Theorem stoweidlem56 46506

Description: This theorem proves Lemma 1 in [BrosowskiDeutsh] p. 90. Here 𝑍 is used to represent t₀ in the paper, 𝑣 is used to represent 𝑉 in the paper, and 𝑒 is used to represent ε. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem56.1	⊢ Ⅎ𝑡𝑈
stoweidlem56.2	⊢ Ⅎ𝑡𝜑
stoweidlem56.3	⊢ 𝐾 = (topGen‘ran (,))
stoweidlem56.4	⊢ (𝜑 → 𝐽 ∈ Comp)
stoweidlem56.5	⊢ 𝑇 = ∪ 𝐽
stoweidlem56.6	⊢ 𝐶 = (𝐽 Cn 𝐾)
stoweidlem56.7	⊢ (𝜑 → 𝐴 ⊆ 𝐶)
stoweidlem56.8	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem56.9	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem56.10	⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)
stoweidlem56.11	⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
stoweidlem56.12	⊢ (𝜑 → 𝑈 ∈ 𝐽)
stoweidlem56.13	⊢ (𝜑 → 𝑍 ∈ 𝑈)

Assertion

Ref	Expression
stoweidlem56	⊢ (𝜑 → ∃𝑣 ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃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:   𝐴,𝑒,𝑡,𝑣,𝑥   𝜑,𝑞,𝑟,𝑔   𝑒,𝑓,𝜑,𝑦   𝑈,𝑓,𝑞,𝑟,𝑦   𝑈,𝑔,𝑒   𝑣,𝑈,𝑥   𝑡,𝑍,𝑦   𝑡,𝐾   𝑔,𝐽,𝑡   𝑇,𝑓,𝑔,𝑞,𝑟,𝑡   𝑦,𝑇   𝐴,𝑔   𝑒,𝑍,𝑣   𝑇,𝑒,𝑣,𝑥   𝑓,𝑍,𝑔,𝑞   𝑣,𝐽   𝐴,𝑓,𝑞,𝑟,𝑦   𝑒,𝑔

Allowed substitution hints:   𝜑(𝑥,𝑣,𝑡)   𝐶(𝑥,𝑦,𝑣,𝑡,𝑒,𝑓,𝑔,𝑟,𝑞)   𝑈(𝑡)   𝐽(𝑥,𝑦,𝑒,𝑓,𝑟,𝑞)   𝐾(𝑥,𝑦,𝑣,𝑒,𝑓,𝑔,𝑟,𝑞)   𝑍(𝑥,𝑟)

