Theorem stoweidlem28 46671

Description: There exists a δ as in Lemma 1 [BrosowskiDeutsh] p. 90: 0 < delta < 1 and p >= delta on 𝑇 ∖ 𝑈. Here 𝑑 is used to represent δ in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem28.1	⊢ Ⅎ𝑡𝑈
stoweidlem28.2	⊢ Ⅎ𝑡𝜑
stoweidlem28.3	⊢ 𝐾 = (topGen‘ran (,))
stoweidlem28.4	⊢ 𝑇 = ∪ 𝐽
stoweidlem28.5	⊢ (𝜑 → 𝐽 ∈ Comp)
stoweidlem28.6	⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
stoweidlem28.7	⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))
stoweidlem28.8	⊢ (𝜑 → 𝑈 ∈ 𝐽)

Assertion

Ref	Expression
stoweidlem28	⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))

Distinct variable groups:   𝑡,𝑑,𝑃   𝑇,𝑑,𝑡   𝑈,𝑑   𝑡,𝐽

Allowed substitution hints:   𝜑(𝑡,𝑑)   𝑈(𝑡)   𝐽(𝑑)   𝐾(𝑡,𝑑)

Proof of Theorem stoweidlem28

Dummy variables 𝑐 𝑥 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		halfre 12459	. . . . 5 ⊢ (1 / 2) ∈ ℝ
2		halfgt0 12461	. . . . 5 ⊢ 0 < (1 / 2)
3	1, 2	elrpii 13021	. . . 4 ⊢ (1 / 2) ∈ ℝ+
4	3	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) ∈ ℝ+)
5		halflt1 12463	. . . 4 ⊢ (1 / 2) < 1
6	5	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) < 1)
7		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑡𝑇
8		stoweidlem28.1	. . . . . . 7 ⊢ Ⅎ𝑡𝑈
9	7, 8	nfdif 4092	. . . . . 6 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈)
10	9	nfeq1 2946	. . . . 5 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈) = ∅
11	10	rzalf 45666	. . . 4 ⊢ ((𝑇 ∖ 𝑈) = ∅ → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡))
12	11	adantl 486	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡))
13		ovex 7446	. . . 4 ⊢ (1 / 2) ∈ V
14		eleq1 2857	. . . . 5 ⊢ (𝑑 = (1 / 2) → (𝑑 ∈ ℝ+ ↔ (1 / 2) ∈ ℝ+))
15		breq1 5116	. . . . 5 ⊢ (𝑑 = (1 / 2) → (𝑑 < 1 ↔ (1 / 2) < 1))
16		breq1 5116	. . . . . 6 ⊢ (𝑑 = (1 / 2) → (𝑑 ≤ (𝑃‘𝑡) ↔ (1 / 2) ≤ (𝑃‘𝑡)))
17	16	ralbidv 3194	. . . . 5 ⊢ (𝑑 = (1 / 2) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡)))
18	14, 15, 17	3anbi123d 1462	. . . 4 ⊢ (𝑑 = (1 / 2) → ((𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)) ↔ ((1 / 2) ∈ ℝ+ ∧ (1 / 2) < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡))))
19	13, 18	spcev 3574	. . 3 ⊢ (((1 / 2) ∈ ℝ+ ∧ (1 / 2) < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
20	4, 6, 12, 19	syl3anc 1396	. 2 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
21		simplll 786	. . . 4 ⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → 𝜑)
22		simplr 780	. . . 4 ⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → 𝑥 ∈ (𝑇 ∖ 𝑈))
23		simpr 489	. . . 4 ⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
24		stoweidlem28.3	. . . . . . . . . . 11 ⊢ 𝐾 = (topGen‘ran (,))
25		stoweidlem28.4	. . . . . . . . . . 11 ⊢ 𝑇 = ∪ 𝐽
26		eqid 2769	. . . . . . . . . . 11 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
27		stoweidlem28.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
28	24, 25, 26, 27	fcnre 45674	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
29	28	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑃:𝑇⟶ℝ)
30		eldifi 4093	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ 𝑈) → 𝑥 ∈ 𝑇)
31	30	adantl 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ 𝑇)
32	29, 31	ffvelcdmd 7083	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ)
33		stoweidlem28.7	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))
34		nfcv 2931	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑇 ∖ 𝑈)
35		nfv 1941	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥0 < (𝑃‘𝑡)
36		nfv 1941	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡0 < (𝑃‘𝑥)
37		fveq2 6884	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (𝑃‘𝑡) = (𝑃‘𝑥))
38	37	breq2d 5125	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (0 < (𝑃‘𝑡) ↔ 0 < (𝑃‘𝑥)))
39	9, 34, 35, 36, 38	cbvralfw 3311	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) ↔ ∀𝑥 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑥))
40	39	biimpi 219	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) → ∀𝑥 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑥))
41	40	r19.21bi 3263	. . . . . . . . 9 ⊢ ((∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑥))
42	33, 41	sylan 591	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑥))
43	32, 42	elrpd 13059	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ+)
44	43	3adant3 1148	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → (𝑃‘𝑥) ∈ ℝ+)
45		stoweidlem28.2	. . . . . . . 8 ⊢ Ⅎ𝑡𝜑
46	9	nfcri 2923	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑥 ∈ (𝑇 ∖ 𝑈)
47		nfra1 3295	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)
48	45, 46, 47	nf3an 1928	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
