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Mirrors > Home > MPE Home > Th. List > pntlema | Structured version Visualization version GIF version |
Description: Lemma for pnt 25303. Closure for the constants used in the proof. The mammoth expression 𝑊 is a number large enough to satisfy all the lower bounds needed for 𝑍. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝑌 is x2, 𝑋 is x1, 𝐶 is the big-O constant in Equation 10.6.29 of [Shapiro], p. 435, and 𝑊 is the unnamed lower bound of "for sufficiently large x" in Equation 10.6.34 of [Shapiro], p. 436. (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))) |
pntlem1.u | ⊢ (𝜑 → 𝑈 ∈ ℝ+) |
pntlem1.u2 | ⊢ (𝜑 → 𝑈 ≤ 𝐴) |
pntlem1.e | ⊢ 𝐸 = (𝑈 / 𝐷) |
pntlem1.k | ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) |
pntlem1.y | ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) |
pntlem1.x | ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) |
pntlem1.c | ⊢ (𝜑 → 𝐶 ∈ ℝ+) |
pntlem1.w | ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) |
Ref | Expression |
---|---|
pntlema | ⊢ (𝜑 → 𝑊 ∈ ℝ+) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pntlem1.w | . 2 ⊢ 𝑊 = (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) | |
2 | pntlem1.y | . . . . . 6 ⊢ (𝜑 → (𝑌 ∈ ℝ+ ∧ 1 ≤ 𝑌)) | |
3 | 2 | simpld 475 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℝ+) |
4 | 4nn 11187 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
5 | nnrp 11842 | . . . . . . 7 ⊢ (4 ∈ ℕ → 4 ∈ ℝ+) | |
6 | 4, 5 | ax-mp 5 | . . . . . 6 ⊢ 4 ∈ ℝ+ |
7 | pntlem1.r | . . . . . . . . 9 ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) | |
8 | pntlem1.a | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
9 | pntlem1.b | . . . . . . . . 9 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
10 | pntlem1.l | . . . . . . . . 9 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) | |
11 | pntlem1.d | . . . . . . . . 9 ⊢ 𝐷 = (𝐴 + 1) | |
12 | pntlem1.f | . . . . . . . . 9 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) | |
13 | 7, 8, 9, 10, 11, 12 | pntlemd 25283 | . . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
14 | 13 | simp1d 1073 | . . . . . . 7 ⊢ (𝜑 → 𝐿 ∈ ℝ+) |
15 | pntlem1.u | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ∈ ℝ+) | |
16 | pntlem1.u2 | . . . . . . . . 9 ⊢ (𝜑 → 𝑈 ≤ 𝐴) | |
17 | pntlem1.e | . . . . . . . . 9 ⊢ 𝐸 = (𝑈 / 𝐷) | |
18 | pntlem1.k | . . . . . . . . 9 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) | |
19 | 7, 8, 9, 10, 11, 12, 15, 16, 17, 18 | pntlemc 25284 | . . . . . . . 8 ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))) |
20 | 19 | simp1d 1073 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
21 | 14, 20 | rpmulcld 11888 | . . . . . 6 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ+) |
22 | rpdivcl 11856 | . . . . . 6 ⊢ ((4 ∈ ℝ+ ∧ (𝐿 · 𝐸) ∈ ℝ+) → (4 / (𝐿 · 𝐸)) ∈ ℝ+) | |
23 | 6, 21, 22 | sylancr 695 | . . . . 5 ⊢ (𝜑 → (4 / (𝐿 · 𝐸)) ∈ ℝ+) |
24 | 3, 23 | rpaddcld 11887 | . . . 4 ⊢ (𝜑 → (𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ+) |
25 | 2z 11409 | . . . 4 ⊢ 2 ∈ ℤ | |
26 | rpexpcl 12879 | . . . 4 ⊢ (((𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ+ ∧ 2 ∈ ℤ) → ((𝑌 + (4 / (𝐿 · 𝐸)))↑2) ∈ ℝ+) | |
27 | 24, 25, 26 | sylancl 694 | . . 3 ⊢ (𝜑 → ((𝑌 + (4 / (𝐿 · 𝐸)))↑2) ∈ ℝ+) |
28 | pntlem1.x | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) | |
29 | 28 | simpld 475 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
30 | 19 | simp2d 1074 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ+) |
31 | rpexpcl 12879 | . . . . . . 7 ⊢ ((𝐾 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐾↑2) ∈ ℝ+) | |
32 | 30, 25, 31 | sylancl 694 | . . . . . 6 ⊢ (𝜑 → (𝐾↑2) ∈ ℝ+) |
33 | 29, 32 | rpmulcld 11888 | . . . . 5 ⊢ (𝜑 → (𝑋 · (𝐾↑2)) ∈ ℝ+) |
34 | 4z 11411 | . . . . 5 ⊢ 4 ∈ ℤ | |
35 | rpexpcl 12879 | . . . . 5 ⊢ (((𝑋 · (𝐾↑2)) ∈ ℝ+ ∧ 4 ∈ ℤ) → ((𝑋 · (𝐾↑2))↑4) ∈ ℝ+) | |
