Theorem stoweidlem28 45984

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 12478	. . . . 5 ⊢ (1 / 2) ∈ ℝ
2		halfgt0 12480	. . . . 5 ⊢ 0 < (1 / 2)
3	1, 2	elrpii 13035	. . . 4 ⊢ (1 / 2) ∈ ℝ+
4	3	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) ∈ ℝ+)
5		halflt1 12482	. . . 4 ⊢ (1 / 2) < 1
6	5	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) < 1)
7		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑡𝑇
8		stoweidlem28.1	. . . . . . 7 ⊢ Ⅎ𝑡𝑈
9	7, 8	nfdif 4139	. . . . . 6 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈)
10	9	nfeq1 2919	. . . . 5 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈) = ∅
11	10	rzalf 44955	. . . 4 ⊢ ((𝑇 ∖ 𝑈) = ∅ → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡))
12	11	adantl 481	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡))
13		ovex 7464	. . . 4 ⊢ (1 / 2) ∈ V
14		eleq1 2827	. . . . 5 ⊢ (𝑑 = (1 / 2) → (𝑑 ∈ ℝ+ ↔ (1 / 2) ∈ ℝ+))
15		breq1 5151	. . . . 5 ⊢ (𝑑 = (1 / 2) → (𝑑 < 1 ↔ (1 / 2) < 1))
16		breq1 5151	. . . . . 6 ⊢ (𝑑 = (1 / 2) → (𝑑 ≤ (𝑃‘𝑡) ↔ (1 / 2) ≤ (𝑃‘𝑡)))
17	16	ralbidv 3176	. . . . 5 ⊢ (𝑑 = (1 / 2) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡)))
18	14, 15, 17	3anbi123d 1435	. . . 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 3606	. . 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 1370	. 2 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
21		simplll 775	. . . 4 ⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → 𝜑)
22		simplr 769	. . . 4 ⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → 𝑥 ∈ (𝑇 ∖ 𝑈))
23		simpr 484	. . . 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 2735	. . . . . . . . . . 11 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
27		stoweidlem28.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
28	24, 25, 26, 27	fcnre 44963	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
29	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑃:𝑇⟶ℝ)
30		eldifi 4141	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ 𝑈) → 𝑥 ∈ 𝑇)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ 𝑇)
32	29, 31	ffvelcdmd 7105	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ)
33		stoweidlem28.7	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))
34		nfcv 2903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑇 ∖ 𝑈)
35		nfv 1912	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥0 < (𝑃‘𝑡)
36		nfv 1912	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡0 < (𝑃‘𝑥)
37		fveq2 6907	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (𝑃‘𝑡) = (𝑃‘𝑥))
38	37	breq2d 5160	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (0 < (𝑃‘𝑡) ↔ 0 < (𝑃‘𝑥)))
39	9, 34, 35, 36, 38	cbvralfw 3302	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) ↔ ∀𝑥 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑥))
40	39	biimpi 216	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) → ∀𝑥 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑥))
41	40	r19.21bi 3249	. . . . . . . . 9 ⊢ ((∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑥))
42	33, 41	sylan 580	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑥))
43	32, 42	elrpd 13072	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ+)
44	43	3adant3 1131	. . . . . 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 2895	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑥 ∈ (𝑇 ∖ 𝑈)
47		nfra1 3282	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)
48	45, 46, 47	nf3an 1899	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
49		rspa 3246	. . . . . . . . . 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 1186	. . . . . . . . 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 1191	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ (𝑇 ∖ 𝑈))
52		fvres 6926	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ 𝑈) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) = (𝑃‘𝑥))
53	51, 52	syl 17	. . . . . . . . 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 6926	. . . . . . . . . 10 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) = (𝑃‘𝑡))
55	54	adantl 481	. . . . . . . . 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 5179	. . . . . . . 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 412	. . . . . . 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 3255	. . . . . 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 2827	. . . . . . . . 9 ⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ∈ ℝ+ ↔ (𝑃‘𝑥) ∈ ℝ+))
