Theorem stoweidlem56 46696

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 2769	. . . . 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 2769	. . . . 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 46695	. . . 4 ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))
17		df-rex 3096	. . . 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 221	. . 3 ⊢ (𝜑 → ∃𝑝(𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))))
19		simpl 487	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑝 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))) → 𝜑)
20		simprl 782	. . . . . . 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 1240	. . . . . . 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 1941	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑝 ∈ 𝐴
23		nfra1 3295	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)
24	2, 22, 23	nf3an 1928	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))
25	4	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → 𝐽 ∈ Comp)
26	7	sselda 3945	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐶)
27	26, 6	eleqtrdi 2879	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (𝐽 Cn 𝐾))
28	27	3adant3 1148	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → 𝑝 ∈ (𝐽 Cn 𝐾))
29		simp3 1154	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))
30	12	3ad2ant1 1149	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → 𝑈 ∈ 𝐽)
31	1, 24, 3, 5, 25, 28, 29, 30	stoweidlem28 46668	. . . . . . 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 1396	. . . . . 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 1211	. . . . . . . . 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 1212	. . . . . . . . 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 788	. . . . . . . . . 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 1238	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 1239	. . . . . . . . . . . 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 485	. . . . . . . . . . 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 1213	. . . . . . . . . . 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 1144	. . . . . . . . . 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 520	. . . . . . . . 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 1144	. . . . . . . 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 417	. . . . . . 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 1944	. . . . . 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 16	. . . . 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 417	. . . 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 1944	. . 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 16	. 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 1941	. . . . . . 7 ⊢ Ⅎ𝑡 𝑑 ∈ ℝ+
51		nfv 1941	. . . . . . 7 ⊢ Ⅎ𝑡 𝑑 < 1
52		nfra1 3295	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1)
53		nfv 1941	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑝‘𝑍) = 0
54		nfra1 3295	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)
55	52, 53, 54	nf3an 1928	. . . . . . . 8 ⊢ Ⅎ𝑡(∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))
56	22, 55	nfan 1926	. . . . . . 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 1928	. . . . . 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 1926	. . . . 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 2931	. . . . 5 ⊢ Ⅎ𝑡𝑝
60		eqid 2769	. . . . 5 ⊢ {𝑡 ∈ 𝑇 ∣ (𝑝‘𝑡) < (𝑑 / 2)} = {𝑡 ∈ 𝑇 ∣ (𝑝‘𝑡) < (𝑑 / 2)}
61	7	adantr 485	. . . . 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 1194	. . . . 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 1194	. . . . 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 727	. . . . 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 1211	. . . . 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 1212	. . . . 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 485	. . . . 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 485	. . . . 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 1251	. . . . 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 1298	. . . . . 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 486	. . . . 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 1299	. . . . . 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 486	. . . . 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 1300	. . . . . 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 486	. . . . 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 46692	. . . 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 417	. . 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 1961	. 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 16	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 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806  Ⅎwnf 1810   ∈ wcel 2149  Ⅎwnfc 2916   ≠ wne 2964  ∀wral 3085  ∃wrex 3095  {crab 3423   ∖ cdif 3910   ⊆ wss 3913  ∪ cuni 4876   class class class wbr 5113   ↦ cmpt 5196  ran crn 5663  ‘cfv 6537  (class class class)co 7411  ℝcr 11099  0cc0 11100  1c1 11101   + caddc 11103   · cmul 11105   < clt 11243   ≤ cle 11244   − cmin 11441   / cdiv 11871  2c2 12295  ℝ⁺crp 13016  (,)cioo 13372  topGenctg 17490   Cn ccn 23350  Compccmp 23512

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5242  ax-sep 5261  ax-nul 5271  ax-pow 5337  ax-pr 5405  ax-un 7733  ax-inf2 9610  ax-cnex 11156  ax-resscn 11157  ax-1cn 11158  ax-icn 11159  ax-addcl 11160  ax-addrcl 11161  ax-mulcl 11162  ax-mulrcl 11163  ax-mulcom 11164  ax-addass 11165  ax-mulass 11166  ax-distr 11167  ax-i2m1 11168  ax-1ne0 11169  ax-1rid 11170  ax-rnegex 11171  ax-rrecex 11172  ax-cnre 11173  ax-pre-lttri 11174  ax-pre-lttrn 11175  ax-pre-ltadd 11176  ax-pre-mulgt0 11177  ax-pre-sup 11178

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-nel 3071  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3933  df-nul 4295  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-tp 4599  df-op 4601  df-uni 4877  df-int 4917  df-iun 4962  df-iin 4963  df-br 5114  df-opab 5178  df-mpt 5197  df-tr 5223  df-id 5557  df-eprel 5562  df-po 5570  df-so 5571  df-fr 5615  df-se 5616  df-we 5617  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-pred 6303  df-ord 6364  df-on 6365  df-lim 6366  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-isom 6546  df-riota 7368  df-ov 7414  df-oprab 7415  df-mpo 7416  df-of 7675  df-om 7863  df-1st 7986  df-2nd 7987  df-supp 8157  df-frecs 8278  df-wrecs 8309  df-recs 8358  df-rdg 8397  df-1o 8453  df-2o 8454  df-er 8694  df-map 8826  df-pm 8827  df-ixp 8896  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-fsupp 9322  df-fi 9371  df-sup 9402  df-inf 9403  df-oi 9472  df-card 9925  df-pnf 11245  df-mnf 11246  df-xr 11247  df-ltxr 11248  df-le 11249  df-sub 11443  df-neg 11444  df-div 11872  df-nn 12234  df-2 12303  df-3 12304  df-4 12305  df-5 12306  df-6 12307  df-7 12308  df-8 12309  df-9 12310  df-n0 12505  df-z 12592  df-dec 12712  df-uz 12863  df-q 12973  df-rp 13017  df-xneg 13137  df-xadd 13138  df-xmul 13139  df-ioo 13376  df-ico 13378  df-icc 13379  df-fz 13536  df-fzo 13683  df-fl 13825  df-seq 14038  df-exp 14098  df-hash 14367  df-cj 15150  df-re 15151  df-im 15152  df-sqrt 15286  df-abs 15287  df-clim 15539  df-rlim 15540  df-sum 15738  df-struct 17207  df-sets 17224  df-slot 17242  df-ndx 17254  df-base 17270  df-ress 17291  df-plusg 17323  df-mulr 17324  df-starv 17325  df-sca 17326  df-vsca 17327  df-ip 17328  df-tset 17329  df-ple 17330  df-ds 17332  df-unif 17333  df-hom 17334  df-cco 17335  df-rest 17475  df-topn 17476  df-0g 17494  df-gsum 17495  df-topgen 17496  df-pt 17497  df-prds 17500  df-xrs 17556  df-qtop 17561  df-imas 17562  df-xps 17564  df-mre 17638  df-mrc 17639  df-acs 17641  df-mgm 18698  df-sgrp 18777  df-mnd 18793  df-submnd 18842  df-mulg 19134  df-cntz 19387  df-cmn 19852  df-psmet 21483  df-xmet 21484  df-met 21485  df-bl 21486  df-mopn 21487  df-cnfld 21492  df-top 23020  df-topon 23037  df-topsp 23059  df-bases 23072  df-cld 23145  df-cn 23353  df-cnp 23354  df-cmp 23513  df-tx 23688  df-hmeo 23881  df-xms 24446  df-ms 24447  df-tms 24448

This theorem is referenced by:  stoweidlem57  46697