Theorem stoweidlem52 47265

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 2926	. . 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 13188	. . . . 5 ⊢ (𝜑 → 𝐷 ∈ ℝ)
9	8	rehalfcld 12617	. . . 4 ⊢ (𝜑 → (𝐷 / 2) ∈ ℝ)
10	9	rexrd 11383	. . 3 ⊢ (𝜑 → (𝐷 / 2) ∈ ℝ*)
11		stoweidlem52.9	. . . . 5 ⊢ (𝜑 → 𝐴 ⊆ 𝐶)
12		stoweidlem52.8	. . . . 5 ⊢ 𝐶 = (𝐽 Cn 𝐾)
13	11, 12	sseqtrdi 3953	. . . 4 ⊢ (𝜑 → 𝐴 ⊆ (𝐽 Cn 𝐾))
14		stoweidlem52.17	. . . 4 ⊢ (𝜑 → 𝑃 ∈ 𝐴)
15	13, 14	sseldd 3914	. . 3 ⊢ (𝜑 → 𝑃 ∈ (𝐽 Cn 𝐾))
16	1, 2, 3, 4, 5, 6, 10, 15	rfcnpre2 46251	. 2 ⊢ (𝜑 → 𝑉 ∈ 𝐽)
17		stoweidlem52.15	. . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ 𝐽)
18		elssuni 4902	. . . . . . . 8 ⊢ (𝑈 ∈ 𝐽 → 𝑈 ⊆ ∪ 𝐽)
19	17, 18	syl 18	. . . . . . 7 ⊢ (𝜑 → 𝑈 ⊆ ∪ 𝐽)
20	19, 5	sseqtrrdi 3954	. . . . . 6 ⊢ (𝜑 → 𝑈 ⊆ 𝑇)
21		stoweidlem52.16	. . . . . 6 ⊢ (𝜑 → 𝑍 ∈ 𝑈)
22	20, 21	sseldd 3914	. . . . 5 ⊢ (𝜑 → 𝑍 ∈ 𝑇)
23		stoweidlem52.19	. . . . . 6 ⊢ (𝜑 → (𝑃‘𝑍) = 0)
24		2re 12441	. . . . . . . 8 ⊢ 2 ∈ ℝ
25	24	a1i 11	. . . . . . 7 ⊢ (𝜑 → 2 ∈ ℝ)
26	7	rpgt0d 13191	. . . . . . 7 ⊢ (𝜑 → 0 < 𝐷)
27		2pos 12471	. . . . . . . 8 ⊢ 0 < 2
28	27	a1i 11	. . . . . . 7 ⊢ (𝜑 → 0 < 2)
29	8, 25, 26, 28	divgt0d 12276	. . . . . 6 ⊢ (𝜑 → 0 < (𝐷 / 2))
30	23, 29	eqbrtrd 5130	. . . . 5 ⊢ (𝜑 → (𝑃‘𝑍) < (𝐷 / 2))
31		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑡𝑍
32		nfcv 2926	. . . . . 6 ⊢ Ⅎ𝑡𝑇
33	2, 31	nffv 6915	. . . . . . 7 ⊢ Ⅎ𝑡(𝑃‘𝑍)
34		nfcv 2926	. . . . . . 7 ⊢ Ⅎ𝑡 <
35	33, 34, 1	nfbr 5155	. . . . . 6 ⊢ Ⅎ𝑡(𝑃‘𝑍) < (𝐷 / 2)
36		fveq2 6905	. . . . . . 7 ⊢ (𝑡 = 𝑍 → (𝑃‘𝑡) = (𝑃‘𝑍))
37	36	breq1d 5116	. . . . . 6 ⊢ (𝑡 = 𝑍 → ((𝑃‘𝑡) < (𝐷 / 2) ↔ (𝑃‘𝑍) < (𝐷 / 2)))
38	31, 32, 35, 37	elrabf 3645	. . . . 5 ⊢ (𝑍 ∈ {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} ↔ (𝑍 ∈ 𝑇 ∧ (𝑃‘𝑍) < (𝐷 / 2)))
39	22, 30, 38	sylanbrc 595	. . . 4 ⊢ (𝜑 → 𝑍 ∈ {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)})
40	39, 6	eleqtrrdi 2875	. . 3 ⊢ (𝜑 → 𝑍 ∈ 𝑉)
41		nfrab1 3435	. . . . 5 ⊢ Ⅎ𝑡{𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)}
42	6, 41	nfcxfr 2924	. . . 4 ⊢ Ⅎ𝑡𝑉
43		stoweidlem52.1	. . . 4 ⊢ Ⅎ𝑡𝑈
44	11, 14	sseldd 3914	. . . . . . . . . . 11 ⊢ (𝜑 → 𝑃 ∈ 𝐶)
45	4, 5, 12, 44	fcnre 46245	. . . . . . . . . 10 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ)
46	45	adantr 486	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝑃:𝑇⟶ℝ)
47	6	reqabi 3438	. . . . . . . . . . 11 ⊢ (𝑡 ∈ 𝑉 ↔ (𝑡 ∈ 𝑇 ∧ (𝑃‘𝑡) < (𝐷 / 2)))
48	47	bilani 510	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑡 ∈ 𝑇 ∧ (𝑃‘𝑡) < (𝐷 / 2)))
49	48	simpld 500	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝑡 ∈ 𝑇)
50	46, 49	ffvelcdmd 7106	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑃‘𝑡) ∈ ℝ)
51	9	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐷 / 2) ∈ ℝ)
52	8	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝐷 ∈ ℝ)
53	48	simprd 501	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑃‘𝑡) < (𝐷 / 2))
