Theorem stoweidlem51 42693

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 4007	. . . 4 ⊢ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ⊆ 𝐴
3	1, 2	eqsstri 3949	. . 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 12001	. . . . . 6 ⊢ (𝜑 → 1 ∈ ℤ)
7		stoweidlem51.10	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℕ)
8	7	nnzd 12074	. . . . . 6 ⊢ (𝜑 → 𝑀 ∈ ℤ)
9	6, 8, 8	3jca 1125	. . . . 5 ⊢ (𝜑 → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
10	7	nnge1d 11673	. . . . . 6 ⊢ (𝜑 → 1 ≤ 𝑀)
11	7	nnred 11640	. . . . . . 7 ⊢ (𝜑 → 𝑀 ∈ ℝ)
12	11	leidd 11195	. . . . . 6 ⊢ (𝜑 → 𝑀 ≤ 𝑀)
13	10, 12	jca 515	. . . . 5 ⊢ (𝜑 → (1 ≤ 𝑀 ∧ 𝑀 ≤ 𝑀))
14		elfz2 12892	. . . . 5 ⊢ (𝑀 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (1 ≤ 𝑀 ∧ 𝑀 ≤ 𝑀)))
15	9, 13, 14	sylanbrc 586	. . . 4 ⊢ (𝜑 → 𝑀 ∈ (1...𝑀))
16		stoweidlem51.12	. . . 4 ⊢ (𝜑 → 𝑈:(1...𝑀)⟶𝑌)
17		stoweidlem51.2	. . . . 5 ⊢ Ⅎ𝑡𝜑
18		eqid 2798	. . . . 5 ⊢ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) = (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
19		stoweidlem51.20	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ)
20		stoweidlem51.19	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐴) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝐴)
21	17, 1, 18, 19, 20	stoweidlem16 42658	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌 ∧ 𝑔 ∈ 𝑌) → (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))) ∈ 𝑌)
22		stoweidlem51.21	. . . 4 ⊢ (𝜑 → 𝑇 ∈ V)
23	4, 5, 15, 16, 21, 22	fmulcl 42223	. . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑌)
24	3, 23	sseldi 3913	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
25	1	eleq2i 2881	. . . . . . 7 ⊢ (𝑋 ∈ 𝑌 ↔ 𝑋 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)})
26		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎℎ1
27		nfrab1 3337	. . . . . . . . . . . . . 14 ⊢ Ⅎℎ{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
28	1, 27	nfcxfr 2953	. . . . . . . . . . . . 13 ⊢ Ⅎℎ𝑌
29		nfcv 2955	. . . . . . . . . . . . 13 ⊢ Ⅎℎ(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
30	28, 28, 29	nfmpo 7215	. . . . . . . . . . . 12 ⊢ Ⅎℎ(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
31	4, 30	nfcxfr 2953	. . . . . . . . . . 11 ⊢ Ⅎℎ𝑃
32		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎℎ𝑈
33	26, 31, 32	nfseq 13374	. . . . . . . . . 10 ⊢ Ⅎℎseq1(𝑃, 𝑈)
34		nfcv 2955	. . . . . . . . . 10 ⊢ Ⅎℎ𝑀
35	33, 34	nffv 6655	. . . . . . . . 9 ⊢ Ⅎℎ(seq1(𝑃, 𝑈)‘𝑀)
36	5, 35	nfcxfr 2953	. . . . . . . 8 ⊢ Ⅎℎ𝑋
37		nfcv 2955	. . . . . . . 8 ⊢ Ⅎℎ𝐴
38		nfcv 2955	. . . . . . . . 9 ⊢ Ⅎℎ𝑇
39		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎℎ0
40		nfcv 2955	. . . . . . . . . . 11 ⊢ Ⅎℎ ≤
41		nfcv 2955	. . . . . . . . . . . 12 ⊢ Ⅎℎ𝑡
42	36, 41	nffv 6655	. . . . . . . . . . 11 ⊢ Ⅎℎ(𝑋‘𝑡)
43	39, 40, 42	nfbr 5077	. . . . . . . . . 10 ⊢ Ⅎℎ0 ≤ (𝑋‘𝑡)
44	42, 40, 26	nfbr 5077	. . . . . . . . . 10 ⊢ Ⅎℎ(𝑋‘𝑡) ≤ 1
45	43, 44	nfan 1900	. . . . . . . . 9 ⊢ Ⅎℎ(0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)
46	38, 45	nfralw 3189	. . . . . . . 8 ⊢ Ⅎℎ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)
