Mathbox for Glauco Siliprandi < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  stoweidlem51 Structured version   Visualization version   GIF version

Theorem stoweidlem51 42484
 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 4035 . . . 4 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ⊆ 𝐴
31, 2eqsstri 3980 . . 3 𝑌𝐴
4 stoweidlem51.6 . . . 4 𝑃 = (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
5 stoweidlem51.7 . . . 4 𝑋 = (seq1(𝑃, 𝑈)‘𝑀)
6 1zzd 11992 . . . . . 6 (𝜑 → 1 ∈ ℤ)
7 stoweidlem51.10 . . . . . . 7 (𝜑𝑀 ∈ ℕ)
87nnzd 12065 . . . . . 6 (𝜑𝑀 ∈ ℤ)
96, 8, 83jca 1124 . . . . 5 (𝜑 → (1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ))
107nnge1d 11664 . . . . . 6 (𝜑 → 1 ≤ 𝑀)
117nnred 11631 . . . . . . 7 (𝜑𝑀 ∈ ℝ)
1211leidd 11184 . . . . . 6 (𝜑𝑀𝑀)
1310, 12jca 514 . . . . 5 (𝜑 → (1 ≤ 𝑀𝑀𝑀))
14 elfz2 12883 . . . . 5 (𝑀 ∈ (1...𝑀) ↔ ((1 ∈ ℤ ∧ 𝑀 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ (1 ≤ 𝑀𝑀𝑀)))
159, 13, 14sylanbrc 585 . . . 4 (𝜑𝑀 ∈ (1...𝑀))
16 stoweidlem51.12 . . . 4 (𝜑𝑈:(1...𝑀)⟶𝑌)
17 stoweidlem51.2 . . . . 5 𝑡𝜑
18 eqid 2820 . . . . 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 42449 . . . 4 ((𝜑𝑓𝑌𝑔𝑌) → (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))) ∈ 𝑌)
22 stoweidlem51.21 . . . 4 (𝜑𝑇 ∈ V)
234, 5, 15, 16, 21, 22fmulcl 42014 . . 3 (𝜑𝑋𝑌)
243, 23sseldi 3944 . 2 (𝜑𝑋𝐴)
251eleq2i 2902 . . . . . . 7 (𝑋𝑌𝑋 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)})
26 nfcv 2973 . . . . . . . . . . 11 1
27 nfrab1 3371 . . . . . . . . . . . . . 14 {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
281, 27nfcxfr 2971 . . . . . . . . . . . . 13 𝑌
29 nfcv 2973 . . . . . . . . . . . . 13 (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
3028, 28, 29nfmpo 7213 . . . . . . . . . . . 12 (𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
314, 30nfcxfr 2971 . . . . . . . . . . 11 𝑃
32 nfcv 2973 . . . . . . . . . . 11 𝑈
3326, 31, 32nfseq 13363 . . . . . . . . . 10 seq1(𝑃, 𝑈)
34 nfcv 2973 . . . . . . . . . 10 𝑀
3533, 34nffv 6656 . . . . . . . . 9 (seq1(𝑃, 𝑈)‘𝑀)
365, 35nfcxfr 2971 . . . . . . . 8 𝑋
37 nfcv 2973 . . . . . . . 8 𝐴
38 nfcv 2973 . . . . . . . . 9 𝑇
39 nfcv 2973 . . . . . . . . . . 11 0
40 nfcv 2973 . . . . . . . . . . 11
41 nfcv 2973 . . . . . . . . . . . 12 𝑡
4236, 41nffv 6656 . . . . . . . . . . 11 (𝑋𝑡)
4339, 40, 42nfbr 5089 . . . . . . . . . 10 0 ≤ (𝑋𝑡)
4442, 40, 26nfbr 5089 . . . . . . . . . 10 (𝑋𝑡) ≤ 1
4543, 44nfan 1900 . . . . . . . . 9 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)
