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Theorem stoweidlem51 41199
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 𝑖𝜑
stoweidlem51.2 𝑡𝜑
stoweidlem51.3 𝑤𝜑
stoweidlem51.4 𝑤𝑉
stoweidlem51.5 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
stoweidlem51.6 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
stoweidlem51.7 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
stoweidlem51.8 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
stoweidlem51.9 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
stoweidlem51.10 (𝜑𝑀 ∈ ℕ)
stoweidlem51.11 (𝜑𝑊:(1...𝑀)⟶𝑉)
stoweidlem51.12 (𝜑𝑈:(1...𝑀)⟶𝑌)
stoweidlem51.13 ((𝜑𝑤𝑉) → 𝑤𝑇)
stoweidlem51.14 (𝜑𝐷 ran 𝑊)
stoweidlem51.15 (𝜑𝐷𝑇)
stoweidlem51.16 (𝜑𝐵𝑇)
stoweidlem51.17 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑈𝑖)‘𝑡) < (𝐸 / 𝑀))
stoweidlem51.18 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈𝑖)‘𝑡))
stoweidlem51.19 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
stoweidlem51.20 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
stoweidlem51.21 (𝜑𝑇 ∈ V)
stoweidlem51.22 (𝜑𝐸 ∈ ℝ+)
stoweidlem51.23 (𝜑𝐸 < (1 / 3))
Assertion
Ref Expression
stoweidlem51 (𝜑 → ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡))))
Distinct variable groups:   𝑓,𝑔,,𝑡,𝐴   𝑓,𝑖,𝑀,,𝑡   𝑓,𝐹,𝑔   𝑇,𝑓,𝑔,,𝑡   𝑈,𝑓,𝑔,,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,,𝑖)   𝐴(𝑤,𝑖)   𝐵(𝑤,𝑡,𝑓,𝑔,)   𝐷(𝑤,𝑡,𝑓,𝑔,)   𝑃(𝑥,𝑤,𝑡,𝑓,𝑔,,𝑖)   𝑈(𝑥,𝑤)   𝐸(𝑤,𝑡,𝑓,𝑔,)   𝐹(𝑥,𝑤,𝑡,,𝑖)   𝑀(𝑥,𝑤)   𝑉(𝑥,𝑤,𝑡,𝑓,𝑔,,𝑖)   𝑊(𝑥,𝑡,𝑓,𝑔,)   𝑋(𝑤,𝑡,𝑓,𝑔,,𝑖)   𝑌(𝑥,𝑤,𝑡,,𝑖)   𝑍(𝑥,𝑤,𝑡,𝑓,𝑔,,𝑖)

Proof of Theorem stoweidlem51
StepHypRef Expression
1 stoweidlem51.5 . . . 4 𝑌 = {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
2 ssrab2 3908 . . . 4 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ⊆ 𝐴
31, 2eqsstri 3854 . . 3 𝑌𝐴
4 stoweidlem51.6 . . . 4 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
5 stoweidlem51.7 . . . 4 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
6 1zzd 11760 . . . . . 6 (𝜑 → 1 ∈ ℤ)
7 stoweidlem51.10 . . . . . . 7 (𝜑𝑀 ∈ ℕ)
87nnzd 11833 . . . . . 6 (𝜑𝑀 ∈ ℤ)
96, 8, 83jca 1119 . . . . 5 (𝜑 → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
107nnge1d 11423 . . . . . 6 (𝜑 → 1 ≤ 𝑀)
117nnred 11391 . . . . . . 7 (𝜑𝑀 ∈ ℝ)
1211leidd 10941 . . . . . 6 (𝜑𝑀𝑀)
1310, 12jca 507 . . . . 5 (𝜑 → (1 ≤ 𝑀𝑀𝑀))
14 elfz2 12650 . . . . 5 (𝑀 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (1 ≤ 𝑀𝑀𝑀)))
159, 13, 14sylanbrc 578 . . . 4 (𝜑𝑀 ∈ (1...𝑀))
16 stoweidlem51.12 . . . 4 (𝜑𝑈:(1...𝑀)⟶𝑌)
17 stoweidlem51.2 . . . . 5 𝑡𝜑
18 eqid 2778 . . . . 5 (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) = (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
19 stoweidlem51.20 . . . . 5 ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ)
20 stoweidlem51.19 . . . . 5 ((𝜑𝑓𝐴𝑔𝐴) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝐴)
2117, 1, 18, 19, 20stoweidlem16 41164 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
22 stoweidlem51.21 . . . 4 (𝜑𝑇 ∈ V)
234, 5, 15, 16, 21, 22fmulcl 40725 . . 3 (𝜑𝑋𝑌)
243, 23sseldi 3819 . 2 (𝜑𝑋𝐴)
251eleq2i 2851 . . . . . . 7 (𝑋𝑌𝑋 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