47		nfcv 2955	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡1
48		nfra1 3183	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)
49		nfcv 2955	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑡𝐴
50	48, 49	nfrabw 3338	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑡{ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)}
51	1, 50	nfcxfr 2953	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡𝑌
52		nfmpt1 5128	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑡(𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡)))
53	51, 51, 52	nfmpo 7215	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑡(𝑓 ∈ 𝑌, 𝑔 ∈ 𝑌 ↦ (𝑡 ∈ 𝑇 ↦ ((𝑓‘𝑡) · (𝑔‘𝑡))))
54	4, 53	nfcxfr 2953	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑃
55		nfcv 2955	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑡𝑈
56	47, 54, 55	nfseq 13374	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡seq1(𝑃, 𝑈)
57		nfcv 2955	. . . . . . . . . . . 12 ⊢ Ⅎ𝑡𝑀
58	56, 57	nffv 6655	. . . . . . . . . . 11 ⊢ Ⅎ𝑡(seq1(𝑃, 𝑈)‘𝑀)
59	5, 58	nfcxfr 2953	. . . . . . . . . 10 ⊢ Ⅎ𝑡𝑋
60	59	nfeq2 2972	. . . . . . . . 9 ⊢ Ⅎ𝑡 ℎ = 𝑋
61		fveq1 6644	. . . . . . . . . . 11 ⊢ (ℎ = 𝑋 → (ℎ‘𝑡) = (𝑋‘𝑡))
62	61	breq2d 5042	. . . . . . . . . 10 ⊢ (ℎ = 𝑋 → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
63	61	breq1d 5040	. . . . . . . . . 10 ⊢ (ℎ = 𝑋 → ((ℎ‘𝑡) ≤ 1 ↔ (𝑋‘𝑡) ≤ 1))
64	62, 63	anbi12d 633	. . . . . . . . 9 ⊢ (ℎ = 𝑋 → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
65	60, 64	ralbid 3195	. . . . . . . 8 ⊢ (ℎ = 𝑋 → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
66	36, 37, 46, 65	elrabf 3624	. . . . . . 7 ⊢ (𝑋 ∈ {ℎ ∈ 𝐴 ∣ ∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1)} ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
67	25, 66	bitri 278	. . . . . 6 ⊢ (𝑋 ∈ 𝑌 ↔ (𝑋 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
68	23, 67	sylib 221	. . . . 5 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
69	68	simprd 499	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1))
70		stoweidlem51.1	. . . . 5 ⊢ Ⅎ𝑖𝜑
71		stoweidlem51.8	. . . . 5 ⊢ 𝐹 = (𝑡 ∈ 𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈‘𝑖)‘𝑡)))
72		stoweidlem51.9	. . . . 5 ⊢ 𝑍 = (𝑡 ∈ 𝑇 ↦ (seq1( · , (𝐹‘𝑡))‘𝑀))
73		stoweidlem51.11	. . . . 5 ⊢ (𝜑 → 𝑊:(1...𝑀)⟶𝑉)
74		stoweidlem51.14	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ ∪ ran 𝑊)
75		stoweidlem51.15	. . . . 5 ⊢ (𝜑 → 𝐷 ⊆ 𝑇)
76		nfv 1915	. . . . . . 7 ⊢ Ⅎ𝑡 𝑖 ∈ (1...𝑀)
77	17, 76	nfan 1900	. . . . . 6 ⊢ Ⅎ𝑡(𝜑 ∧ 𝑖 ∈ (1...𝑀))
78	16	ffvelrnda 6828	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝑌)
79		fveq1 6644	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑈‘𝑖) → (ℎ‘𝑡) = ((𝑈‘𝑖)‘𝑡))
80	79	breq2d 5042	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑈‘𝑖) → (0 ≤ (ℎ‘𝑡) ↔ 0 ≤ ((𝑈‘𝑖)‘𝑡)))
81	79	breq1d 5040	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑈‘𝑖) → ((ℎ‘𝑡) ≤ 1 ↔ ((𝑈‘𝑖)‘𝑡) ≤ 1))
82	80, 81	anbi12d 633	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑈‘𝑖) → ((0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
