Theorem stoweidlem28 46456

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 12390	. . . . 5 ⊢ (1 / 2) ∈ ℝ
2		halfgt0 12392	. . . . 5 ⊢ 0 < (1 / 2)
3	1, 2	elrpii 12945	. . . 4 ⊢ (1 / 2) ∈ ℝ+
4	3	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) ∈ ℝ+)
5		halflt1 12394	. . . 4 ⊢ (1 / 2) < 1
6	5	a1i 11	. . 3 ⊢ ((𝜑 ∧ (𝑇 ∖ 𝑈) = ∅) → (1 / 2) < 1)
7		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑡𝑇
8		stoweidlem28.1	. . . . . . 7 ⊢ Ⅎ𝑡𝑈
9	7, 8	nfdif 4069	. . . . . 6 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈)
10	9	nfeq1 2914	. . . . 5 ⊢ Ⅎ𝑡(𝑇 ∖ 𝑈) = ∅
11	10	rzalf 45448	. . . 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 7400	. . . 4 ⊢ (1 / 2) ∈ V
14		eleq1 2824	. . . . 5 ⊢ (𝑑 = (1 / 2) → (𝑑 ∈ ℝ+ ↔ (1 / 2) ∈ ℝ+))
15		breq1 5088	. . . . 5 ⊢ (𝑑 = (1 / 2) → (𝑑 < 1 ↔ (1 / 2) < 1))
16		breq1 5088	. . . . . 6 ⊢ (𝑑 = (1 / 2) → (𝑑 ≤ (𝑃‘𝑡) ↔ (1 / 2) ≤ (𝑃‘𝑡)))
17	16	ralbidv 3160	. . . . 5 ⊢ (𝑑 = (1 / 2) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑑 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 / 2) ≤ (𝑃‘𝑡)))
18	14, 15, 17	3anbi123d 1439	. . . 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 3548	. . 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 1374	. 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 2736	. . . . . . . . . . 11 ⊢ (𝐽 Cn 𝐾) = (𝐽 Cn 𝐾)
27		stoweidlem28.6	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
28	24, 25, 26, 27	fcnre 45456	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
29	28	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑃:𝑇⟶ℝ)
30		eldifi 4071	. . . . . . . . . 10 ⊢ (𝑥 ∈ (𝑇 ∖ 𝑈) → 𝑥 ∈ 𝑇)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ 𝑇)
32	29, 31	ffvelcdmd 7037	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ)
33		stoweidlem28.7	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡))
34		nfcv 2898	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥(𝑇 ∖ 𝑈)
35		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥0 < (𝑃‘𝑡)
36		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡0 < (𝑃‘𝑥)
37		fveq2 6840	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑥 → (𝑃‘𝑡) = (𝑃‘𝑥))
38	37	breq2d 5097	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑥 → (0 < (𝑃‘𝑡) ↔ 0 < (𝑃‘𝑥)))
39	9, 34, 35, 36, 38	cbvralfw 3277	. . . . . . . . . . 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 3229	. . . . . . . . 9 ⊢ ((∀𝑡 ∈ (𝑇 ∖ 𝑈)0 < (𝑃‘𝑡) ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑥))
42	33, 41	sylan 581	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → 0 < (𝑃‘𝑥))
43	32, 42	elrpd 12983	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈)) → (𝑃‘𝑥) ∈ ℝ+)
44	43	3adant3 1133	. . . . . 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 2890	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑥 ∈ (𝑇 ∖ 𝑈)
47		nfra1 3261	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)
48	45, 46, 47	nf3an 1903	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
49		rspa 3226	. . . . . . . . . 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 1189	. . . . . . . . 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 1194	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑥 ∈ (𝑇 ∖ 𝑈) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)) ∧ 𝑡 ∈ (𝑇 ∖ 𝑈)) → 𝑥 ∈ (𝑇 ∖ 𝑈))
52		fvres 6859	. . . . . . . . . 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 6859	. . . . . . . . . 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 5116	. . . . . . . 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 3235	. . . . . 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 2824	. . . . . . . . 9 ⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ∈ ℝ+ ↔ (𝑃‘𝑥) ∈ ℝ+))
60		breq1 5088	. . . . . . . . . 10 ⊢ (𝑐 = (𝑃‘𝑥) → (𝑐 ≤ (𝑃‘𝑡) ↔ (𝑃‘𝑥) ≤ (𝑃‘𝑡)))
61	60	ralbidv 3160	. . . . . . . . 9 ⊢ (𝑐 = (𝑃‘𝑥) → (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑃‘𝑥) ≤ (𝑃‘𝑡)))