54		halfpos 12600	. . . . . . . . . . 11 ⊢ (𝐷 ∈ ℝ → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
55	8, 54	syl 18	. . . . . . . . . 10 ⊢ (𝜑 → (0 < 𝐷 ↔ (𝐷 / 2) < 𝐷))
56	26, 55	mpbid 235	. . . . . . . . 9 ⊢ (𝜑 → (𝐷 / 2) < 𝐷)
57	56	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝐷 / 2) < 𝐷)
58	50, 51, 52, 53, 57	lttrd 11495	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → (𝑃‘𝑡) < 𝐷)
59	58	adantr 486	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → (𝑃‘𝑡) < 𝐷)
60	8	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → 𝐷 ∈ ℝ)
61	50	adantr 486	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → (𝑃‘𝑡) ∈ ℝ)
62		stoweidlem52.20	. . . . . . . . 9 ⊢ (𝜑 → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡))
63	62	ad2antrr 739	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡))
64	49	anim1i 627	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → (𝑡 ∈ 𝑇 ∧ ¬ 𝑡 ∈ 𝑈))
65		eldif 4052	. . . . . . . . 9 ⊢ (𝑡 ∈ (𝑇 ∖ 𝑈) ↔ (𝑡 ∈ 𝑇 ∧ ¬ 𝑡 ∈ 𝑈))
66	64, 65	sylibr 237	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → 𝑡 ∈ (𝑇 ∖ 𝑈))
67		rsp 3254	. . . . . . . 8 ⊢ (∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡) → (𝑡 ∈ (𝑇 ∖ 𝑈) → 𝐷 ≤ (𝑃‘𝑡)))
68	63, 66, 67	sylc 66	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → 𝐷 ≤ (𝑃‘𝑡))
69	60, 61, 68	lensymd 11485	. . . . . 6 ⊢ (((𝜑 ∧ 𝑡 ∈ 𝑉) ∧ ¬ 𝑡 ∈ 𝑈) → ¬ (𝑃‘𝑡) < 𝐷)
70	59, 69	condan 830	. . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑉) → 𝑡 ∈ 𝑈)
71	70	ex 418	. . . 4 ⊢ (𝜑 → (𝑡 ∈ 𝑉 → 𝑡 ∈ 𝑈))
72	3, 42, 43, 71	ssrd 3918	. . 3 ⊢ (𝜑 → 𝑉 ⊆ 𝑈)
73		nfv 1949	. . . . . . . . 9 ⊢ Ⅎ𝑡 𝑒 ∈ ℝ+
74	3, 73	nfan 1934	. . . . . . . 8 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑒 ∈ ℝ+)
75		nfv 1949	. . . . . . . 8 ⊢ Ⅎ𝑡 𝑦 ∈ 𝐴
76	74, 75	nfan 1934	. . . . . . 7 ⊢ Ⅎ𝑡((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑦 ∈ 𝐴)
77		nfra1 3290	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1)
78		nfra1 3290	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡)
79		nfra1 3290	. . . . . . . 8 ⊢ Ⅎ𝑡∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒
80	77, 78, 79	nf3an 1936	. . . . . . 7 ⊢ Ⅎ𝑡(∀𝑡 ∈ 𝑇 (0 ≤ (𝑦‘𝑡) ∧ (𝑦‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝑉 (1 − 𝑒) < (𝑦‘𝑡) ∧ ∀𝑡 ∈ (𝑇 ∖ 𝑈)(𝑦‘𝑡) < 𝑒)
81	76, 80	nfan 1934	. . . . . 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))
82		eqid 2763	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡))) = (𝑡 ∈ 𝑇 ↦ (1 − (𝑦‘𝑡)))
83		eqid 2763	. . . . . 6 ⊢ (𝑡 ∈ 𝑇 ↦ 1) = (𝑡 ∈ 𝑇 ↦ 1)
84		ssrab2 4008	. . . . . . 7 ⊢ {𝑡 ∈ 𝑇 ∣ (𝑃‘𝑡) < (𝐷 / 2)} ⊆ 𝑇
85	6, 84	eqsstri 3959	. . . . . 6 ⊢ 𝑉 ⊆ 𝑇
86		simplr 781	. . . . . 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)
87		simplll 787	. . . . . . 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)
88	11	sselda 3913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦 ∈ 𝐶)
