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Mirrors > Home > MPE Home > Th. List > pntlemd | Structured version Visualization version GIF version |
Description: Lemma for pnt 26914. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝐴 is C^*, 𝐵 is c1, 𝐿 is λ, 𝐷 is c2, and 𝐹 is c3. (Contributed by Mario Carneiro, 13-Apr-2016.) |
Ref | Expression |
---|---|
pntlem1.r | ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) |
pntlem1.a | ⊢ (𝜑 → 𝐴 ∈ ℝ+) |
pntlem1.b | ⊢ (𝜑 → 𝐵 ∈ ℝ+) |
pntlem1.l | ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) |
pntlem1.d | ⊢ 𝐷 = (𝐴 + 1) |
pntlem1.f | ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) |
Ref | Expression |
---|---|
pntlemd | ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioossre 13279 | . . . 4 ⊢ (0(,)1) ⊆ ℝ | |
2 | pntlem1.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) | |
3 | 1, 2 | sselid 3940 | . . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
4 | eliooord 13277 | . . . . 5 ⊢ (𝐿 ∈ (0(,)1) → (0 < 𝐿 ∧ 𝐿 < 1)) | |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (0 < 𝐿 ∧ 𝐿 < 1)) |
6 | 5 | simpld 495 | . . 3 ⊢ (𝜑 → 0 < 𝐿) |
7 | 3, 6 | elrpd 12908 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℝ+) |
8 | pntlem1.d | . . 3 ⊢ 𝐷 = (𝐴 + 1) | |
9 | pntlem1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
10 | 1rp 12873 | . . . 4 ⊢ 1 ∈ ℝ+ | |
11 | rpaddcl 12891 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝐴 + 1) ∈ ℝ+) | |
12 | 9, 10, 11 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
13 | 8, 12 | eqeltrid 2842 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
14 | pntlem1.f | . . 3 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) | |
15 | 1re 11113 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
16 | ltaddrp 12906 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
17 | 15, 9, 16 | sylancr 587 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
18 | 9 | rpcnd 12913 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | ax-1cn 11067 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
20 | addcom 11299 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
21 | 18, 19, 20 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
22 | 8, 21 | eqtrid 2789 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
23 | 17, 22 | breqtrrd 5131 | . . . . . 6 ⊢ (𝜑 → 1 < 𝐷) |
24 | 13 | recgt1d 12925 | . . . . . 6 ⊢ (𝜑 → (1 < 𝐷 ↔ (1 / 𝐷) < 1)) |
25 | 23, 24 | mpbid 231 | . . . . 5 ⊢ (𝜑 → (1 / 𝐷) < 1) |
26 | 13 | rprecred 12922 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐷) ∈ ℝ) |
27 | difrp 12907 | . . . . . 6 ⊢ (((1 / 𝐷) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 𝐷) < 1 ↔ (1 − (1 / 𝐷)) ∈ ℝ+)) | |
28 | 26, 15, 27 | sylancl 586 | . . . . 5 ⊢ (𝜑 → ((1 / 𝐷) < 1 ↔ (1 − (1 / 𝐷)) ∈ ℝ+)) |
29 | 25, 28 | mpbid 231 | . . . 4 ⊢ (𝜑 → (1 − (1 / 𝐷)) ∈ ℝ+) |
30 | 3nn0 12389 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
31 | 2nn 12184 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
32 | 30, 31 | decnncl 12596 | . . . . . . . 8 ⊢ ;32 ∈ ℕ |
33 | nnrp 12880 | . . . . . . . 8 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
34 | 32, 33 | ax-mp 5 | . . . . . . 7 ⊢ ;32 ∈ ℝ+ |
35 | pntlem1.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
36 | rpmulcl 12892 | . . . . . . 7 ⊢ ((;32 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (;32 · 𝐵) ∈ ℝ+) | |
37 | 34, 35, 36 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ+) |
38 | 7, 37 | rpdivcld 12928 | . . . . 5 ⊢ (𝜑 → (𝐿 / (;32 · 𝐵)) ∈ ℝ+) |
39 | 2z 12493 | . . . . . 6 ⊢ 2 ∈ ℤ | |
40 | rpexpcl 13940 | . . . . . 6 ⊢ ((𝐷 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐷↑2) ∈ ℝ+) | |
41 | 13, 39, 40 | sylancl 586 | . . . . 5 ⊢ (𝜑 → (𝐷↑2) ∈ ℝ+) |
42 | 38, 41 | rpdivcld 12928 | . . . 4 ⊢ (𝜑 → ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) ∈ ℝ+) |
43 | 29, 42 | rpmulcld 12927 | . . 3 ⊢ (𝜑 → ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) ∈ ℝ+) |
44 | 14, 43 | eqeltrid 2842 | . 2 ⊢ (𝜑 → 𝐹 ∈ ℝ+) |
45 | 7, 13, 44 | 3jca 1128 | 1 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5103 ↦ cmpt 5186 ‘cfv 6493 (class class class)co 7351 ℂcc 11007 ℝcr 11008 0cc0 11009 1c1 11010 + caddc 11012 · cmul 11014 < clt 11147 − cmin 11343 / cdiv 11770 ℕcn 12111 2c2 12166 3c3 12167 ℤcz 12457 ;cdc 12576 ℝ+crp 12869 (,)cioo 13218 ↑cexp 13921 ψcchp 26394 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2708 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7664 ax-cnex 11065 ax-resscn 11066 ax-1cn 11067 ax-icn 11068 ax-addcl 11069 ax-addrcl 11070 ax-mulcl 11071 ax-mulrcl 11072 ax-mulcom 11073 ax-addass 11074 ax-mulass 11075 ax-distr 11076 ax-i2m1 11077 ax-1ne0 11078 ax-1rid 11079 ax-rnegex 11080 ax-rrecex 11081 ax-cnre 11082 ax-pre-lttri 11083 ax-pre-lttrn 11084 ax-pre-ltadd 11085 ax-pre-mulgt0 11086 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2887 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-rmo 3351 df-reu 3352 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-pss 3927 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-tr 5221 df-id 5529 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6251 df-ord 6318 df-on 6319 df-lim 6320 df-suc 6321 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7307 df-ov 7354 df-oprab 7355 df-mpo 7356 df-om 7795 df-1st 7913 df-2nd 7914 df-frecs 8204 df-wrecs 8235 df-recs 8309 df-rdg 8348 df-er 8606 df-en 8842 df-dom 8843 df-sdom 8844 df-pnf 11149 df-mnf 11150 df-xr 11151 df-ltxr 11152 df-le 11153 df-sub 11345 df-neg 11346 df-div 11771 df-nn 12112 df-2 12174 df-3 12175 df-4 12176 df-5 12177 df-6 12178 df-7 12179 df-8 12180 df-9 12181 df-n0 12372 df-z 12458 df-dec 12577 df-uz 12722 df-rp 12870 df-ioo 13222 df-seq 13861 df-exp 13922 |
This theorem is referenced by: pntlemc 26895 pntlema 26896 pntlemb 26897 pntlemq 26901 pntlemr 26902 pntlemj 26903 pntlemf 26905 pntlemo 26907 pntleml 26911 |
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