Theorem stoweidlem52 46149

Description: There exists a neighborhood V as in Lemma 1 of [BrosowskiDeutsh] p. 90. Here Z is used to represent t₀ in the paper, and v is used to represent V in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

Hypotheses

Ref	Expression
stoweidlem52.1	⊢ Ⅎ𝑡𝑈
stoweidlem52.2	⊢ Ⅎ𝑡𝜑
stoweidlem52.3	⊢ Ⅎ𝑡𝑃
stoweidlem52.4	⊢ 𝐾 = (topGen‘ran (,))
stoweidlem52.5	⊢ 𝑉 = {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)}
stoweidlem52.7	⊢ 𝑇 = ∪ 𝐽
stoweidlem52.8	⊢ 𝐶 = (𝐽 Cn 𝐾)
stoweidlem52.9	⊢ (𝜑 → 𝐴 ⊆ 𝐶)
stoweidlem52.10	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem52.11	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem52.12	⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
stoweidlem52.13	⊢ (𝜑 → 𝐷 ∈ ℝ+)
stoweidlem52.14	⊢ (𝜑 → 𝐷 < 1)
stoweidlem52.15	⊢ (𝜑 → 𝑈 ∈ 𝐽)
stoweidlem52.16	⊢ (𝜑 → 𝑍 ∈ 𝑈)
stoweidlem52.17	⊢ (𝜑 → 𝑃 ∈ 𝐴)
stoweidlem52.18	⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
stoweidlem52.19	⊢ (𝜑 → (𝑃‘𝑍) = 0)
stoweidlem52.20	⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡))

Assertion

Ref	Expression
stoweidlem52	⊢ (𝜑 → ∃𝑣 ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡))))

Distinct variable groups:   𝑒,𝑎,𝑡   𝐴,𝑎,𝑡   𝐷,𝑎,𝑡   𝑇,𝑎,𝑡   𝑈,𝑎   𝑉,𝑎,𝑒   𝜑,𝑎,𝑒   𝑒,𝑓,𝑔,𝑡   𝑣,𝑒,𝑥,𝑡   𝐴,𝑓,𝑔   𝐷,𝑓,𝑔   𝑃,𝑓,𝑔   𝑇,𝑓,𝑔   𝑈,𝑓,𝑔   𝑓,𝑉,𝑔   𝜑,𝑓,𝑔   𝑡,𝑍,𝑣   𝑣,𝐴   𝑣,𝐽   𝑣,𝑇,𝑥   𝑣,𝑈,𝑥   𝑣,𝑉,𝑥   𝑥,𝐴

Allowed substitution hints:   𝜑(𝑥,𝑣,𝑡)   𝐴(𝑒)   𝐶(𝑥,𝑣,𝑡,𝑒,𝑓,𝑔,𝑎)   𝐷(𝑥,𝑣,𝑒)   𝑃(𝑥,𝑣,𝑡,𝑒,𝑎)   𝑇(𝑒)   𝑈(𝑡,𝑒)   𝐽(𝑥,𝑡,𝑒,𝑓,𝑔,𝑎)   𝐾(𝑥,𝑣,𝑡,𝑒,𝑓,𝑔,𝑎)   𝑉(𝑡)   𝑍(𝑥,𝑒,𝑓,𝑔,𝑎)

