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Theorem stoweidlem51 40908
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 3849 . . . 4 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ⊆ 𝐴
31, 2eqsstri 3797 . . 3 𝑌𝐴
4 stoweidlem51.6 . . . 4 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
5 stoweidlem51.7 . . . 4 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
6 1zzd 11658 . . . . . 6 (𝜑 → 1 ∈ ℤ)
7 stoweidlem51.10 . . . . . . 7 (𝜑𝑀 ∈ ℕ)
87nnzd 11731 . . . . . 6 (𝜑𝑀 ∈ ℤ)
96, 8, 83jca 1158 . . . . 5 (𝜑 → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
107nnge1d 11322 . . . . . 6 (𝜑 → 1 ≤ 𝑀)
117nnred 11293 . . . . . . 7 (𝜑𝑀 ∈ ℝ)
1211leidd 10850 . . . . . 6 (𝜑𝑀𝑀)
1310, 12jca 507 . . . . 5 (𝜑 → (1 ≤ 𝑀𝑀𝑀))
14 elfz2 12543 . . . . 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 2765 . . . . 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 40873 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
22 stoweidlem51.21 . . . 4 (𝜑𝑇 ∈ V)
234, 5, 15, 16, 21, 22fmulcl 40454 . . 3 (𝜑𝑋𝑌)
243, 23sseldi 3761 . 2 (𝜑𝑋𝐴)
251eleq2i 2836 . . . . . . 7 (𝑋𝑌𝑋 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
26 nfcv 2907 . . . . . . . . . . 11 1
27 nfrab1 3270 . . . . . . . . . . . . . 14 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
281, 27nfcxfr 2905 . . . . . . . . . . . . 13 𝑌
29 nfcv 2907 . . . . . . . . . . . . 13 (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
3028, 28, 29nfmpt2 6924 . . . . . . . . . . . 12 (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
314, 30nfcxfr 2905 . . . . . . . . . . 11 𝑃
32 nfcv 2907 . . . . . . . . . . 11 𝑈
3326, 31, 32nfseq 13021 . . . . . . . . . 10 seq1(𝑃, 𝑈)
34 nfcv 2907 . . . . . . . . . 10 𝑀
3533, 34nffv 6387 . . . . . . . . 9 (seq1(𝑃, 𝑈)‘𝑀)
365, 35nfcxfr 2905 . . . . . . . 8 𝑋
37 nfcv 2907 . . . . . . . 8 𝐴
38 nfcv 2907 . . . . . . . . 9 𝑇
39 nfcv 2907 . . . . . . . . . . 11 0
40 nfcv 2907 . . . . . . . . . . 11
41 nfcv 2907 . . . . . . . . . . . 12 𝑡
4236, 41nffv 6387 . . . . . . . . . . 11 (𝑋𝑡)
4339, 40, 42nfbr 4858 . . . . . . . . . 10 0 ≤ (𝑋𝑡)
4442, 40, 26nfbr 4858 . . . . . . . . . 10 (𝑋𝑡) ≤ 1
4543, 44nfan 1998 . . . . . . . . 9 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)
4638, 45nfral 3092 . . . . . . . 8 𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)
47 nfcv 2907 . . . . . . . . . . . . 13 𝑡1
48 nfra1 3088 . . . . . . . . . . . . . . . . 17 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
49 nfcv 2907 . . . . . . . . . . . . . . . . 17 𝑡𝐴
5048, 49nfrab 3271 . . . . . . . . . . . . . . . 16 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
511, 50nfcxfr 2905 . . . . . . . . . . . . . . 15 𝑡𝑌
52 nfmpt1 4908 . . . . . . . . . . . . . . 15 𝑡(𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
5351, 51, 52nfmpt2 6924 . . . . . . . . . . . . . 14 𝑡(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
544, 53nfcxfr 2905 . . . . . . . . . . . . 13 𝑡𝑃
55 nfcv 2907 . . . . . . . . . . . . 13 𝑡𝑈
5647, 54, 55nfseq 13021 . . . . . . . . . . . 12 𝑡seq1(𝑃, 𝑈)
57 nfcv 2907 . . . . . . . . . . . 12 𝑡𝑀
5856, 57nffv 6387 . . . . . . . . . . 11 𝑡(seq1(𝑃, 𝑈)‘𝑀)
595, 58nfcxfr 2905 . . . . . . . . . 10 𝑡𝑋
6059nfeq2 2923 . . . . . . . . 9 𝑡 = 𝑋
61 fveq1 6376 . . . . . . . . . . 11 ( = 𝑋 → (𝑡) = (𝑋𝑡))
6261breq2d 4823 . . . . . . . . . 10 ( = 𝑋 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑋𝑡)))
6361breq1d 4821 . . . . . . . . . 10 ( = 𝑋 → ((𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
6462, 63anbi12d 624 . . . . . . . . 9 ( = 𝑋 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6560, 64ralbid 3130 . . . . . . . 8 ( = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6636, 37, 46, 65elrabf 3517 . . . . . . 7 (𝑋 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6725, 66bitri 266 . . . . . 6 (𝑋𝑌 ↔ (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6823, 67sylib 209 . . . . 5 (𝜑 → (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6968simprd 489 . . . 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 2009 . . . . . . 7 𝑡 𝑖 ∈ (1...𝑀)
