Theorem stoweidlem28 46214

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 12352	. . . . 5 ⊢ (1 / 2) ∈ ℝ
2		halfgt0 12354	. . . . 5 ⊢ 0 < (1 / 2)
3	1, 2	elrpii 12906	. . . 4 ⊢ (1 / 2) ∈ ℝ+
4	3	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) ∈ ℝ+)
5		halflt1 12356	. . . 4 ⊢ (1 / 2) < 1
6	5	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) < 1)
7		nfcv 2896	. . . . . . 7 ⊢ Ⅎ𝑡𝑇
8		stoweidlem28.1	. . . . . . 7 ⊢ Ⅎ𝑡𝑈
9	7, 8	nfdif 4079	. . . . . 6 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈)
10	9	nfeq1 2912	. . . . 5 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈) = ∅
11	10	rzalf 45204	. . . 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 7389	. . . 4 ⊢ (1 / 2) ∈ V
14		eleq1 2822	. . . . 5 ⊢ (𝑑 = (1 / 2) → (𝑑 ∈ ℝ+ ↔ (1 / 2) ∈ ℝ+))
15		breq1 5099	. . . . 5 ⊢ (𝑑 = (1 / 2) → (𝑑 < 1 ↔ (1 / 2) < 1))
16		breq1 5099	. . . . . 6 ⊢ (𝑑 = (1 / 2) → (𝑑 ≤ (𝑃‘𝑡) ↔ (1 / 2) ≤ (𝑃‘𝑡)))
17	16	ralbidv 3157	. . . . 5 ⊢ (𝑑 = (1 / 2) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡)))
18	14, 15, 17	3anbi123d 1438	. . . 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 3558	. . 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 1373	. 2 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
21		simplll 774	. . . 4 ⊢ ((((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) → 𝜑)
22		simplr 768	. . . 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 2734	. . . . . . . . . . 11 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
27		stoweidlem28.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
28	24, 25, 26, 27	fcnre 45212	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
29	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑃:𝑇⟶ℝ)
30		eldifi 4081	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ 𝑈) → 𝑥 ∈ 𝑇)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ 𝑇)
32	29, 31	ffvelcdmd 7028	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ)
33		stoweidlem28.7	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))
34		nfcv 2896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑇 ∖ 𝑈)
35		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥0 < (𝑃‘𝑡)
36		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡0 < (𝑃‘𝑥)
37		fveq2 6832	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (𝑃‘𝑡) = (𝑃‘𝑥))
38	37	breq2d 5108	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (0 < (𝑃‘𝑡) ↔ 0 < (𝑃‘𝑥)))
39	9, 34, 35, 36, 38	cbvralfw 3274	. . . . . . . . . . 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 3226	. . . . . . . . 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 12944	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ+)
44	43	3adant3 1132	. . . . . 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 2888	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑥 ∈ (𝑇 ∖ 𝑈)
47		nfra1 3258	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)
48	45, 46, 47	nf3an 1902	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
49		rspa 3223	. . . . . . . . . 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 1188	. . . . . . . . 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 1193	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ (𝑇 ∖ 𝑈))
52		fvres 6851	. . . . . . . . . 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 6851	. . . . . . . . . 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 5127	. . . . . . . 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 3232	. . . . . 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 2822	. . . . . . . . 9 ⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ∈ ℝ+ ↔ (𝑃‘𝑥) ∈ ℝ+))
60		breq1 5099	. . . . . . . . . 10 ⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ≤ (𝑃‘𝑡) ↔ (𝑃‘𝑥) ≤ (𝑃‘𝑡)))
61	60	ralbidv 3157	. . . . . . . . 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 3549	. . . . . . 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 1192	. . . . . 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 770	. . . . . 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 772	. . . . . 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 1915	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑐 ∈ ℝ+
70		nfra1 3258	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)
71	45, 69, 70	nf3an 1902	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
72		eqid 2734	. . . . . . 7 ⊢ if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2)) = if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2))
73	28	3ad2ant1 1133	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑃:𝑇⟶ℝ)
74		difssd 4087	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → (𝑇 ∖ 𝑈) ⊆ 𝑇)
75		simp2 1137	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑐 ∈ ℝ+)
76		simp3 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
77	71, 72, 73, 74, 75, 76	stoweidlem5 46191	. . . . . 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 1373	. . . . 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 1936	. . . 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 1373	. . 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 2734	. . . . . 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 23337	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
85	82, 84	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ Top)
86		elssuni 4892	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽)
87	83, 86	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ ∪ 𝐽)
88	87, 25	sseqtrrdi 3973	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑇)
89	25	isopn2 22974	. . . . . . . . . 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 23344	. . . . . . . 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 4087	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑇 ∖ 𝑈) ⊆ 𝑇)
97	25	cnrest 23227	. . . . . . 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 4087	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ 𝑇)
100	25	restuni 23104	. . . . . . . . . 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 2989	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∖ 𝑈) ≠ ∅ ↔ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅))
103		df-ne 2931	. . . . . . . 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 24913	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑠 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠))
107		nfcv 2896	. . . . . . 7 ⊢ Ⅎ𝑠∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
108		nfcv 2896	. . . . . . . . 9 ⊢ Ⅎ𝑡𝐽
109		nfcv 2896	. . . . . . . . 9 ⊢ Ⅎ𝑡 ↾t
110	108, 109, 9	nfov 7386	. . . . . . . 8 ⊢ Ⅎ𝑡(𝐽 ↾t (𝑇 ∖ 𝑈))
111	110	nfuni 4868	. . . . . . 7 ⊢ Ⅎ𝑡∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
112		nfcv 2896	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑃
113	112, 9	nfres 5938	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑃 ↾ (𝑇 ∖ 𝑈))
114		nfcv 2896	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑥
115	113, 114	nffv 6842	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥)
116		nfcv 2896	. . . . . . . 8 ⊢ Ⅎ𝑡 ≤
117		nfcv 2896	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑠
118	113, 117	nffv 6842	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)
119	115, 116, 118	nfbr 5143	. . . . . . 7 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)
120		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑠((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)
121		fveq2 6832	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) = ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
122	121	breq2d 5108	. . . . . . 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 3274	. . . . . 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 3081	. . . . 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 3323	. . . . . . 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 3307	. . . . . 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 3142	. 2 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡)))
132	20, 131	pm2.61dan 812	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 1541  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2113  Ⅎwnfc 2881   ≠ wne 2930  ∀wral 3049  ∃wrex 3058   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4283  ifcif 4477  ∪ cuni 4861   class class class wbr 5096  ran crn 5623   ↾ cres 5624  ⟶wf 6486  ‘cfv 6490  (class class class)co 7356  ℝcr 11023  0cc0 11024  1c1 11025   < clt 11164   ≤ cle 11165   / cdiv 11792  2c2 12198  ℝ⁺crp 12903  (,)cioo 13259   ↾_t crest 17338  topGenctg 17355  Topctop 22835  Clsdccld 22958   Cn ccn 23166  Compccmp 23328