49		rspa 3260	. . . . . . . . . 10 ⊢ ((∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
50	49	3ad2antl3 1204	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
51		simpl2 1209	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ (𝑇 ∖ 𝑈))
52		fvres 6903	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ 𝑈) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) = (𝑃‘𝑥))
53	51, 52	syl 18	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) = (𝑃‘𝑥))
54		fvres 6903	. . . . . . . . . 10 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) = (𝑃‘𝑡))
55	54	adantl 486	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) = (𝑃‘𝑡))
56	50, 53, 55	3brtr3d 5146	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ≤ (𝑃‘𝑡))
57	56	ex 417	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → (𝑡 ∈ (𝑇 ∖ 𝑈) → (𝑃‘𝑥) ≤ (𝑃‘𝑡)))
58	48, 57	ralrimi 3269	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡))
59		eleq1 2857	. . . . . . . . 9 ⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ∈ ℝ+ ↔ (𝑃‘𝑥) ∈ ℝ+))
60		breq1 5116	. . . . . . . . . 10 ⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ≤ (𝑃‘𝑡) ↔ (𝑃‘𝑥) ≤ (𝑃‘𝑡)))
61	60	ralbidv 3194	. . . . . . . . 9 ⊢ (𝑐 = (𝑃‘𝑥) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡)))
62	59, 61	anbi12d 643	. . . . . . . 8 ⊢ (𝑐 = (𝑃‘𝑥) → ((𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) ↔ ((𝑃‘𝑥) ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡))))
63	62	spcegv 3565	. . . . . . 7 ⊢ ((𝑃‘𝑥) ∈ ℝ+ → (((𝑃‘𝑥) ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡)) → ∃𝑐(𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))))
64	44, 63	syl 18	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → (((𝑃‘𝑥) ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡)) → ∃𝑐(𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))))
65	44, 58, 64	mp2and 711	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → ∃𝑐(𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)))
66		simpl1 1208	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))) → 𝜑)
67		simprl 782	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))) → 𝑐 ∈ ℝ+)
68		simprr 784	. . . . . 6 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
69		nfv 1941	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑐 ∈ ℝ+
70		nfra1 3295	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)
71	45, 69, 70	nf3an 1928	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
72		eqid 2769	. . . . . . 7 ⊢ if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2)) = if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2))
73	28	3ad2ant1 1149	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑃:𝑇⟶ℝ)
74		difssd 4099	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → (𝑇 ∖ 𝑈) ⊆ 𝑇)
75		simp2 1153	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑐 ∈ ℝ+)
76		simp3 1154	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
77	71, 72, 73, 74, 75, 76	stoweidlem5 46648	. . . . . 6 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
78	66, 67, 68, 77	syl3anc 1396	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ (𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
79	65, 78	exlimddv 1962	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
80	21, 22, 23, 79	syl3anc 1396	. . 3 ⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
81		eqid 2769	. . . . . 6 ⊢ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
82		stoweidlem28.5	. . . . . . . 8 ⊢ (𝜑 → 𝐽 ∈ Comp)
83		stoweidlem28.8	. . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
84		cmptop 23523	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
85	82, 84	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ Top)
86		elssuni 4908	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽)
87	83, 86	syl 18	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ ∪ 𝐽)
88	87, 25	sseqtrrdi 3986	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑇)
89	25	isopn2 23160	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑈 ⊆ 𝑇) → (𝑈 ∈ 𝐽 ↔ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)))
90	85, 88, 89	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∈ 𝐽 ↔ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)))
91	83, 90	mpbid 235	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽))
92		cmpcld 23530	. . . . . . . 8 ⊢ ((𝐽 ∈ Comp ∧ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)) → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp)
93	82, 91, 92	syl2anc 595	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp)
94	93	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp)
95	27	adantr 485	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → 𝑃 ∈ (𝐽 Cn 𝐾))
