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Theorem stoweidlem51 39572
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 3666 . . . 4 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ⊆ 𝐴
31, 2eqsstri 3614 . . 3 𝑌𝐴
4 stoweidlem51.6 . . . 4 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
5 stoweidlem51.7 . . . 4 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
6 1zzd 11352 . . . . . 6 (𝜑 → 1 ∈ ℤ)
7 stoweidlem51.10 . . . . . . 7 (𝜑𝑀 ∈ ℕ)
87nnzd 11425 . . . . . 6 (𝜑𝑀 ∈ ℤ)
96, 8, 83jca 1240 . . . . 5 (𝜑 → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
107nnge1d 11007 . . . . . 6 (𝜑 → 1 ≤ 𝑀)
117nnred 10979 . . . . . . 7 (𝜑𝑀 ∈ ℝ)
1211leidd 10538 . . . . . 6 (𝜑𝑀𝑀)
1310, 12jca 554 . . . . 5 (𝜑 → (1 ≤ 𝑀𝑀𝑀))
14 elfz2 12275 . . . . 5 (𝑀 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (1 ≤ 𝑀𝑀𝑀)))
159, 13, 14sylanbrc 697 . . . 4 (𝜑𝑀 ∈ (1...𝑀))
16 stoweidlem51.12 . . . 4 (𝜑𝑈:(1...𝑀)⟶𝑌)
17 stoweidlem51.2 . . . . 5 𝑡𝜑
18 eqid 2621 . . . . 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 39537 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
22 stoweidlem51.21 . . . 4 (𝜑𝑇 ∈ V)
234, 5, 15, 16, 21, 22fmulcl 39214 . . 3 (𝜑𝑋𝑌)
243, 23sseldi 3581 . 2 (𝜑𝑋𝐴)
251eleq2i 2690 . . . . . . 7 (𝑋𝑌𝑋 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
26 nfcv 2761 . . . . . . . . . . 11 1
27 nfrab1 3111 . . . . . . . . . . . . . 14 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
281, 27nfcxfr 2759 . . . . . . . . . . . . 13 𝑌
29 nfcv 2761 . . . . . . . . . . . . 13 (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
3028, 28, 29nfmpt2 6677 . . . . . . . . . . . 12 (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
314, 30nfcxfr 2759 . . . . . . . . . . 11 𝑃
32 nfcv 2761 . . . . . . . . . . 11 𝑈
3326, 31, 32nfseq 12751 . . . . . . . . . 10 seq1(𝑃, 𝑈)
34 nfcv 2761 . . . . . . . . . 10 𝑀
3533, 34nffv 6155 . . . . . . . . 9 (seq1(𝑃, 𝑈)‘𝑀)
365, 35nfcxfr 2759 . . . . . . . 8 𝑋
37 nfcv 2761 . . . . . . . 8 𝐴
38 nfcv 2761 . . . . . . . . 9 𝑇
39 nfcv 2761 . . . . . . . . . . 11 0
40 nfcv 2761 . . . . . . . . . . 11
41 nfcv 2761 . . . . . . . . . . . 12 𝑡
4236, 41nffv 6155 . . . . . . . . . . 11 (𝑋𝑡)
4339, 40, 42nfbr 4659 . . . . . . . . . 10 0 ≤ (𝑋𝑡)
4442, 40, 26nfbr 4659 . . . . . . . . . 10 (𝑋𝑡) ≤ 1
4543, 44nfan 1825 . . . . . . . . 9 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)
4638, 45nfral 2940 . . . . . . . 8 𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)
47 nfcv 2761 . . . . . . . . . . . . 13 𝑡1
48 nfra1 2936 . . . . . . . . . . . . . . . . 17 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
49 nfcv 2761 . . . . . . . . . . . . . . . . 17 𝑡𝐴
5048, 49nfrab 3112 . . . . . . . . . . . . . . . 16 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
511, 50nfcxfr 2759 . . . . . . . . . . . . . . 15 𝑡𝑌
52 nfmpt1 4707 . . . . . . . . . . . . . . 15 𝑡(𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
5351, 51, 52nfmpt2 6677 . . . . . . . . . . . . . 14 𝑡(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
544, 53nfcxfr 2759 . . . . . . . . . . . . 13 𝑡𝑃
55 nfcv 2761 . . . . . . . . . . . . 13 𝑡𝑈
5647, 54, 55nfseq 12751 . . . . . . . . . . . 12 𝑡seq1(𝑃, 𝑈)
57 nfcv 2761 . . . . . . . . . . . 12 𝑡𝑀
5856, 57nffv 6155 . . . . . . . . . . 11 𝑡(seq1(𝑃, 𝑈)‘𝑀)
595, 58nfcxfr 2759 . . . . . . . . . 10 𝑡𝑋
6059nfeq2 2776 . . . . . . . . 9 𝑡 = 𝑋
61 fveq1 6147 . . . . . . . . . . 11 ( = 𝑋 → (𝑡) = (𝑋𝑡))
6261breq2d 4625 . . . . . . . . . 10 ( = 𝑋 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑋𝑡)))
6361breq1d 4623 . . . . . . . . . 10 ( = 𝑋 → ((𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
