Theorem stoweidlem51 46056

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 4046	. . . 4 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ⊆ 𝐴
3	1, 2	eqsstri 3996	. . 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 12571	. . . . 5 ⊢ (𝜑 → 1 ∈ ℤ)
7		stoweidlem51.10	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8	7	nnzd 12563	. . . . 5 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	7	nnge1d 12241	. . . . 5 ⊢ (𝜑 → 1 ≤ 𝑀)
10	7	nnred 12208	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℝ)
11	10	leidd 11751	. . . . 5 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
12	6, 8, 8, 9, 11	elfzd 13483	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
13		stoweidlem51.12	. . . 4 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
14		stoweidlem51.2	. . . . 5 ⊢ Ⅎ𝑡𝜑
15		eqid 2730	. . . . 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 46021	. . . 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 45586	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
21	3, 20	sselid 3947	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
22	1	eleq2i 2821	. . . . . . 7 ⊢ (𝑋 ∈ 𝑌 ↔ 𝑋 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
23		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎℎ1
24		nfrab1 3429	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
25	1, 24	nfcxfr 2890	. . . . . . . . . . . . 13 ⊢ Ⅎℎ𝑌
26		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
27	25, 25, 26	nfmpo 7474	. . . . . . . . . . . 12 ⊢ Ⅎℎ(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
28	4, 27	nfcxfr 2890	. . . . . . . . . . 11 ⊢ Ⅎℎ𝑃
29		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎℎ𝑈
30	23, 28, 29	nfseq 13983	. . . . . . . . . 10 ⊢ Ⅎℎseq1(𝑃, 𝑈)
31		nfcv 2892	. . . . . . . . . 10 ⊢ Ⅎℎ𝑀
32	30, 31	nffv 6871	. . . . . . . . 9 ⊢ Ⅎℎ(seq1(𝑃, 𝑈)‘𝑀)
33	5, 32	nfcxfr 2890	. . . . . . . 8 ⊢ Ⅎℎ𝑋
34		nfcv 2892	. . . . . . . 8 ⊢ Ⅎℎ𝐴
35		nfcv 2892	. . . . . . . . 9 ⊢ Ⅎℎ𝑇
36		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎℎ0
37		nfcv 2892	. . . . . . . . . . 11 ⊢ Ⅎℎ ≤
38		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎℎ𝑡
39	33, 38	nffv 6871	. . . . . . . . . . 11 ⊢ Ⅎℎ(𝑋‘𝑡)
40	36, 37, 39	nfbr 5157	. . . . . . . . . 10 ⊢ Ⅎℎ0 ≤ (𝑋‘𝑡)
41	39, 37, 23	nfbr 5157	. . . . . . . . . 10 ⊢ Ⅎℎ(𝑋‘𝑡) ≤ 1
42	40, 41	nfan 1899	. . . . . . . . 9 ⊢ Ⅎℎ(0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)
43	35, 42	nfralw 3287	. . . . . . . 8 ⊢ Ⅎℎ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)
44		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡1
45		nfra1 3262	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
46		nfcv 2892	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝐴
47	45, 46	nfrabw 3446	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
48	1, 47	nfcxfr 2890	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝑌
49		nfmpt1 5209	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
50	48, 48, 49	nfmpo 7474	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
51	4, 50	nfcxfr 2890	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑃
52		nfcv 2892	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑈
53	44, 51, 52	nfseq 13983	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡seq1(𝑃, 𝑈)
54		nfcv 2892	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝑀
55	53, 54	nffv 6871	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(seq1(𝑃, 𝑈)‘𝑀)
56	5, 55	nfcxfr 2890	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑋
57	56	nfeq2 2910	. . . . . . . . 9 ⊢ Ⅎ𝑡 ℎ = 𝑋
58		fveq1 6860	. . . . . . . . . . 11 ⊢ (ℎ = 𝑋 → (ℎ‘𝑡) = (𝑋‘𝑡))
59	58	breq2d 5122	. . . . . . . . . 10 ⊢ (ℎ = 𝑋 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
60	58	breq1d 5120	. . . . . . . . . 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 3251	. . . . . . . 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 3658	. . . . . . 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 7059	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
76		fveq1 6860	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑈‘𝑖) → (ℎ‘𝑡) = ((𝑈‘𝑖)‘𝑡))
77	76	breq2d 5122	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑈‘𝑖) → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ ((𝑈‘𝑖)‘𝑡)))
78	76	breq1d 5120	. . . . . . . . . . . . . . . 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 3157	. . . . . . . . . . . . . 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 3665	. . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . . 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 6669	. . . . . . . . . . . . . 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 3546	. . . . . . . . . . . 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 7059	. . . . . . . . . . 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 2911	. . . . . . . . . . . . . 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 2817	. . . . . . . . . . . . . 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 3975	. . . . . . . . . . . . 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 3539	. . . . . . . . . . 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 3949	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑡 ∈ 𝑇)
110	92, 109	ffvelcdmd 7060	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
111		stoweidlem51.22	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
112	111	rpred 13002	. . . . . . . . . 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 12243	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≠ 0)
116	115	ad2antrr 726	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑀 ≠ 0)
117	113, 114, 116	redivcld 12017	. . . . . . . 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 3230	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < (𝐸 / 𝑀))
120		1red 11182	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
121		0lt1 11707	. . . . . . . . . . . . 13 ⊢ 0 < 1
122	121	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
123	7	nngt0d 12242	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑀)
124	111	rpregt0d 13008	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
125		lediv2 12080	. . . . . . . . . . . 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 13004	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℂ)
129	128	div1d 11957	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 1) = 𝐸)
130	127, 129	breqtrd 5136	. . . . . . . . 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 11341	. . . . . . 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 3236	. . . . 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 46053	. . . 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 3945	. . . . . 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 46047	. . . 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 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 2817	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
145	56	nfeq2 2910	. . . . . 6 ⊢ Ⅎ𝑡 𝑥 = 𝑋
146		fveq1 6860	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑡) = (𝑋‘𝑡))
147	146	breq2d 5122	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
148	146	breq1d 5120	. . . . . . 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 3251	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
151	146	breq1d 5120	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) < 𝐸 ↔ (𝑋‘𝑡) < 𝐸))
152	145, 151	ralbid 3251	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸))
153	146	breq2d 5122	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝑋‘𝑡)))
154	145, 153	ralbid 3251	. . . . 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 3566	. 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 1086   = wceq 1540  ∃wex 1779  Ⅎwnf 1783   ∈ wcel 2109  Ⅎwnfc 2877   ≠ wne 2926  ∀wral 3045  {crab 3408  Vcvv 3450   ⊆ wss 3917  ∪ cuni 4874   class class class wbr 5110   ↦ cmpt 5191  ran crn 5642  ⟶wf 6510  ‘cfv 6514  (class class class)co 7390   ∈ cmpo 7392  ℝcr 11074  0cc0 11075  1c1 11076   · cmul 11080   < clt 11215   ≤ cle 11216   − cmin 11412   / cdiv 11842  ℕcn 12193  3c3 12249  ℝ⁺crp 12958  ...cfz 13475  seqcseq 13973

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-div 11843  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-rp 12959  df-fz 13476  df-fzo 13623  df-seq 13974  df-exp 14034

This theorem is referenced by:  stoweidlem54  46059