89	4, 5, 12, 88	fcnre 46245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → 𝑦:𝑇⟶ℝ)
90	87, 86, 89	syl2anc 596	. . . . . 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)
91	11	sselda 3913	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐶)
92	4, 5, 12, 91	fcnre 46245	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
93	87, 92	sylan 592	. . . . . 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)
94		stoweidlem52.10	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
95	87, 94	syl3an1 1181	. . . . . 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)
96		stoweidlem52.11	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
97	87, 96	syl3an1 1181	. . . . . 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)
98		stoweidlem52.12	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
99	87, 98	sylan 592	. . . . . 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)
100		simpllr 788	. . . . . 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+)
101		simpr1 1213	. . . . . 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))
102		simpr2 1214	. . . . . 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))
103		simpr3 1215	. . . . . 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)
104	81, 82, 83, 85, 86, 90, 93, 95, 97, 99, 100, 101, 102, 103	stoweidlem41 47254	. . . . 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)))
105	7	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝐷 ∈ ℝ+)
106		stoweidlem52.14	. . . . . . 7 ⊢ (𝜑 → 𝐷 < 1)
107	106	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝐷 < 1)
108	14	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝑃 ∈ 𝐴)
109	45	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝑃:𝑇⟶ℝ)
110		stoweidlem52.18	. . . . . . 7 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
111	110	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑃‘𝑡) ∧ (𝑃‘𝑡) ≤ 1))
112	62	adantr 486	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → ∀𝑡 ∈ (𝑇 ∖ 𝑈)𝐷 ≤ (𝑃‘𝑡))
113	92	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
114	94	3adant1r 1196	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) + (𝑔‘𝑡))) ∈ 𝐴)
115	96	3adant1r 1196	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
116	98	adantlr 728	. . . . . 6 ⊢ (((𝜑 ∧ 𝑒 ∈ ℝ+) ∧ 𝑎 ∈ ℝ) → (𝑡 ∈ 𝑇 ↦ 𝑎) ∈ 𝐴)
117		simpr 490	. . . . . 6 ⊢ ((𝜑 ∧ 𝑒 ∈ ℝ+) → 𝑒 ∈ ℝ+)
118	2, 74, 6, 105, 107, 108, 109, 111, 112, 113, 114, 115, 116, 117	stoweidlem49 47262	. . . . 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))
119	104, 118	r19.29a 3174	. . . 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)))
120	119	ralrimiva 3158	. . 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)))
121	40, 72, 120	jca31 524	. 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))))
122		eleq2 2853	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑍 ∈ 𝑣 ↔ 𝑍 ∈ 𝑉))
123		sseq1 3938	. . . . 5 ⊢ (𝑣 = 𝑉 → (𝑣 ⊆ 𝑈 ↔ 𝑉 ⊆ 𝑈))
