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Mirrors > Home > MPE Home > Th. List > pntlemd | Structured version Visualization version GIF version |
Description: Lemma for pnt 27672. Closure for the constants used in the proof. For comparison with Equation 10.6.27 of [Shapiro], p. 434, 𝐴 is C^*, 𝐵 is c1, 𝐿 is λ, 𝐷 is c2, and 𝐹 is c3. (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))) |
Ref | Expression |
---|---|
pntlemd | ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ioossre 13444 | . . . 4 ⊢ (0(,)1) ⊆ ℝ | |
2 | pntlem1.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) | |
3 | 1, 2 | sselid 3992 | . . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
4 | eliooord 13442 | . . . . 5 ⊢ (𝐿 ∈ (0(,)1) → (0 < 𝐿 ∧ 𝐿 < 1)) | |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (0 < 𝐿 ∧ 𝐿 < 1)) |
6 | 5 | simpld 494 | . . 3 ⊢ (𝜑 → 0 < 𝐿) |
7 | 3, 6 | elrpd 13071 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℝ+) |
8 | pntlem1.d | . . 3 ⊢ 𝐷 = (𝐴 + 1) | |
9 | pntlem1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
10 | 1rp 13035 | . . . 4 ⊢ 1 ∈ ℝ+ | |
11 | rpaddcl 13054 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝐴 + 1) ∈ ℝ+) | |
12 | 9, 10, 11 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
13 | 8, 12 | eqeltrid 2842 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
14 | pntlem1.f | . . 3 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) | |
15 | 1re 11258 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
16 | ltaddrp 13069 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
17 | 15, 9, 16 | sylancr 587 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
18 | 9 | rpcnd 13076 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | ax-1cn 11210 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
20 | addcom 11444 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
21 | 18, 19, 20 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
22 | 8, 21 | eqtrid 2786 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
23 | 17, 22 | breqtrrd 5175 | . . . . . 6 ⊢ (𝜑 → 1 < 𝐷) |
24 | 13 | recgt1d 13088 | . . . . . 6 ⊢ (𝜑 → (1 < 𝐷 ↔ (1 / 𝐷) < 1)) |
25 | 23, 24 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (1 / 𝐷) < 1) |
26 | 13 | rprecred 13085 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐷) ∈ ℝ) |
27 | difrp 13070 | . . . . . 6 ⊢ (((1 / 𝐷) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 𝐷) < 1 ↔ (1 − (1 / 𝐷)) ∈ ℝ+)) | |
28 | 26, 15, 27 | sylancl 586 | . . . . 5 ⊢ (𝜑 → ((1 / 𝐷) < 1 ↔ (1 − (1 / 𝐷)) ∈ ℝ+)) |
29 | 25, 28 | mpbid 232 | . . . 4 ⊢ (𝜑 → (1 − (1 / 𝐷)) ∈ ℝ+) |
30 | 3nn0 12541 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
31 | 2nn 12336 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
32 | 30, 31 | decnncl 12750 | . . . . . . . 8 ⊢ ;32 ∈ ℕ |
33 | nnrp 13043 | . . . . . . . 8 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
34 | 32, 33 | ax-mp 5 | . . . . . . 7 ⊢ ;32 ∈ ℝ+ |
35 | pntlem1.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
36 | rpmulcl 13055 | . . . . . . 7 ⊢ ((;32 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (;32 · 𝐵) ∈ ℝ+) | |
37 | 34, 35, 36 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ+) |
38 | 7, 37 | rpdivcld 13091 | . . . . 5 ⊢ (𝜑 → (𝐿 / (;32 · 𝐵)) ∈ ℝ+) |
39 | 2z 12646 | . . . . . 6 ⊢ 2 ∈ ℤ | |
40 | rpexpcl 14117 | . . . . . 6 ⊢ ((𝐷 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐷↑2) ∈ ℝ+) | |
41 | 13, 39, 40 | sylancl 586 | . . . . 5 ⊢ (𝜑 → (𝐷↑2) ∈ ℝ+) |
42 | 38, 41 | rpdivcld 13091 | . . . 4 ⊢ (𝜑 → ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) ∈ ℝ+) |
43 | 29, 42 | rpmulcld 13090 | . . 3 ⊢ (𝜑 → ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) ∈ ℝ+) |
44 | 14, 43 | eqeltrid 2842 | . 2 ⊢ (𝜑 → 𝐹 ∈ ℝ+) |
45 | 7, 13, 44 | 3jca 1127 | 1 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 < clt 11292 − cmin 11489 / cdiv 11917 ℕcn 12263 2c2 12318 3c3 12319 ℤcz 12610 ;cdc 12730 ℝ+crp 13031 (,)cioo 13383 ↑cexp 14098 ψcchp 27150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-er 8743 df-en 8984 df-dom 8985 df-sdom 8986 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-ioo 13387 df-seq 14039 df-exp 14099 |
This theorem is referenced by: pntlemc 27653 pntlema 27654 pntlemb 27655 pntlemq 27659 pntlemr 27660 pntlemj 27661 pntlemf 27663 pntlemo 27665 pntleml 27669 |
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