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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 11246 | . . . . 5 ⊢ (1 / 2) ∈ ℝ | |
4 | halfgt0 11248 | . . . . 5 ⊢ 0 < (1 / 2) | |
5 | 3, 4 | elrpii 11835 | . . . 4 ⊢ (1 / 2) ∈ ℝ+ |
6 | ifcl 4130 | . . . 4 ⊢ ((𝐶 ∈ ℝ+ ∧ (1 / 2) ∈ ℝ+) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ+) | |
7 | 2, 5, 6 | sylancl 694 | . . 3 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ∈ ℝ+) |
8 | 1, 7 | syl5eqel 2705 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
9 | 8 | rpred 11872 | . . 3 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
10 | 3 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / 2) ∈ ℝ) |
11 | 1red 10055 | . . 3 ⊢ (𝜑 → 1 ∈ ℝ) | |
12 | 2 | rpred 11872 | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) |
13 | min2 12021 | . . . . 5 ⊢ ((𝐶 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (1 / 2)) | |
14 | 12, 3, 13 | sylancl 694 | . . . 4 ⊢ (𝜑 → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (1 / 2)) |
15 | 1, 14 | syl5eqbr 4688 | . . 3 ⊢ (𝜑 → 𝐷 ≤ (1 / 2)) |
16 | halflt1 11250 | . . . 4 ⊢ (1 / 2) < 1 | |
17 | 16 | a1i 11 | . . 3 ⊢ (𝜑 → (1 / 2) < 1) |
18 | 9, 10, 11, 15, 17 | lelttrd 10195 | . 2 ⊢ (𝜑 → 𝐷 < 1) |
19 | stoweidlem5.1 | . . 3 ⊢ Ⅎ𝑡𝜑 | |
20 | 7 | rpred 11872 | . . . . . . 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 3603 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝑡 ∈ 𝑇) |
27 | 24, 26 | ffvelrnd 6360 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → (𝑃‘𝑡) ∈ ℝ) |
28 | min1 12020 | . . . . . . . 8 ⊢ ((𝐶 ∈ ℝ ∧ (1 / 2) ∈ ℝ) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ 𝐶) | |
29 | 12, 3, 28 | sylancl 694 | . . . . . . 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 2932 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐶 ≤ (𝑃‘𝑡)) |
33 | 21, 22, 27, 30, 32 | letrd 10194 | . . . . 5 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → if(𝐶 ≤ (1 / 2), 𝐶, (1 / 2)) ≤ (𝑃‘𝑡)) |
34 | 1, 33 | syl5eqbr 4688 | . . . 4 ⊢ ((𝜑 ∧ 𝑡 ∈ 𝑄) → 𝐷 ≤ (𝑃‘𝑡)) |
35 | 34 | ex 450 | . . 3 ⊢ (𝜑 → (𝑡 ∈ 𝑄 → 𝐷 ≤ (𝑃‘𝑡))) |
36 | 19, 35 | ralrimi 2957 | . 2 ⊢ (𝜑 → ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)) |
37 | eleq1 2689 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 ∈ ℝ+ ↔ 𝐷 ∈ ℝ+)) | |
38 | breq1 4656 | . . . . 5 ⊢ (𝑑 = 𝐷 → (𝑑 < 1 ↔ 𝐷 < 1)) | |
39 | breq1 4656 | . . . . . 6 ⊢ (𝑑 = 𝐷 → (𝑑 ≤ (𝑃‘𝑡) ↔ 𝐷 ≤ (𝑃‘𝑡))) | |
40 | 39 | ralbidv 2986 | . . . . 5 ⊢ (𝑑 = 𝐷 → (∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡) ↔ ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡))) |
41 | 37, 38, 40 | 3anbi123d 1399 | . . . 4 ⊢ (𝑑 = 𝐷 → ((𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡)) ↔ (𝐷 ∈ ℝ+ ∧ 𝐷 < 1 ∧ ∀𝑡 ∈ 𝑄 𝐷 ≤ (𝑃‘𝑡)))) |
42 | 41 | spcegv 3294 | . . 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 1427 | 1 ⊢ (𝜑 → ∃𝑑(𝑑 ∈ ℝ+ ∧ 𝑑 < 1 ∧ ∀𝑡 ∈ 𝑄 𝑑 ≤ (𝑃‘𝑡))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∃wex 1704 Ⅎwnf 1708 ∈ wcel 1990 ∀wral 2912 ⊆ wss 3574 ifcif 4086 class class class wbr 4653 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 ℝcr 9935 1c1 9937 < clt 10074 ≤ cle 10075 / cdiv 10684 2c2 11070 ℝ+crp 11832 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-2 11079 df-rp 11833 |
This theorem is referenced by: stoweidlem28 40245 |
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