Theorem stoweidlem41 38934

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 9918	. . . . . . . 8 ⊢ 1 ∈ ℝ
3		stoweidlem41.3	. . . . . . . . 9 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ 1)
4	3	fvmpt2 6200	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐹‘𝑡) = 1)
5	2, 4	mpan2 703	. . . . . . 7 ⊢ (𝑡 ∈ 𝑇 → (𝐹‘𝑡) = 1)
6	5	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = 1)
7	6	oveq1d 6564	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) − (𝑦‘𝑡)) = (1 − (𝑦‘𝑡)))
8	1, 7	mpteq2da 4671	. . . 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	syl6eqr 2662	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝑦‘𝑡))) = 𝑋)
11		stoweidlem41.10	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑤) ∈ 𝐴)
12	11	stoweidlem4 38897	. . . . . 6 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
13	2, 12	mpan2 703	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
14	3, 13	syl5eqel 2692	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
15		stoweidlem41.5	. . . 4 ⊢ (𝜑 → 𝑦 ∈ 𝐴)
16		nfmpt1 4675	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
17	3, 16	nfcxfr 2749	. . . . 5 ⊢ Ⅎ𝑡𝐹
18		nfcv 2751	. . . . 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 38926	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝑦‘𝑡))) ∈ 𝐴)
23	14, 15, 22	mpd3an23 1418	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝑦‘𝑡))) ∈ 𝐴)
24	10, 23	eqeltrrd 2689	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
25		stoweidlem41.6	. . . . . . . 8 ⊢ (𝜑 → 𝑦:𝑇⟶ℝ)
26	25	fnvinran 38196	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦‘𝑡) ∈ ℝ)
27		1red 9934	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
28		0red 9920	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ)
29		stoweidlem41.12	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1))
30	29	r19.21bi 2916	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1))
31	30	simprd 478	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦‘𝑡) ≤ 1)
32		1m0e1 11008	. . . . . . . 8 ⊢ (1 − 0) = 1
33	31, 32	syl6breqr 4625	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦‘𝑡) ≤ (1 − 0))
34	26, 27, 28, 33	lesubd 10510	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (1 − (𝑦‘𝑡)))
35		simpr 476	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
36	27, 26	resubcld 10337	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − (𝑦‘𝑡)) ∈ ℝ)
37	9	fvmpt2 6200	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ (1 − (𝑦‘𝑡)) ∈ ℝ) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
38	35, 36, 37	syl2anc 691	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
39	34, 38	breqtrrd 4611	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑋‘𝑡))
40	30	simpld 474	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑦‘𝑡))
41	28, 26, 27, 40	lesub2dd 10523	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − (𝑦‘𝑡)) ≤ (1 − 0))
42	41, 32	syl6breq 4624	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − (𝑦‘𝑡)) ≤ 1)
43	38, 42	eqbrtrd 4605	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) ≤ 1)
44	39, 43	jca 553	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1))
45	44	ex 449	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
46	1, 45	ralrimi 2940	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1))
47		stoweidlem41.4	. . . . . . 7 ⊢ 𝑉 ⊆ 𝑇
48	47	sseli 3564	. . . . . 6 ⊢ (𝑡 ∈ 𝑉 → 𝑡 ∈ 𝑇)
49	48, 38	sylan2 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
50		1red 9934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 1 ∈ ℝ)
51		stoweidlem41.11	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
52	51	rpred 11748	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
53	52	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝐸 ∈ ℝ)
54	48, 26	sylan2 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑦‘𝑡) ∈ ℝ)
55		stoweidlem41.13	. . . . . . 7 ⊢ (𝜑 → ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡))
56	55	r19.21bi 2916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − 𝐸) < (𝑦‘𝑡))
57	50, 53, 54, 56	ltsub23d 10511	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − (𝑦‘𝑡)) < 𝐸)
58	49, 57	eqbrtrd 4605	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑋‘𝑡) < 𝐸)
59	58	ex 449	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑉 → (𝑋‘𝑡) < 𝐸))
60	1, 59	ralrimi 2940	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑉 (𝑋‘𝑡) < 𝐸)
61		eldifi 3694	. . . . . . 7 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) → 𝑡 ∈ 𝑇)
62	61, 26	sylan2 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑦‘𝑡) ∈ ℝ)
63	52	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 𝐸 ∈ ℝ)
64		1red 9934	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 1 ∈ ℝ)
65		stoweidlem41.14	. . . . . . 7 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸)
66	65	r19.21bi 2916	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑦‘𝑡) < 𝐸)
67	62, 63, 64, 66	ltsub2dd 10519	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (1 − 𝐸) < (1 − (𝑦‘𝑡)))
68	61, 38	sylan2 490	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
69	67, 68	breqtrrd 4611	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (1 − 𝐸) < (𝑋‘𝑡))
70	69	ex 449	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → (1 − 𝐸) < (𝑋‘𝑡)))
71	1, 70	ralrimi 2940	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑋‘𝑡))
72		nfmpt1 4675	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡)))
73	9, 72	nfcxfr 2749	. . . . . 6 ⊢ Ⅎ𝑡𝑋
74	73	nfeq2 2766	. . . . 5 ⊢ Ⅎ𝑡 𝑥 = 𝑋
75		fveq1 6102	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑡) = (𝑋‘𝑡))
76	75	breq2d 4595	. . . . . 6 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
77	75	breq1d 4593	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) ≤ 1 ↔ (𝑋‘𝑡) ≤ 1))
78	76, 77	anbi12d 743	. . . . 5 ⊢ (𝑥 = 𝑋 → ((0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
79	74, 78	ralbid 2966	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
80	75	breq1d 4593	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) < 𝐸 ↔ (𝑋‘𝑡) < 𝐸))
81	74, 80	ralbid 2966	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝑉 (𝑋‘𝑡) < 𝐸))
82	75	breq2d 4595	. . . . 5 ⊢ (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝑋‘𝑡)))
83	74, 82	ralbid 2966	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑋‘𝑡)))
84	79, 81, 83	3anbi123d 1391	. . 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 3282	. 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 1320	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 383   ∧ w3a 1031   = wceq 1475  Ⅎwnf 1699   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∖ cdif 3537   ⊆ wss 3540   class class class wbr 4583   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  ℝcr 9814  0cc0 9815  1c1 9816   + caddc 9818   · cmul 9820   < clt 9953   ≤ cle 9954   − cmin 10145  ℝ⁺crp 11708

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-resscn 9872  ax-1cn 9873  ax-icn 9874  ax-addcl 9875  ax-addrcl 9876  ax-mulcl 9877  ax-mulrcl 9878  ax-mulcom 9879  ax-addass 9880  ax-mulass 9881  ax-distr 9882  ax-i2m1 9883  ax-1ne0 9884  ax-1rid 9885  ax-rnegex 9886  ax-rrecex 9887  ax-cnre 9888  ax-pre-lttri 9889  ax-pre-lttrn 9890  ax-pre-ltadd 9891

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-sub 10147  df-neg 10148  df-rp 11709

This theorem is referenced by:  stoweidlem52  38945