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| Mirrors > Home > MPE Home > Th. List > pntlema | Structured version Visualization version GIF version | ||
| Description: Lemma for pnt 27560. 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 494 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ ℝ+) |
| 4 | 4nn 12326 | . . . . . . 7 ⊢ 4 ∈ ℕ | |
| 5 | nnrp 13018 | . . . . . . 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 27540 | . . . . . . . 8 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
| 14 | 13 | simp1d 1140 | . . . . . . 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 27541 | . . . . . . . 8 ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))) |
| 20 | 19 | simp1d 1140 | . . . . . . 7 ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
| 21 | 14, 20 | rpmulcld 13065 | . . . . . 6 ⊢ (𝜑 → (𝐿 · 𝐸) ∈ ℝ+) |
| 22 | rpdivcl 13032 | . . . . . 6 ⊢ ((4 ∈ ℝ+ ∧ (𝐿 · 𝐸) ∈ ℝ+) → (4 / (𝐿 · 𝐸)) ∈ ℝ+) | |
| 23 | 6, 21, 22 | sylancr 586 | . . . . 5 ⊢ (𝜑 → (4 / (𝐿 · 𝐸)) ∈ ℝ+) |
| 24 | 3, 23 | rpaddcld 13064 | . . . 4 ⊢ (𝜑 → (𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ+) |
| 25 | 2z 12625 | . . . 4 ⊢ 2 ∈ ℤ | |
| 26 | rpexpcl 14078 | . . . 4 ⊢ (((𝑌 + (4 / (𝐿 · 𝐸))) ∈ ℝ+ ∧ 2 ∈ ℤ) → ((𝑌 + (4 / (𝐿 · 𝐸)))↑2) ∈ ℝ+) | |
| 27 | 24, 25, 26 | sylancl 585 | . . 3 ⊢ (𝜑 → ((𝑌 + (4 / (𝐿 · 𝐸)))↑2) ∈ ℝ+) |
| 28 | pntlem1.x | . . . . . . 7 ⊢ (𝜑 → (𝑋 ∈ ℝ+ ∧ 𝑌 < 𝑋)) | |
| 29 | 28 | simpld 494 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ ℝ+) |
| 30 | 19 | simp2d 1141 | . . . . . . 7 ⊢ (𝜑 → 𝐾 ∈ ℝ+) |
| 31 | rpexpcl 14078 | . . . . . . 7 ⊢ ((𝐾 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐾↑2) ∈ ℝ+) | |
| 32 | 30, 25, 31 | sylancl 585 | . . . . . 6 ⊢ (𝜑 → (𝐾↑2) ∈ ℝ+) |
| 33 | 29, 32 | rpmulcld 13065 | . . . . 5 ⊢ (𝜑 → (𝑋 · (𝐾↑2)) ∈ ℝ+) |
| 34 | 4z 12627 | . . . . 5 ⊢ 4 ∈ ℤ | |
| 35 | rpexpcl 14078 | . . . . 5 ⊢ (((𝑋 · (𝐾↑2)) ∈ ℝ+ ∧ 4 ∈ ℤ) → ((𝑋 · (𝐾↑2))↑4) ∈ ℝ+) | |
| 36 | 33, 34, 35 | sylancl 585 | . . . 4 ⊢ (𝜑 → ((𝑋 · (𝐾↑2))↑4) ∈ ℝ+) |
| 37 | 3nn0 12521 | . . . . . . . . . . 11 ⊢ 3 ∈ ℕ0 | |
| 38 | 2nn 12316 | . . . . . . . . . . 11 ⊢ 2 ∈ ℕ | |
| 39 | 37, 38 | decnncl 12728 | . . . . . . . . . 10 ⊢ ;32 ∈ ℕ |
| 40 | nnrp 13018 | . . . . . . . . . 10 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
| 41 | 39, 40 | ax-mp 5 | . . . . . . . . 9 ⊢ ;32 ∈ ℝ+ |
| 42 | rpmulcl 13030 | . . . . . . . . 9 ⊢ ((;32 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (;32 · 𝐵) ∈ ℝ+) | |
| 43 | 41, 9, 42 | sylancr 586 | . . . . . . . 8 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ+) |
| 44 | 19 | simp3d 1142 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+)) |
| 45 | 44 | simp3d 1142 | . . . . . . . . 9 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ+) |
| 46 | rpexpcl 14078 | . . . . . . . . . . 11 ⊢ ((𝐸 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐸↑2) ∈ ℝ+) | |
