Theorem stoweidlem52 43050

Description: There exists a neighborood 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 2920	. . 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 12462	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ)
9	8	rehalfcld 11911	. . . 4 ⊢ (𝜑 → (𝐷 / 2) ∈ ℝ)
10	9	rexrd 10719	. . 3 ⊢ (𝜑 → (𝐷 / 2) ∈ ℝ*)
11		stoweidlem52.9	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
12		stoweidlem52.8	. . . . 5 ⊢ 𝐶 = (𝐽 Cn 𝐾)
13	11, 12	sseqtrdi 3943	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
14		stoweidlem52.17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
15	13, 14	sseldd 3894	. . 3 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
16	1, 2, 3, 4, 5, 6, 10, 15	rfcnpre2 42023	. 2 ⊢ (𝜑 → 𝑉 ∈ 𝐽)
17		stoweidlem52.15	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
18		elssuni 4828	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽)
19	17, 18	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ ∪ 𝐽)
20	19, 5	sseqtrrdi 3944	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ 𝑇)
21		stoweidlem52.16	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
22	20, 21	sseldd 3894	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
23		stoweidlem52.19	. . . . . 6 ⊢ (𝜑 → (𝑃‘𝑍) = 0)
24		2re 11738	. . . . . . . 8 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℝ)
26	7	rpgt0d 12465	. . . . . . 7 ⊢ (𝜑 → 0 < 𝐷)
27		2pos 11767	. . . . . . . 8 ⊢ 0 < 2
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 2)
29	8, 25, 26, 28	divgt0d 11603	. . . . . 6 ⊢ (𝜑 → 0 < (𝐷 / 2))
30	23, 29	eqbrtrd 5052	. . . . 5 ⊢ (𝜑 → (𝑃‘𝑍) < (𝐷 / 2))
31		nfcv 2920	. . . . . 6 ⊢ Ⅎ𝑡𝑍
32		nfcv 2920	. . . . . 6 ⊢ Ⅎ𝑡𝑇
33	2, 31	nffv 6666	. . . . . . 7 ⊢ Ⅎ𝑡(𝑃‘𝑍)
34		nfcv 2920	. . . . . . 7 ⊢ Ⅎ𝑡 <
35	33, 34, 1	nfbr 5077	. . . . . 6 ⊢ Ⅎ𝑡(𝑃‘𝑍) < (𝐷 / 2)
36		fveq2 6656	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (𝑃‘𝑡) = (𝑃‘𝑍))
37	36	breq1d 5040	. . . . . 6 ⊢ (𝑡 = 𝑍 → ((𝑃‘𝑡) < (𝐷 / 2) ↔ (𝑃‘𝑍) < (𝐷 / 2)))
38	31, 32, 35, 37	elrabf 3599	. . . . 5 ⊢ (𝑍 ∈ {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} ↔ (𝑍 ∈ 𝑇 ∧ (𝑃‘𝑍) < (𝐷 / 2)))
39	22, 30, 38	sylanbrc 587	. . . 4 ⊢ (𝜑 → 𝑍 ∈ {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)})
40	39, 6	eleqtrrdi 2864	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
41		nfrab1 3303	. . . . 5 ⊢ Ⅎ𝑡{𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)}
42	6, 41	nfcxfr 2918	. . . 4 ⊢ Ⅎ𝑡𝑉
43		stoweidlem52.1	. . . 4 ⊢ Ⅎ𝑡𝑈
44	11, 14	sseldd 3894	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ 𝐶)
45	4, 5, 12, 44	fcnre 42017	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
46	45	adantr 485	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝑃:𝑇⟶ℝ)
47	6	rabeq2i 3401	. . . . . . . . . . . 12 ⊢ (𝑡 ∈ 𝑉 ↔ (𝑡 ∈ 𝑇 ∧ (𝑃‘𝑡) < (𝐷 / 2)))
48	47	biimpi 219	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑉 → (𝑡 ∈ 𝑇 ∧ (𝑃‘𝑡) < (𝐷 / 2)))
49	48	adantl 486	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑡 ∈ 𝑇 ∧ (𝑃‘𝑡) < (𝐷 / 2)))
50	49	simpld 499	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝑡 ∈ 𝑇)
51	46, 50	ffvelrnd 6841	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑃‘𝑡) ∈ ℝ)
52	9	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐷 / 2) ∈ ℝ)
53	8	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝐷 ∈ ℝ)
54	49	simprd 500	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑃‘𝑡) < (𝐷 / 2))
55		halfpos 11894	. . . . . . . . . . 11 ⊢ (𝐷 ∈ ℝ → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
56	8, 55	syl 17	. . . . . . . . . 10 ⊢ (𝜑 → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
57	26, 56	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 / 2) < 𝐷)
58	57	adantr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐷 / 2) < 𝐷)
59	51, 52, 53, 54, 58	lttrd 10829	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑃‘𝑡) < 𝐷)
60	59	adantr 485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → (𝑃‘𝑡) < 𝐷)
61	8	ad2antrr 726	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → 𝐷 ∈ ℝ)
62	51	adantr 485	. . . . . . 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 618	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → (𝑡 ∈ 𝑇 ∧ ¬ 𝑡 ∈ 𝑈))
66		eldif 3869	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) ↔ (𝑡 ∈ 𝑇 ∧ ¬ 𝑡 ∈ 𝑈))
67	65, 66	sylibr 237	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → 𝑡 ∈ (𝑇 ∖ 𝑈))
68		rsp 3135	. . . . . . . 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 10819	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → ¬ (𝑃‘𝑡) < 𝐷)
71	60, 70	condan 818	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝑡 ∈ 𝑈)
72	71	ex 417	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑉 → 𝑡 ∈ 𝑈))
73	3, 42, 43, 72	ssrd 3898	. . 3 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
74		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑒 ∈ ℝ+
75	3, 74	nfan 1901	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑒 ∈ ℝ+)
76		nfv 1916	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑦 ∈ 𝐴
77	75, 76	nfan 1901	. . . . . . 7 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴)
78		nfra1 3148	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)
79		nfra1 3148	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡)
80		nfra1 3148	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒
81	78, 79, 80	nf3an 1903	. . . . . . 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 1901	. . . . . 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 2759	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡)))
84		eqid 2759	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ 1)
85		ssrab2 3985	. . . . . . 7 ⊢ {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} ⊆ 𝑇
86	6, 85	eqsstri 3927	. . . . . 6 ⊢ 𝑉 ⊆ 𝑇
87		simplr 769	. . . . . 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 775	. . . . . . 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 3893	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐶)
90	4, 5, 12, 89	fcnre 42017	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦:𝑇⟶ℝ)
91	88, 87, 90	syl2anc 588	. . . . . 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 3893	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶)
93	4, 5, 12, 92	fcnre 42017	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
94	88, 93	sylan 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)) ∧ 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 1161	. . . . . 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 1161	. . . . . 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 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)) ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
101		simpllr 776	. . . . . 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 1192	. . . . . 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 1193	. . . . . 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 1194	. . . . . 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 43039	. . . . 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 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝐷 ∈ ℝ+)
107		stoweidlem52.14	. . . . . . 7 ⊢ (𝜑 → 𝐷 < 1)
108	107	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝐷 < 1)
109	14	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝑃 ∈ 𝐴)
110	45	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝑃:𝑇⟶ℝ)
111		stoweidlem52.18	. . . . . . 7 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
112	111	adantr 485	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
113	63	adantr 485	. . . . . 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 1175	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
116	97	3adant1r 1175	. . . . . 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 489	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝑒 ∈ ℝ+)
119	2, 75, 6, 106, 108, 109, 110, 112, 113, 114, 115, 116, 117, 118	stoweidlem49 43047	. . . . 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 3214	. . . 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 3114	. . 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 519	. 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 2841	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑍 ∈ 𝑣 ↔ 𝑍 ∈ 𝑉))
124		sseq1 3918	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ 𝑈 ↔ 𝑉 ⊆ 𝑈))
125	123, 124	anbi12d 634	. . . 4 ⊢ (𝑣 = 𝑉 → ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ↔ (𝑍 ∈ 𝑉 ∧ 𝑉 ⊆ 𝑈)))
126		nfcv 2920	. . . . . . . 8 ⊢ Ⅎ𝑡𝑣
127	126, 42	raleqf 3316	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ↔ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒))
128	127	3anbi2d 1439	. . . . . 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 3222	. . . . 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 3127	. . . 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 634	. . 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 3542	. 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 588	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 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539  Ⅎwnf 1786   ∈ wcel 2112  Ⅎwnfc 2900  ∀wral 3071  ∃wrex 3072  {crab 3075   ∖ cdif 3856   ⊆ wss 3859  ∪ cuni 4796   class class class wbr 5030   ↦ cmpt 5110  ran crn 5523  ⟶wf 6329  ‘cfv 6333  (class class class)co 7148  ℝcr 10564  0cc0 10565  1c1 10566   + caddc 10568   · cmul 10570   < clt 10703   ≤ cle 10704   − cmin 10898   / cdiv 11325  2c2 11719  ℝ⁺crp 12420  (,)cioo 12769  topGenctg 16759   Cn ccn 21914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7457  ax-cnex 10621  ax-resscn 10622  ax-1cn 10623  ax-icn 10624  ax-addcl 10625  ax-addrcl 10626  ax-mulcl 10627  ax-mulrcl 10628  ax-mulcom 10629  ax-addass 10630  ax-mulass 10631  ax-distr 10632  ax-i2m1 10633  ax-1ne0 10634  ax-1rid 10635  ax-rnegex 10636  ax-rrecex 10637  ax-cnre 10638  ax-pre-lttri 10639  ax-pre-lttrn 10640  ax-pre-ltadd 10641  ax-pre-mulgt0 10642  ax-pre-sup 10643

