Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > stoweidlem5 | Structured version Visualization version GIF version |
Description: There exists a δ as in the proof of Lemma 1 in [BrosowskiDeutsh] p. 90: 0 < δ < 1 , p >= δ on 𝑇 ∖ 𝑈. Here 𝐷 is used to represent δ in the paper and 𝑄 to represent 𝑇 ∖ 𝑈 in the paper. (Contributed by Glauco Siliprandi, 20-Apr-2017.) |
Ref | Expression |
---|---|
stoweidlem5.1 | ⊢ Ⅎ𝑡𝜑 |
stoweidlem5.2 | ⊢ 𝐷 = if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) |
stoweidlem5.3 | ⊢ (𝜑 → 𝑃:𝑇⟶ℝ) |
stoweidlem5.4 | ⊢ (𝜑 → 𝑄 ⊆ 𝑇) |
stoweidlem5.5 | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
stoweidlem5.6 | ⊢ (𝜑 → ∀𝑡 ∈ 𝑄 𝐶 ≤ (𝑃‘𝑡)) |
Ref | Expression |
---|---|
stoweidlem5 | ⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | stoweidlem5.2 | . . 3 ⊢ 𝐷 = if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) | |
2 | stoweidlem5.5 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
3 | halfre 11839 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
4 | halfgt0 11841 | . . . . 5 ⊢ 0 < (1 / 2) | |
5 | 3, 4 | elrpii 12380 | . . . 4 ⊢ (1 / 2) ∈ ℝ+ |
6 | ifcl 4507 | . . . 4 ⊢ ((𝐶 ∈ ℝ+ ∧ (1 / 2) ∈ ℝ+) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ+) | |
7 | 2, 5, 6 | sylancl 586 | . . 3 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ+) |
8 | 1, 7 | eqeltrid 2914 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
9 | 8 | rpred 12419 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
10 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
11 | 1red 10630 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
12 | 2 | rpred 12419 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
13 | min2 12571 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (1 / 2)) | |
14 | 12, 3, 13 | sylancl 586 | . . . 4 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (1 / 2)) |
15 | 1, 14 | eqbrtrid 5092 | . . 3 ⊢ (𝜑 → 𝐷 ≤ (1 / 2)) |
16 | halflt1 11843 | . . . 4 ⊢ (1 / 2) < 1 | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / 2) < 1) |
18 | 9, 10, 11, 15, 17 | lelttrd 10786 | . 2 ⊢ (𝜑 → 𝐷 < 1) |
19 | stoweidlem5.1 | . . 3 ⊢ Ⅎ𝑡𝜑 | |
20 | 7 | rpred 12419 | . . . . . . 7 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ) |
21 | 20 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ) |
22 | 12 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐶 ∈ ℝ) |
23 | stoweidlem5.3 | . . . . . . . 8 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ) | |
24 | 23 | adantr 481 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝑃:𝑇⟶ℝ) |
25 | stoweidlem5.4 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ 𝑇) | |
26 | 25 | sselda 3964 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝑡 ∈ 𝑇) |
27 | 24, 26 | ffvelrnd 6844 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → (𝑃‘𝑡) ∈ ℝ) |
28 | min1 12570 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ 𝐶) | |
29 | 12, 3, 28 | sylancl 586 | . . . . . . 7 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ 𝐶) |
30 | 29 | adantr 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ 𝐶) |
31 | stoweidlem5.6 | . . . . . . 7 ⊢ (𝜑 → ∀𝑡 ∈ 𝑄 𝐶 ≤ (𝑃‘𝑡)) | |
32 | 31 | r19.21bi 3205 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐶 ≤ (𝑃‘𝑡)) |
33 | 21, 22, 27, 30, 32 | letrd 10785 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (𝑃‘𝑡)) |
34 | 1, 33 | eqbrtrid 5092 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐷 ≤ (𝑃‘𝑡)) |
35 | 34 | ex 413 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑄 → 𝐷 ≤ (𝑃‘𝑡))) |
36 | 19, 35 | ralrimi 3213 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)) |
37 | eleq1 2897 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 ∈ ℝ+ ↔ 𝐷 ∈ ℝ+)) | |
38 | breq1 5060 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 < 1 ↔ 𝐷 < 1)) | |
39 | breq1 5060 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (𝑑 ≤ (𝑃‘𝑡) ↔ 𝐷 ≤ (𝑃‘𝑡))) | |
40 | 39 | ralbidv 3194 | . . . . 5 ⊢ (𝑑 = 𝐷 → (∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡))) |
41 | 37, 38, 40 | 3anbi123d 1427 | . . . 4 ⊢ (𝑑 = 𝐷 → ((𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡)) ↔ (𝐷 ∈ ℝ+ ∧ 𝐷 < 1 ∧ ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)))) |
42 | 41 | spcegv 3594 | . . 3 ⊢ (𝐷 ∈ ℝ+ → ((𝐷 ∈ ℝ+ ∧ 𝐷 < 1 ∧ ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡)))) |
43 | 8, 42 | syl 17 | . 2 ⊢ (𝜑 → ((𝐷 ∈ ℝ+ ∧ 𝐷 < 1 ∧ ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)) → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡)))) |
44 | 8, 18, 36, 43 | mp3and 1455 | 1 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∃wex 1771 Ⅎwnf 1775 ∈ wcel 2105 ∀wral 3135 ⊆ wss 3933 ifcif 4463 class class class wbr 5057 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 ℝcr 10524 1c1 10526 < clt 10663 ≤ cle 10664 / cdiv 11285 2c2 11680 ℝ+crp 12377 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-1cn 10583 ax-icn 10584 ax-addcl 10585 ax-addrcl 10586 ax-mulcl 10587 ax-mulrcl 10588 ax-mulcom 10589 ax-addass 10590 ax-mulass 10591 ax-distr 10592 ax-i2m1 10593 ax-1ne0 10594 ax-1rid 10595 ax-rnegex 10596 ax-rrecex 10597 ax-cnre 10598 ax-pre-lttri 10599 ax-pre-lttrn 10600 ax-pre-ltadd 10601 ax-pre-mulgt0 10602 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-reu 3142 df-rmo 3143 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-sub 10860 df-neg 10861 df-div 11286 df-2 11688 df-rp 12378 |
This theorem is referenced by: stoweidlem28 42190 |
Copyright terms: Public domain | W3C validator |