60		breq1 5151	. . . . . . . . . 10 ⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ≤ (𝑃‘𝑡) ↔ (𝑃‘𝑥) ≤ (𝑃‘𝑡)))
61	60	ralbidv 3176	. . . . . . . . 9 ⊢ (𝑐 = (𝑃‘𝑥) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡)))
62	59, 61	anbi12d 632	. . . . . . . 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 3597	. . . . . . 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 17	. . . . . 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 699	. . . . 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 1190	. . . . . 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 771	. . . . . 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 773	. . . . . 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 1912	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑐 ∈ ℝ+
70		nfra1 3282	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)
71	45, 69, 70	nf3an 1899	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
72		eqid 2735	. . . . . . 7 ⊢ if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2)) = if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2))
73	28	3ad2ant1 1132	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑃:𝑇⟶ℝ)
74		difssd 4147	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → (𝑇 ∖ 𝑈) ⊆ 𝑇)
75		simp2 1136	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑐 ∈ ℝ+)
76		simp3 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
77	71, 72, 73, 74, 75, 76	stoweidlem5 45961	. . . . . 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 1370	. . . . 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 1933	. . . 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 1370	. . 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 2735	. . . . . 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 23419	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
85	82, 84	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ Top)
86		elssuni 4942	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽)
87	83, 86	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ ∪ 𝐽)
88	87, 25	sseqtrrdi 4047	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑇)
89	25	isopn2 23056	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑈 ⊆ 𝑇) → (𝑈 ∈ 𝐽 ↔ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)))
90	85, 88, 89	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∈ 𝐽 ↔ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)))
91	83, 90	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽))
92		cmpcld 23426	. . . . . . . 8 ⊢ ((𝐽 ∈ Comp ∧ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)) → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp)
93	82, 91, 92	syl2anc 584	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp)
94	93	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp)
95	27	adantr 480	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → 𝑃 ∈ (𝐽 Cn 𝐾))
96		difssd 4147	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑇 ∖ 𝑈) ⊆ 𝑇)
97	25	cnrest 23309	. . . . . . 7 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ (𝑇 ∖ 𝑈) ⊆ 𝑇) → (𝑃 ↾ (𝑇 ∖ 𝑈)) ∈ ((𝐽 ↾t (𝑇 ∖ 𝑈)) Cn 𝐾))
98	95, 96, 97	syl2anc 584	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑃 ↾ (𝑇 ∖ 𝑈)) ∈ ((𝐽 ↾t (𝑇 ∖ 𝑈)) Cn 𝐾))
99		difssd 4147	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ 𝑇)
100	25	restuni 23186	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑇 ∖ 𝑈) ⊆ 𝑇) → (𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)))
101	85, 99, 100	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)))
102	101	neeq1d 2998	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∖ 𝑈) ≠ ∅ ↔ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅))
103		df-ne 2939	. . . . . . . 8 ⊢ ((𝑇 ∖ 𝑈) ≠ ∅ ↔ ¬ (𝑇 ∖ 𝑈) = ∅)
104	102, 103	bitr3di 286	. . . . . . 7 ⊢ (𝜑 → (∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅ ↔ ¬ (𝑇 ∖ 𝑈) = ∅))
105	104	biimpar 477	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅)
106	81, 24, 94, 98, 105	evth2 25006	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑠 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠))
107		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑠∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
108		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑡𝐽
109		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑡 ↾t
110	108, 109, 9	nfov 7461	. . . . . . . 8 ⊢ Ⅎ𝑡(𝐽 ↾t (𝑇 ∖ 𝑈))
111	110	nfuni 4919	. . . . . . 7 ⊢ Ⅎ𝑡∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
