Theorem stoweidlem51 46066

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 4080	. . . 4 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ⊆ 𝐴
3	1, 2	eqsstri 4030	. . 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 12648	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
7		stoweidlem51.10	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8	7	nnzd 12640	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	7	nnge1d 12314	. . . . 5 ⊢ (𝜑 → 1 ≤ 𝑀)
10	7	nnred 12281	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
11	10	leidd 11829	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
12	6, 8, 8, 9, 11	elfzd 13555	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
13		stoweidlem51.12	. . . 4 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
14		stoweidlem51.2	. . . . 5 ⊢ Ⅎ𝑡𝜑
15		eqid 2737	. . . . 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 46031	. . . 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 45596	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
21	3, 20	sselid 3981	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
22	1	eleq2i 2833	. . . . . . 7 ⊢ (𝑋 ∈ 𝑌 ↔ 𝑋 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
23		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎℎ1
24		nfrab1 3457	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
25	1, 24	nfcxfr 2903	. . . . . . . . . . . . 13 ⊢ Ⅎℎ𝑌
26		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
27	25, 25, 26	nfmpo 7515	. . . . . . . . . . . 12 ⊢ Ⅎℎ(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
28	4, 27	nfcxfr 2903	. . . . . . . . . . 11 ⊢ Ⅎℎ𝑃
29		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎℎ𝑈
30	23, 28, 29	nfseq 14052	. . . . . . . . . 10 ⊢ Ⅎℎseq1(𝑃, 𝑈)
31		nfcv 2905	. . . . . . . . . 10 ⊢ Ⅎℎ𝑀
32	30, 31	nffv 6916	. . . . . . . . 9 ⊢ Ⅎℎ(seq1(𝑃, 𝑈)‘𝑀)
33	5, 32	nfcxfr 2903	. . . . . . . 8 ⊢ Ⅎℎ𝑋
34		nfcv 2905	. . . . . . . 8 ⊢ Ⅎℎ𝐴
35		nfcv 2905	. . . . . . . . 9 ⊢ Ⅎℎ𝑇
36		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎℎ0
37		nfcv 2905	. . . . . . . . . . 11 ⊢ Ⅎℎ ≤
38		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎℎ𝑡
39	33, 38	nffv 6916	. . . . . . . . . . 11 ⊢ Ⅎℎ(𝑋‘𝑡)
40	36, 37, 39	nfbr 5190	. . . . . . . . . 10 ⊢ Ⅎℎ0 ≤ (𝑋‘𝑡)
41	39, 37, 23	nfbr 5190	. . . . . . . . . 10 ⊢ Ⅎℎ(𝑋‘𝑡) ≤ 1
42	40, 41	nfan 1899	. . . . . . . . 9 ⊢ Ⅎℎ(0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)
43	35, 42	nfralw 3311	. . . . . . . 8 ⊢ Ⅎℎ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)
44		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡1
45		nfra1 3284	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
46		nfcv 2905	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝐴
47	45, 46	nfrabw 3475	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
48	1, 47	nfcxfr 2903	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝑌
49		nfmpt1 5250	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
50	48, 48, 49	nfmpo 7515	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
51	4, 50	nfcxfr 2903	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑃
52		nfcv 2905	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑈
53	44, 51, 52	nfseq 14052	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡seq1(𝑃, 𝑈)
54		nfcv 2905	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝑀
55	53, 54	nffv 6916	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(seq1(𝑃, 𝑈)‘𝑀)
56	5, 55	nfcxfr 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑋
57	56	nfeq2 2923	. . . . . . . . 9 ⊢ Ⅎ𝑡 ℎ = 𝑋
58		fveq1 6905	. . . . . . . . . . 11 ⊢ (ℎ = 𝑋 → (ℎ‘𝑡) = (𝑋‘𝑡))
59	58	breq2d 5155	. . . . . . . . . 10 ⊢ (ℎ = 𝑋 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
60	58	breq1d 5153	. . . . . . . . . 10 ⊢ (ℎ = 𝑋 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑋‘𝑡) ≤ 1))
61	59, 60	anbi12d 632	. . . . . . . . 9 ⊢ (ℎ = 𝑋 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
62	57, 61	ralbid 3273	. . . . . . . 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 3688	. . . . . . 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 275	. . . . . 6 ⊢ (𝑋 ∈ 𝑌 ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
65	20, 64	sylib 218	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
66	65	simprd 495	. . . 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 1914	. . . . . . 7 ⊢ Ⅎ𝑡 𝑖 ∈ (1...𝑀)
74	14, 73	nfan 1899	. . . . . 6 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑖 ∈ (1...𝑀))
75	13	ffvelcdmda 7104	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
76		fveq1 6905	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑈‘𝑖) → (ℎ‘𝑡) = ((𝑈‘𝑖)‘𝑡))
77	76	breq2d 5155	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑈‘𝑖) → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ ((𝑈‘𝑖)‘𝑡)))
78	76	breq1d 5153	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑈‘𝑖) → ((ℎ‘𝑡) ≤ 1 ↔ ((𝑈‘𝑖)‘𝑡) ≤ 1))
79	77, 78	anbi12d 632	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑈‘𝑖) → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
80	79	ralbidv 3178	. . . . . . . . . . . . . 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 3695	. . . . . . . . . . . . 13 ⊢ ((𝑈‘𝑖) ∈ 𝑌 ↔ ((𝑈‘𝑖) ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
82	81	simplbi 497	. . . . . . . . . . . 12 ⊢ ((𝑈‘𝑖) ∈ 𝑌 → (𝑈‘𝑖) ∈ 𝐴)
83	75, 82	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝐴)
84		eleq1 2829	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝐴 ↔ (𝑈‘𝑖) ∈ 𝐴))
85	84	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴)))
86		feq1 6716	. . . . . . . . . . . . . 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 3577	. . . . . . . . . . . 12 ⊢ ((𝑈‘𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴) → (𝑈‘𝑖):𝑇⟶ℝ))
90	89	anabsi7 671	. . . . . . . . . . 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 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → (𝑈‘𝑖):𝑇⟶ℝ)