4638, 45nfralw 3212 . . . . . . . 8 𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)
47 nfcv 2973 . . . . . . . . . . . . 13 𝑡1
48 nfra1 3206 . . . . . . . . . . . . . . . . 17 𝑡𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)
49 nfcv 2973 . . . . . . . . . . . . . . . . 17 𝑡𝐴
5048, 49nfrabw 3372 . . . . . . . . . . . . . . . 16 𝑡{𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)}
511, 50nfcxfr 2971 . . . . . . . . . . . . . . 15 𝑡𝑌
52 nfmpt1 5140 . . . . . . . . . . . . . . 15 𝑡(𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡)))
5351, 51, 52nfmpo 7213 . . . . . . . . . . . . . 14 𝑡(𝑓𝑌, 𝑔𝑌 ↦ (𝑡𝑇 ↦ ((𝑓𝑡) · (𝑔𝑡))))
544, 53nfcxfr 2971 . . . . . . . . . . . . 13 𝑡𝑃
55 nfcv 2973 . . . . . . . . . . . . 13 𝑡𝑈
5647, 54, 55nfseq 13363 . . . . . . . . . . . 12 𝑡seq1(𝑃, 𝑈)
57 nfcv 2973 . . . . . . . . . . . 12 𝑡𝑀
5856, 57nffv 6656 . . . . . . . . . . 11 𝑡(seq1(𝑃, 𝑈)‘𝑀)
595, 58nfcxfr 2971 . . . . . . . . . 10 𝑡𝑋
6059nfeq2 2990 . . . . . . . . 9 𝑡 = 𝑋
61 fveq1 6645 . . . . . . . . . . 11 ( = 𝑋 → (𝑡) = (𝑋𝑡))
6261breq2d 5054 . . . . . . . . . 10 ( = 𝑋 → (0 ≤ (𝑡) ↔ 0 ≤ (𝑋𝑡)))
6361breq1d 5052 . . . . . . . . . 10 ( = 𝑋 → ((𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
6462, 63anbi12d 632 . . . . . . . . 9 ( = 𝑋 → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6560, 64ralbid 3218 . . . . . . . 8 ( = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6636, 37, 46, 65elrabf 3656 . . . . . . 7 (𝑋 ∈ {𝐴 ∣ ∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1)} ↔ (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6725, 66bitri 277 . . . . . 6 (𝑋𝑌 ↔ (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6823, 67sylib 220 . . . . 5 (𝜑 → (𝑋𝐴 ∧ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
6968simprd 498 . . . 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 1915 . . . . . . 7 𝑡 𝑖 ∈ (1...𝑀)
7717, 76nfan 1900 . . . . . 6 𝑡(𝜑𝑖 ∈ (1...𝑀))
7816ffvelrnda 6827 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝑌)
79 fveq1 6645 . . . . . . . . . . . . . . . . 17 ( = (𝑈𝑖) → (𝑡) = ((𝑈𝑖)‘𝑡))
8079breq2d 5054 . . . . . . . . . . . . . . . 16 ( = (𝑈𝑖) → (0 ≤ (𝑡) ↔ 0 ≤ ((𝑈𝑖)‘𝑡)))
8179breq1d 5052 . . . . . . . . . . . . . . . 16 ( = (𝑈𝑖) → ((𝑡) ≤ 1 ↔ ((𝑈𝑖)‘𝑡) ≤ 1))
8280, 81anbi12d 632 . . . . . . . . . . . . . . 15 ( = (𝑈𝑖) → ((0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8382ralbidv 3184 . . . . . . . . . . . . . 14 ( = (𝑈𝑖) → (∀𝑡𝑇 (0 ≤ (𝑡) ∧ (𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8483, 1elrab2 3663 . . . . . . . . . . . . 13 ((𝑈𝑖) ∈ 𝑌 ↔ ((𝑈𝑖) ∈ 𝐴 ∧ ∀𝑡𝑇 (0 ≤ ((𝑈𝑖)‘𝑡) ∧ ((𝑈𝑖)‘𝑡) ≤ 1)))