Proof of Theorem stoweidlem52

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfcv 2894	. . 3 ⊢ Ⅎ𝑡(𝐷 / 2)
2		stoweidlem52.3	. . 3 ⊢ Ⅎ𝑡𝑃
3		stoweidlem52.2	. . 3 ⊢ Ⅎ𝑡𝜑
4		stoweidlem52.4	. . 3 ⊢ 𝐾 = (topGen‘ran (,))
5		stoweidlem52.7	. . 3 ⊢ 𝑇 = ∪ 𝐽
6		stoweidlem52.5	. . 3 ⊢ 𝑉 = {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)}
7		stoweidlem52.13	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ ℝ+)
8	7	rpred 12934	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ)
9	8	rehalfcld 12368	. . . 4 ⊢ (𝜑 → (𝐷 / 2) ∈ ℝ)
10	9	rexrd 11162	. . 3 ⊢ (𝜑 → (𝐷 / 2) ∈ ℝ*)
11		stoweidlem52.9	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
12		stoweidlem52.8	. . . . 5 ⊢ 𝐶 = (𝐽 Cn 𝐾)
13	11, 12	sseqtrdi 3970	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
14		stoweidlem52.17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
15	13, 14	sseldd 3930	. . 3 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
16	1, 2, 3, 4, 5, 6, 10, 15	rfcnpre2 45127	. 2 ⊢ (𝜑 → 𝑉 ∈ 𝐽)
17		stoweidlem52.15	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
18		elssuni 4887	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ ∪ 𝐽)
20	19, 5	sseqtrrdi 3971	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ 𝑇)
21		stoweidlem52.16	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
22	20, 21	sseldd 3930	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
23		stoweidlem52.19	. . . . . 6 ⊢ (𝜑 → (𝑃‘𝑍) = 0)
24		2re 12199	. . . . . . . 8 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℝ)
26	7	rpgt0d 12937	. . . . . . 7 ⊢ (𝜑 → 0 < 𝐷)
27		2pos 12228	. . . . . . . 8 ⊢ 0 < 2
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 2)
29	8, 25, 26, 28	divgt0d 12057	. . . . . 6 ⊢ (𝜑 → 0 < (𝐷 / 2))
30	23, 29	eqbrtrd 5111	. . . . 5 ⊢ (𝜑 → (𝑃‘𝑍) < (𝐷 / 2))
31		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑡𝑍
32		nfcv 2894	. . . . . 6 ⊢ Ⅎ𝑡𝑇
33	2, 31	nffv 6832	. . . . . . 7 ⊢ Ⅎ𝑡(𝑃‘𝑍)
34		nfcv 2894	. . . . . . 7 ⊢ Ⅎ𝑡 <
35	33, 34, 1	nfbr 5136	. . . . . 6 ⊢ Ⅎ𝑡(𝑃‘𝑍) < (𝐷 / 2)
36		fveq2 6822	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (𝑃‘𝑡) = (𝑃‘𝑍))
37	36	breq1d 5099	. . . . . 6 ⊢ (𝑡 = 𝑍 → ((𝑃‘𝑡) < (𝐷 / 2) ↔ (𝑃‘𝑍) < (𝐷 / 2)))
38	31, 32, 35, 37	elrabf 3639	. . . . 5 ⊢ (𝑍 ∈ {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} ↔ (𝑍 ∈ 𝑇 ∧ (𝑃‘𝑍) < (𝐷 / 2)))
39	22, 30, 38	sylanbrc 583	. . . 4 ⊢ (𝜑 → 𝑍 ∈ {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)})
40	39, 6	eleqtrrdi 2842	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
41		nfrab1 3415	. . . . 5 ⊢ Ⅎ𝑡{𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)}
42	6, 41	nfcxfr 2892	. . . 4 ⊢ Ⅎ𝑡𝑉
43		stoweidlem52.1	. . . 4 ⊢ Ⅎ𝑡𝑈
44	11, 14	sseldd 3930	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ 𝐶)
45	4, 5, 12, 44	fcnre 45121	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
46	45	adantr 480	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝑃:𝑇⟶ℝ)
47	6	reqabi 3418	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝑉 ↔ (𝑡 ∈ 𝑇 ∧ (𝑃‘𝑡) < (𝐷 / 2)))
48	47	biimpi 216	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑉 → (𝑡 ∈ 𝑇 ∧ (𝑃‘𝑡) < (𝐷 / 2)))
49	48	adantl 481	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑡 ∈ 𝑇 ∧ (𝑃‘𝑡) < (𝐷 / 2)))
50	49	simpld 494	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝑡 ∈ 𝑇)
51	46, 50	ffvelcdmd 7018	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑃‘𝑡) ∈ ℝ)
52	9	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐷 / 2) ∈ ℝ)
53	8	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝐷 ∈ ℝ)
54	49	simprd 495	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑃‘𝑡) < (𝐷 / 2))
55		halfpos 12351	. . . . . . . . . . 11 ⊢ (𝐷 ∈ ℝ → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
56	8, 55	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
57	26, 56	mpbid 232	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 / 2) < 𝐷)
58	57	adantr 480	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐷 / 2) < 𝐷)
59	51, 52, 53, 54, 58	lttrd 11274	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑃‘𝑡) < 𝐷)
60	59	adantr 480	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → (𝑃‘𝑡) < 𝐷)
61	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → 𝐷 ∈ ℝ)
62	51	adantr 480	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → (𝑃‘𝑡) ∈ ℝ)
63		stoweidlem52.20	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡))
64	63	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡))
65	50	anim1i 615	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → (𝑡 ∈ 𝑇 ∧ ¬ 𝑡 ∈ 𝑈))
66		eldif 3907	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) ↔ (𝑡 ∈ 𝑇 ∧ ¬ 𝑡 ∈ 𝑈))
67	65, 66	sylibr 234	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → 𝑡 ∈ (𝑇 ∖ 𝑈))
68		rsp 3220	. . . . . . . 8 ⊢ (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡) → (𝑡 ∈ (𝑇 ∖ 𝑈) → 𝐷 ≤ (𝑃‘𝑡)))
69	64, 67, 68	sylc 65	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → 𝐷 ≤ (𝑃‘𝑡))
70	61, 62, 69	lensymd 11264	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → ¬ (𝑃‘𝑡) < 𝐷)
71	60, 70	condan 817	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝑡 ∈ 𝑈)
72	71	ex 412	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑉 → 𝑡 ∈ 𝑈))
73	3, 42, 43, 72	ssrd 3934	. . 3 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
74		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑒 ∈ ℝ+
75	3, 74	nfan 1900	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑒 ∈ ℝ+)
76		nfv 1915	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑦 ∈ 𝐴
77	75, 76	nfan 1900	. . . . . . 7 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴)
78		nfra1 3256	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)
79		nfra1 3256	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡)