83	82	ralbidv 3162	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑈‘𝑖) → (∀𝑡 ∈ 𝑇 (0 ≤ (ℎ‘𝑡) ∧ (ℎ‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
84	83, 1	elrab2 3631	. . . . . . . . . . . . 13 ⊢ ((𝑈‘𝑖) ∈ 𝑌 ↔ ((𝑈‘𝑖) ∈ 𝐴 ∧ ∀𝑡 ∈ 𝑇 (0 ≤ ((𝑈‘𝑖)‘𝑡) ∧ ((𝑈‘𝑖)‘𝑡) ≤ 1)))
85	84	simplbi 501	. . . . . . . . . . . 12 ⊢ ((𝑈‘𝑖) ∈ 𝑌 → (𝑈‘𝑖) ∈ 𝐴)
86	78, 85	syl 17	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖) ∈ 𝐴)
87		eleq1 2877	. . . . . . . . . . . . . . 15 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓 ∈ 𝐴 ↔ (𝑈‘𝑖) ∈ 𝐴))
88	87	anbi2d 631	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑈‘𝑖) → ((𝜑 ∧ 𝑓 ∈ 𝐴) ↔ (𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴)))
89		feq1 6468	. . . . . . . . . . . . . 14 ⊢ (𝑓 = (𝑈‘𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈‘𝑖):𝑇⟶ℝ))
90	88, 89	imbi12d 348	. . . . . . . . . . . . 13 ⊢ (𝑓 = (𝑈‘𝑖) → (((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴) → (𝑈‘𝑖):𝑇⟶ℝ)))
91	19	a1i 11	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐴 → ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝑇⟶ℝ))
92	90, 91	vtoclga 3522	. . . . . . . . . . . 12 ⊢ ((𝑈‘𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴) → (𝑈‘𝑖):𝑇⟶ℝ))
93	92	anabsi7 670	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑈‘𝑖) ∈ 𝐴) → (𝑈‘𝑖):𝑇⟶ℝ)
94	86, 93	syldan 594	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑈‘𝑖):𝑇⟶ℝ)
95	94	adantr 484	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → (𝑈‘𝑖):𝑇⟶ℝ)
96	73	ffvelrnda 6828	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑊‘𝑖) ∈ 𝑉)
97		simpl 486	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → 𝜑)
98	97, 96	jca 515	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉))
99		stoweidlem51.3	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤𝜑
100		stoweidlem51.4	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑤𝑉
101	100	nfel2 2973	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑤(𝑊‘𝑖) ∈ 𝑉
102	99, 101	nfan 1900	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉)
103		nfv 1915	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑤(𝑊‘𝑖) ⊆ 𝑇
104	102, 103	nfim 1897	. . . . . . . . . . . 12 ⊢ Ⅎ𝑤((𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉) → (𝑊‘𝑖) ⊆ 𝑇)
105		eleq1 2877	. . . . . . . . . . . . . 14 ⊢ (𝑤 = (𝑊‘𝑖) → (𝑤 ∈ 𝑉 ↔ (𝑊‘𝑖) ∈ 𝑉))
106	105	anbi2d 631	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊‘𝑖) → ((𝜑 ∧ 𝑤 ∈ 𝑉) ↔ (𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉)))
107		sseq1 3940	. . . . . . . . . . . . 13 ⊢ (𝑤 = (𝑊‘𝑖) → (𝑤 ⊆ 𝑇 ↔ (𝑊‘𝑖) ⊆ 𝑇))
108	106, 107	imbi12d 348	. . . . . . . . . . . 12 ⊢ (𝑤 = (𝑊‘𝑖) → (((𝜑 ∧ 𝑤 ∈ 𝑉) → 𝑤 ⊆ 𝑇) ↔ ((𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉) → (𝑊‘𝑖) ⊆ 𝑇)))
109		stoweidlem51.13	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑤 ∈ 𝑉) → 𝑤 ⊆ 𝑇)
110	104, 108, 109	vtoclg1f 3514	. . . . . . . . . . 11 ⊢ ((𝑊‘𝑖) ∈ 𝑉 → ((𝜑 ∧ (𝑊‘𝑖) ∈ 𝑉) → (𝑊‘𝑖) ⊆ 𝑇))
111	96, 98, 110	sylc 65	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑊‘𝑖) ⊆ 𝑇)
