stoweidlem51 - Mathbox for Glauco Siliprandi

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Theorem stoweidlem51 44753

Description: There exists a function x as in the proof of Lemma 2 in [BrosowskiDeutsh] p. 91. Here 𝐷 is used to represent 𝐴 in the paper, because here 𝐴 is used for the subalgebra of functions. 𝐸 is used to represent ε in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.)

**Hypotheses**
Ref	Expression
stoweidlem51.1	⊢ Ⅎ𝑖𝜑
stoweidlem51.2	⊢ Ⅎ𝑡𝜑
stoweidlem51.3	⊢ Ⅎ𝑤𝜑
stoweidlem51.4	⊢ Ⅎ𝑤𝑉
stoweidlem51.5	⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
stoweidlem51.6	⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
stoweidlem51.7	⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
stoweidlem51.8	⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
stoweidlem51.9	⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
stoweidlem51.10	⊢ (𝜑 → 𝑀 ∈ ℕ)
stoweidlem51.11	⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉)
stoweidlem51.12	⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
stoweidlem51.13	⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → 𝑤 ⊆ 𝑇)
stoweidlem51.14	⊢ (𝜑 → 𝐷 ⊆ ∪ ran 𝑊)
stoweidlem51.15	⊢ (𝜑 → 𝐷 ⊆ 𝑇)
stoweidlem51.16	⊢ (𝜑 → 𝐵 ⊆ 𝑇)
stoweidlem51.17	⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊‘𝑖)((𝑈‘𝑖)‘𝑡) < (𝐸 / 𝑀))
stoweidlem51.18	⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
stoweidlem51.19	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
stoweidlem51.20	⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem51.21	⊢ (𝜑 → 𝑇 ∈ V)
stoweidlem51.22	⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
stoweidlem51.23	⊢ (𝜑 → 𝐸 < (1 / 3))

**Assertion**
Ref	Expression
stoweidlem51	⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))))

Distinct variable groups: 𝑓,𝑔,ℎ,𝑡,𝐴 𝑓,𝑖,𝑀,ℎ,𝑡 𝑓,𝐹,𝑔 𝑇,𝑓,𝑔,ℎ,𝑡 𝑈,𝑓,𝑔,ℎ,𝑡 𝑓,𝑌,𝑔 𝜑,𝑓,𝑔 𝑔,𝑀 𝑤,𝑖,𝑇 𝐵,𝑖 𝐷,𝑖 𝑖,𝐸 𝑈,𝑖 𝑖,𝑊,𝑤 𝑥,𝑡,𝐴 𝑥,𝐵 𝑥,𝐷 𝑥,𝐸 𝑥,𝑇 𝑥,𝑋 Allowed substitution hints: 𝜑(𝑥,𝑤,𝑡,ℎ,𝑖) 𝐴(𝑤,𝑖) 𝐵(𝑤,𝑡,𝑓,𝑔,ℎ) 𝐷(𝑤,𝑡,𝑓,𝑔,ℎ) 𝑃(𝑥,𝑤,𝑡,𝑓,𝑔,ℎ,𝑖) 𝑈(𝑥,𝑤) 𝐸(𝑤,𝑡,𝑓,𝑔,ℎ) 𝐹(𝑥,𝑤,𝑡,ℎ,𝑖) 𝑀(𝑥,𝑤) 𝑉(𝑥,𝑤,𝑡,𝑓,𝑔,ℎ,𝑖) 𝑊(𝑥,𝑡,𝑓,𝑔,ℎ) 𝑋(𝑤,𝑡,𝑓,𝑔,ℎ,𝑖) 𝑌(𝑥,𝑤,𝑡,ℎ,𝑖) 𝑍(𝑥,𝑤,𝑡,𝑓,𝑔,ℎ,𝑖)