80		nfra1 3256	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒
81	78, 79, 80	nf3an 1902	. . . . . . 7 ⊢ Ⅎ𝑡(∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)
82	77, 81	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑡(((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒))
83		eqid 2731	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡)))
84		eqid 2731	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ 1)
85		ssrab2 4027	. . . . . . 7 ⊢ {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} ⊆ 𝑇
86	6, 85	eqsstri 3976	. . . . . 6 ⊢ 𝑉 ⊆ 𝑇
87		simplr 768	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) → 𝑦 ∈ 𝐴)
88		simplll 774	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) → 𝜑)
89	11	sselda 3929	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐶)
90	4, 5, 12, 89	fcnre 45121	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦:𝑇⟶ℝ)
91	88, 87, 90	syl2anc 584	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) → 𝑦:𝑇⟶ℝ)
92	11	sselda 3929	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶)
93	4, 5, 12, 92	fcnre 45121	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
94	88, 93	sylan 580	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
95		stoweidlem52.10	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
96	88, 95	syl3an1 1163	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
97		stoweidlem52.11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
98	88, 97	syl3an1 1163	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
99		stoweidlem52.12	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
100	88, 99	sylan 580	. . . . . 6 ⊢ (((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
101		simpllr 775	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) → 𝑒 ∈ ℝ+)
102		simpr1 1195	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1))
103		simpr2 1196	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) → ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡))
104		simpr3 1197	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)
105	82, 83, 84, 86, 87, 91, 94, 96, 98, 100, 101, 102, 103, 104	stoweidlem41 46138	. . . . 5 ⊢ ((((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴) ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)) → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)))
106	7	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝐷 ∈ ℝ+)
107		stoweidlem52.14	. . . . . . 7 ⊢ (𝜑 → 𝐷 < 1)
108	107	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝐷 < 1)
109	14	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝑃 ∈ 𝐴)
110	45	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝑃:𝑇⟶ℝ)
111		stoweidlem52.18	. . . . . . 7 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
112	111	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
113	63	adantr 480	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡))
114	93	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
115	95	3adant1r 1178	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
116	97	3adant1r 1178	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
117	99	adantlr 715	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
118		simpr 484	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝑒 ∈ ℝ+)
119	2, 75, 6, 106, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118	stoweidlem49 46146	. . . . 5 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → ∃𝑦 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒))
120	105, 119	r19.29a 3140	. . . 4 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)))
121	120	ralrimiva 3124	. . 3 ⊢ (𝜑 → ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)))
122	40, 73, 121	jca31 514	. 2 ⊢ (𝜑 → ((𝑍 ∈ 𝑉 ∧ 𝑉 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡))))
123		eleq2 2820	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑍 ∈ 𝑣 ↔ 𝑍 ∈ 𝑉))
124		sseq1 3955	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ 𝑈 ↔ 𝑉 ⊆ 𝑈))
125	123, 124	anbi12d 632	. . . 4 ⊢ (𝑣 = 𝑉 → ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ↔ (𝑍 ∈ 𝑉 ∧ 𝑉 ⊆ 𝑈)))
126		nfcv 2894	. . . . . . . 8 ⊢ Ⅎ𝑡𝑣
127	126, 42	raleqf 3321	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ↔ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒))
128	127	3anbi2d 1443	. . . . . 6 ⊢ (𝑣 = 𝑉 → ((∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡))))
129	128	rexbidv 3156	. . . . 5 ⊢ (𝑣 = 𝑉 → (∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)) ↔ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡))))
130	129	ralbidv 3155	. . . 4 ⊢ (𝑣 = 𝑉 → (∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)) ↔ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡))))
131	125, 130	anbi12d 632	. . 3 ⊢ (𝑣 = 𝑉 → (((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡))) ↔ ((𝑍 ∈ 𝑉 ∧ 𝑉 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)))))
132	131	rspcev 3572	. 2 ⊢ ((𝑉 ∈ 𝐽 ∧ ((𝑍 ∈ 𝑉 ∧ 𝑉 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡)))) → ∃𝑣 ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡))))
133	16, 122, 132	syl2anc 584	1 ⊢ (𝜑 → ∃𝑣 ∈ 𝐽 ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ∧ ∀𝑒 ∈ ℝ+ ∃𝑥 ∈ 𝐴 (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(1 − 𝑒) < (𝑥‘𝑡))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  Ⅎwnf 1784   ∈ wcel 2111  Ⅎwnfc 2879  ∀wral 3047  ∃wrex 3056  {crab 3395   ∖ cdif 3894   ⊆ wss 3897  ∪ cuni 4856   class class class wbr 5089   ↦ cmpt 5170  ran crn 5615  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  ℝcr 11005  0cc0 11006  1c1 11007   + caddc 11009   · cmul 11011   < clt 11146   ≤ cle 11147   − cmin 11344   / cdiv 11774  2c2 12180  ℝ⁺crp 12890  (,)cioo 13245  topGenctg 17341   Cn ccn 23139