7717, 76nfan 1998 . . . . . 6 𝑡(𝜑𝑖 ∈ (1...𝑀))
7816ffvelrnda 6551 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
79 fveq1 6376 . . . . . . . . . . . . . . . . 17 ( = (𝑈𝑖) → (𝑡) = ((𝑈𝑖)‘𝑡))
8079breq2d 4823 . . . . . . . . . . . . . . . 16 ( = (𝑈𝑖) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝑈𝑖)‘𝑡)))
8179breq1d 4821 . . . . . . . . . . . . . . . 16 ( = (𝑈𝑖) → ((𝑡) ≤ 1 ↔ ((𝑈𝑖)‘𝑡) ≤ 1))
8280, 81anbi12d 624 . . . . . . . . . . . . . . 15 ( = (𝑈𝑖) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8382ralbidv 3133 . . . . . . . . . . . . . 14 ( = (𝑈𝑖) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8483, 1elrab2 3525 . . . . . . . . . . . . 13 ((𝑈𝑖) ∈ 𝑌 ↔ ((𝑈𝑖) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8584simplbi 491 . . . . . . . . . . . 12 ((𝑈𝑖) ∈ 𝑌 → (𝑈𝑖) ∈ 𝐴)
8678, 85syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝐴)
87 eleq1 2832 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈𝑖) → (𝑓𝐴 ↔ (𝑈𝑖) ∈ 𝐴))
8887anbi2d 622 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝐴)))
89 feq1 6206 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈𝑖):𝑇⟶ℝ))
9088, 89imbi12d 335 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ)))
9119a1i 11 . . . . . . . . . . . . 13 (𝑓𝐴 → ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ))
9290, 91vtoclga 3425 . . . . . . . . . . . 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 472 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝑈𝑖):𝑇⟶ℝ)
9673ffvelrnda 6551 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊𝑖) ∈ 𝑉)
97 simpl 474 . . . . . . . . . . . 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 2924 . . . . . . . . . . . . . 14 𝑤(𝑊𝑖) ∈ 𝑉
10299, 101nfan 1998 . . . . . . . . . . . . 13 𝑤(𝜑 ∧ (𝑊𝑖) ∈ 𝑉)
103 nfv 2009 . . . . . . . . . . . . 13 𝑤(𝑊𝑖) ⊆ 𝑇
104102, 103nfim 1995 . . . . . . . . . . . 12 𝑤((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇)
105 eleq1 2832 . . . . . . . . . . . . . 14 (𝑤 = (𝑊𝑖) → (𝑤𝑉 ↔ (𝑊𝑖) ∈ 𝑉))
106105anbi2d 622 . . . . . . . . . . . . 13 (𝑤 = (𝑊𝑖) → ((𝜑𝑤𝑉) ↔ (𝜑 ∧ (𝑊𝑖) ∈ 𝑉)))
107 sseq1 3788 . . . . . . . . . . . . 13 (𝑤 = (𝑊𝑖) → (𝑤𝑇 ↔ (𝑊𝑖) ⊆ 𝑇))
108106, 107imbi12d 335 . . . . . . . . . . . 12 (𝑤 = (𝑊𝑖) → (((𝜑𝑤𝑉) → 𝑤𝑇) ↔ ((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇)))
109 stoweidlem51.13 . . . . . . . . . . . 12 ((𝜑𝑤𝑉) → 𝑤𝑇)
110104, 108, 109vtoclg1f 3417 . . . . . . . . . . 11 ((𝑊𝑖) ∈ 𝑉 → ((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇))
11196, 98, 110sylc 65 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊𝑖) ⊆ 𝑇)
112111sselda 3763 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑡𝑇)
11395, 112ffvelrnd 6552 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
114 stoweidlem51.22 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℝ+)
115114rpred 12073 . . . . . . . . . 10 (𝜑𝐸 ∈ ℝ)
116115ad2antrr 717 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝐸 ∈ ℝ)
11711ad2antrr 717 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑀 ∈ ℝ)
1187nnne0d 11324 . . . . . . . . . 10 (𝜑𝑀 ≠ 0)
119118ad2antrr 717 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑀 ≠ 0)
120116, 117, 119redivcld 11109 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝐸 / 𝑀) ∈ ℝ)
121 stoweidlem51.17 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑈𝑖)‘𝑡) < (𝐸 / 𝑀))
122121r19.21bi 3079 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < (𝐸 / 𝑀))
123 1red 10296 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℝ)
124 0lt1 10806 . . . . . . . . . . . . 13 0 < 1
125124a1i 11 . . . . . . . . . . . 12 (𝜑 → 0 < 1)
1267nngt0d 11323 . . . . . . . . . . . 12 (𝜑 → 0 < 𝑀)
