Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pntlema | Structured version Visualization version GIF version |
Description: Lemma for pnt 26192. 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 11723 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
5 | nnrp 12403 | . . . . . . 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 26172 | . . . . . . . 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 26173 | . . . . . . . 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 12450 | . . . . . 6 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ+) |
22 | rpdivcl 12417 | . . . . . 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 12449 | . . . 4 ⊢ (𝜑 → (𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ+) |
25 | 2z 12017 | . . . 4 ⊢ 2 ∈ ℤ | |
26 | rpexpcl 13451 | . . . 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 13451 | . . . . . . 7 ⊢ ((𝐾 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐾↑2) ∈ ℝ+) | |
32 | 30, 25, 31 | sylancl 588 | . . . . . 6 ⊢ (𝜑 → (𝐾↑2) ∈ ℝ+) |
33 | 29, 32 | rpmulcld 12450 | . . . . 5 ⊢ (𝜑 → (𝑋 · (𝐾↑2)) ∈ ℝ+) |
34 | 4z 12019 | . . . . 5 ⊢ 4 ∈ ℤ | |
35 | rpexpcl 13451 | . . . . 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 11918 | . . . . . . . . . . 11 ⊢ 3 ∈ ℕ0 | |
38 | 2nn 11713 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ | |
39 | 37, 38 | decnncl 12121 | . . . . . . . . . 10 ⊢ ;32 ∈ ℕ |
40 | nnrp 12403 | . . . . . . . . . 10 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
41 | 39, 40 | ax-mp 5 | . . . . . . . . 9 ⊢ ;32 ∈ ℝ+ |
42 | rpmulcl 12415 | . . . . . . . . 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 13451 | . . . . . . . . . . 11 ⊢ ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+) | |
47 | 20, 25, 46 | sylancl 588 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ+) |
48 | 14, 47 | rpmulcld 12450 | . . . . . . . . 9 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+) |
49 | 45, 48 | rpmulcld 12450 | . . . . . . . 8 ⊢ (𝜑 → ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2))) ∈ ℝ+) |
50 | 43, 49 | rpdivcld 12451 | . . . . . . 7 ⊢ (𝜑 → ((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) ∈ ℝ+) |
51 | 3rp 12398 | . . . . . . . . 9 ⊢ 3 ∈ ℝ+ | |
52 | rpmulcl 12415 | . . . . . . . . 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 12449 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 · 3) + 𝐶) ∈ ℝ+) |
56 | 50, 55 | rpmulcld 12450 | . . . . . 6 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ+) |
57 | 56 | rpred 12434 | . . . . 5 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ) |
58 | 57 | rpefcld 15460 | . . . 4 ⊢ (𝜑 → (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))) ∈ ℝ+) |
59 | 36, 58 | rpaddcld 12449 | . . 3 ⊢ (𝜑 → (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))) ∈ ℝ+) |
60 | 27, 59 | rpaddcld 12449 | . 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 2919 | 1 ⊢ (𝜑 → 𝑊 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ↦ cmpt 5148 ‘cfv 6357 (class class class)co 7158 0cc0 10539 1c1 10540 + caddc 10542 · cmul 10544 < clt 10677 ≤ cle 10678 − cmin 10872 / cdiv 11299 ℕcn 11640 2c2 11695 3c3 11696 4c4 11697 ℤcz 11984 ;cdc 12101 ℝ+crp 12392 (,)cioo 12741 ↑cexp 13432 expce 15417 ψcchp 25672 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-inf2 9106 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 ax-pre-sup 10617 ax-addf 10618 ax-mulf 10619 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rmo 3148 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-se 5517 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-isom 6366 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-oadd 8108 df-er 8291 df-pm 8411 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-sup 8908 df-inf 8909 df-oi 8976 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-div 11300 df-nn 11641 df-2 11703 df-3 11704 df-4 11705 df-5 11706 df-6 11707 df-7 11708 df-8 11709 df-9 11710 df-n0 11901 df-z 11985 df-dec 12102 df-uz 12247 df-rp 12393 df-ioo 12745 df-ico 12747 df-fz 12896 df-fzo 13037 df-fl 13165 df-seq 13373 df-exp 13433 df-fac 13637 df-bc 13666 df-hash 13694 df-shft 14428 df-cj 14460 df-re 14461 df-im 14462 df-sqrt 14596 df-abs 14597 df-limsup 14830 df-clim 14847 df-rlim 14848 df-sum 15045 df-ef 15423 |
This theorem is referenced by: pntlemb 26175 pntleme 26186 |
Copyright terms: Public domain | W3C validator |