Theorem stoweidlem28 40245

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

Step	Hyp	Ref	Expression
1		halfre 11246	. . . . 5 ⊢ (1 / 2) ∈ ℝ
2		halfgt0 11248	. . . . 5 ⊢ 0 < (1 / 2)
3	1, 2	elrpii 11835	. . . 4 ⊢ (1 / 2) ∈ ℝ+
4	3	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) ∈ ℝ+)
5		halflt1 11250	. . . 4 ⊢ (1 / 2) < 1
6	5	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) < 1)
7		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑡𝑇
8		stoweidlem28.1	. . . . . . 7 ⊢ Ⅎ𝑡𝑈
9	7, 8	nfdif 3731	. . . . . 6 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈)
10	9	nfeq1 2778	. . . . 5 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈) = ∅
11	10	rzalf 39176	. . . 4 ⊢ ((𝑇 ∖ 𝑈) = ∅ → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡))
12	11	adantl 482	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡))
13		ovex 6678	. . . 4 ⊢ (1 / 2) ∈ V
14		eleq1 2689	. . . . 5 ⊢ (𝑑 = (1 / 2) → (𝑑 ∈ ℝ+ ↔ (1 / 2) ∈ ℝ+))
15		breq1 4656	. . . . 5 ⊢ (𝑑 = (1 / 2) → (𝑑 < 1 ↔ (1 / 2) < 1))
16		breq1 4656	. . . . . 6 ⊢ (𝑑 = (1 / 2) → (𝑑 ≤ (𝑃‘𝑡) ↔ (1 / 2) ≤ (𝑃‘𝑡)))
17	16	ralbidv 2986	. . . . 5 ⊢ (𝑑 = (1 / 2) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡)))
18	14, 15, 17	3anbi123d 1399	. . . 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 3300	. . 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 1326	. 2 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
21		simplll 798	. . . 4 ⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → 𝜑)
22		simplr 792	. . . 4 ⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → 𝑥 ∈ (𝑇 ∖ 𝑈))
23		simpr 477	. . . 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 2622	. . . . . . . . . . 11 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
27		stoweidlem28.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
28	24, 25, 26, 27	fcnre 39184	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
29	28	adantr 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑃:𝑇⟶ℝ)
30		eldifi 3732	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ 𝑈) → 𝑥 ∈ 𝑇)
31	30	adantl 482	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ 𝑇)
32	29, 31	ffvelrnd 6360	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ)
33		stoweidlem28.7	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))
34		nfcv 2764	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑇 ∖ 𝑈)
35		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥0 < (𝑃‘𝑡)
36		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡0 < (𝑃‘𝑥)
37		fveq2 6191	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (𝑃‘𝑡) = (𝑃‘𝑥))
38	37	breq2d 4665	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (0 < (𝑃‘𝑡) ↔ 0 < (𝑃‘𝑥)))
39	9, 34, 35, 36, 38	cbvralf 3165	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) ↔ ∀𝑥 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑥))
40	39	biimpi 206	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) → ∀𝑥 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑥))
41	40	r19.21bi 2932	. . . . . . . . 9 ⊢ ((∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑥))
42	33, 41	sylan 488	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑥))
43	32, 42	elrpd 11869	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ+)
44	43	3adant3 1081	. . . . . 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 2758	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑥 ∈ (𝑇 ∖ 𝑈)
47		nfra1 2941	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)
48	45, 46, 47	nf3an 1831	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
49		rspa 2930	. . . . . . . . . 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 1225	. . . . . . . . 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 1065	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ (𝑇 ∖ 𝑈))
52		fvres 6207	. . . . . . . . . 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 6207	. . . . . . . . . 10 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡) = (𝑃‘𝑡))
55	54	adantl 482	. . . . . . . . 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 4684	. . . . . . . 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 450	. . . . . . 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 2957	. . . . . 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 2689	. . . . . . . . 9 ⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ∈ ℝ+ ↔ (𝑃‘𝑥) ∈ ℝ+))
60		breq1 4656	. . . . . . . . . 10 ⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ≤ (𝑃‘𝑡) ↔ (𝑃‘𝑥) ≤ (𝑃‘𝑡)))
61	60	ralbidv 2986	. . . . . . . . 9 ⊢ (𝑐 = (𝑃‘𝑥) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡)))
62	59, 61	anbi12d 747	. . . . . . . 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 3294	. . . . . . 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 715	. . . . 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 1064	. . . . . 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 794	. . . . . 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 796	. . . . . 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 1843	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑐 ∈ ℝ+
70		nfra1 2941	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)
71	45, 69, 70	nf3an 1831	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
72		eqid 2622	. . . . . . 7 ⊢ if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2)) = if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2))
73	28	3ad2ant1 1082	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑃:𝑇⟶ℝ)