6462, 63anbi12d 746 . . . . . . . . 9 ( = 𝑋 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6560, 64ralbid 2977 . . . . . . . 8 ( = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6636, 37, 46, 65elrabf 3343 . . . . . . 7 (𝑋 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6725, 66bitri 264 . . . . . 6 (𝑋𝑌 ↔ (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6823, 67sylib 208 . . . . 5 (𝜑 → (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6968simprd 479 . . . 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 1840 . . . . . . 7 𝑡 𝑖 ∈ (1...𝑀)
7717, 76nfan 1825 . . . . . 6 𝑡(𝜑𝑖 ∈ (1...𝑀))
7816ffvelrnda 6315 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
79 fveq1 6147 . . . . . . . . . . . . . . . . 17 ( = (𝑈𝑖) → (𝑡) = ((𝑈𝑖)‘𝑡))
8079breq2d 4625 . . . . . . . . . . . . . . . 16 ( = (𝑈𝑖) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝑈𝑖)‘𝑡)))
8179breq1d 4623 . . . . . . . . . . . . . . . 16 ( = (𝑈𝑖) → ((𝑡) ≤ 1 ↔ ((𝑈𝑖)‘𝑡) ≤ 1))
8280, 81anbi12d 746 . . . . . . . . . . . . . . 15 ( = (𝑈𝑖) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8382ralbidv 2980 . . . . . . . . . . . . . 14 ( = (𝑈𝑖) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8483, 1elrab2 3348 . . . . . . . . . . . . 13 ((𝑈𝑖) ∈ 𝑌 ↔ ((𝑈𝑖) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8584simplbi 476 . . . . . . . . . . . 12 ((𝑈𝑖) ∈ 𝑌 → (𝑈𝑖) ∈ 𝐴)
8678, 85syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝐴)
87 eleq1 2686 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈𝑖) → (𝑓𝐴 ↔ (𝑈𝑖) ∈ 𝐴))
8887anbi2d 739 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝐴)))
89 feq1 5983 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈𝑖):𝑇⟶ℝ))
9088, 89imbi12d 334 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ)))
9119a1i 11 . . . . . . . . . . . . 13 (𝑓𝐴 → ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ))
9290, 91vtoclga 3258 . . . . . . . . . . . 12 ((𝑈𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ))
9392anabsi7 859 . . . . . . . . . . 11 ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ)
9486, 93syldan 487 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
9594adantr 481 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝑈𝑖):𝑇⟶ℝ)
9673ffvelrnda 6315 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊𝑖) ∈ 𝑉)
97 simpl 473 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → 𝜑)
9897, 96jca 554 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑊𝑖) ∈ 𝑉))
99 stoweidlem51.3 . . . . . . . . . . . . . 14 𝑤𝜑
100 stoweidlem51.4 . . . . . . . . . . . . . . 15 𝑤𝑉
101100nfel2 2777 . . . . . . . . . . . . . 14 𝑤(𝑊𝑖) ∈ 𝑉
10299, 101nfan 1825 . . . . . . . . . . . . 13 𝑤(𝜑 ∧ (𝑊𝑖) ∈ 𝑉)
103 nfv 1840 . . . . . . . . . . . . 13 𝑤(𝑊𝑖) ⊆ 𝑇
104102, 103nfim 1822 . . . . . . . . . . . 12 𝑤((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇)
105 eleq1 2686 . . . . . . . . . . . . . 14 (𝑤 = (𝑊𝑖) → (𝑤𝑉 ↔ (𝑊𝑖) ∈ 𝑉))
106105anbi2d 739 . . . . . . . . . . . . 13 (𝑤 = (𝑊𝑖) → ((𝜑𝑤𝑉) ↔ (𝜑 ∧ (𝑊𝑖) ∈ 𝑉)))
107 sseq1 3605 . . . . . . . . . . . . 13 (𝑤 = (𝑊𝑖) → (𝑤𝑇 ↔ (𝑊𝑖) ⊆ 𝑇))
108106, 107imbi12d 334 . . . . . . . . . . . 12 (𝑤 = (𝑊𝑖) → (((𝜑𝑤𝑉) → 𝑤𝑇) ↔ ((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇)))
109 stoweidlem51.13 . . . . . . . . . . . 12 ((𝜑𝑤𝑉) → 𝑤𝑇)
110104, 108, 109vtoclg1f 3251 . . . . . . . . . . 11 ((𝑊𝑖) ∈ 𝑉 → ((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇))
11196, 98, 110sylc 65 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊𝑖) ⊆ 𝑇)
112111sselda 3583 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑡𝑇)
11395, 112ffvelrnd 6316 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
114 stoweidlem51.22 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℝ+)
115114rpred 11816 . . . . . . . . . 10 (𝜑𝐸 ∈ ℝ)
116115ad2antrr 761 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝐸 ∈ ℝ)
11711ad2antrr 761 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑀 ∈ ℝ)
1187nnne0d 11009 . . . . . . . . . 10 (𝜑𝑀 ≠ 0)
119118ad2antrr 761 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑀 ≠ 0)
120116, 117, 119redivcld 10797 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝐸 / 𝑀) ∈ ℝ)
