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Mirrors > Home > MPE Home > Th. List > pntlema | Structured version Visualization version GIF version |
Description: Lemma for pnt 26184. 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 497 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℝ+) |
4 | 4nn 11714 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
5 | nnrp 12394 | . . . . . . 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 26164 | . . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
14 | 13 | simp1d 1138 | . . . . . . 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 26165 | . . . . . . . 8 ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))) |
20 | 19 | simp1d 1138 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
21 | 14, 20 | rpmulcld 12441 | . . . . . 6 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ+) |
22 | rpdivcl 12408 | . . . . . 6 ⊢ ((4 ∈ ℝ+ ∧ (𝐿 · 𝐸) ∈ ℝ+) → (4 / (𝐿 · 𝐸)) ∈ ℝ+) | |
23 | 6, 21, 22 | sylancr 589 | . . . . 5 ⊢ (𝜑 → (4 / (𝐿 · 𝐸)) ∈ ℝ+) |
24 | 3, 23 | rpaddcld 12440 | . . . 4 ⊢ (𝜑 → (𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ+) |
25 | 2z 12008 | . . . 4 ⊢ 2 ∈ ℤ | |
26 | rpexpcl 13442 | . . . 4 ⊢ (((𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ+ ∧ 2 ∈ ℤ) → ((𝑌 + (4 / (𝐿 · 𝐸)))↑2) ∈ ℝ+) | |
27 | 24, 25, 26 | sylancl 588 | . . 3 ⊢ (𝜑 → ((𝑌 + (4 / (𝐿 · 𝐸)))↑2) ∈ ℝ+) |
28 | pntlem1.x | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) | |
29 | 28 | simpld 497 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
30 | 19 | simp2d 1139 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ+) |
31 | rpexpcl 13442 | . . . . . . 7 ⊢ ((𝐾 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐾↑2) ∈ ℝ+) | |
32 | 30, 25, 31 | sylancl 588 | . . . . . 6 ⊢ (𝜑 → (𝐾↑2) ∈ ℝ+) |
33 | 29, 32 | rpmulcld 12441 | . . . . 5 ⊢ (𝜑 → (𝑋 · (𝐾↑2)) ∈ ℝ+) |
34 | 4z 12010 | . . . . 5 ⊢ 4 ∈ ℤ | |
35 | rpexpcl 13442 | . . . . 5 ⊢ (((𝑋 · (𝐾↑2)) ∈ ℝ+ ∧ 4 ∈ ℤ) → ((𝑋 · (𝐾↑2))↑4) ∈ ℝ+) | |
36 | 33, 34, 35 | sylancl 588 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝐾↑2))↑4) ∈ ℝ+) |
37 | 3nn0 11909 | . . . . . . . . . . 11 ⊢ 3 ∈ ℕ0 | |
38 | 2nn 11704 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ | |
39 | 37, 38 | decnncl 12112 | . . . . . . . . . 10 ⊢ ;32 ∈ ℕ |
40 | nnrp 12394 | . . . . . . . . . 10 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
41 | 39, 40 | ax-mp 5 | . . . . . . . . 9 ⊢ ;32 ∈ ℝ+ |
42 | rpmulcl 12406 | . . . . . . . . 9 ⊢ ((;32 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (;32 · 𝐵) ∈ ℝ+) | |
43 | 41, 9, 42 | sylancr 589 | . . . . . . . 8 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ+) |
44 | 19 | simp3d 1140 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+)) |
45 | 44 | simp3d 1140 | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ+) |
46 | rpexpcl 13442 | . . . . . . . . . . 11 ⊢ ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+) | |
47 | 20, 25, 46 | sylancl 588 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ+) |
48 | 14, 47 | rpmulcld 12441 | . . . . . . . . 9 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+) |
49 | 45, 48 | rpmulcld 12441 | . . . . . . . 8 ⊢ (𝜑 → ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2))) ∈ ℝ+) |
50 | 43, 49 | rpdivcld 12442 | . . . . . . 7 ⊢ (𝜑 → ((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) ∈ ℝ+) |
51 | 3rp 12389 | . . . . . . . . 9 ⊢ 3 ∈ ℝ+ | |
52 | rpmulcl 12406 | . . . . . . . . 9 ⊢ ((𝑈 ∈ ℝ+ ∧ 3 ∈ ℝ+) → (𝑈 · 3) ∈ ℝ+) | |
53 | 15, 51, 52 | sylancl 588 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 · 3) ∈ ℝ+) |
54 | pntlem1.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
55 | 53, 54 | rpaddcld 12440 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 · 3) + 𝐶) ∈ ℝ+) |
56 | 50, 55 | rpmulcld 12441 | . . . . . 6 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ+) |
57 | 56 | rpred 12425 | . . . . 5 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ) |
58 | 57 | rpefcld 15452 | . . . 4 ⊢ (𝜑 → (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))) ∈ ℝ+) |
59 | 36, 58 | rpaddcld 12440 | . . 3 ⊢ (𝜑 → (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))) ∈ ℝ+) |
60 | 27, 59 | rpaddcld 12440 | . 2 ⊢ (𝜑 → (((𝑌 + (4 / (𝐿 · 𝐸)))↑2) + (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))))) ∈ ℝ+) |
61 | 1, 60 | eqeltrid 2917 | 1 ⊢ (𝜑 → 𝑊 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 0cc0 10531 1c1 10532 + caddc 10534 · cmul 10536 < clt 10669 ≤ cle 10670 − cmin 10864 / cdiv 11291 ℕcn 11632 2c2 11686 3c3 11687 4c4 11688 ℤcz 11975 ;cdc 12092 ℝ+crp 12383 (,)cioo 12732 ↑cexp 13423 expce 15409 ψcchp 25664 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-inf2 9098 ax-cnex 10587 ax-resscn 10588 ax-1cn 10589 ax-icn 10590 ax-addcl 10591 ax-addrcl 10592 ax-mulcl 10593 ax-mulrcl 10594 ax-mulcom 10595 ax-addass 10596 ax-mulass 10597 ax-distr 10598 ax-i2m1 10599 ax-1ne0 10600 ax-1rid 10601 ax-rnegex 10602 ax-rrecex 10603 ax-cnre 10604 ax-pre-lttri 10605 ax-pre-lttrn 10606 ax-pre-ltadd 10607 ax-pre-mulgt0 10608 ax-pre-sup 10609 ax-addf 10610 ax-mulf 10611 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-fal 1546 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-int 4869 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-se 5509 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-isom 6358 df-riota 7108 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-1st 7683 df-2nd 7684 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-oadd 8100 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 df-sup 8900 df-inf 8901 df-oi 8968 df-card 9362 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-sub 10866 df-neg 10867 df-div 11292 df-nn 11633 df-2 11694 df-3 11695 df-4 11696 df-5 11697 df-6 11698 df-7 11699 df-8 11700 df-9 11701 df-n0 11892 df-z 11976 df-dec 12093 df-uz 12238 df-rp 12384 df-ioo 12736 df-ico 12738 df-fz 12887 df-fzo 13028 df-fl 13156 df-seq 13364 df-exp 13424 df-fac 13628 df-bc 13657 df-hash 13685 df-shft 14420 df-cj 14452 df-re 14453 df-im 14454 df-sqrt 14588 df-abs 14589 df-limsup 14822 df-clim 14839 df-rlim 14840 df-sum 15037 df-ef 15415 |
This theorem is referenced by: pntlemb 26167 pntleme 26178 |
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