Theorem stoweidlem56 43487

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:   𝑒,𝑓,𝑔,𝑡,𝐴   𝑣,𝑒,𝑥,𝑡,𝐴   𝑦,𝑒,𝑓,𝑡,𝐴   𝑔,𝐽,𝑡   𝑇,𝑒,𝑓,𝑔,𝑡   𝑈,𝑒,𝑓,𝑔   𝑒,𝑍,𝑓,𝑔,𝑡   𝜑,𝑒,𝑓,𝑔   𝑓,𝑞,𝑔,𝑡,𝐴,𝑟   𝑦,𝑞,𝑇   𝑈,𝑞,𝑦   𝑍,𝑞,𝑦   𝜑,𝑞,𝑦,𝑟   𝑇,𝑟   𝑈,𝑟   𝜑,𝑟   𝑡,𝐾   𝑣,𝐽   𝑣,𝑇,𝑥   𝑣,𝑈,𝑥   𝑣,𝑍

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 2738	. . . . 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 2738	. . . . 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 43486	. . . 4 ⊢ (𝜑 → ∃𝑝 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)))
17		df-rex 3069	. . . 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 217	. . 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 767	. . . . . . 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 1221	. . . . . . 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 1918	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑝 ∈ 𝐴
23		nfra1 3142	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)
24	2, 22, 23	nf3an 1905	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))
25	4	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → 𝐽 ∈ Comp)
26	7	sselda 3917	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ 𝐶)
27	26, 6	eleqtrdi 2849	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴) → 𝑝 ∈ (𝐽 Cn 𝐾))
28	27	3adant3 1130	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → 𝑝 ∈ (𝐽 Cn 𝐾))
29		simp3 1136	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡))
30	12	3ad2ant1 1131	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑝 ∈ 𝐴 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑝‘𝑡)) → 𝑈 ∈ 𝐽)
31	1, 24, 3, 5, 25, 28, 29, 30	stoweidlem28 43459	. . . . . . 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 1369	. . . . . 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 1192	. . . . . . . . 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 1193	. . . . . . . . 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 773	. . . . . . . . . 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 1219	. . . . . . . . . . . 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 1220	. . . . . . . . . . . 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 1194	. . . . . . . . . . 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 1126	. . . . . . . . . 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 1126	. . . . . . . 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 1921	. . . . . 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 1921	. . 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 1918	. . . . . . 7 ⊢ Ⅎ𝑡 𝑑 ∈ ℝ+
51		nfv 1918	. . . . . . 7 ⊢ Ⅎ𝑡 𝑑 < 1
52		nfra1 3142	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1)
53		nfv 1918	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑝‘𝑍) = 0
54		nfra1 3142	. . . . . . . . 9 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡)
55	52, 53, 54	nf3an 1905	. . . . . . . 8 ⊢ Ⅎ𝑡(∀𝑡 ∈ 𝑇 (0 ≤ (𝑝‘𝑡) ∧ (𝑝‘𝑡) ≤ 1) ∧ (𝑝‘𝑍) = 0 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑝‘𝑡))
56	22, 55	nfan 1903	. . . . . . 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 1905	. . . . . 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 1903	. . . . 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 2906	. . . . 5 ⊢ Ⅎ𝑡𝑝
60		eqid 2738	. . . . 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 1175	. . . . 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 1175	. . . . 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 711	. . . . 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 1192	. . . . 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 1193	. . . . 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 1232	. . . . 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 1279	. . . . . 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 1280	. . . . . 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 1281	. . . . . 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 43483	. . . 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 1938	. 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 1085   = wceq 1539  ∃wex 1783  Ⅎwnf 1787   ∈ wcel 2108  Ⅎwnfc 2886   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067   ∖ cdif 3880   ⊆ wss 3883  ∪ cuni 4836   class class class wbr 5070   ↦ cmpt 5153  ran crn 5581  ‘cfv 6418  (class class class)co 7255  ℝcr 10801  0cc0 10802  1c1 10803   + caddc 10805   · cmul 10807   < clt 10940   ≤ cle 10941   − cmin 11135   / cdiv 11562  2c2 11958  ℝ⁺crp 12659  (,)cioo 13008  topGenctg 17065   Cn ccn 22283  Compccmp 22445