8584simplbi 500 . . . . . . . . . . . 12 ((𝑈𝑖) ∈ 𝑌 → (𝑈𝑖) ∈ 𝐴)
8678, 85syl 17 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖) ∈ 𝐴)
87 eleq1 2898 . . . . . . . . . . . . . . 15 (𝑓 = (𝑈𝑖) → (𝑓𝐴 ↔ (𝑈𝑖) ∈ 𝐴))
8887anbi2d 630 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → ((𝜑𝑓𝐴) ↔ (𝜑 ∧ (𝑈𝑖) ∈ 𝐴)))
89 feq1 6471 . . . . . . . . . . . . . 14 (𝑓 = (𝑈𝑖) → (𝑓:𝑇⟶ℝ ↔ (𝑈𝑖):𝑇⟶ℝ))
9088, 89imbi12d 347 . . . . . . . . . . . . 13 (𝑓 = (𝑈𝑖) → (((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ) ↔ ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ)))
9119a1i 11 . . . . . . . . . . . . 13 (𝑓𝐴 → ((𝜑𝑓𝐴) → 𝑓:𝑇⟶ℝ))
9290, 91vtoclga 3553 . . . . . . . . . . . 12 ((𝑈𝑖) ∈ 𝐴 → ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ))
9392anabsi7 669 . . . . . . . . . . 11 ((𝜑 ∧ (𝑈𝑖) ∈ 𝐴) → (𝑈𝑖):𝑇⟶ℝ)
9486, 93syldan 593 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑈𝑖):𝑇⟶ℝ)
9594adantr 483 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝑈𝑖):𝑇⟶ℝ)
9673ffvelrnda 6827 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊𝑖) ∈ 𝑉)
97 simpl 485 . . . . . . . . . . . 12 ((𝜑𝑖 ∈ (1...𝑀)) → 𝜑)
9897, 96jca 514 . . . . . . . . . . 11 ((𝜑𝑖 ∈ (1...𝑀)) → (𝜑 ∧ (𝑊𝑖) ∈ 𝑉))
99 stoweidlem51.3 . . . . . . . . . . . . . 14 𝑤𝜑
100 stoweidlem51.4 . . . . . . . . . . . . . . 15 𝑤𝑉
101100nfel2 2991 . . . . . . . . . . . . . 14 𝑤(𝑊𝑖) ∈ 𝑉
10299, 101nfan 1900 . . . . . . . . . . . . 13 𝑤(𝜑 ∧ (𝑊𝑖) ∈ 𝑉)
103 nfv 1915 . . . . . . . . . . . . 13 𝑤(𝑊𝑖) ⊆ 𝑇
104102, 103nfim 1897 . . . . . . . . . . . 12 𝑤((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇)
105 eleq1 2898 . . . . . . . . . . . . . 14 (𝑤 = (𝑊𝑖) → (𝑤𝑉 ↔ (𝑊𝑖) ∈ 𝑉))
106105anbi2d 630 . . . . . . . . . . . . 13 (𝑤 = (𝑊𝑖) → ((𝜑𝑤𝑉) ↔ (𝜑 ∧ (𝑊𝑖) ∈ 𝑉)))
107 sseq1 3971 . . . . . . . . . . . . 13 (𝑤 = (𝑊𝑖) → (𝑤𝑇 ↔ (𝑊𝑖) ⊆ 𝑇))
108106, 107imbi12d 347 . . . . . . . . . . . 12 (𝑤 = (𝑊𝑖) → (((𝜑𝑤𝑉) → 𝑤𝑇) ↔ ((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇)))
109 stoweidlem51.13 . . . . . . . . . . . 12 ((𝜑𝑤𝑉) → 𝑤𝑇)
110104, 108, 109vtoclg1f 3545 . . . . . . . . . . 11 ((𝑊𝑖) ∈ 𝑉 → ((𝜑 ∧ (𝑊𝑖) ∈ 𝑉) → (𝑊𝑖) ⊆ 𝑇))
11196, 98, 110sylc 65 . . . . . . . . . 10 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑊𝑖) ⊆ 𝑇)
112111sselda 3946 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑡𝑇)
11395, 112ffvelrnd 6828 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) ∈ ℝ)
114 stoweidlem51.22 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℝ+)