96		difssd 4099	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑇 ∖ 𝑈) ⊆ 𝑇)
97	25	cnrest 23413	. . . . . . 7 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ (𝑇 ∖ 𝑈) ⊆ 𝑇) → (𝑃 ↾ (𝑇 ∖ 𝑈)) ∈ ((𝐽 ↾t (𝑇 ∖ 𝑈)) Cn 𝐾))
98	95, 96, 97	syl2anc 595	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑃 ↾ (𝑇 ∖ 𝑈)) ∈ ((𝐽 ↾t (𝑇 ∖ 𝑈)) Cn 𝐾))
99		difssd 4099	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ 𝑇)
100	25	restuni 23290	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑇 ∖ 𝑈) ⊆ 𝑇) → (𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)))
101	85, 99, 100	syl2anc 595	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)))
102	101	neeq1d 3023	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∖ 𝑈) ≠ ∅ ↔ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅))
103		df-ne 2965	. . . . . . . 8 ⊢ ((𝑇 ∖ 𝑈) ≠ ∅ ↔ ¬ (𝑇 ∖ 𝑈) = ∅)
104	102, 103	bitr3di 289	. . . . . . 7 ⊢ (𝜑 → (∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅ ↔ ¬ (𝑇 ∖ 𝑈) = ∅))
105	104	biimpar 482	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅)
106	81, 24, 94, 98, 105	evth2 25090	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑠 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠))
107		nfcv 2931	. . . . . . 7 ⊢ Ⅎ𝑠∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
108		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑡𝐽
109		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑡 ↾t
110	108, 109, 9	nfov 7443	. . . . . . . 8 ⊢ Ⅎ𝑡(𝐽 ↾t (𝑇 ∖ 𝑈))
111	110	nfuni 4883	. . . . . . 7 ⊢ Ⅎ𝑡∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
112		nfcv 2931	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑃
113	112, 9	nfres 5983	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑃 ↾ (𝑇 ∖ 𝑈))
114		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑥
115	113, 114	nffv 6894	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥)
116		nfcv 2931	. . . . . . . 8 ⊢ Ⅎ𝑡 ≤
117		nfcv 2931	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑠
118	113, 117	nffv 6894	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)
119	115, 116, 118	nfbr 5162	. . . . . . 7 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)
120		nfv 1941	. . . . . . 7 ⊢ Ⅎ𝑠((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)
121		fveq2 6884	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) = ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
122	121	breq2d 5125	. . . . . . 7 ⊢ (𝑠 = 𝑡 → (((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) ↔ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)))
123	107, 111, 119, 120, 122	cbvralfw 3311	. . . . . 6 ⊢ (∀𝑠 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) ↔ ∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
124	123	rexbii 3118	. . . . 5 ⊢ (∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑠 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) ↔ ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
125	106, 124	sylib 221	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
126	9, 111	raleqf 3352	. . . . . . 7 ⊢ ((𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) ↔ ∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)))
127	126	rexeqbi1dv 3340	. . . . . 6 ⊢ ((𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) → (∃𝑥 ∈ (𝑇 ∖ 𝑈)∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) ↔ ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)))
128	101, 127	syl 18	. . . . 5 ⊢ (𝜑 → (∃𝑥 ∈ (𝑇 ∖ 𝑈)∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) ↔ ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)))
129	128	adantr 485	. . . 4 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (∃𝑥 ∈ (𝑇 ∖ 𝑈)∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) ↔ ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑡 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)))
130	125, 129	mpbird 260	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ (𝑇 ∖ 𝑈)∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
131	80, 130	r19.29a 3179	. 2 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
132	20, 131	pm2.61dan 824	1 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567  ∃wex 1806  Ⅎwnf 1810   ∈ wcel 2149  Ⅎwnfc 2916   ≠ wne 2964  ∀wral 3085  ∃wrex 3095   ∖ cdif 3910   ⊆ wss 3913  ∅c0 4294  ifcif 4492  ∪ cuni 4876   class class class wbr 5113  ran crn 5665   ↾ cres 5666  ⟶wf 6535  ‘cfv 6539  (class class class)co 7413  ℝcr 11101  0cc0 11102  1c1 11103   < clt 11245   ≤ cle 11246   / cdiv 11873  2c2 12297  ℝ⁺crp 13018  (,)cioo 13374   ↾_t crest 17475  topGenctg 17492  Topctop 23021  Clsdccld 23144   Cn ccn 23352  Compccmp 23514