62	59, 61	anbi12d 633	. . . . . . . 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 3539	. . . . . . 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 700	. . . . 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 1193	. . . . . 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 1916	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑐 ∈ ℝ+
70		nfra1 3261	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)
71	45, 69, 70	nf3an 1903	. . . . . . 7 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
72		eqid 2736	. . . . . . 7 ⊢ if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2)) = if(𝑐 ≤ (1 / 2), 𝑐, (1 / 2))
73	28	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑃:𝑇⟶ℝ)
74		difssd 4077	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → (𝑇 ∖ 𝑈) ⊆ 𝑇)
75		simp2 1138	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → 𝑐 ∈ ℝ+)
76		simp3 1139	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑐 ∈ ℝ+ ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝑐 ≤ (𝑃‘𝑡))
77	71, 72, 73, 74, 75, 76	stoweidlem5 46433	. . . . . 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 1374	. . . . 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 1937	. . . 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 1374	. . 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 2736	. . . . . 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 23360	. . . . . . . . . . 11 ⊢ (𝐽 ∈ Comp → 𝐽 ∈ Top)
85	82, 84	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → 𝐽 ∈ Top)
86		elssuni 4881	. . . . . . . . . . . 12 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽)
87	83, 86	syl 17	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑈 ⊆ ∪ 𝐽)
88	87, 25	sseqtrrdi 3963	. . . . . . . . . 10 ⊢ (𝜑 → 𝑈 ⊆ 𝑇)
89	25	isopn2 22997	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ 𝑈 ⊆ 𝑇) → (𝑈 ∈ 𝐽 ↔ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)))
90	85, 88, 89	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑈 ∈ 𝐽 ↔ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)))
91	83, 90	mpbid 232	. . . . . . . 8 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽))
92		cmpcld 23367	. . . . . . . 8 ⊢ ((𝐽 ∈ Comp ∧ (𝑇 ∖ 𝑈) ∈ (Clsd‘𝐽)) → (𝐽 ↾t (𝑇 ∖ 𝑈)) ∈ Comp)
93	82, 91, 92	syl2anc 585	. . . . . . 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 4077	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑇 ∖ 𝑈) ⊆ 𝑇)
97	25	cnrest 23250	. . . . . . 7 ⊢ ((𝑃 ∈ (𝐽 Cn 𝐾) ∧ (𝑇 ∖ 𝑈) ⊆ 𝑇) → (𝑃 ↾ (𝑇 ∖ 𝑈)) ∈ ((𝐽 ↾t (𝑇 ∖ 𝑈)) Cn 𝐾))
98	95, 96, 97	syl2anc 585	. . . . . 6 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → (𝑃 ↾ (𝑇 ∖ 𝑈)) ∈ ((𝐽 ↾t (𝑇 ∖ 𝑈)) Cn 𝐾))
99		difssd 4077	. . . . . . . . . 10 ⊢ (𝜑 → (𝑇 ∖ 𝑈) ⊆ 𝑇)
100	25	restuni 23127	. . . . . . . . . 10 ⊢ ((𝐽 ∈ Top ∧ (𝑇 ∖ 𝑈) ⊆ 𝑇) → (𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)))
101	85, 99, 100	syl2anc 585	. . . . . . . . 9 ⊢ (𝜑 → (𝑇 ∖ 𝑈) = ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)))
102	101	neeq1d 2991	. . . . . . . 8 ⊢ (𝜑 → ((𝑇 ∖ 𝑈) ≠ ∅ ↔ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈)) ≠ ∅))
103		df-ne 2933	. . . . . . . 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 24927	. . . . 5 ⊢ ((𝜑 ∧ ¬ (𝑇 ∖ 𝑈) = ∅) → ∃𝑥 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))∀𝑠 ∈ ∪ (𝐽 ↾t (𝑇 ∖ 𝑈))((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠))
107		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑠∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
108		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑡𝐽
109		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑡 ↾t
110	108, 109, 9	nfov 7397	. . . . . . . 8 ⊢ Ⅎ𝑡(𝐽 ↾t (𝑇 ∖ 𝑈))
111	110	nfuni 4857	. . . . . . 7 ⊢ Ⅎ𝑡∪ (𝐽 ↾t (𝑇 ∖ 𝑈))
112		nfcv 2898	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑃
113	112, 9	nfres 5946	. . . . . . . . 9 ⊢ Ⅎ𝑡(𝑃 ↾ (𝑇 ∖ 𝑈))
114		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑥
115	113, 114	nffv 6850	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥)
116		nfcv 2898	. . . . . . . 8 ⊢ Ⅎ𝑡 ≤
117		nfcv 2898	. . . . . . . . 9 ⊢ Ⅎ𝑡𝑠
118	113, 117	nffv 6850	. . . . . . . 8 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)
119	115, 116, 118	nfbr 5132	. . . . . . 7 ⊢ Ⅎ𝑡((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠)