**Proof of Theorem stoweidlem51**
Step	Hyp	Ref	Expression
1		stoweidlem51.5	. . . 4 ⊢ 𝑌 = {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
2		ssrab2 4076	. . . 4 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ⊆ 𝐴
3	1, 2	eqsstri 4015	. . 3 ⊢ 𝑌 ⊆ 𝐴
4		stoweidlem51.6	. . . 4 ⊢ 𝑃 = (𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
5		stoweidlem51.7	. . . 4 ⊢ 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
6		1zzd 12589	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
7		stoweidlem51.10	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8	7	nnzd 12581	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	7	nnge1d 12256	. . . . 5 ⊢ (𝜑 → 1 ≤ 𝑀)
10	7	nnred 12223	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
11	10	leidd 11776	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
12	6, 8, 8, 9, 11	elfzd 13488	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
13		stoweidlem51.12	. . . 4 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
14		stoweidlem51.2	. . . . 5 ⊢ Ⅎ𝑡𝜑
15		eqid 2732	. . . . 5 ⊢ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
16		stoweidlem51.20	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
17		stoweidlem51.19	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
18	14, 1, 15, 16, 17	stoweidlem16 44718	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)
19		stoweidlem51.21	. . . 4 ⊢ (𝜑 → 𝑇 ∈ V)
20	4, 5, 12, 13, 18, 19	fmulcl 44283	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
21	3, 20	sselid 3979	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
22	1	eleq2i 2825	. . . . . . 7 ⊢ (𝑋 ∈ 𝑌 ↔ 𝑋 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
23		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎℎ1
24		nfrab1 3451	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
25	1, 24	nfcxfr 2901	. . . . . . . . . . . . 13 ⊢ Ⅎℎ𝑌
26		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
27	25, 25, 26	nfmpo 7487	. . . . . . . . . . . 12 ⊢ Ⅎℎ(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
28	4, 27	nfcxfr 2901	. . . . . . . . . . 11 ⊢ Ⅎℎ𝑃
29		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎℎ𝑈
30	23, 28, 29	nfseq 13972	. . . . . . . . . 10 ⊢ Ⅎℎseq1(𝑃, 𝑈)
31		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎℎ𝑀
32	30, 31	nffv 6898	. . . . . . . . 9 ⊢ Ⅎℎ(seq1(𝑃, 𝑈)‘𝑀)
33	5, 32	nfcxfr 2901	. . . . . . . 8 ⊢ Ⅎℎ𝑋
34		nfcv 2903	. . . . . . . 8 ⊢ Ⅎℎ𝐴
35		nfcv 2903	. . . . . . . . 9 ⊢ Ⅎℎ𝑇
36		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎℎ0
37		nfcv 2903	. . . . . . . . . . 11 ⊢ Ⅎℎ ≤
38		nfcv 2903	. . . . . . . . . . . 12 ⊢ Ⅎℎ𝑡
39	33, 38	nffv 6898	. . . . . . . . . . 11 ⊢ Ⅎℎ(𝑋‘𝑡)
40	36, 37, 39	nfbr 5194	. . . . . . . . . 10 ⊢ Ⅎℎ0 ≤ (𝑋‘𝑡)
41	39, 37, 23	nfbr 5194	. . . . . . . . . 10 ⊢ Ⅎℎ(𝑋‘𝑡) ≤ 1
42	40, 41	nfan 1902	. . . . . . . . 9 ⊢ Ⅎℎ(0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)
43	35, 42	nfralw 3308	. . . . . . . 8 ⊢ Ⅎℎ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)
44		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡1
45		nfra1 3281	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
46		nfcv 2903	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝐴
47	45, 46	nfrabw 3468	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
48	1, 47	nfcxfr 2901	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝑌
49		nfmpt1 5255	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
50	48, 48, 49	nfmpo 7487	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
51	4, 50	nfcxfr 2901	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑃
52		nfcv 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑈
53	44, 51, 52	nfseq 13972	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡seq1(𝑃, 𝑈)
54		nfcv 2903	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝑀
55	53, 54	nffv 6898	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(seq1(𝑃, 𝑈)‘𝑀)
56	5, 55	nfcxfr 2901	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑋
57	56	nfeq2 2920	. . . . . . . . 9 ⊢ Ⅎ𝑡 ℎ = 𝑋
58		fveq1 6887	. . . . . . . . . . 11 ⊢ (ℎ = 𝑋 → (ℎ‘𝑡) = (𝑋‘𝑡))
59	58	breq2d 5159	. . . . . . . . . 10 ⊢ (ℎ = 𝑋 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
60	58	breq1d 5157	. . . . . . . . . 10 ⊢ (ℎ = 𝑋 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑋‘𝑡) ≤ 1))
61	59, 60	anbi12d 631	. . . . . . . . 9 ⊢ (ℎ = 𝑋 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
62	57, 61	ralbid 3270	. . . . . . . 8 ⊢ (ℎ = 𝑋 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
63	33, 34, 43, 62	elrabf 3678	. . . . . . 7 ⊢ (𝑋 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
64	22, 63	bitri 274	. . . . . 6 ⊢ (𝑋 ∈ 𝑌 ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