74		difssd 3738	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → (𝑇 ∖ 𝑈) ⊆ 𝑇)
75		simp2 1062	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑐 ∈ ℝ+)
76		simp3 1063	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
77	71, 72, 73, 74, 75, 76	stoweidlem5 40222	. . . . . 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 1326	. . . . 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 1863	. . . 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 1326	. . 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 2622	. . . . . 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 21198	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
85	82, 84	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ Top)
86		elssuni 4467	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽)
87	83, 86	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ ∪ 𝐽)
88	87, 25	syl6sseqr 3652	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑇)
89	25	isopn2 20836	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑈 ⊆ 𝑇) → (𝑈 ∈ 𝐽 ↔ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)))
90	85, 88, 89	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∈ 𝐽 ↔ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)))
91	83, 90	mpbid 222	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽))
92		cmpcld 21205	. . . . . . . 8 ⊢ ((𝐽 ∈ Comp ∧ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)) → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp)
93	82, 91, 92	syl2anc 693	. . . . . . 7 ⊢ (𝜑 → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp)
94	93	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp)
95	27	adantr 481	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → 𝑃 ∈ (𝐽 Cn 𝐾))
96		difssd 3738	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑇 ∖ 𝑈) ⊆ 𝑇)
97	25	cnrest 21089	. . . . . . 7 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ (𝑇 ∖ 𝑈) ⊆ 𝑇) → (𝑃 ↾ (𝑇 ∖ 𝑈)) ∈ ((𝐽 ↾t (𝑇 ∖ 𝑈)) Cn 𝐾))
98	95, 96, 97	syl2anc 693	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑃 ↾ (𝑇 ∖ 𝑈)) ∈ ((𝐽 ↾t (𝑇 ∖ 𝑈)) Cn 𝐾))
99		df-ne 2795	. . . . . . . 8 ⊢ ((𝑇 ∖ 𝑈) ≠ ∅ ↔ ¬ (𝑇 ∖ 𝑈) = ∅)
100		difssd 3738	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ 𝑇)
101	25	restuni 20966	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑇 ∖ 𝑈) ⊆ 𝑇) → (𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)))
102	85, 100, 101	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)))
103	102	neeq1d 2853	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∖ 𝑈) ≠ ∅ ↔ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅))
104	99, 103	syl5rbbr 275	. . . . . . 7 ⊢ (𝜑 → (∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅ ↔ ¬ (𝑇 ∖ 𝑈) = ∅))
105	104	biimpar 502	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅)
106	81, 24, 94, 98, 105	evth2 22759	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑠 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠))
107		nfcv 2764	. . . . . . 7 ⊢ Ⅎ𝑠∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
108		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑡𝐽
109		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑡 ↾t
110	108, 109, 9	nfov 6676	. . . . . . . 8 ⊢ Ⅎ𝑡(𝐽 ↾t (𝑇 ∖ 𝑈))
111	110	nfuni 4442	. . . . . . 7 ⊢ Ⅎ𝑡∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
112		nfcv 2764	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑃
113	112, 9	nfres 5398	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑃 ↾ (𝑇 ∖ 𝑈))
114		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑥
115	113, 114	nffv 6198	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥)
116		nfcv 2764	. . . . . . . 8 ⊢ Ⅎ𝑡 ≤
117		nfcv 2764	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑠
118	113, 117	nffv 6198	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)
119	115, 116, 118	nfbr 4699	. . . . . . 7 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)
120		nfv 1843	. . . . . . 7 ⊢ Ⅎ𝑠((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)
121		fveq2 6191	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) = ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
122	121	breq2d 4665	. . . . . . 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	cbvralf 3165	. . . . . 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 3041	. . . . 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 208	. . . 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 3134	. . . . . . 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 3147	. . . . . 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	102, 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 481	. . . 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 247	. . 3 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ (𝑇 ∖ 𝑈)∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
131	80, 130	r19.29a 3078	. 2 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
132	20, 131	pm2.61dan 832	1 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704  Ⅎwnf 1708   ∈ wcel 1990  Ⅎwnfc 2751   ≠ wne 2794  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  ifcif 4086  ∪ cuni 4436   class class class wbr 4653  ran crn 5115   ↾ cres 5116  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  ℝcr 9935  0cc0 9936  1c1 9937   < clt 10074   ≤ cle 10075   / cdiv 10684  2c2 11070  ℝ⁺crp 11832  (,)cioo 12175   ↾_t crest 16081  topGenctg 16098  Topctop 20698  Clsdccld 20820   Cn ccn 21028  Compccmp 21189