121 stoweidlem51.17 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑈𝑖)‘𝑡) < (𝐸 / 𝑀))
122121r19.21bi 2927 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < (𝐸 / 𝑀))
123 1red 9999 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℝ)
124 0lt1 10494 . . . . . . . . . . . . 13 0 < 1
125124a1i 11 . . . . . . . . . . . 12 (𝜑 → 0 < 1)
1267nngt0d 11008 . . . . . . . . . . . 12 (𝜑 → 0 < 𝑀)
127114rpregt0d 11822 . . . . . . . . . . . 12 (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
128 lediv2 10857 . . . . . . . . . . . 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 1334 . . . . . . . . . . 11 (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
13010, 129mpbid 222 . . . . . . . . . 10 (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
131114rpcnd 11818 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℂ)
132131div1d 10737 . . . . . . . . . 10 (𝜑 → (𝐸 / 1) = 𝐸)
133130, 132breqtrd 4639 . . . . . . . . 9 (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
134133ad2antrr 761 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝐸 / 𝑀) ≤ 𝐸)
135113, 120, 116, 122, 134ltletrd 10141 . . . . . . 7 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸)
136135ex 450 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊𝑖) → ((𝑈𝑖)‘𝑡) < 𝐸))
13777, 136ralrimi 2951 . . . . 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 39569 . . . 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 3579 . . . . . 6 (𝑓𝑌𝑓𝐴)
142141, 19sylan2 491 . . . . 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 39563 . . . 4 (𝜑 → ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡))
14569, 138, 1443jca 1240 . . 3 (𝜑 → (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑋𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡)))
14624, 145jca 554 . 2 (𝜑 → (𝑋𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑋𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡))))
147 eleq1 2686 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
14859nfeq2 2776 . . . . . 6 𝑡 𝑥 = 𝑋
149 fveq1 6147 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥𝑡) = (𝑋𝑡))
150149breq2d 4625 . . . . . . 7 (𝑥 = 𝑋 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝑋𝑡)))
151149breq1d 4623 . . . . . . 7 (𝑥 = 𝑋 → ((𝑥𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
152150, 151anbi12d 746 . . . . . 6 (𝑥 = 𝑋 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
153148, 152ralbid 2977 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
154149breq1d 4623 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑡) < 𝐸 ↔ (𝑋𝑡) < 𝐸))
155148, 154ralbid 2977 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝐷 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝐷 (𝑋𝑡) < 𝐸))
156149breq2d 4625 . . . . . 6 (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝑋𝑡)))
157148, 156ralbid 2977 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡)))
158153, 155, 1573anbi123d 1396 . . . 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 746 . . 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 3280 . 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 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wnf 1705  wcel 1987  wnfc 2748  wne 2790  wral 2907  {crab 2911  Vcvv 3186  wss 3555   cuni 4402   class class class wbr 4613  cmpt 4673  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  cmpt2 6606  cr 9879  0cc0 9880  1c1 9881   · cmul 9885   < clt 10018  cle 10019  cmin 10210   / cdiv 10628  cn 10964  3c3 11015  cz 11321  +crp 11776  ...cfz 12268  seqcseq 12741
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-div 10629  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-rp 11777  df-fz 12269  df-fzo 12407  df-seq 12742  df-exp 12801
This theorem is referenced by:  stoweidlem54  39575
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