124	122, 123	anbi12d 644	. . . 4 ⊢ (𝑣 = 𝑉 → ((𝑍 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑈) ↔ (𝑍 ∈ 𝑉 ∧ 𝑉 ⊆ 𝑈)))
125		nfcv 2926	. . . . . . . 8 ⊢ Ⅎ𝑡𝑣
126	125, 42	raleqf 3345	. . . . . . 7 ⊢ (𝑣 = 𝑉 → (∀𝑡 ∈ 𝑣 (𝑥‘𝑡) < 𝑒 ↔ ∀𝑡 ∈ 𝑉 (𝑥‘𝑡) < 𝑒))
127	126	3anbi2d 1470	. . . . . 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))))
128	127	rexbidv 3190	. . . . 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))))
129	128	ralbidv 3189	. . . 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))))
130	124, 129	anbi12d 644	. . 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)))))
131	130	rspcev 3580	. 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))))
132	16, 121, 131	syl2anc 596	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))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  Ⅎwnf 1816   ∈ wcel 2147  Ⅎwnfc 2911  ∀wral 3080  ∃wrex 3090  {crab 3416   ⊆ wss 3900   ∖ cdif 4048  ∪ cuni 4870   class class class wbr 5106   ↦ cmpt 5189  ran crn 5656  ⟶wf 6551  ‘cfv 6555  (class class class)co 7440  ℝcr 11223  0cc0 11224  1c1 11225   + caddc 11227   · cmul 11229   < clt 11367   ≤ cle 11368   − cmin 11565   / cdiv 11997  2c2 12421  ℝ⁺crp 13144  (,)cioo 13500  topGenctg 17634   Cn ccn 23569

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5235  ax-sep 5252  ax-nul 5263  ax-pow 5331  ax-pr 5395  ax-un 7772  ax-cnex 11280  ax-resscn 11281  ax-1cn 11282  ax-icn 11283  ax-addcl 11284  ax-addrcl 11285  ax-mulcl 11286  ax-mulrcl 11287  ax-mulcom 11288  ax-addass 11289  ax-mulass 11290  ax-distr 11291  ax-i2m1 11292  ax-1ne0 11293  ax-1rid 11294  ax-rnegex 11295  ax-rrecex 11296  ax-cnre 11297  ax-pre-lttri 11298  ax-pre-lttrn 11299  ax-pre-ltadd 11300  ax-pre-mulgt0 11301  ax-pre-sup 11302

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-nel 3066  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-mpt 5190  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6319  df-ord 6380  df-on 6381  df-lim 6382  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-f 6559  df-f1 6560  df-fo 6561  df-f1o 6562  df-fv 6563  df-riota 7397  df-ov 7443  df-oprab 7444  df-mpo 7445  df-om 7899  df-1st 8022  df-2nd 8023  df-frecs 8324  df-wrecs 8355  df-recs 8404  df-rdg 8443  df-er 8742  df-map 8874  df-pm 8875  df-en 8999  df-dom 9000  df-sdom 9001  df-sup 9458  df-inf 9459  df-pnf 11369  df-mnf 11370  df-xr 11371  df-ltxr 11372  df-le 11373  df-sub 11567  df-neg 11568  df-div 11998  df-nn 12360  df-2 12429  df-3 12430  df-n0 12631  df-z 12718  df-uz 12990  df-q 13100  df-rp 13145  df-ioo 13504  df-fl 13956  df-seq 14169  df-exp 14229  df-cj 15292  df-re 15293  df-im 15294  df-sqrt 15428  df-abs 15429  df-clim 15681  df-rlim 15682  df-topgen 17640  df-top 23239  df-topon 23256  df-bases 23291  df-cn 23572

This theorem is used by:  stoweidlem56  47269