Proof of Theorem stoweidlem56

Dummy variables 𝑑 𝑝 ℎ 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		stoweidlem56.1	. . . . 5 ⊢ Ⅎ𝑡𝑈
2		stoweidlem56.2	. . . . 5 ⊢ Ⅎ𝑡𝜑
3		stoweidlem56.3	. . . . 5 ⊢ 𝐾 = (topGen‘ran (,))
4		stoweidlem56.4	. . . . 5 ⊢ (𝜑 → 𝐽 ∈ Comp)
5		stoweidlem56.5	. . . . 5 ⊢ 𝑇 = ∪ 𝐽
6		stoweidlem56.6	. . . . 5 ⊢ 𝐶 = (𝐽 Cn 𝐾)
7		stoweidlem56.7	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
8		stoweidlem56.8	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
9		stoweidlem56.9	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
10		stoweidlem56.10	. . . . 5 ⊢ ((𝜑 ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)
11		stoweidlem56.11	. . . . 5 ⊢ ((𝜑 ∧ (𝑟 ∈ 𝑇 ∧ 𝑡 ∈ 𝑇 ∧ 𝑟 ≠ 𝑡)) → ∃𝑞 ∈ 𝐴 (𝑞‘𝑟) ≠ (𝑞‘𝑡))
12		stoweidlem56.12	. . . . 5 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
13		stoweidlem56.13	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
14		eqid 2737	. . . . 5 ⊢ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))} = {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}
15		eqid 2737	. . . . 5 ⊢ {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}} = {𝑤 ∈ 𝐽 ∣ ∃ℎ ∈ {ℎ ∈ 𝐴 ∣ ((ℎ‘𝑍) = 0 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1))}𝑤 = {𝑡 ∈ 𝑇 ∣ 0 < (ℎ‘𝑡)}}
16	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15	stoweidlem55 46505	. . . 4 ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))
17		df-rex 3063	. . . 4 ⊢ (∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))))
18	16, 17	sylib 218	. . 3 ⊢ (𝜑 → ∃𝑝(𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))))
19		simpl 482	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) → 𝜑)
20		simprl 771	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) → 𝑝 ∈ 𝐴)
21		simprr3 1225	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))
22		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑝 ∈ 𝐴
23		nfra1 3262	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)
24	2, 22, 23	nf3an 1903	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))
25	4	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → 𝐽 ∈ Comp)
26	7	sselda 3922	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐶)
27	26, 6	eleqtrdi 2847	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (𝐽 Cn 𝐾))
28	27	3adant3 1133	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → 𝑝 ∈ (𝐽 Cn 𝐾))
29		simp3 1139	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))
30	12	3ad2ant1 1134	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → 𝑈 ∈ 𝐽)
31	1, 24, 3, 5, 25, 28, 29, 30	stoweidlem28 46478	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))
32	19, 20, 21, 31	syl3anc 1374	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))
33		simpr1 1196	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))) → 𝑑 ∈ ℝ+)
34		simpr2 1197	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))) → 𝑑 < 1)
35		simplrl 777	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))) → 𝑝 ∈ 𝐴)
36		simprr1 1223	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1))
37	36	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1))
38		simprr2 1224	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) → (𝑝‘𝑍) = 0)
39	38	adantr 480	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))) → (𝑝‘𝑍) = 0)
40		simpr3 1198	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))
41	37, 39, 40	3jca 1129	. . . . . . . . . 10 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))) → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))
42	35, 41	jca 511	. . . . . . . . 9 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))) → (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))
43	33, 34, 42	3jca 1129	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))) → (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))))
44	43	ex 412	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) → ((𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)) → (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))))
45	44	eximdv 1919	. . . . . 6 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) → (∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))))
46	32, 45	mpd 15	. . . . 5 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))))
47	46	ex 412	. . . 4 ⊢ (𝜑 → ((𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))))
48	47	eximdv 1919	. . 3 ⊢ (𝜑 → (∃𝑝(𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))) → ∃𝑝∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))))
49	18, 48	mpd 15	. 2 ⊢ (𝜑 → ∃𝑝∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))))
50		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑡 𝑑 ∈ ℝ+
51		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑡 𝑑 < 1
52		nfra1 3262	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1)
53		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑝‘𝑍) = 0
54		nfra1 3262	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)
55	52, 53, 54	nf3an 1903	. . . . . . . 8 ⊢ Ⅎ𝑡(∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))
56	22, 55	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑡(𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))
57	50, 51, 56	nf3an 1903	. . . . . 6 ⊢ Ⅎ𝑡(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))
58	2, 57	nfan 1901	. . . . 5 ⊢ Ⅎ𝑡(𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))))
59		nfcv 2899	. . . . 5 ⊢ Ⅎ𝑡𝑝
60		eqid 2737	. . . . 5 ⊢ {𝑡 ∈ 𝑇 ∣ (𝑝‘𝑡) < (𝑑 / 2)} = {𝑡 ∈ 𝑇 ∣ (𝑝‘𝑡) < (𝑑 / 2)}
61	7	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) → 𝐴 ⊆ 𝐶)
62	8	3adant1r 1179	. . . . 5 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
63	9	3adant1r 1179	. . . . 5 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
64	10	adantlr 716	. . . . 5 ⊢ (((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) ∧ 𝑦 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑦) ∈ 𝐴)
65		simpr1 1196	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) → 𝑑 ∈ ℝ+)
66		simpr2 1197	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) → 𝑑 < 1)
67	12	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) → 𝑈 ∈ 𝐽)
68	13	adantr 480	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) → 𝑍 ∈ 𝑈)
69		simpr3l 1236	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) → 𝑝 ∈ 𝐴)
70		simp3r1 1283	. . . . . 6 ⊢ ((𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1))
71	70	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1))
72		simp3r2 1284	. . . . . 6 ⊢ ((𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))) → (𝑝‘𝑍) = 0)
73	72	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) → (𝑝‘𝑍) = 0)
74		simp3r3 1285	. . . . . 6 ⊢ ((𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))
75	74	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))
76	1, 58, 59, 3, 60, 5, 6, 61, 62, 63, 64, 65, 66, 67, 68, 69, 71, 73, 75	stoweidlem52 46502	. . . 4 ⊢ ((𝜑 ∧ (𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))))) → ∃𝑣 ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡))))
77	76	ex 412	. . 3 ⊢ (𝜑 → ((𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))) → ∃𝑣 ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)))))
78	77	exlimdvv 1936	. 2 ⊢ (𝜑 → (∃𝑝∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)))) → ∃𝑣 ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)))))
79	49, 78	mpd 15	1 ⊢ (𝜑 → ∃𝑣 ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃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 1542  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2884   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390   ∖ cdif 3887   ⊆ wss 3890  ∪ cuni 4851   class class class wbr 5086   ↦ cmpt 5167  ran crn 5627  ‘cfv 6494  (class class class)co 7362  ℝcr 11032  0cc0 11033  1c1 11034   + caddc 11036   · cmul 11038   < clt 11174   ≤ cle 11175   − cmin 11372   / cdiv 11802  2c2 12231  ℝ⁺crp 12937  (,)cioo 13293  topGenctg 17395   Cn ccn 23203  Compccmp 23365

