13	. . . . . . 7 ⊢ (𝜑 → ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡))
56	55	r19.21bi 3173	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − 𝐸) < (𝑦‘𝑡))
57	50, 53, 54, 56	ltsub23d 11234	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − (𝑦‘𝑡)) < 𝐸)
58	49, 57	eqbrtrd 5052	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑋‘𝑡) < 𝐸)
59	58	ex 416	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑉 → (𝑋‘𝑡) < 𝐸))
60	1, 59	ralrimi 3180	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑉 (𝑋‘𝑡) < 𝐸)
61		eldifi 4054	. . . . . . 7 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) → 𝑡 ∈ 𝑇)
62	61, 26	sylan2 595	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑦‘𝑡) ∈ ℝ)
63	52	adantr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 𝐸 ∈ ℝ)
64		1red 10631	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 1 ∈ ℝ)
65		stoweidlem41.14	. . . . . . 7 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸)
66	65	r19.21bi 3173	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑦‘𝑡) < 𝐸)
67	62, 63, 64, 66	ltsub2dd 11242	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (1 − 𝐸) < (1 − (𝑦‘𝑡)))
68	61, 38	sylan2 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
69	67, 68	breqtrrd 5058	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (1 − 𝐸) < (𝑋‘𝑡))
70	69	ex 416	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → (1 − 𝐸) < (𝑋‘𝑡)))
71	1, 70	ralrimi 3180	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑋‘𝑡))
72		nfmpt1 5128	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡)))
73	9, 72	nfcxfr 2953	. . . . . 6 ⊢ Ⅎ𝑡𝑋
74	73	nfeq2 2972	. . . . 5 ⊢ Ⅎ𝑡 𝑥 = 𝑋
75		fveq1 6644	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑡) = (𝑋‘𝑡))
76	75	breq2d 5042	. . . . . 6 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
77	75	breq1d 5040	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) ≤ 1 ↔ (𝑋‘𝑡) ≤ 1))
78	76, 77	anbi12d 633	. . . . 5 ⊢ (𝑥 = 𝑋 → ((0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
79	74, 78	ralbid 3195	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
80	75	breq1d 5040	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) < 𝐸 ↔ (𝑋‘𝑡) < 𝐸))
81	74, 80	ralbid 3195	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝑉 (𝑋‘𝑡) < 𝐸))
82	75	breq2d 5042	. . . . 5 ⊢ (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝑋‘𝑡)))
83	74, 82	ralbid 3195	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑋‘𝑡)))
84	79, 81, 83	3anbi123d 1433	. . 3 ⊢ (𝑥 = 𝑋 → ((∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑥‘𝑡)) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑋‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑋‘𝑡))))
85	84	rspcev 3571	. 2 ⊢ ((𝑋 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑋‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑋‘𝑡))) → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑥‘𝑡)))
86	24, 46, 60, 71, 85	syl13anc 1369	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑥‘𝑡)))

Syntax hints:   → wi 4   ∧ wa 399   ∧ w3a 1084   = wceq 1538  Ⅎwnf 1785   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107   ∖ cdif 3878   ⊆ wss 3881   class class class wbr 5030   ↦ cmpt 5110  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135  ℝcr 10525  0cc0 10526  1c1 10527   + caddc 10529   · cmul 10531   < clt 10664   ≤ cle 10665   − cmin 10859  ℝ⁺crp 12377

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-po 5438  df-so 5439  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-rp 12378

This theorem is referenced by:  stoweidlem52  42694