![]() |
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 12478 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
4 | halfgt0 12480 | . . . . 5 ⊢ 0 < (1 / 2) | |
5 | 3, 4 | elrpii 13035 | . . . 4 ⊢ (1 / 2) ∈ ℝ+ |
6 | ifcl 4576 | . . . 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 2843 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
9 | 8 | rpred 13075 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
10 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
11 | 1red 11260 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
12 | 2 | rpred 13075 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
13 | min2 13229 | . . . . 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 5183 | . . 3 ⊢ (𝜑 → 𝐷 ≤ (1 / 2)) |
16 | halflt1 12482 | . . . 4 ⊢ (1 / 2) < 1 | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / 2) < 1) |
18 | 9, 10, 11, 15, 17 | lelttrd 11417 | . 2 ⊢ (𝜑 → 𝐷 < 1) |
19 | stoweidlem5.1 | . . 3 ⊢ Ⅎ𝑡𝜑 | |
20 | 7 | rpred 13075 | . . . . . . 7 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ) |
21 | 20 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ) |
22 | 12 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐶 ∈ ℝ) |
23 | stoweidlem5.3 | . . . . . . . 8 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ) | |
24 | 23 | adantr 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝑃:𝑇⟶ℝ) |
25 | stoweidlem5.4 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ 𝑇) | |
26 | 25 | sselda 3995 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝑡 ∈ 𝑇) |
27 | 24, 26 | ffvelcdmd 7105 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → (𝑃‘𝑡) ∈ ℝ) |
28 | min1 13228 | . . . . . . . 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 480 | . . . . . 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 3249 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐶 ≤ (𝑃‘𝑡)) |
33 | 21, 22, 27, 30, 32 | letrd 11416 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (𝑃‘𝑡)) |
34 | 1, 33 | eqbrtrid 5183 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐷 ≤ (𝑃‘𝑡)) |
35 | 34 | ex 412 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑄 → 𝐷 ≤ (𝑃‘𝑡))) |
36 | 19, 35 | ralrimi 3255 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)) |
37 | eleq1 2827 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 ∈ ℝ+ ↔ 𝐷 ∈ ℝ+)) | |
38 | breq1 5151 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 < 1 ↔ 𝐷 < 1)) | |
39 | breq1 5151 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (𝑑 ≤ (𝑃‘𝑡) ↔ 𝐷 ≤ (𝑃‘𝑡))) | |
40 | 39 | ralbidv 3176 | . . . . 5 ⊢ (𝑑 = 𝐷 → (∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡))) |
41 | 37, 38, 40 | 3anbi123d 1435 | . . . 4 ⊢ (𝑑 = 𝐷 → ((𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡)) ↔ (𝐷 ∈ ℝ+ ∧ 𝐷 < 1 ∧ ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)))) |
42 | 41 | spcegv 3597 | . . 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 1463 | 1 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∃wex 1776 Ⅎwnf 1780 ∈ wcel 2106 ∀wral 3059 ⊆ wss 3963 ifcif 4531 class class class wbr 5148 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 1c1 11154 < clt 11293 ≤ cle 11294 / cdiv 11918 2c2 12319 ℝ+crp 13032 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-er 8744 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-div 11919 df-2 12327 df-rp 13033 |
This theorem is referenced by: stoweidlem28 45984 |
Copyright terms: Public domain | W3C validator |