36 | 33, 34, 35 | sylancl 694 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝐾↑2))↑4) ∈ ℝ+) |
37 | 3nn0 11310 | . . . . . . . . . . 11 ⊢ 3 ∈ ℕ0 | |
38 | 2nn 11185 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ | |
39 | 37, 38 | decnncl 11518 | . . . . . . . . . 10 ⊢ ;32 ∈ ℕ |
40 | nnrp 11842 | . . . . . . . . . 10 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
41 | 39, 40 | ax-mp 5 | . . . . . . . . 9 ⊢ ;32 ∈ ℝ+ |
42 | rpmulcl 11855 | . . . . . . . . 9 ⊢ ((;32 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (;32 · 𝐵) ∈ ℝ+) | |
43 | 41, 9, 42 | sylancr 695 | . . . . . . . 8 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ+) |
44 | 19 | simp3d 1075 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+)) |
45 | 44 | simp3d 1075 | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ+) |
46 | rpexpcl 12879 | . . . . . . . . . . 11 ⊢ ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+) | |
47 | 20, 25, 46 | sylancl 694 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ+) |
48 | 14, 47 | rpmulcld 11888 | . . . . . . . . 9 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+) |
49 | 45, 48 | rpmulcld 11888 | . . . . . . . 8 ⊢ (𝜑 → ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2))) ∈ ℝ+) |
50 | 43, 49 | rpdivcld 11889 | . . . . . . 7 ⊢ (𝜑 → ((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) ∈ ℝ+) |
51 | 3nn 11186 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ | |
52 | nnrp 11842 | . . . . . . . . . 10 ⊢ (3 ∈ ℕ → 3 ∈ ℝ+) | |
53 | 51, 52 | ax-mp 5 | . . . . . . . . 9 ⊢ 3 ∈ ℝ+ |
54 | rpmulcl 11855 | . . . . . . . . 9 ⊢ ((𝑈 ∈ ℝ+ ∧ 3 ∈ ℝ+) → (𝑈 · 3) ∈ ℝ+) | |
55 | 15, 53, 54 | sylancl 694 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 · 3) ∈ ℝ+) |
56 | pntlem1.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
57 | 55, 56 | rpaddcld 11887 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 · 3) + 𝐶) ∈ ℝ+) |
58 | 50, 57 | rpmulcld 11888 | . . . . . 6 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ+) |
59 | 58 | rpred 11872 | . . . . 5 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ) |
60 | 59 | rpefcld 14835 | . . . 4 ⊢ (𝜑 → (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))) ∈ ℝ+) |
61 | 36, 60 | rpaddcld 11887 | . . 3 ⊢ (𝜑 → (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))) ∈ ℝ+) |
62 | 27, 61 | rpaddcld 11887 | . 2 ⊢ (𝜑 → (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) ∈ ℝ+) |
63 | 1, 62 | syl5eqel 2705 | 1 ⊢ (𝜑 → 𝑊 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 ≤ cle 10075 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 3c3 11071 4c4 11072 ℤcz 11377 ;cdc 11493 ℝ+crp 11832 (,)cioo 12175 ↑cexp 12860 expce 14792 ψcchp 24819 |
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-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-inf2 8538 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 ax-pre-sup 10014 ax-addf 10015 ax-mulf 10016 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-fal 1489 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-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 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-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 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-isom 5897 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-oadd 7564 df-er 7742 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-sup 8348 df-inf 8349 df-oi 8415 df-card 8765 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-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-ioo 12179 df-ico 12181 df-fz 12327 df-fzo 12466 df-fl 12593 df-seq 12802 df-exp 12861 df-fac 13061 df-bc 13090 df-hash 13118 df-shft 13807 df-cj 13839 df-re 13840 df-im 13841 df-sqrt 13975 df-abs 13976 df-limsup 14202 df-clim 14219 df-rlim 14220 df-sum 14417 df-ef 14798 |
This theorem is referenced by: pntlemb 25286 pntleme 25297 |
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