112	111	sselda 3915	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑡 ∈ 𝑇)
113	95, 112	ffvelrnd 6829	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) ∈ ℝ)
114		stoweidlem51.22	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℝ+)
115	114	rpred 12419	. . . . . . . . . 10 ⊢ (𝜑 → 𝐸 ∈ ℝ)
116	115	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝐸 ∈ ℝ)
117	11	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑀 ∈ ℝ)
118	7	nnne0d 11675	. . . . . . . . . 10 ⊢ (𝜑 → 𝑀 ≠ 0)
119	118	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → 𝑀 ≠ 0)
120	116, 117, 119	redivcld 11457	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → (𝐸 / 𝑀) ∈ ℝ)
121		stoweidlem51.17	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊‘𝑖)((𝑈‘𝑖)‘𝑡) < (𝐸 / 𝑀))
122	121	r19.21bi 3173	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < (𝐸 / 𝑀))
123		1red 10631	. . . . . . . . . . . 12 ⊢ (𝜑 → 1 ∈ ℝ)
124		0lt1 11151	. . . . . . . . . . . . 13 ⊢ 0 < 1
125	124	a1i 11	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 1)
126	7	nngt0d 11674	. . . . . . . . . . . 12 ⊢ (𝜑 → 0 < 𝑀)
127	114	rpregt0d 12425	. . . . . . . . . . . 12 ⊢ (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
128		lediv2 11519	. . . . . . . . . . . 12 ⊢ (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀) ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
129	123, 125, 11, 126, 127, 128	syl221anc 1378	. . . . . . . . . . 11 ⊢ (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
130	10, 129	mpbid 235	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
131	114	rpcnd 12421	. . . . . . . . . . 11 ⊢ (𝜑 → 𝐸 ∈ ℂ)
132	131	div1d 11397	. . . . . . . . . 10 ⊢ (𝜑 → (𝐸 / 1) = 𝐸)
133	130, 132	breqtrd 5056	. . . . . . . . 9 ⊢ (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
134	133	ad2antrr 725	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → (𝐸 / 𝑀) ≤ 𝐸)
135	113, 120, 116, 122, 134	ltletrd 10789	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊‘𝑖)) → ((𝑈‘𝑖)‘𝑡) < 𝐸)
136	135	ex 416	. . . . . 6 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊‘𝑖) → ((𝑈‘𝑖)‘𝑡) < 𝐸))
137	77, 136	ralrimi 3180	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊‘𝑖)((𝑈‘𝑖)‘𝑡) < 𝐸)
138	70, 17, 1, 4, 5, 71, 72, 7, 73, 16, 74, 75, 137, 22, 19, 20, 114	stoweidlem48 42690	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸)
139		stoweidlem51.18	. . . . 5 ⊢ ((𝜑 ∧ 𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ 𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈‘𝑖)‘𝑡))
140		stoweidlem51.23	. . . . 5 ⊢ (𝜑 → 𝐸 < (1 / 3))
141	3	sseli 3911	. . . . . 6 ⊢ (𝑓 ∈ 𝑌 → 𝑓 ∈ 𝐴)
142	141, 19	sylan2 595	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑌) → 𝑓:𝑇⟶ℝ)
143		stoweidlem51.16	. . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝑇)
144	70, 17, 51, 4, 5, 71, 72, 7, 16, 139, 114, 140, 142, 21, 22, 143	stoweidlem42 42684	. . . 4 ⊢ (𝜑 → ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))
145	69, 138, 144	3jca 1125	. . 3 ⊢ (𝜑 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡)))
