Theorem stoweidlem41 46056

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 11261	. . . . . . . 8 ⊢ 1 ∈ ℝ
3		stoweidlem41.3	. . . . . . . . 9 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ 1)
4	3	fvmpt2 7027	. . . . . . . 8 ⊢ ((𝑡 ∈ 𝑇 ∧ 1 ∈ ℝ) → (𝐹‘𝑡) = 1)
5	2, 4	mpan2 691	. . . . . . 7 ⊢ (𝑡 ∈ 𝑇 → (𝐹‘𝑡) = 1)
6	5	adantl 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝐹‘𝑡) = 1)
7	6	oveq1d 7446	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → ((𝐹‘𝑡) − (𝑦‘𝑡)) = (1 − (𝑦‘𝑡)))
8	1, 7	mpteq2da 5240	. . . 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 2795	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝑦‘𝑡))) = 𝑋)
11		stoweidlem41.10	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑤 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑤) ∈ 𝐴)
12	11	stoweidlem4 46019	. . . . . 6 ⊢ ((𝜑 ∧ 1 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
13	2, 12	mpan2 691	. . . . 5 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ 1) ∈ 𝐴)
14	3, 13	eqeltrid 2845	. . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐴)
15		stoweidlem41.5	. . . 4 ⊢ (𝜑 → 𝑦 ∈ 𝐴)
16		nfmpt1 5250	. . . . . 6 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ 1)
17	3, 16	nfcxfr 2903	. . . . 5 ⊢ Ⅎ𝑡𝐹
18		nfcv 2905	. . . . 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 46048	. . . 4 ⊢ ((𝜑 ∧ 𝐹 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝑦‘𝑡))) ∈ 𝐴)
23	14, 15, 22	mpd3an23 1465	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 ↦ ((𝐹‘𝑡) − (𝑦‘𝑡))) ∈ 𝐴)
24	10, 23	eqeltrrd 2842	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
25		stoweidlem41.6	. . . . . . . 8 ⊢ (𝜑 → 𝑦:𝑇⟶ℝ)
26	25	ffvelcdmda 7104	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦‘𝑡) ∈ ℝ)
27		1red 11262	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 1 ∈ ℝ)
28		0red 11264	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ∈ ℝ)
29		stoweidlem41.12	. . . . . . . . . 10 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1))
30	29	r19.21bi 3251	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1))
31	30	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦‘𝑡) ≤ 1)
32		1m0e1 12387	. . . . . . . 8 ⊢ (1 − 0) = 1
33	31, 32	breqtrrdi 5185	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑦‘𝑡) ≤ (1 − 0))
34	26, 27, 28, 33	lesubd 11867	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (1 − (𝑦‘𝑡)))
35		simpr 484	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 𝑡 ∈ 𝑇)
36	27, 26	resubcld 11691	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − (𝑦‘𝑡)) ∈ ℝ)
37	9	fvmpt2 7027	. . . . . . 7 ⊢ ((𝑡 ∈ 𝑇 ∧ (1 − (𝑦‘𝑡)) ∈ ℝ) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
38	35, 36, 37	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
39	34, 38	breqtrrd 5171	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑋‘𝑡))
40	30	simpld 494	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → 0 ≤ (𝑦‘𝑡))
41	28, 26, 27, 40	lesub2dd 11880	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − (𝑦‘𝑡)) ≤ (1 − 0))
42	41, 32	breqtrdi 5184	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (1 − (𝑦‘𝑡)) ≤ 1)
43	38, 42	eqbrtrd 5165	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (𝑋‘𝑡) ≤ 1)
44	39, 43	jca 511	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑇) → (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1))
45	44	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑇 → (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
46	1, 45	ralrimi 3257	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1))
47		stoweidlem41.4	. . . . . . 7 ⊢ 𝑉 ⊆ 𝑇
48	47	sseli 3979	. . . . . 6 ⊢ (𝑡 ∈ 𝑉 → 𝑡 ∈ 𝑇)