127114rpregt0d 12079 . . . . . . . . . . . 12 (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
128 lediv2 11169 . . . . . . . . . . . 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 1500 . . . . . . . . . . 11 (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
13010, 129mpbid 223 . . . . . . . . . 10 (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
131114rpcnd 12075 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℂ)
132131div1d 11049 . . . . . . . . . 10 (𝜑 → (𝐸 / 1) = 𝐸)
133130, 132breqtrd 4837 . . . . . . . . 9 (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
134133ad2antrr 717 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝐸 / 𝑀) ≤ 𝐸)
135113, 120, 116, 122, 134ltletrd 10453 . . . . . . 7 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸)
136135ex 401 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊𝑖) → ((𝑈𝑖)‘𝑡) < 𝐸))
13777, 136ralrimi 3104 . . . . 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 40905 . . . 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 3759 . . . . . 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 40899 . . . 4 (𝜑 → ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡))
14569, 138, 1443jca 1158 . . 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 2832 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
14859nfeq2 2923 . . . . . 6 𝑡 𝑥 = 𝑋
149 fveq1 6376 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥𝑡) = (𝑋𝑡))
150149breq2d 4823 . . . . . . 7 (𝑥 = 𝑋 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝑋𝑡)))
151149breq1d 4821 . . . . . . 7 (𝑥 = 𝑋 → ((𝑥𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
152150, 151anbi12d 624 . . . . . 6 (𝑥 = 𝑋 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
153148, 152ralbid 3130 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
154149breq1d 4821 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑡) < 𝐸 ↔ (𝑋𝑡) < 𝐸))
155148, 154ralbid 3130 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝐷 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝐷 (𝑋𝑡) < 𝐸))
156149breq2d 4823 . . . . . 6 (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝑋𝑡)))
157148, 156ralbid 3130 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡)))
158153, 155, 1573anbi123d 1560 . . . 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 3447 . 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 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wnf 1878  wcel 2155  wnfc 2894  wne 2937  wral 3055  {crab 3059  Vcvv 3350  wss 3734   cuni 4596   class class class wbr 4811  cmpt 4890  ran crn 5280  wf 6066  cfv 6070  (class class class)co 6844  cmpt2 6846  cr 10190  0cc0 10191  1c1 10192   · cmul 10196   < clt 10330  cle 10331  cmin 10522   / cdiv 10940  cn 11276  3c3 11330  cz 11626  +crp 12031  ...cfz 12536  seqcseq 13011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149  ax-cnex 10247  ax-resscn 10248  ax-1cn 10249  ax-icn 10250  ax-addcl 10251  ax-addrcl 10252  ax-mulcl 10253  ax-mulrcl 10254  ax-mulcom 10255  ax-addass 10256  ax-mulass 10257  ax-distr 10258  ax-i2m1 10259  ax-1ne0 10260  ax-1rid 10261  ax-rnegex 10262  ax-rrecex 10263  ax-cnre 10264  ax-pre-lttri 10265  ax-pre-lttrn 10266  ax-pre-ltadd 10267  ax-pre-mulgt0 10268
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-nel 3041  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-om 7266  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-recs 7674  df-rdg 7712  df-er 7949  df-en 8163  df-dom 8164  df-sdom 8165  df-pnf 10332  df-mnf 10333  df-xr 10334  df-ltxr 10335  df-le 10336  df-sub 10524  df-neg 10525  df-div 10941  df-nn 11277  df-2 11337  df-3 11338  df-n0 11541  df-z 11627  df-uz 11890  df-rp 12032  df-fz 12537  df-fzo 12677  df-seq 13012  df-exp 13071
This theorem is referenced by:  stoweidlem54  40911
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