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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 11660 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
4 | halfgt0 11662 | . . . . 5 ⊢ 0 < (1 / 2) | |
5 | 3, 4 | elrpii 12206 | . . . 4 ⊢ (1 / 2) ∈ ℝ+ |
6 | ifcl 4389 | . . . 4 ⊢ ((𝐶 ∈ ℝ+ ∧ (1 / 2) ∈ ℝ+) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ+) | |
7 | 2, 5, 6 | sylancl 578 | . . 3 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ+) |
8 | 1, 7 | syl5eqel 2865 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
9 | 8 | rpred 12247 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
10 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
11 | 1red 10439 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
12 | 2 | rpred 12247 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
13 | min2 12399 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (1 / 2)) | |
14 | 12, 3, 13 | sylancl 578 | . . . 4 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (1 / 2)) |
15 | 1, 14 | syl5eqbr 4961 | . . 3 ⊢ (𝜑 → 𝐷 ≤ (1 / 2)) |
16 | halflt1 11664 | . . . 4 ⊢ (1 / 2) < 1 | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / 2) < 1) |
18 | 9, 10, 11, 15, 17 | lelttrd 10597 | . 2 ⊢ (𝜑 → 𝐷 < 1) |
19 | stoweidlem5.1 | . . 3 ⊢ Ⅎ𝑡𝜑 | |
20 | 7 | rpred 12247 | . . . . . . 7 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ) |
21 | 20 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ) |
22 | 12 | adantr 473 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐶 ∈ ℝ) |
23 | stoweidlem5.3 | . . . . . . . 8 ⊢ (𝜑 → 𝑃:𝑇⟶ℝ) | |
24 | 23 | adantr 473 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝑃:𝑇⟶ℝ) |
25 | stoweidlem5.4 | . . . . . . . 8 ⊢ (𝜑 → 𝑄 ⊆ 𝑇) | |
26 | 25 | sselda 3853 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝑡 ∈ 𝑇) |
27 | 24, 26 | ffvelrnd 6676 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → (𝑃‘𝑡) ∈ ℝ) |
28 | min1 12398 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ 𝐶) | |
29 | 12, 3, 28 | sylancl 578 | . . . . . . 7 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ 𝐶) |
30 | 29 | adantr 473 | . . . . . 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 3153 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐶 ≤ (𝑃‘𝑡)) |
33 | 21, 22, 27, 30, 32 | letrd 10596 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (𝑃‘𝑡)) |
34 | 1, 33 | syl5eqbr 4961 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐷 ≤ (𝑃‘𝑡)) |
35 | 34 | ex 405 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑄 → 𝐷 ≤ (𝑃‘𝑡))) |
36 | 19, 35 | ralrimi 3161 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)) |
37 | eleq1 2848 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 ∈ ℝ+ ↔ 𝐷 ∈ ℝ+)) | |
38 | breq1 4929 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 < 1 ↔ 𝐷 < 1)) | |
39 | breq1 4929 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (𝑑 ≤ (𝑃‘𝑡) ↔ 𝐷 ≤ (𝑃‘𝑡))) | |
40 | 39 | ralbidv 3142 | . . . . 5 ⊢ (𝑑 = 𝐷 → (∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡))) |
41 | 37, 38, 40 | 3anbi123d 1416 | . . . 4 ⊢ (𝑑 = 𝐷 → ((𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡)) ↔ (𝐷 ∈ ℝ+ ∧ 𝐷 < 1 ∧ ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)))) |
42 | 41 | spcegv 3511 | . . 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 1444 | 1 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 ∧ w3a 1069 = wceq 1508 ∃wex 1743 Ⅎwnf 1747 ∈ wcel 2051 ∀wral 3083 ⊆ wss 3824 ifcif 4345 class class class wbr 4926 ⟶wf 6182 ‘cfv 6186 (class class class)co 6975 ℝcr 10333 1c1 10335 < clt 10473 ≤ cle 10474 / cdiv 11097 2c2 11494 ℝ+crp 12203 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2745 ax-sep 5057 ax-nul 5064 ax-pow 5116 ax-pr 5183 ax-un 7278 ax-cnex 10390 ax-resscn 10391 ax-1cn 10392 ax-icn 10393 ax-addcl 10394 ax-addrcl 10395 ax-mulcl 10396 ax-mulrcl 10397 ax-mulcom 10398 ax-addass 10399 ax-mulass 10400 ax-distr 10401 ax-i2m1 10402 ax-1ne0 10403 ax-1rid 10404 ax-rnegex 10405 ax-rrecex 10406 ax-cnre 10407 ax-pre-lttri 10408 ax-pre-lttrn 10409 ax-pre-ltadd 10410 ax-pre-mulgt0 10411 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3or 1070 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2754 df-cleq 2766 df-clel 2841 df-nfc 2913 df-ne 2963 df-nel 3069 df-ral 3088 df-rex 3089 df-reu 3090 df-rmo 3091 df-rab 3092 df-v 3412 df-sbc 3677 df-csb 3782 df-dif 3827 df-un 3829 df-in 3831 df-ss 3838 df-nul 4174 df-if 4346 df-pw 4419 df-sn 4437 df-pr 4439 df-op 4443 df-uni 4710 df-br 4927 df-opab 4989 df-mpt 5006 df-id 5309 df-po 5323 df-so 5324 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-iota 6150 df-fun 6188 df-fn 6189 df-f 6190 df-f1 6191 df-fo 6192 df-f1o 6193 df-fv 6194 df-riota 6936 df-ov 6978 df-oprab 6979 df-mpo 6980 df-er 8088 df-en 8306 df-dom 8307 df-sdom 8308 df-pnf 10475 df-mnf 10476 df-xr 10477 df-ltxr 10478 df-le 10479 df-sub 10671 df-neg 10672 df-div 11098 df-2 11502 df-rp 12204 |
This theorem is referenced by: stoweidlem28 41774 |
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