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-rep 5205  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566  ax-inf2 9329  ax-cnex 10858  ax-resscn 10859  ax-1cn 10860  ax-icn 10861  ax-addcl 10862  ax-addrcl 10863  ax-mulcl 10864  ax-mulrcl 10865  ax-mulcom 10866  ax-addass 10867  ax-mulass 10868  ax-distr 10869  ax-i2m1 10870  ax-1ne0 10871  ax-1rid 10872  ax-rnegex 10873  ax-rrecex 10874  ax-cnre 10875  ax-pre-lttri 10876  ax-pre-lttrn 10877  ax-pre-ltadd 10878  ax-pre-mulgt0 10879  ax-pre-sup 10880  ax-mulf 10882

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3068  df-rex 3069  df-reu 3070  df-rmo 3071  df-rab 3072  df-v 3424  df-sbc 3712  df-csb 3829  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-int 4877  df-iun 4923  df-iin 4924  df-br 5071  df-opab 5133  df-mpt 5154  df-tr 5188  df-id 5480  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-se 5536  df-we 5537  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-pred 6191  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-isom 6427  df-riota 7212  df-ov 7258  df-oprab 7259  df-mpo 7260  df-of 7511  df-om 7688  df-1st 7804  df-2nd 7805  df-supp 7949  df-frecs 8068  df-wrecs 8099  df-recs 8173  df-rdg 8212  df-1o 8267  df-2o 8268  df-er 8456  df-map 8575  df-pm 8576  df-ixp 8644  df-en 8692  df-dom 8693  df-sdom 8694  df-fin 8695  df-fsupp 9059  df-fi 9100  df-sup 9131  df-inf 9132  df-oi 9199  df-card 9628  df-pnf 10942  df-mnf 10943  df-xr 10944  df-ltxr 10945  df-le 10946  df-sub 11137  df-neg 11138  df-div 11563  df-nn 11904  df-2 11966  df-3 11967  df-4 11968  df-5 11969  df-6 11970  df-7 11971  df-8 11972  df-9 11973  df-n0 12164  df-z 12250  df-dec 12367  df-uz 12512  df-q 12618  df-rp 12660  df-xneg 12777  df-xadd 12778  df-xmul 12779  df-ioo 13012  df-ico 13014  df-icc 13015  df-fz 13169  df-fzo 13312  df-fl 13440  df-seq 13650  df-exp 13711  df-hash 13973  df-cj 14738  df-re 14739  df-im 14740  df-sqrt 14874  df-abs 14875  df-clim 15125  df-rlim 15126  df-sum 15326  df-struct 16776  df-sets 16793  df-slot 16811  df-ndx 16823  df-base 16841  df-ress 16868  df-plusg 16901  df-mulr 16902  df-starv 16903  df-sca 16904  df-vsca 16905  df-ip 16906  df-tset 16907  df-ple 16908  df-ds 16910  df-unif 16911  df-hom 16912  df-cco 16913  df-rest 17050  df-topn 17051  df-0g 17069  df-gsum 17070  df-topgen 17071  df-pt 17072  df-prds 17075  df-xrs 17130  df-qtop 17135  df-imas 17136  df-xps 17138  df-mre 17212  df-mrc 17213  df-acs 17215  df-mgm 18241  df-sgrp 18290  df-mnd 18301  df-submnd 18346  df-mulg 18616  df-cntz 18838  df-cmn 19303  df-psmet 20502  df-xmet 20503  df-met 20504  df-bl 20505  df-mopn 20506  df-cnfld 20511  df-top 21951  df-topon 21968  df-topsp 21990  df-bases 22004  df-cld 22078  df-cn 22286  df-cnp 22287  df-cmp 22446  df-tx 22621  df-hmeo 22814  df-xms 23381  df-ms 23382  df-tms 23383

This theorem is referenced by:  stoweidlem57  43488