115114rpred 12410 . . . . . . . . . 10 (𝜑𝐸 ∈ ℝ)
116115ad2antrr 724 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝐸 ∈ ℝ)
11711ad2antrr 724 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑀 ∈ ℝ)
1187nnne0d 11666 . . . . . . . . . 10 (𝜑𝑀 ≠ 0)
119118ad2antrr 724 . . . . . . . . 9 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → 𝑀 ≠ 0)
120116, 117, 119redivcld 11446 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝐸 / 𝑀) ∈ ℝ)
121 stoweidlem51.17 . . . . . . . . 9 ((𝜑𝑖 ∈ (1...𝑀)) → ∀𝑡 ∈ (𝑊𝑖)((𝑈𝑖)‘𝑡) < (𝐸 / 𝑀))
122121r19.21bi 3195 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < (𝐸 / 𝑀))
123 1red 10620 . . . . . . . . . . . 12 (𝜑 → 1 ∈ ℝ)
124 0lt1 11140 . . . . . . . . . . . . 13 0 < 1
125124a1i 11 . . . . . . . . . . . 12 (𝜑 → 0 < 1)
1267nngt0d 11665 . . . . . . . . . . . 12 (𝜑 → 0 < 𝑀)
127114rpregt0d 12416 . . . . . . . . . . . 12 (𝜑 → (𝐸 ∈ ℝ ∧ 0 < 𝐸))
128 lediv2 11508 . . . . . . . . . . . 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 1377 . . . . . . . . . . 11 (𝜑 → (1 ≤ 𝑀 ↔ (𝐸 / 𝑀) ≤ (𝐸 / 1)))
13010, 129mpbid 234 . . . . . . . . . 10 (𝜑 → (𝐸 / 𝑀) ≤ (𝐸 / 1))
131114rpcnd 12412 . . . . . . . . . . 11 (𝜑𝐸 ∈ ℂ)
132131div1d 11386 . . . . . . . . . 10 (𝜑 → (𝐸 / 1) = 𝐸)
133130, 132breqtrd 5068 . . . . . . . . 9 (𝜑 → (𝐸 / 𝑀) ≤ 𝐸)
134133ad2antrr 724 . . . . . . . 8 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → (𝐸 / 𝑀) ≤ 𝐸)
135113, 120, 116, 122, 134ltletrd 10778 . . . . . . 7 (((𝜑𝑖 ∈ (1...𝑀)) ∧ 𝑡 ∈ (𝑊𝑖)) → ((𝑈𝑖)‘𝑡) < 𝐸)
136135ex 415 . . . . . 6 ((𝜑𝑖 ∈ (1...𝑀)) → (𝑡 ∈ (𝑊𝑖) → ((𝑈𝑖)‘𝑡) < 𝐸))
13777, 136ralrimi 3203 . . . . 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 42481 . . . 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 3942 . . . . . 6 (𝑓𝑌𝑓𝐴)
142141, 19sylan2 594 . . . . 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 42475 . . . 4 (𝜑 → ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡))
14569, 138, 1443jca 1124 . . 3 (𝜑 → (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑋𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡)))
14624, 145jca 514 . 2 (𝜑 → (𝑋𝐴 ∧ (∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1) ∧ ∀𝑡𝐷 (𝑋𝑡) < 𝐸 ∧ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡))))
147 eleq1 2898 . . . 4 (𝑥 = 𝑋 → (𝑥𝐴𝑋𝐴))
14859nfeq2 2990 . . . . . 6 𝑡 𝑥 = 𝑋
149 fveq1 6645 . . . . . . . 8 (𝑥 = 𝑋 → (𝑥𝑡) = (𝑋𝑡))
150149breq2d 5054 . . . . . . 7 (𝑥 = 𝑋 → (0 ≤ (𝑥𝑡) ↔ 0 ≤ (𝑋𝑡)))
151149breq1d 5052 . . . . . . 7 (𝑥 = 𝑋 → ((𝑥𝑡) ≤ 1 ↔ (𝑋𝑡) ≤ 1))
152150, 151anbi12d 632 . . . . . 6 (𝑥 = 𝑋 → ((0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