26 nfcv 2934 . . . . . . . . . . 11 1
27 nfrab1 3309 . . . . . . . . . . . . . 14 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
281, 27nfcxfr 2932 . . . . . . . . . . . . 13 𝑌
29 nfcv 2934 . . . . . . . . . . . . 13 (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
3028, 28, 29nfmpt2 7001 . . . . . . . . . . . 12 (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
314, 30nfcxfr 2932 . . . . . . . . . . 11 𝑃
32 nfcv 2934 . . . . . . . . . . 11 𝑈
3326, 31, 32nfseq 13129 . . . . . . . . . 10 seq1(𝑃, 𝑈)
34 nfcv 2934 . . . . . . . . . 10 𝑀
3533, 34nffv 6456 . . . . . . . . 9 (seq1(𝑃, 𝑈)‘𝑀)
365, 35nfcxfr 2932 . . . . . . . 8 𝑋
37 nfcv 2934 . . . . . . . 8 𝐴
38 nfcv 2934 . . . . . . . . 9 𝑇
39 nfcv 2934 . . . . . . . . . . 11 0
40 nfcv 2934 . . . . . . . . . . 11
41 nfcv 2934 . . . . . . . . . . . 12 𝑡
4236, 41nffv 6456 . . . . . . . . . . 11 (𝑋𝑡)
4339, 40, 42nfbr 4933 . . . . . . . . . 10 0 ≤ (𝑋𝑡)
4442, 40, 26nfbr 4933 . . . . . . . . . 10 (𝑋𝑡) ≤ 1
4543, 44nfan 1946 . . . . . . . . 9 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)
4638, 45nfral 3127 . . . . . . . 8 𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)
47 nfcv 2934 . . . . . . . . . . . . 13 𝑡1
48 nfra1 3123 . . . . . . . . . . . . . . . . 17 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
49 nfcv 2934 . . . . . . . . . . . . . . . . 17 𝑡𝐴
5048, 49nfrab 3310 . . . . . . . . . . . . . . . 16 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
511, 50nfcxfr 2932 . . . . . . . . . . . . . . 15 𝑡𝑌
52 nfmpt1 4982 . . . . . . . . . . . . . . 15 𝑡(𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
5351, 51, 52nfmpt2 7001 . . . . . . . . . . . . . 14 𝑡(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
544, 53nfcxfr 2932 . . . . . . . . . . . . 13 𝑡𝑃
55 nfcv 2934 . . . . . . . . . . . . 13 𝑡𝑈
5647, 54, 55nfseq 13129 . . . . . . . . . . . 12 𝑡seq1(𝑃, 𝑈)
57 nfcv 2934 . . . . . . . . . . . 12 𝑡𝑀
5856, 57nffv 6456 . . . . . . . . . . 11 𝑡(seq1(𝑃, 𝑈)‘𝑀)
595, 58nfcxfr 2932 . . . . . . . . . 10 𝑡𝑋
6059nfeq2 2949 . . . . . . . . 9 𝑡 = 𝑋
61 fveq1 6445 . . . . . . . . . . 11 ( = 𝑋 → (𝑡) = (𝑋𝑡))
6261breq2d 4898 . . . . . . . . . 10 ( = 𝑋 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑋𝑡)))
6361breq1d 4896 . . . . . . . . . 10 ( = 𝑋 → ((𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
6462, 63anbi12d 624 . . . . . . . . 9 ( = 𝑋 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6560, 64ralbid 3165 . . . . . . . 8 ( = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6636, 37, 46, 65elrabf 3568 . . . . . . 7 (𝑋 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6725, 66bitri 267 . . . . . 6 (𝑋𝑌 ↔ (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6823, 67sylib 210 . . . . 5 (𝜑 → (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6968simprd 491 . . . 4 (𝜑 → ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1))
70 stoweidlem51.1 . . . . 5 𝑖𝜑
71 stoweidlem51.8 . . . . 5 𝐹 = (𝑡𝑇 ↦ (𝑖 ∈ (1...𝑀) ↦ ((𝑈𝑖)‘𝑡)))
72 stoweidlem51.9 . . . . 5 𝑍 = (𝑡𝑇 ↦ (seq1( · , (𝐹𝑡))‘𝑀))
73 stoweidlem51.11 . . . . 5 (𝜑𝑊:(1...𝑀)⟶𝑉)