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-rep 5222  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678  ax-cnex 11080  ax-resscn 11081  ax-1cn 11082  ax-icn 11083  ax-addcl 11084  ax-addrcl 11085  ax-mulcl 11086  ax-mulrcl 11087  ax-mulcom 11088  ax-addass 11089  ax-mulass 11090  ax-distr 11091  ax-i2m1 11092  ax-1ne0 11093  ax-1rid 11094  ax-rnegex 11095  ax-rrecex 11096  ax-cnre 11097  ax-pre-lttri 11098  ax-pre-lttrn 11099  ax-pre-ltadd 11100  ax-pre-mulgt0 11101  ax-pre-sup 11102

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-nel 3035  df-ral 3050  df-rex 3059  df-rmo 3348  df-reu 3349  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-tp 4583  df-op 4585  df-uni 4862  df-int 4901  df-iun 4946  df-iin 4947  df-br 5097  df-opab 5159  df-mpt 5178  df-tr 5204  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-se 5576  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-isom 6499  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-of 7620  df-om 7807  df-1st 7931  df-2nd 7932  df-supp 8101  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-2o 8396  df-er 8633  df-map 8763  df-ixp 8834  df-en 8882  df-dom 8883  df-sdom 8884  df-fin 8885  df-fsupp 9263  df-fi 9312  df-sup 9343  df-inf 9344  df-oi 9413  df-card 9849  df-pnf 11166  df-mnf 11167  df-xr 11168  df-ltxr 11169  df-le 11170  df-sub 11364  df-neg 11365  df-div 11793  df-nn 12144  df-2 12206  df-3 12207  df-4 12208  df-5 12209  df-6 12210  df-7 12211  df-8 12212  df-9 12213  df-n0 12400  df-z 12487  df-dec 12606  df-uz 12750  df-q 12860  df-rp 12904  df-xneg 13024  df-xadd 13025  df-xmul 13026  df-ioo 13263  df-icc 13266  df-fz 13422  df-fzo 13569  df-seq 13923  df-exp 13983  df-hash 14252  df-cj 15020  df-re 15021  df-im 15022  df-sqrt 15156  df-abs 15157  df-struct 17072  df-sets 17089  df-slot 17107  df-ndx 17119  df-base 17135  df-ress 17156  df-plusg 17188  df-mulr 17189  df-starv 17190  df-sca 17191  df-vsca 17192  df-ip 17193  df-tset 17194  df-ple 17195  df-ds 17197  df-unif 17198  df-hom 17199  df-cco 17200  df-rest 17340  df-topn 17341  df-0g 17359  df-gsum 17360  df-topgen 17361  df-pt 17362  df-prds 17365  df-xrs 17421  df-qtop 17426  df-imas 17427  df-xps 17429  df-mre 17503  df-mrc 17504  df-acs 17506  df-mgm 18563  df-sgrp 18642  df-mnd 18658  df-submnd 18707  df-mulg 18996  df-cntz 19244  df-cmn 19709  df-psmet 21299  df-xmet 21300  df-met 21301  df-bl 21302  df-mopn 21303  df-cnfld 21308  df-top 22836  df-topon 22853  df-topsp 22875  df-bases 22888  df-cld 22961  df-cn 23169  df-cnp 23170  df-cmp 23329  df-tx 23504  df-hmeo 23697  df-xms 24262  df-ms 24263  df-tms 24264

This theorem is referenced by:  stoweidlem56  46242