93	70	ffvelcdmda 7104	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑊‘𝑖) ∈ 𝑉)
94		simpl 482	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
95	94, 93	jca 511	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉))
96		stoweidlem51.3	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤𝜑
97		stoweidlem51.4	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤𝑉
98	97	nfel2 2924	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(𝑊‘𝑖) ∈ 𝑉
99	96, 98	nfan 1899	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉)
100		nfv 1914	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(𝑊‘𝑖) ⊆ 𝑇
101	99, 100	nfim 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤((𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉) → (𝑊‘𝑖) ⊆ 𝑇)
102		eleq1 2829	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑊‘𝑖) → (𝑤 ∈ 𝑉 ↔ (𝑊‘𝑖) ∈ 𝑉))
103	102	anbi2d 630	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊‘𝑖) → ((𝜑 ∧ 𝑤 ∈ 𝑉) ↔ (𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉)))
104		sseq1 4009	. . . . . . . . . . . . 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 3570	. . . . . . . . . . 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 3983	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑡 ∈ 𝑇)
110	92, 109	ffvelcdmd 7105	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
111		stoweidlem51.22	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
112	111	rpred 13077	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℝ)
113	112	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝐸 ∈ ℝ)
114	10	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑀 ∈ ℝ)
115	7	nnne0d 12316	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≠ 0)
116	115	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑀 ≠ 0)
117	113, 114, 116	redivcld 12095	. . . . . . . 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 3251	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < (𝐸 / 𝑀))
120		1red 11262	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
121		0lt1 11785	. . . . . . . . . . . . 13 ⊢ 0 < 1
122	121	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
123	7	nngt0d 12315	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑀)
124	111	rpregt0d 13083	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
125		lediv2 12158	. . . . . . . . . . . 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 1383	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
127	9, 126	mpbid 232	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
128	111	rpcnd 13079	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℂ)
129	128	div1d 12035	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 1) = 𝐸)
130	127, 129	breqtrd 5169	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
131	130	ad2antrr 726	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → (𝐸 / 𝑀) ≤ 𝐸)
132	110, 117, 113, 119, 131	ltletrd 11421	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < 𝐸)
133	132	ex 412	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊‘𝑖) → ((𝑈‘𝑖)‘𝑡) < 𝐸))
134	74, 133	ralrimi 3257	. . . . 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 46063	. . . 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 3979	. . . . . 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 46057	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))
142	66, 135, 141	3jca 1129	. . 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 511	. 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 2829	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
145	56	nfeq2 2923	. . . . . 6 ⊢ Ⅎ𝑡 𝑥 = 𝑋
146		fveq1 6905	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑡) = (𝑋‘𝑡))
147	146	breq2d 5155	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
148	146	breq1d 5153	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) ≤ 1 ↔ (𝑋‘𝑡) ≤ 1))
149	147, 148	anbi12d 632	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
150	145, 149	ralbid 3273	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
151	146	breq1d 5153	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) < 𝐸 ↔ (𝑋‘𝑡) < 𝐸))
152	145, 151	ralbid 3273	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸))
153	146	breq2d 5155	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝑋‘𝑡)))
154	145, 153	ralbid 3273	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡)))
155	150, 152, 154	3anbi123d 1438	. . . 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 632	. . 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 3597	. 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))))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2108  Ⅎwnfc 2890   ≠ wne 2940  ∀wral 3061  {crab 3436  Vcvv 3480   ⊆ wss 3951  ∪ cuni 4907   class class class wbr 5143   ↦ cmpt 5225  ran crn 5686  ⟶wf 6557  ‘cfv 6561  (class class class)co 7431   ∈ cmpo 7433  ℝcr 11154  0cc0 11155  1c1 11156   · cmul 11160   < clt 11295   ≤ cle 11296   − cmin 11492   / cdiv 11920  ℕcn 12266  3c3 12322  ℝ⁺crp 13034  ...cfz 13547  seqcseq 14042

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755  ax-cnex 11211  ax-resscn 11212  ax-1cn 11213  ax-icn 11214  ax-addcl 11215  ax-addrcl 11216  ax-mulcl 11217  ax-mulrcl 11218  ax-mulcom 11219  ax-addass 11220  ax-mulass 11221  ax-distr 11222  ax-i2m1 11223  ax-1ne0 11224  ax-1rid 11225  ax-rnegex 11226  ax-rrecex 11227  ax-cnre 11228  ax-pre-lttri 11229  ax-pre-lttrn 11230  ax-pre-ltadd 11231  ax-pre-mulgt0 11232

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-1st 8014  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-er 8745  df-en 8986  df-dom 8987  df-sdom 8988  df-pnf 11297  df-mnf 11298  df-xr 11299  df-ltxr 11300  df-le 11301  df-sub 11494  df-neg 11495  df-div 11921  df-nn 12267  df-2 12329  df-3 12330  df-n0 12527  df-z 12614  df-uz 12879  df-rp 13035  df-fz 13548  df-fzo 13695  df-seq 14043  df-exp 14103

This theorem is referenced by:  stoweidlem54  46069