Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > pntlema | Structured version Visualization version GIF version |
Description: Lemma for pnt 25103. 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 474 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℝ+) |
4 | 4nn 11064 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
5 | nnrp 11718 | . . . . . . 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 25083 | . . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
14 | 13 | simp1d 1066 | . . . . . . 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 25084 | . . . . . . . 8 ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))) |
20 | 19 | simp1d 1066 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
21 | 14, 20 | rpmulcld 11764 | . . . . . 6 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ+) |
22 | rpdivcl 11732 | . . . . . 6 ⊢ ((4 ∈ ℝ+ ∧ (𝐿 · 𝐸) ∈ ℝ+) → (4 / (𝐿 · 𝐸)) ∈ ℝ+) | |
23 | 6, 21, 22 | sylancr 694 | . . . . 5 ⊢ (𝜑 → (4 / (𝐿 · 𝐸)) ∈ ℝ+) |
24 | 3, 23 | rpaddcld 11763 | . . . 4 ⊢ (𝜑 → (𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ+) |
25 | 2z 11286 | . . . 4 ⊢ 2 ∈ ℤ | |
26 | rpexpcl 12741 | . . . 4 ⊢ (((𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ+ ∧ 2 ∈ ℤ) → ((𝑌 + (4 / (𝐿 · 𝐸)))↑2) ∈ ℝ+) | |
27 | 24, 25, 26 | sylancl 693 | . . 3 ⊢ (𝜑 → ((𝑌 + (4 / (𝐿 · 𝐸)))↑2) ∈ ℝ+) |
28 | pntlem1.x | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) | |
29 | 28 | simpld 474 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
30 | 19 | simp2d 1067 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ+) |
31 | rpexpcl 12741 | . . . . . . 7 ⊢ ((𝐾 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐾↑2) ∈ ℝ+) | |
32 | 30, 25, 31 | sylancl 693 | . . . . . 6 ⊢ (𝜑 → (𝐾↑2) ∈ ℝ+) |
33 | 29, 32 | rpmulcld 11764 | . . . . 5 ⊢ (𝜑 → (𝑋 · (𝐾↑2)) ∈ ℝ+) |
34 | 4z 11288 | . . . . 5 ⊢ 4 ∈ ℤ | |
35 | rpexpcl 12741 | . . . . 5 ⊢ (((𝑋 · (𝐾↑2)) ∈ ℝ+ ∧ 4 ∈ ℤ) → ((𝑋 · (𝐾↑2))↑4) ∈ ℝ+) | |
36 | 33, 34, 35 | sylancl 693 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝐾↑2))↑4) ∈ ℝ+) |
37 | 3nn0 11187 | . . . . . . . . . . 11 ⊢ 3 ∈ ℕ0 | |
38 | 2nn 11062 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ | |
39 | 37, 38 | decnncl 11394 | . . . . . . . . . 10 ⊢ ;32 ∈ ℕ |
40 | nnrp 11718 | . . . . . . . . . 10 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
41 | 39, 40 | ax-mp 5 | . . . . . . . . 9 ⊢ ;32 ∈ ℝ+ |
42 | rpmulcl 11731 | . . . . . . . . 9 ⊢ ((;32 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (;32 · 𝐵) ∈ ℝ+) | |
43 | 41, 9, 42 | sylancr 694 | . . . . . . . 8 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ+) |
44 | 19 | simp3d 1068 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+)) |
45 | 44 | simp3d 1068 | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ+) |
46 | rpexpcl 12741 | . . . . . . . . . . 11 ⊢ ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+) | |
47 | 20, 25, 46 | sylancl 693 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ+) |
48 | 14, 47 | rpmulcld 11764 | . . . . . . . . 9 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+) |
49 | 45, 48 | rpmulcld 11764 | . . . . . . . 8 ⊢ (𝜑 → ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2))) ∈ ℝ+) |
50 | 43, 49 | rpdivcld 11765 | . . . . . . 7 ⊢ (𝜑 → ((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) ∈ ℝ+) |
51 | 3nn 11063 | . . . . . . . . . 10 ⊢ 3 ∈ ℕ | |
52 | nnrp 11718 | . . . . . . . . . 10 ⊢ (3 ∈ ℕ → 3 ∈ ℝ+) | |
53 | 51, 52 | ax-mp 5 | . . . . . . . . 9 ⊢ 3 ∈ ℝ+ |
54 | rpmulcl 11731 | . . . . . . . . 9 ⊢ ((𝑈 ∈ ℝ+ ∧ 3 ∈ ℝ+) → (𝑈 · 3) ∈ ℝ+) | |
55 | 15, 53, 54 | sylancl 693 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 · 3) ∈ ℝ+) |
56 | pntlem1.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
57 | 55, 56 | rpaddcld 11763 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 · 3) + 𝐶) ∈ ℝ+) |
58 | 50, 57 | rpmulcld 11764 | . . . . . 6 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ+) |
59 | 58 | rpred 11748 | . . . . 5 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ) |
60 | 59 | rpefcld 14674 | . . . 4 ⊢ (𝜑 → (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))) ∈ ℝ+) |
61 | 36, 60 | rpaddcld 11763 | . . 3 ⊢ (𝜑 → (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))) ∈ ℝ+) |
62 | 27, 61 | rpaddcld 11763 | . 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 2692 | 1 ⊢ (𝜑 → 𝑊 ∈ ℝ+) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ↦ cmpt 4643 ‘cfv 5804 (class class class)co 6549 0cc0 9815 1c1 9816 + caddc 9818 · cmul 9820 < clt 9953 ≤ cle 9954 − cmin 10145 / cdiv 10563 ℕcn 10897 2c2 10947 3c3 10948 4c4 10949 ℤcz 11254 ;cdc 11369 ℝ+crp 11708 (,)cioo 12046 ↑cexp 12722 expce 14631 ψcchp 24619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-inf2 8421 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 ax-pre-sup 9893 ax-addf 9894 ax-mulf 9895 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-sup 8231 df-inf 8232 df-oi 8298 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-div 10564 df-nn 10898 df-2 10956 df-3 10957 df-4 10958 df-5 10959 df-6 10960 df-7 10961 df-8 10962 df-9 10963 df-n0 11170 df-z 11255 df-dec 11370 df-uz 11564 df-rp 11709 df-ioo 12050 df-ico 12052 df-fz 12198 df-fzo 12335 df-fl 12455 df-seq 12664 df-exp 12723 df-fac 12923 df-bc 12952 df-hash 12980 df-shft 13655 df-cj 13687 df-re 13688 df-im 13689 df-sqrt 13823 df-abs 13824 df-limsup 14050 df-clim 14067 df-rlim 14068 df-sum 14265 df-ef 14637 |
This theorem is referenced by: pntlemb 25086 pntleme 25097 |
Copyright terms: Public domain | W3C validator |