Users' Mathboxes 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  wnf 1784  wcel 2114  wnfc 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  cr 10514  0cc0 10515  1c1 10516   · cmul 10520   < clt 10653  cle 10654  cmin 10848   / cdiv 11275  cn 11616  3c3 11672  cz 11960  +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
  Copyright terms: Public domain W3C validator