74 stoweidlem51.14 . . . . 5 (𝜑𝐷 ran 𝑊)
75 stoweidlem51.15 . . . . 5 (𝜑𝐷𝑇)
76 nfv 1957 . . . . . . 7 𝑡 𝑖 ∈ (1...𝑀)
7717, 76nfan 1946 . . . . . 6 𝑡(𝜑𝑖 ∈ (1...𝑀))
7816ffvelrnda 6623 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
79 fveq1 6445 . . . . . . . . . . . . . . . . 17 ( = (𝑈𝑖) → (𝑡) = ((𝑈𝑖)‘𝑡))
8079breq2d 4898 . . . . . . . . . . . . . . . 16 ( = (𝑈𝑖) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝑈𝑖)‘𝑡)))
8179breq1d 4896 . . . . . . . . . . . . . . . 16 ( = (𝑈𝑖) → ((𝑡) ≤ 1 ↔ ((𝑈𝑖)‘𝑡) ≤ 1))
8280, 81anbi12d 624 . . . . . . . . . . . . . . 15 ( = (𝑈𝑖) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8382ralbidv 3168 . . . . . . . . . . . . . 14 ( = (𝑈𝑖) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8483, 1elrab2 3576 . . . . . . . . . . . . 13 ((𝑈𝑖) ∈ 𝑌 ↔ ((𝑈𝑖) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8584simplbi 493 . . . . . . . . . . . 12 ((𝑈𝑖) ∈ 𝑌 → (𝑈𝑖) ∈ 𝐴)
8678, 85syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝐴)
87 eleq1 2847 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈𝑖) → (𝑓𝐴 ↔ (𝑈𝑖) ∈ 𝐴))
8887anbi2d 622 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝐴)))
89 feq1 6272 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈𝑖):𝑇⟶ℝ))
9088, 89imbi12d 336 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ)))
9119a1i 11 . . . . . . . . . . . . 13 (𝑓𝐴 → ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ))
9290, 91vtoclga 3474 . . . . . . . . . . . 12 ((𝑈𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ))
9392anabsi7 661 . . . . . . . . . . 11 ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ)
9486, 93syldan 585 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
9594adantr 474 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝑈𝑖):𝑇⟶ℝ)
9673ffvelrnda 6623 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊𝑖) ∈ 𝑉)
97 simpl 476 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → 𝜑)
9897, 96jca 507 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑊𝑖) ∈ 𝑉))
99 stoweidlem51.3 . . . . . . . . . . . . . 14 𝑤𝜑
100 stoweidlem51.4 . . . . . . . . . . . . . . 15 𝑤𝑉
101100nfel2 2950 . . . . . . . . . . . . . 14 𝑤(𝑊𝑖) ∈ 𝑉
10299, 101nfan 1946 . . . . . . . . . . . . 13 𝑤(𝜑 ∧ (𝑊𝑖) ∈ 𝑉)
103 nfv 1957 . . . . . . . . . . . . 13 𝑤(𝑊𝑖) ⊆ 𝑇
104102, 103nfim 1943 . . . . . . . . . . . 12 𝑤((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇)
105 eleq1 2847 . . . . . . . . . . . . . 14 (𝑤 = (𝑊𝑖) → (𝑤𝑉 ↔ (𝑊𝑖) ∈ 𝑉))
106105anbi2d 622 . . . . . . . . . . . . 13 (𝑤 = (𝑊𝑖) → ((𝜑𝑤𝑉) ↔ (𝜑 ∧ (𝑊𝑖) ∈ 𝑉)))
107 sseq1 3845 . . . . . . . . . . . . 13 (𝑤 = (𝑊𝑖) → (𝑤𝑇 ↔ (𝑊𝑖) ⊆ 𝑇))
108106, 107imbi12d 336 . . . . . . . . . . . 12 (𝑤 = (𝑊𝑖) → (((𝜑𝑤𝑉) → 𝑤𝑇) ↔ ((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇)))
109 stoweidlem51.13 . . . . . . . . . . . 12 ((𝜑𝑤𝑉) → 𝑤𝑇)
110104, 108, 109vtoclg1f 3466 . . . . . . . . . . 11 ((𝑊𝑖) ∈ 𝑉 → ((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇))
11196, 98, 110sylc 65 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊𝑖) ⊆ 𝑇)
112111sselda 3821 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑡𝑇)
11395, 112ffvelrnd 6624 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