153148, 152ralbid 3218 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝑇 (0 ≤ (𝑥𝑡) ∧ (𝑥𝑡) ≤ 1) ↔ ∀𝑡𝑇 (0 ≤ (𝑋𝑡) ∧ (𝑋𝑡) ≤ 1)))
154149breq1d 5052 . . . . . 6 (𝑥 = 𝑋 → ((𝑥𝑡) < 𝐸 ↔ (𝑋𝑡) < 𝐸))
155148, 154ralbid 3218 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝐷 (𝑥𝑡) < 𝐸 ↔ ∀𝑡𝐷 (𝑋𝑡) < 𝐸))
156149breq2d 5054 . . . . . 6 (𝑥 = 𝑋 → ((1 − 𝐸) < (𝑥𝑡) ↔ (1 − 𝐸) < (𝑋𝑡)))
157148, 156ralbid 3218 . . . . 5 (𝑥 = 𝑋 → (∀𝑡𝐵 (1 − 𝐸) < (𝑥𝑡) ↔ ∀𝑡𝐵 (1 − 𝐸) < (𝑋𝑡)))
158153, 155, 1573anbi123d 1432 . . . 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 632 . . 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 3576 . 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 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780  Ⅎwnf 1784   ∈ wcel 2114  Ⅎwnfc 2957   ≠ wne 3006  ∀wral 3125  {crab 3129  Vcvv 3473   ⊆ wss 3913  ∪ cuni 4814   class class class wbr 5042   ↦ cmpt 5122  ran crn 5532  ⟶wf 6327  ‘cfv 6331  (class class class)co 7133   ∈ cmpo 7135  ℝcr 10514  0cc0 10515  1c1 10516   · cmul 10520   < clt 10653   ≤ cle 10654   − cmin 10848   / cdiv 11275  ℕcn 11616  3c3 11672  ℤcz 11960  ℝ+crp 12368  ...cfz 12876  seqcseq 13353 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-1cn 10573  ax-icn 10574  ax-addcl 10575  ax-addrcl 10576  ax-mulcl 10577  ax-mulrcl 10578  ax-mulcom 10579  ax-addass 10580  ax-mulass 10581  ax-distr 10582  ax-i2m1 10583  ax-1ne0 10584  ax-1rid 10585  ax-rnegex 10586  ax-rrecex 10587  ax-cnre 10588  ax-pre-lttri 10589  ax-pre-lttrn 10590  ax-pre-ltadd 10591  ax-pre-mulgt0 10592 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-reu 3132  df-rmo 3133  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-tp 4548  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-tr 5149  df-id 5436  df-eprel 5441  df-po 5450  df-so 5451  df-fr 5490  df-we 5492  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-pred 6124  df-ord 6170  df-on 6171  df-lim 6172  df-suc 6173  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-riota 7091  df-ov 7136  df-oprab 7137  df-mpo 7138  df-om 7559  df-1st 7667  df-2nd 7668  df-wrecs 7925  df-recs 7986  df-rdg 8024  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-sub 10850  df-neg 10851  df-div 11276  df-nn 11617  df-2 11679  df-3 11680  df-n0 11877  df-z 11961  df-uz 12223  df-rp 12369  df-fz 12877  df-fzo 13018  df-seq 13354  df-exp 13415 This theorem is referenced by:  stoweidlem54  42487
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