120		nfv 1916	. . . . . . 7 ⊢ Ⅎ𝑠((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑥) ≤ ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡)
121		fveq2 6840	. . . . . . . 8 ⊢ (𝑠 = 𝑡 → ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑠) = ((𝑃 ↾ (𝑇 ∖ 𝑈))‘𝑡))
122	121	breq2d 5097	. . . . . . 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 3277	. . . . . 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 3084	. . . . 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 3318	. . . . . . 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 3306	. . . . . 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 3145	. 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 1087   = wceq 1542  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2114  Ⅎwnfc 2883   ≠ wne 2932  ∀wral 3051  ∃wrex 3061   ∖ cdif 3886   ⊆ wss 3889  ∅c0 4273  ifcif 4466  ∪ cuni 4850   class class class wbr 5085  ran crn 5632   ↾ cres 5633  ⟶wf 6494  ‘cfv 6498  (class class class)co 7367  ℝcr 11037  0cc0 11038  1c1 11039   < clt 11179   ≤ cle 11180   / cdiv 11807  2c2 12236  ℝ⁺crp 12942  (,)cioo 13298   ↾_t crest 17383  topGenctg 17400  Topctop 22858  Clsdccld 22981   Cn ccn 23189  Compccmp 23351

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5307  ax-pr 5375  ax-un 7689  ax-cnex 11094  ax-resscn 11095  ax-1cn 11096  ax-icn 11097  ax-addcl 11098  ax-addrcl 11099  ax-mulcl 11100  ax-mulrcl 11101  ax-mulcom 11102  ax-addass 11103  ax-mulass 11104  ax-distr 11105  ax-i2m1 11106  ax-1ne0 11107  ax-1rid 11108  ax-rnegex 11109  ax-rrecex 11110  ax-cnre 11111  ax-pre-lttri 11112  ax-pre-lttrn 11113  ax-pre-ltadd 11114  ax-pre-mulgt0 11115  ax-pre-sup 11116

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-nel 3037  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-pss 3909  df-nul 4274  df-if 4467  df-pw 4543  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4851  df-int 4890  df-iun 4935  df-iin 4936  df-br 5086  df-opab 5148  df-mpt 5167  df-tr 5193  df-id 5526  df-eprel 5531  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6265  df-ord 6326  df-on 6327  df-lim 6328  df-suc 6329  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-isom 6507  df-riota 7324  df-ov 7370  df-oprab 7371  df-mpo 7372  df-of 7631  df-om 7818  df-1st 7942  df-2nd 7943  df-supp 8111  df-frecs 8231  df-wrecs 8262  df-recs 8311  df-rdg 8349  df-1o 8405  df-2o 8406  df-er 8643  df-map 8775  df-ixp 8846  df-en 8894  df-dom 8895  df-sdom 8896  df-fin 8897  df-fsupp 9275  df-fi 9324  df-sup 9355  df-inf 9356  df-oi 9425  df-card 9863  df-pnf 11181  df-mnf 11182  df-xr 11183  df-ltxr 11184  df-le 11185  df-sub 11379  df-neg 11380  df-div 11808  df-nn 12175  df-2 12244  df-3 12245  df-4 12246  df-5 12247  df-6 12248  df-7 12249  df-8 12250  df-9 12251  df-n0 12438  df-z 12525  df-dec 12645  df-uz 12789  df-q 12899  df-rp 12943  df-xneg 13063  df-xadd 13064  df-xmul 13065  df-ioo 13302  df-icc 13305  df-fz 13462  df-fzo 13609  df-seq 13964  df-exp 14024  df-hash 14293  df-cj 15061  df-re 15062  df-im 15063  df-sqrt 15197  df-abs 15198  df-struct 17117  df-sets 17134  df-slot 17152  df-ndx 17164  df-base 17180  df-ress 17201  df-plusg 17233  df-mulr 17234  df-starv 17235  df-sca 17236  df-vsca 17237  df-ip 17238  df-tset 17239  df-ple 17240  df-ds 17242  df-unif 17243  df-hom 17244  df-cco 17245  df-rest 17385  df-topn 17386  df-0g 17404  df-gsum 17405  df-topgen 17406  df-pt 17407  df-prds 17410  df-xrs 17466  df-qtop 17471  df-imas 17472  df-xps 17474  df-mre 17548  df-mrc 17549  df-acs 17551  df-mgm 18608  df-sgrp 18687  df-mnd 18703  df-submnd 18752  df-mulg 19044  df-cntz 19292  df-cmn 19757  df-psmet 21344  df-xmet 21345  df-met 21346  df-bl 21347  df-mopn 21348  df-cnfld 21353  df-top 22859  df-topon 22876  df-topsp 22898  df-bases 22911  df-cld 22984  df-cn 23192  df-cnp 23193  df-cmp 23352  df-tx 23527  df-hmeo 23720  df-xms 24285  df-ms 24286  df-tms 24287

This theorem is referenced by:  stoweidlem56  46484