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 5273  ax-pow 5339  ax-pr 5407  ax-un 7735  ax-cnex 11158  ax-resscn 11159  ax-1cn 11160  ax-icn 11161  ax-addcl 11162  ax-addrcl 11163  ax-mulcl 11164  ax-mulrcl 11165  ax-mulcom 11166  ax-addass 11167  ax-mulass 11168  ax-distr 11169  ax-i2m1 11170  ax-1ne0 11171  ax-1rid 11172  ax-rnegex 11173  ax-rrecex 11174  ax-cnre 11175  ax-pre-lttri 11176  ax-pre-lttrn 11177  ax-pre-ltadd 11178  ax-pre-mulgt0 11179  ax-pre-sup 11180

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 5559  df-eprel 5564  df-po 5572  df-so 5573  df-fr 5617  df-se 5618  df-we 5619  df-xp 5670  df-rel 5671  df-cnv 5672  df-co 5673  df-dm 5674  df-rn 5675  df-res 5676  df-ima 5677  df-pred 6305  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6495  df-fun 6541  df-fn 6542  df-f 6543  df-f1 6544  df-fo 6545  df-f1o 6546  df-fv 6547  df-isom 6548  df-riota 7370  df-ov 7416  df-oprab 7417  df-mpo 7418  df-of 7677  df-om 7865  df-1st 7988  df-2nd 7989  df-supp 8159  df-frecs 8280  df-wrecs 8311  df-recs 8360  df-rdg 8399  df-1o 8455  df-2o 8456  df-er 8696  df-map 8828  df-ixp 8898  df-en 8946  df-dom 8947  df-sdom 8948  df-fin 8949  df-fsupp 9324  df-fi 9373  df-sup 9404  df-inf 9405  df-oi 9474  df-card 9927  df-pnf 11247  df-mnf 11248  df-xr 11249  df-ltxr 11250  df-le 11251  df-sub 11445  df-neg 11446  df-div 11874  df-nn 12236  df-2 12305  df-3 12306  df-4 12307  df-5 12308  df-6 12309  df-7 12310  df-8 12311  df-9 12312  df-n0 12507  df-z 12594  df-dec 12714  df-uz 12865  df-q 12975  df-rp 13019  df-xneg 13139  df-xadd 13140  df-xmul 13141  df-ioo 13378  df-icc 13381  df-fz 13538  df-fzo 13685  df-seq 14040  df-exp 14100  df-hash 14369  df-cj 15152  df-re 15153  df-im 15154  df-sqrt 15288  df-abs 15289  df-struct 17209  df-sets 17226  df-slot 17244  df-ndx 17256  df-base 17272  df-ress 17293  df-plusg 17325  df-mulr 17326  df-starv 17327  df-sca 17328  df-vsca 17329  df-ip 17330  df-tset 17331  df-ple 17332  df-ds 17334  df-unif 17335  df-hom 17336  df-cco 17337  df-rest 17477  df-topn 17478  df-0g 17496  df-gsum 17497  df-topgen 17498  df-pt 17499  df-prds 17502  df-xrs 17558  df-qtop 17563  df-imas 17564  df-xps 17566  df-mre 17640  df-mrc 17641  df-acs 17643  df-mgm 18700  df-sgrp 18779  df-mnd 18795  df-submnd 18844  df-mulg 19136  df-cntz 19389  df-cmn 19854  df-psmet 21485  df-xmet 21486  df-met 21487  df-bl 21488  df-mopn 21489  df-cnfld 21494  df-top 23022  df-topon 23039  df-topsp 23061  df-bases 23074  df-cld 23147  df-cn 23355  df-cnp 23356  df-cmp 23515  df-tx 23690  df-hmeo 23883  df-xms 24448  df-ms 24449  df-tms 24450

This theorem is referenced by:  stoweidlem56  46699