114 stoweidlem51.22 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℝ+)
115114rpred 12181 . . . . . . . . . 10 (𝜑𝐸 ∈ ℝ)
116115ad2antrr 716 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝐸 ∈ ℝ)
11711ad2antrr 716 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑀 ∈ ℝ)
1187nnne0d 11425 . . . . . . . . . 10 (𝜑𝑀 ≠ 0)
119118ad2antrr 716 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑀 ≠ 0)
120116, 117, 119redivcld 11203 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝐸 / 𝑀) ∈ ℝ)
121 stoweidlem51.17 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑈𝑖)‘𝑡) < (𝐸 / 𝑀))
122121r19.21bi 3114 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < (𝐸 / 𝑀))
123 1red 10377 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℝ)
124 0lt1 10897 . . . . . . . . . . . . 13 0 < 1
125124a1i 11 . . . . . . . . . . . 12 (𝜑 → 0 < 1)
1267nngt0d 11424 . . . . . . . . . . . 12 (𝜑 → 0 < 𝑀)
127114rpregt0d 12187 . . . . . . . . . . . 12 (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
128 lediv2 11267 . . . . . . . . . . . 12 (((1 ∈ ℝ ∧ 0 < 1) ∧ (𝑀 ∈ ℝ ∧ 0 < 𝑀) ∧ (𝐸 ∈ ℝ ∧ 0 < 𝐸)) → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
129123, 125, 11, 126, 127, 128syl221anc 1449 . . . . . . . . . . 11 (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
13010, 129mpbid 224 . . . . . . . . . 10 (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
131114rpcnd 12183 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℂ)
132131div1d 11143 . . . . . . . . . 10 (𝜑 → (𝐸 / 1) = 𝐸)
133130, 132breqtrd 4912 . . . . . . . . 9 (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
134133ad2antrr 716 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝐸 / 𝑀) ≤ 𝐸)
135113, 120, 116, 122, 134ltletrd 10536 . . . . . . 7 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸)
136135ex 403 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊𝑖) → ((𝑈𝑖)‘𝑡) < 𝐸))
13777, 136ralrimi 3139 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑈𝑖)‘𝑡) < 𝐸)
13870, 17, 1, 4, 5, 71, 72, 7, 73, 16, 74, 75, 137, 22, 19, 20, 114stoweidlem48 41196 . . . 4 (𝜑 → ∀𝑡𝐷 (𝑋𝑡) < 𝐸)
139 stoweidlem51.18 . . . . 5 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡𝐵 (1 − (𝐸 / 𝑀)) < ((𝑈𝑖)‘𝑡))
140 stoweidlem51.23 . . . . 5 (𝜑𝐸 < (1 / 3))
1413sseli 3817 . . . . . 6 (𝑓𝑌𝑓𝐴)
142141, 19sylan2 586 . . . . 5 ((𝜑𝑓𝑌) → 𝑓:𝑇⟶ℝ)
143 stoweidlem51.16 . . . . 5 (𝜑𝐵𝑇)
14470, 17, 51, 4, 5, 71, 72, 7, 16, 139, 114, 140, 142, 21, 22, 143stoweidlem42 41190 . . . 4 (𝜑 → ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡))
14569, 138, 1443jca 1119 . . 3 (𝜑 → (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑋𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡)))
14624, 145jca 507 . 2 (𝜑 → (𝑋𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑋𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡))))
147 eleq1 2847 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
14859nfeq2 2949 . . . . . 6 𝑡 𝑥 = 𝑋
149 fveq1 6445 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥𝑡) = (𝑋𝑡))
150149breq2d 4898 . . . . . . 7 (𝑥 = 𝑋 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝑋𝑡)))