112		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑃
113	112, 9	nfres 6002	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑃 ↾ (𝑇 ∖ 𝑈))
114		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑥
115	113, 114	nffv 6917	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥)
116		nfcv 2903	. . . . . . . 8 ⊢ Ⅎ𝑡 ≤
117		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑠
118	113, 117	nffv 6917	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)
119	115, 116, 118	nfbr 5195	. . . . . . 7 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)
120		nfv 1912	. . . . . . 7 ⊢ Ⅎ𝑠((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)
121		fveq2 6907	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) = ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
122	121	breq2d 5160	. . . . . . 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 3302	. . . . . 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 3092	. . . . 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 218	. . . 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 3351	. . . . . . 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 3337	. . . . . 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 17	. . . . 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 480	. . . 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 257	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ (𝑇 ∖ 𝑈)∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
131	80, 130	r19.29a 3160	. 2 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
132	20, 131	pm2.61dan 813	1 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776  Ⅎwnf 1780   ∈ wcel 2106  Ⅎwnfc 2888   ≠ wne 2938  ∀wral 3059  ∃wrex 3068   ∖ cdif 3960   ⊆ wss 3963  ∅c0 4339  ifcif 4531  ∪ cuni 4912   class class class wbr 5148  ran crn 5690   ↾ cres 5691  ⟶wf 6559  ‘cfv 6563  (class class class)co 7431  ℝcr 11152  0cc0 11153  1c1 11154   < clt 11293   ≤ cle 11294   / cdiv 11918  2c2 12319  ℝ⁺crp 13032  (,)cioo 13384   ↾_t crest 17467  topGenctg 17484  Topctop 22915  Clsdccld 23040   Cn ccn 23248  Compccmp 23410

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-1cn 11211  ax-icn 11212  ax-addcl 11213  ax-addrcl 11214  ax-mulcl 11215  ax-mulrcl 11216  ax-mulcom 11217  ax-addass 11218  ax-mulass 11219  ax-distr 11220  ax-i2m1 11221  ax-1ne0 11222  ax-1rid 11223  ax-rnegex 11224  ax-rrecex 11225  ax-cnre 11226  ax-pre-lttri 11227  ax-pre-lttrn 11228  ax-pre-ltadd 11229  ax-pre-mulgt0 11230  ax-pre-sup 11231

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-iin 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-se 5642  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-om 7888  df-1st 8013  df-2nd 8014  df-supp 8185  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449  df-1o 8505  df-2o 8506  df-er 8744  df-map 8867  df-ixp 8937  df-en 8985  df-dom 8986  df-sdom 8987  df-fin 8988  df-fsupp 9400  df-fi 9449  df-sup 9480  df-inf 9481  df-oi 9548  df-card 9977  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-sub 11492  df-neg 11493  df-div 11919  df-nn 12265  df-2 12327  df-3 12328  df-4 12329  df-5 12330  df-6 12331  df-7 12332  df-8 12333  df-9 12334  df-n0 12525  df-z 12612  df-dec 12732  df-uz 12877  df-q 12989  df-rp 13033  df-xneg 13152  df-xadd 13153  df-xmul 13154  df-ioo 13388  df-icc 13391  df-fz 13545  df-fzo 13692  df-seq 14040  df-exp 14100  df-hash 14367  df-cj 15135  df-re 15136  df-im 15137  df-sqrt 15271  df-abs 15272  df-struct 17181  df-sets 17198  df-slot 17216  df-ndx 17228  df-base 17246  df-ress 17275  df-plusg 17311  df-mulr 17312  df-starv 17313  df-sca 17314  df-vsca 17315  df-ip 17316  df-tset 17317  df-ple 17318  df-ds 17320  df-unif 17321  df-hom 17322  df-cco 17323  df-rest 17469  df-topn 17470  df-0g 17488  df-gsum 17489  df-topgen 17490  df-pt 17491  df-prds 17494  df-xrs 17549  df-qtop 17554  df-imas 17555  df-xps 17557  df-mre 17631  df-mrc 17632  df-acs 17634  df-mgm 18666  df-sgrp 18745  df-mnd 18761  df-submnd 18810  df-mulg 19099  df-cntz 19348  df-cmn 19815  df-psmet 21374  df-xmet 21375  df-met 21376  df-bl 21377  df-mopn 21378  df-cnfld 21383  df-top 22916  df-topon 22933  df-topsp 22955  df-bases 22969  df-cld 23043  df-cn 23251  df-cnp 23252  df-cmp 23411  df-tx 23586  df-hmeo 23779  df-xms 24346  df-ms 24347  df-tms 24348

This theorem is referenced by:  stoweidlem56  46012