146	24, 145	jca 515	. 2 ⊢ (𝜑 → (𝑋 ∈ 𝐴 ∧ (∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1) ∧ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸 ∧ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡))))
147		eleq1 2877	. . . 4 ⊢ (𝑥 = 𝑋 → (𝑥 ∈ 𝐴 ↔ 𝑋 ∈ 𝐴))
148	59	nfeq2 2972	. . . . . 6 ⊢ Ⅎ𝑡 𝑥 = 𝑋
149		fveq1 6644	. . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑥‘𝑡) = (𝑋‘𝑡))
150	149	breq2d 5042	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (0 ≤ (𝑥‘𝑡) ↔ 0 ≤ (𝑋‘𝑡)))
151	149	breq1d 5040	. . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) ≤ 1 ↔ (𝑋‘𝑡) ≤ 1))
152	150, 151	anbi12d 633	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
153	148, 152	ralbid 3195	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝑇 (0 ≤ (𝑥‘𝑡) ∧ (𝑥‘𝑡) ≤ 1) ↔ ∀𝑡 ∈ 𝑇 (0 ≤ (𝑋‘𝑡) ∧ (𝑋‘𝑡) ≤ 1)))
154	149	breq1d 5040	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑥‘𝑡) < 𝐸 ↔ (𝑋‘𝑡) < 𝐸))
155	148, 154	ralbid 3195	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝐷 (𝑥‘𝑡) < 𝐸 ↔ ∀𝑡 ∈ 𝐷 (𝑋‘𝑡) < 𝐸))
156	149	breq2d 5042	. . . . . 6 ⊢ (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥‘𝑡) ↔ (1 − 𝐸) < (𝑋‘𝑡)))
157	148, 156	ralbid 3195	. . . . 5 ⊢ (𝑥 = 𝑋 → (∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑥‘𝑡) ↔ ∀𝑡 ∈ 𝐵 (1 − 𝐸) < (𝑋‘𝑡)))
158	153, 155, 157	3anbi123d 1433	. . . 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))))
159	147, 158	anbi12d 633	. . 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)))))
160	159	spcegv 3545	. 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)))))
161	24, 146, 160	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781  Ⅎwnf 1785   ∈ wcel 2111  Ⅎwnfc 2936   ≠ wne 2987  ∀wral 3106  {crab 3110  Vcvv 3441   ⊆ wss 3881  ∪ cuni 4800   class class class wbr 5030   ↦ cmpt 5110  ran crn 5520  ⟶wf 6320  ‘cfv 6324  (class class class)co 7135   ∈ cmpo 7137  ℝcr 10525  0cc0 10526  1c1 10527   · cmul 10531   < clt 10664   ≤ cle 10665   − cmin 10859   / cdiv 11286  ℕcn 11625  3c3 11681  ℤcz 11969  ℝ⁺crp 12377  ...cfz 12885  seqcseq 13364

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441  ax-cnex 10582  ax-resscn 10583  ax-1cn 10584  ax-icn 10585  ax-addcl 10586  ax-addrcl 10587  ax-mulcl 10588  ax-mulrcl 10589  ax-mulcom 10590  ax-addass 10591  ax-mulass 10592  ax-distr 10593  ax-i2m1 10594  ax-1ne0 10595  ax-1rid 10596  ax-rnegex 10597  ax-rrecex 10598  ax-cnre 10599  ax-pre-lttri 10600  ax-pre-lttrn 10601  ax-pre-ltadd 10602  ax-pre-mulgt0 10603

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-nel 3092  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-pred 6116  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-riota 7093  df-ov 7138  df-oprab 7139  df-mpo 7140  df-om 7561  df-1st 7671  df-2nd 7672  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-pnf 10666  df-mnf 10667  df-xr 10668  df-ltxr 10669  df-le 10670  df-sub 10861  df-neg 10862  df-div 11287  df-nn 11626  df-2 11688  df-3 11689  df-n0 11886  df-z 11970  df-uz 12232  df-rp 12378  df-fz 12886  df-fzo 13029  df-seq 13365  df-exp 13426

This theorem is referenced by:  stoweidlem54  42696