49	48, 38	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
50		1red 11262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 1 ∈ ℝ)
51		stoweidlem41.11	. . . . . . . 8 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
52	51	rpred 13077	. . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ)
53	52	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝐸 ∈ ℝ)
54	48, 26	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑦‘𝑡) ∈ ℝ)
55		stoweidlem41.13	. . . . . . 7 ⊢ (𝜑 → ∀𝑡 ∈ 𝑉 (1 − 𝐸) < (𝑦‘𝑡))
56	55	r19.21bi 3251	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − 𝐸) < (𝑦‘𝑡))
57	50, 53, 54, 56	ltsub23d 11868	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (1 − (𝑦‘𝑡)) < 𝐸)
58	49, 57	eqbrtrd 5165	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑋‘𝑡) < 𝐸)
59	58	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑉 → (𝑋‘𝑡) < 𝐸))
60	1, 59	ralrimi 3257	. 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑉 (𝑋‘𝑡) < 𝐸)
61		eldifi 4131	. . . . . . 7 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) → 𝑡 ∈ 𝑇)
62	61, 26	sylan2 593	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑦‘𝑡) ∈ ℝ)
63	52	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 𝐸 ∈ ℝ)
64		1red 11262	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 1 ∈ ℝ)
65		stoweidlem41.14	. . . . . . 7 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝐸)
66	65	r19.21bi 3251	. . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑦‘𝑡) < 𝐸)
67	62, 63, 64, 66	ltsub2dd 11876	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (1 − 𝐸) < (1 − (𝑦‘𝑡)))
68	61, 38	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑋‘𝑡) = (1 − (𝑦‘𝑡)))
69	67, 68	breqtrrd 5171	. . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (1 − 𝐸) < (𝑋‘𝑡))
70	69	ex 412	. . 3 ⊢ (𝜑 → (𝑡 ∈ (𝑇 ∖ 𝑈) → (1 − 𝐸) < (𝑋‘𝑡)))
71	1, 70	ralrimi 3257	. 2 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑋‘𝑡))
72		nfmpt1 5250	. . . . . . 7 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡)))
73	9, 72	nfcxfr 2903	. . . . . 6 ⊢ Ⅎ𝑡𝑋
74	73	nfeq2 2923	. . . . 5 ⊢ Ⅎ𝑡 𝑥 = 𝑋
75		fveq1 6905	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑡) = (𝑋‘𝑡))
76	75	breq2d 5155	. . . . . 6 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
77	75	breq1d 5153	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) ≤ 1 ↔ (𝑋‘𝑡) ≤ 1))
78	76, 77	anbi12d 632	. . . . 5 ⊢ (𝑥 = 𝑋 → ((0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
79	74, 78	ralbid 3273	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
80	75	breq1d 5153	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) < 𝐸 ↔ (𝑋‘𝑡) < 𝐸))
81	74, 80	ralbid 3273	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝑉 (𝑋‘𝑡) < 𝐸))
82	75	breq2d 5155	. . . . 5 ⊢ (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝑋‘𝑡)))
83	74, 82	ralbid 3273	. . . 4 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝐸) < (𝑋‘𝑡)))
84	79, 81, 83	3anbi123d 1438	. . 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 3622	. 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 1374	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 395   ∧ w3a 1087   = wceq 1540  Ⅎwnf 1783   ∈ wcel 2108  ∀wral 3061  ∃wrex 3070   ∖ cdif 3948   ⊆ wss 3951   class class class wbr 5143   ↦ cmpt 5225  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431  ℝcr 11154  0cc0 11155  1c1 11156   + caddc 11158   · cmul 11160   < clt 11295   ≤ cle 11296   − cmin 11492  ℝ⁺crp 13034

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-po 5592  df-so 5593  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-rp 13035

This theorem is referenced by:  stoweidlem52  46067