| 47 | 20, 25, 46 | sylancl 585 | . . . . . . . . . 10 ⊢ (𝜑 → (𝐸↑2) ∈ ℝ+) |
| 48 | 14, 47 | rpmulcld 13065 | . . . . . . . . 9 ⊢ (𝜑 → (𝐿 · (𝐸↑2)) ∈ ℝ+) |
| 49 | 45, 48 | rpmulcld 13065 | . . . . . . . 8 ⊢ (𝜑 → ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2))) ∈ ℝ+) |
| 50 | 43, 49 | rpdivcld 13066 | . . . . . . 7 ⊢ (𝜑 → ((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) ∈ ℝ+) |
| 51 | 3rp 13013 | . . . . . . . . 9 ⊢ 3 ∈ ℝ+ | |
| 52 | rpmulcl 13030 | . . . . . . . . 9 ⊢ ((𝑈 ∈ ℝ+ ∧ 3 ∈ ℝ+) → (𝑈 · 3) ∈ ℝ+) | |
| 53 | 15, 51, 52 | sylancl 585 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 · 3) ∈ ℝ+) |
| 54 | pntlem1.c | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ ℝ+) | |
| 55 | 53, 54 | rpaddcld 13064 | . . . . . . 7 ⊢ (𝜑 → ((𝑈 · 3) + 𝐶) ∈ ℝ+) |
| 56 | 50, 55 | rpmulcld 13065 | . . . . . 6 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ+) |
| 57 | 56 | rpred 13049 | . . . . 5 ⊢ (𝜑 → (((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)) ∈ ℝ) |
| 58 | 57 | rpefcld 16082 | . . . 4 ⊢ (𝜑 → (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶))) ∈ ℝ+) |
| 59 | 36, 58 | rpaddcld 13064 | . . 3 ⊢ (𝜑 → (((𝑋 · (𝐾↑2))↑4) + (exp‘(((;32 · 𝐵) / ((𝑈 − 𝐸) · (𝐿 · (𝐸↑2)))) · ((𝑈 · 3) + 𝐶)))) ∈ ℝ+) |
| 60 | 27, 59 | rpaddcld 13064 | . 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 2833 | 1 ⊢ (𝜑 → 𝑊 ∈ ℝ+) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1085 = wceq 1534 ∈ wcel 2099 class class class wbr 5148 ↦ cmpt 5231 ‘cfv 6548 (class class class)co 7420 0cc0 11139 1c1 11140 + caddc 11142 · cmul 11144 < clt 11279 ≤ cle 11280 − cmin 11475 / cdiv 11902 ℕcn 12243 2c2 12298 3c3 12299 4c4 12300 ℤcz 12589 ;cdc 12708 ℝ+crp 13007 (,)cioo 13357 ↑cexp 14059 expce 16038 ψcchp 27038 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-inf2 9665 ax-cnex 11195 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217 |
| This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3373 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-se 5634 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6305 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-isom 6557 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-om 7871 df-1st 7993 df-2nd 7994 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-sup 9466 df-inf 9467 df-oi 9534 df-card 9963 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478 df-div 11903 df-nn 12244 df-2 12306 df-3 12307 df-4 12308 df-5 12309 df-6 12310 df-7 12311 df-8 12312 df-9 12313 df-n0 12504 df-z 12590 df-dec 12709 df-uz 12854 df-rp 13008 df-ioo 13361 df-ico 13363 df-fz 13518 df-fzo 13661 df-fl 13790 df-seq 14000 df-exp 14060 df-fac 14266 df-bc 14295 df-hash 14323 df-shft 15047 df-cj 15079 df-re 15080 df-im 15081 df-sqrt 15215 df-abs 15216 df-limsup 15448 df-clim 15465 df-rlim 15466 df-sum 15666 df-ef 16044 |
| This theorem is referenced by: pntlemb 27543 pntleme 27554 |
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