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-cnex 11062  ax-resscn 11063  ax-1cn 11064  ax-icn 11065  ax-addcl 11066  ax-addrcl 11067  ax-mulcl 11068  ax-mulrcl 11069  ax-mulcom 11070  ax-addass 11071  ax-mulass 11072  ax-distr 11073  ax-i2m1 11074  ax-1ne0 11075  ax-1rid 11076  ax-rnegex 11077  ax-rrecex 11078  ax-cnre 11079  ax-pre-lttri 11080  ax-pre-lttrn 11081  ax-pre-ltadd 11082  ax-pre-mulgt0 11083  ax-pre-sup 11084

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 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rmo 3346  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-riota 7303  df-ov 7349  df-oprab 7350  df-mpo 7351  df-om 7797  df-1st 7921  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291  df-rdg 8329  df-er 8622  df-map 8752  df-pm 8753  df-en 8870  df-dom 8871  df-sdom 8872  df-sup 9326  df-inf 9327  df-pnf 11148  df-mnf 11149  df-xr 11150  df-ltxr 11151  df-le 11152  df-sub 11346  df-neg 11347  df-div 11775  df-nn 12126  df-2 12188  df-3 12189  df-n0 12382  df-z 12469  df-uz 12733  df-q 12847  df-rp 12891  df-ioo 13249  df-fl 13696  df-seq 13909  df-exp 13969  df-cj 15006  df-re 15007  df-im 15008  df-sqrt 15142  df-abs 15143  df-clim 15395  df-rlim 15396  df-topgen 17347  df-top 22809  df-topon 22826  df-bases 22861  df-cn 23142

This theorem is referenced by:  stoweidlem56  46153