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Mathbox for Glauco Siliprandi |
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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 12433 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
4 | halfgt0 12435 | . . . . 5 ⊢ 0 < (1 / 2) | |
5 | 3, 4 | elrpii 12984 | . . . 4 ⊢ (1 / 2) ∈ ℝ+ |
6 | ifcl 4573 | . . . 4 ⊢ ((𝐶 ∈ ℝ+ ∧ (1 / 2) ∈ ℝ+) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ+) | |
7 | 2, 5, 6 | sylancl 585 | . . 3 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ+) |
8 | 1, 7 | eqeltrid 2836 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
9 | 8 | rpred 13023 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
10 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
11 | 1red 11222 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
12 | 2 | rpred 13023 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
13 | min2 13176 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (1 / 2)) | |
14 | 12, 3, 13 | sylancl 585 | . . . 4 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (1 / 2)) |
15 | 1, 14 | eqbrtrid 5183 | . . 3 ⊢ (𝜑 → 𝐷 ≤ (1 / 2)) |
16 | halflt1 12437 | . . . 4 ⊢ (1 / 2) < 1 | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / 2) < 1) |
18 | 9, 10, 11, 15, 17 | lelttrd 11379 | . 2 ⊢ (𝜑 → 𝐷 < 1) |
19 | stoweidlem5.1 | . . 3 ⊢ Ⅎ𝑡𝜑 | |
20 | 7 | rpred 13023 | . . . . . . 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 3982 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝑡 ∈ 𝑇) |
27 | 24, 26 | ffvelcdmd 7087 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → (𝑃‘𝑡) ∈ ℝ) |
28 | min1 13175 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ 𝐶) | |
29 | 12, 3, 28 | sylancl 585 | . . . . . . 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 3247 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐶 ≤ (𝑃‘𝑡)) |
33 | 21, 22, 27, 30, 32 | letrd 11378 | . . . . 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 3253 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)) |
37 | eleq1 2820 | . . . . 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 3587 | . . 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 1540 ∃wex 1780 Ⅎwnf 1784 ∈ wcel 2105 ∀wral 3060 ⊆ wss 3948 ifcif 4528 class class class wbr 5148 ⟶wf 6539 ‘cfv 6543 (class class class)co 7412 ℝcr 11115 1c1 11117 < clt 11255 ≤ cle 11256 / cdiv 11878 2c2 12274 ℝ+crp 12981 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 ax-cnex 11172 ax-resscn 11173 ax-1cn 11174 ax-icn 11175 ax-addcl 11176 ax-addrcl 11177 ax-mulcl 11178 ax-mulrcl 11179 ax-mulcom 11180 ax-addass 11181 ax-mulass 11182 ax-distr 11183 ax-i2m1 11184 ax-1ne0 11185 ax-1rid 11186 ax-rnegex 11187 ax-rrecex 11188 ax-cnre 11189 ax-pre-lttri 11190 ax-pre-lttrn 11191 ax-pre-ltadd 11192 ax-pre-mulgt0 11193 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3375 df-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-riota 7368 df-ov 7415 df-oprab 7416 df-mpo 7417 df-er 8709 df-en 8946 df-dom 8947 df-sdom 8948 df-pnf 11257 df-mnf 11258 df-xr 11259 df-ltxr 11260 df-le 11261 df-sub 11453 df-neg 11454 df-div 11879 df-2 12282 df-rp 12982 |
This theorem is referenced by: stoweidlem28 45202 |
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