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 2709  ax-rep 5213  ax-sep 5232  ax-nul 5242  ax-pow 5304  ax-pr 5372  ax-un 7684  ax-inf2 9557  ax-cnex 11089  ax-resscn 11090  ax-1cn 11091  ax-icn 11092  ax-addcl 11093  ax-addrcl 11094  ax-mulcl 11095  ax-mulrcl 11096  ax-mulcom 11097  ax-addass 11098  ax-mulass 11099  ax-distr 11100  ax-i2m1 11101  ax-1ne0 11102  ax-1rid 11103  ax-rnegex 11104  ax-rrecex 11105  ax-cnre 11106  ax-pre-lttri 11107  ax-pre-lttrn 11108  ax-pre-ltadd 11109  ax-pre-mulgt0 11110  ax-pre-sup 11111

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 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rmo 3343  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-tp 4573  df-op 4575  df-uni 4852  df-int 4891  df-iun 4936  df-iin 4937  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5521  df-eprel 5526  df-po 5534  df-so 5535  df-fr 5579  df-se 5580  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639  df-pred 6261  df-ord 6322  df-on 6323  df-lim 6324  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-f 6498  df-f1 6499  df-fo 6500  df-f1o 6501  df-fv 6502  df-isom 6503  df-riota 7319  df-ov 7365  df-oprab 7366  df-mpo 7367  df-of 7626  df-om 7813  df-1st 7937  df-2nd 7938  df-supp 8106  df-frecs 8226  df-wrecs 8257  df-recs 8306  df-rdg 8344  df-1o 8400  df-2o 8401  df-er 8638  df-map 8770  df-pm 8771  df-ixp 8841  df-en 8889  df-dom 8890  df-sdom 8891  df-fin 8892  df-fsupp 9270  df-fi 9319  df-sup 9350  df-inf 9351  df-oi 9420  df-card 9858  df-pnf 11176  df-mnf 11177  df-xr 11178  df-ltxr 11179  df-le 11180  df-sub 11374  df-neg 11375  df-div 11803  df-nn 12170  df-2 12239  df-3 12240  df-4 12241  df-5 12242  df-6 12243  df-7 12244  df-8 12245  df-9 12246  df-n0 12433  df-z 12520  df-dec 12640  df-uz 12784  df-q 12894  df-rp 12938  df-xneg 13058  df-xadd 13059  df-xmul 13060  df-ioo 13297  df-ico 13299  df-icc 13300  df-fz 13457  df-fzo 13604  df-fl 13746  df-seq 13959  df-exp 14019  df-hash 14288  df-cj 15056  df-re 15057  df-im 15058  df-sqrt 15192  df-abs 15193  df-clim 15445  df-rlim 15446  df-sum 15644  df-struct 17112  df-sets 17129  df-slot 17147  df-ndx 17159  df-base 17175  df-ress 17196  df-plusg 17228  df-mulr 17229  df-starv 17230  df-sca 17231  df-vsca 17232  df-ip 17233  df-tset 17234  df-ple 17235  df-ds 17237  df-unif 17238  df-hom 17239  df-cco 17240  df-rest 17380  df-topn 17381  df-0g 17399  df-gsum 17400  df-topgen 17401  df-pt 17402  df-prds 17405  df-xrs 17461  df-qtop 17466  df-imas 17467  df-xps 17469  df-mre 17543  df-mrc 17544  df-acs 17546  df-mgm 18603  df-sgrp 18682  df-mnd 18698  df-submnd 18747  df-mulg 19039  df-cntz 19287  df-cmn 19752  df-psmet 21340  df-xmet 21341  df-met 21342  df-bl 21343  df-mopn 21344  df-cnfld 21349  df-top 22873  df-topon 22890  df-topsp 22912  df-bases 22925  df-cld 22998  df-cn 23206  df-cnp 23207  df-cmp 23366  df-tx 23541  df-hmeo 23734  df-xms 24299  df-ms 24300  df-tms 24301

This theorem is referenced by:  stoweidlem57  46507