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-nel 3057  df-ral 3076  df-rex 3077  df-reu 3078  df-rmo 3079  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4419  df-pw 4494  df-sn 4521  df-pr 4523  df-tp 4525  df-op 4527  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5428  df-eprel 5433  df-po 5441  df-so 5442  df-fr 5481  df-we 5483  df-xp 5528  df-rel 5529  df-cnv 5530  df-co 5531  df-dm 5532  df-rn 5533  df-res 5534  df-ima 5535  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6292  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7578  df-1st 7691  df-2nd 7692  df-wrecs 7955  df-recs 8016  df-rdg 8054  df-er 8297  df-map 8416  df-pm 8417  df-en 8526  df-dom 8527  df-sdom 8528  df-sup 8929  df-inf 8930  df-pnf 10705  df-mnf 10706  df-xr 10707  df-ltxr 10708  df-le 10709  df-sub 10900  df-neg 10901  df-div 11326  df-nn 11665  df-2 11727  df-3 11728  df-n0 11925  df-z 12011  df-uz 12273  df-q 12379  df-rp 12421  df-ioo 12773  df-fl 13201  df-seq 13409  df-exp 13470  df-cj 14496  df-re 14497  df-im 14498  df-sqrt 14632  df-abs 14633  df-clim 14883  df-rlim 14884  df-topgen 16765  df-top 21584  df-topon 21601  df-bases 21636  df-cn 21917

This theorem is referenced by:  stoweidlem56  43054