65	20, 64	sylib 217	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
66	65	simprd 496	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1))
67		stoweidlem51.1	. . . . 5 ⊢ Ⅎ𝑖𝜑
68		stoweidlem51.8	. . . . 5 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
69		stoweidlem51.9	. . . . 5 ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
70		stoweidlem51.11	. . . . 5 ⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉)
71		stoweidlem51.14	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ∪ ran 𝑊)
72		stoweidlem51.15	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝑇)
73		nfv 1917	. . . . . . 7 ⊢ Ⅎ𝑡 𝑖 ∈ (1...𝑀)
74	14, 73	nfan 1902	. . . . . 6 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑖 ∈ (1...𝑀))
75	13	ffvelcdmda 7083	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
76		fveq1 6887	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑈‘𝑖) → (ℎ‘𝑡) = ((𝑈‘𝑖)‘𝑡))
77	76	breq2d 5159	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑈‘𝑖) → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ ((𝑈‘𝑖)‘𝑡)))
78	76	breq1d 5157	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑈‘𝑖) → ((ℎ‘𝑡) ≤ 1 ↔ ((𝑈‘𝑖)‘𝑡) ≤ 1))
79	77, 78	anbi12d 631	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑈‘𝑖) → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
80	79	ralbidv 3177	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑈‘𝑖) → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
81	80, 1	elrab2 3685	. . . . . . . . . . . . 13 ⊢ ((𝑈‘𝑖) ∈ 𝑌 ↔ ((𝑈‘𝑖) ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
82	81	simplbi 498	. . . . . . . . . . . 12 ⊢ ((𝑈‘𝑖) ∈ 𝑌 → (𝑈‘𝑖) ∈ 𝐴)
83	75, 82	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝐴)
84		eleq1 2821	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝐴 ↔ (𝑈‘𝑖) ∈ 𝐴))
85	84	anbi2d 629	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴)))
86		feq1 6695	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘𝑖):𝑇⟶ℝ))
87	85, 86	imbi12d 344	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑈‘𝑖) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴) → (𝑈‘𝑖):𝑇⟶ℝ)))
88	16	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐴 → ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ))
89	87, 88	vtoclga 3565	. . . . . . . . . . . 12 ⊢ ((𝑈‘𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴) → (𝑈‘𝑖):𝑇⟶ℝ))
90	89	anabsi7 669	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴) → (𝑈‘𝑖):𝑇⟶ℝ)
91	83, 90	syldan 591	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
92	91	adantr 481	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → (𝑈‘𝑖):𝑇⟶ℝ)
93	70	ffvelcdmda 7083	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑊‘𝑖) ∈ 𝑉)
94		simpl 483	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
95	94, 93	jca 512	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉))
96		stoweidlem51.3	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤𝜑
97		stoweidlem51.4	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤𝑉
98	97	nfel2 2921	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(𝑊‘𝑖) ∈ 𝑉
99	96, 98	nfan 1902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉)
100		nfv 1917	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(𝑊‘𝑖) ⊆ 𝑇
101	99, 100	nfim 1899	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤((𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉) → (𝑊‘𝑖) ⊆ 𝑇)
102		eleq1 2821	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑊‘𝑖) → (𝑤 ∈ 𝑉 ↔ (𝑊‘𝑖) ∈ 𝑉))
103	102	anbi2d 629	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊‘𝑖) → ((𝜑 ∧ 𝑤 ∈ 𝑉) ↔ (𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉)))
104		sseq1 4006	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊‘𝑖) → (𝑤 ⊆ 𝑇 ↔ (𝑊‘𝑖) ⊆ 𝑇))
105	103, 104	imbi12d 344	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑊‘𝑖) → (((𝜑 ∧ 𝑤 ∈ 𝑉) → 𝑤 ⊆ 𝑇) ↔ ((𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉) → (𝑊‘𝑖) ⊆ 𝑇)))
106		stoweidlem51.13	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → 𝑤 ⊆ 𝑇)
107	101, 105, 106	vtoclg1f 3555	. . . . . . . . . . 11 ⊢ ((𝑊‘𝑖) ∈ 𝑉 → ((𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉) → (𝑊‘𝑖) ⊆ 𝑇))
108	93, 95, 107	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑊‘𝑖) ⊆ 𝑇)
109	108	sselda 3981	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑡 ∈ 𝑇)
110	92, 109	ffvelcdmd 7084	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
111		stoweidlem51.22	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ⁺)
112	111	rpred 13012	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℝ)
113	112	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝐸 ∈ ℝ)
114	10	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑀 ∈ ℝ)