151149breq1d 4896 . . . . . . 7 (𝑥 = 𝑋 → ((𝑥𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
152150, 151anbi12d 624 . . . . . 6 (𝑥 = 𝑋 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
153148, 152ralbid 3165 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
154149breq1d 4896 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑡) < 𝐸 ↔ (𝑋𝑡) < 𝐸))
155148, 154ralbid 3165 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝐷 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝐷 (𝑋𝑡) < 𝐸))
156149breq2d 4898 . . . . . 6 (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝑋𝑡)))
157148, 156ralbid 3165 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡)))
158153, 155, 1573anbi123d 1509 . . . 4 (𝑥 = 𝑋 → ((∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)) ↔ (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑋𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡))))
159147, 158anbi12d 624 . . 3 (𝑥 = 𝑋 → ((𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡))) ↔ (𝑋𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑋𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡)))))
160159spcegv 3496 . 2 (𝑋𝐴 → ((𝑋𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑋𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡))) → ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡)))))
16124, 146, 160sylc 65 1 (𝜑 → ∃𝑥(𝑥𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑥𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 386  w3a 1071   = wceq 1601  wex 1823  wnf 1827  wcel 2107  wnfc 2919  wne 2969  wral 3090  {crab 3094  Vcvv 3398  wss 3792   cuni 4671   class class class wbr 4886  cmpt 4965  ran crn 5356  wf 6131  cfv 6135  (class class class)co 6922  cmpt2 6924  cr 10271  0cc0 10272  1c1 10273   · cmul 10277   < clt 10411  cle 10412  cmin 10606   / cdiv 11032  cn 11374  3c3 11431  cz 11728  +crp 12137  ...cfz 12643  seqcseq 13119
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226  ax-cnex 10328  ax-resscn 10329  ax-1cn 10330  ax-icn 10331  ax-addcl 10332  ax-addrcl 10333  ax-mulcl 10334  ax-mulrcl 10335  ax-mulcom 10336  ax-addass 10337  ax-mulass 10338  ax-distr 10339  ax-i2m1 10340  ax-1ne0 10341  ax-1rid 10342  ax-rnegex 10343  ax-rrecex 10344  ax-cnre 10345  ax-pre-lttri 10346  ax-pre-lttrn 10347  ax-pre-ltadd 10348  ax-pre-mulgt0 10349
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-nel 3076  df-ral 3095  df-rex 3096  df-reu 3097  df-rmo 3098  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-riota 6883  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-1st 7445  df-2nd 7446  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-er 8026  df-en 8242  df-dom 8243  df-sdom 8244  df-pnf 10413  df-mnf 10414  df-xr 10415  df-ltxr 10416  df-le 10417  df-sub 10608  df-neg 10609  df-div 11033  df-nn 11375  df-2 11438  df-3 11439  df-n0 11643  df-z 11729  df-uz 11993  df-rp 12138  df-fz 12644  df-fzo 12785  df-seq 13120  df-exp 13179
This theorem is referenced by:  stoweidlem54  41202
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