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013  ax-pre-sup 10014  ax-mulf 10016

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-of 6897  df-om 7066  df-1st 7168  df-2nd 7169  df-supp 7296  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-er 7742  df-map 7859  df-ixp 7909  df-en 7956  df-dom 7957  df-sdom 7958  df-fin 7959  df-fsupp 8276  df-fi 8317  df-sup 8348  df-inf 8349  df-oi 8415  df-card 8765  df-cda 8990  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-div 10685  df-nn 11021  df-2 11079  df-3 11080  df-4 11081  df-5 11082  df-6 11083  df-7 11084  df-8 11085  df-9 11086  df-n0 11293  df-z 11378  df-dec 11494  df-uz 11688  df-q 11789  df-rp 11833  df-xneg 11946  df-xadd 11947  df-xmul 11948  df-ioo 12179  df-icc 12182  df-fz 12327  df-fzo 12466  df-seq 12802  df-exp 12861  df-hash 13118  df-cj 13839  df-re 13840  df-im 13841  df-sqrt 13975  df-abs 13976  df-struct 15859  df-ndx 15860  df-slot 15861  df-base 15863  df-sets 15864  df-ress 15865  df-plusg 15954  df-mulr 15955  df-starv 15956  df-sca 15957  df-vsca 15958  df-ip 15959  df-tset 15960  df-ple 15961  df-ds 15964  df-unif 15965  df-hom 15966  df-cco 15967  df-rest 16083  df-topn 16084  df-0g 16102  df-gsum 16103  df-topgen 16104  df-pt 16105  df-prds 16108  df-xrs 16162  df-qtop 16167  df-imas 16168  df-xps 16170  df-mre 16246  df-mrc 16247  df-acs 16249  df-mgm 17242  df-sgrp 17284  df-mnd 17295  df-submnd 17336  df-mulg 17541  df-cntz 17750  df-cmn 18195  df-psmet 19738  df-xmet 19739  df-met 19740  df-bl 19741  df-mopn 19742  df-cnfld 19747  df-top 20699  df-topon 20716  df-topsp 20737  df-bases 20750  df-cld 20823  df-cn 21031  df-cnp 21032  df-cmp 21190  df-tx 21365  df-hmeo 21558  df-xms 22125  df-ms 22126  df-tms 22127

This theorem is referenced by:  stoweidlem56  40273