115	7	nnne0d 12258	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≠ 0)
116	115	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑀 ≠ 0)
117	113, 114, 116	redivcld 12038	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → (𝐸 / 𝑀) ∈ ℝ)
118		stoweidlem51.17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊‘𝑖)((𝑈‘𝑖)‘𝑡) < (𝐸 / 𝑀))
119	118	r19.21bi 3248	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < (𝐸 / 𝑀))
120		1red 11211	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
121		0lt1 11732	. . . . . . . . . . . . 13 ⊢ 0 < 1
122	121	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
123	7	nngt0d 12257	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑀)
124	111	rpregt0d 13018	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
125		lediv2 12100	. . . . . . . . . . . 12 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀) ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
126	120, 122, 10, 123, 124, 125	syl221anc 1381	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
127	9, 126	mpbid 231	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
128	111	rpcnd 13014	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℂ)
129	128	div1d 11978	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 1) = 𝐸)
130	127, 129	breqtrd 5173	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
131	130	ad2antrr 724	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → (𝐸 / 𝑀) ≤ 𝐸)
132	110, 117, 113, 119, 131	ltletrd 11370	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < 𝐸)
133	132	ex 413	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊‘𝑖) → ((𝑈‘𝑖)‘𝑡) < 𝐸))
134	74, 133	ralrimi 3254	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊‘𝑖)((𝑈‘𝑖)‘𝑡) < 𝐸)
135	67, 14, 1, 4, 5, 68, 69, 7, 70, 13, 71, 72, 134, 19, 16, 17, 111	stoweidlem48 44750	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸)
136		stoweidlem51.18	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
137		stoweidlem51.23	. . . . 5 ⊢ (𝜑 → 𝐸 < (1 / 3))
138	3	sseli 3977	. . . . . 6 ⊢ (𝑓 ∈ 𝑌 → 𝑓 ∈ 𝐴)
139	138, 16	sylan2 593	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)
140		stoweidlem51.16	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
141	67, 14, 48, 4, 5, 68, 69, 7, 13, 136, 111, 137, 139, 18, 19, 140	stoweidlem42 44744	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))
142	66, 135, 141	3jca 1128	. . 3 ⊢ (𝜑 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡)))
143	21, 142	jca 512	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))))
144		eleq1 2821	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
145	56	nfeq2 2920	. . . . . 6 ⊢ Ⅎ𝑡 𝑥 = 𝑋
146		fveq1 6887	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑡) = (𝑋‘𝑡))
147	146	breq2d 5159	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
148	146	breq1d 5157	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) ≤ 1 ↔ (𝑋‘𝑡) ≤ 1))
149	147, 148	anbi12d 631	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
150	145, 149	ralbid 3270	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
151	146	breq1d 5157	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) < 𝐸 ↔ (𝑋‘𝑡) < 𝐸))
152	145, 151	ralbid 3270	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸))
153	146	breq2d 5159	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝑋‘𝑡)))
154	145, 153	ralbid 3270	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡)))
155	150, 152, 154	3anbi123d 1436	. . . 4 ⊢ (𝑥 = 𝑋 → ((∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)) ↔ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))))
156	144, 155	anbi12d 631	. . 3 ⊢ (𝑥 = 𝑋 → ((𝑥 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))) ↔ (𝑋 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡)))))
157	156	spcegv 3587	. 2 ⊢ (𝑋 ∈ 𝐴 → ((𝑋 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))) → ∃𝑥(𝑥 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡)))))
158	21, 143, 157	sylc 65	1 ⊢ (𝜑 → ∃𝑥(𝑥 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∃wex 1781 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 ≠ wne 2940 ∀wral 3061 {crab 3432 Vcvv 3474 ⊆ wss 3947 ∪ cuni 4907 class class class wbr 5147 ↦ cmpt 5230 ran crn 5676 ⟶wf 6536 ‘cfv 6540 (class class class)co 7405 ∈ cmpo 7407 ℝcr 11105 0cc0 11106 1c1 11107 · cmul 11111 < clt 11244 ≤ cle 11245 − cmin 11440 / cdiv 11867 ℕcn 12208 3c3 12264 ℝ⁺crp 12970 ...cfz 13480 seqcseq 13962

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-er 8699 df-en 8936 df-dom 8937 df-sdom 8938 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-2 12271 df-3 12272 df-n0 12469 df-z 12555 df-uz 12819 df-rp 12971 df-fz 13481 df-fzo 13624 df-seq 13963 df-exp 14024

This theorem is referenced by: stoweidlem54 44756

W3C validator