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Mirrors > Home > MPE Home > Th. List > pntlemd | Structured version Visualization version GIF version |
Description: Lemma for pnt 25303. 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 12235 | . . . 4 ⊢ (0(,)1) ⊆ ℝ | |
2 | pntlem1.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) | |
3 | 1, 2 | sseldi 3601 | . . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
4 | eliooord 12233 | . . . . 5 ⊢ (𝐿 ∈ (0(,)1) → (0 < 𝐿 ∧ 𝐿 < 1)) | |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (0 < 𝐿 ∧ 𝐿 < 1)) |
6 | 5 | simpld 475 | . . 3 ⊢ (𝜑 → 0 < 𝐿) |
7 | 3, 6 | elrpd 11869 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℝ+) |
8 | pntlem1.d | . . 3 ⊢ 𝐷 = (𝐴 + 1) | |
9 | pntlem1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
10 | 1rp 11836 | . . . 4 ⊢ 1 ∈ ℝ+ | |
11 | rpaddcl 11854 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝐴 + 1) ∈ ℝ+) | |
12 | 9, 10, 11 | sylancl 694 | . . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
13 | 8, 12 | syl5eqel 2705 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
14 | pntlem1.f | . . 3 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) | |
15 | 1re 10039 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
16 | ltaddrp 11867 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
17 | 15, 9, 16 | sylancr 695 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
18 | 9 | rpcnd 11874 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | ax-1cn 9994 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
20 | addcom 10222 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
21 | 18, 19, 20 | sylancl 694 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
22 | 8, 21 | syl5eq 2668 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
23 | 17, 22 | breqtrrd 4681 | . . . . . 6 ⊢ (𝜑 → 1 < 𝐷) |
24 | 13 | recgt1d 11886 | . . . . . 6 ⊢ (𝜑 → (1 < 𝐷 ↔ (1 / 𝐷) < 1)) |
25 | 23, 24 | mpbid 222 | . . . . 5 ⊢ (𝜑 → (1 / 𝐷) < 1) |
26 | 13 | rprecred 11883 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐷) ∈ ℝ) |
27 | difrp 11868 | . . . . . 6 ⊢ (((1 / 𝐷) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 𝐷) < 1 ↔ (1 − (1 / 𝐷)) ∈ ℝ+)) | |
28 | 26, 15, 27 | sylancl 694 | . . . . 5 ⊢ (𝜑 → ((1 / 𝐷) < 1 ↔ (1 − (1 / 𝐷)) ∈ ℝ+)) |
29 | 25, 28 | mpbid 222 | . . . 4 ⊢ (𝜑 → (1 − (1 / 𝐷)) ∈ ℝ+) |
30 | 3nn0 11310 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
31 | 2nn 11185 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
32 | 30, 31 | decnncl 11518 | . . . . . . . 8 ⊢ ;32 ∈ ℕ |
33 | nnrp 11842 | . . . . . . . 8 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
34 | 32, 33 | ax-mp 5 | . . . . . . 7 ⊢ ;32 ∈ ℝ+ |
35 | pntlem1.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
36 | rpmulcl 11855 | . . . . . . 7 ⊢ ((;32 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (;32 · 𝐵) ∈ ℝ+) | |
37 | 34, 35, 36 | sylancr 695 | . . . . . 6 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ+) |
38 | 7, 37 | rpdivcld 11889 | . . . . 5 ⊢ (𝜑 → (𝐿 / (;32 · 𝐵)) ∈ ℝ+) |
39 | 2z 11409 | . . . . . 6 ⊢ 2 ∈ ℤ | |
40 | rpexpcl 12879 | . . . . . 6 ⊢ ((𝐷 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐷↑2) ∈ ℝ+) | |
41 | 13, 39, 40 | sylancl 694 | . . . . 5 ⊢ (𝜑 → (𝐷↑2) ∈ ℝ+) |
42 | 38, 41 | rpdivcld 11889 | . . . 4 ⊢ (𝜑 → ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) ∈ ℝ+) |
43 | 29, 42 | rpmulcld 11888 | . . 3 ⊢ (𝜑 → ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) ∈ ℝ+) |
44 | 14, 43 | syl5eqel 2705 | . 2 ⊢ (𝜑 → 𝐹 ∈ ℝ+) |
45 | 7, 13, 44 | 3jca 1242 | 1 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 class class class wbr 4653 ↦ cmpt 4729 ‘cfv 5888 (class class class)co 6650 ℂcc 9934 ℝcr 9935 0cc0 9936 1c1 9937 + caddc 9939 · cmul 9941 < clt 10074 − cmin 10266 / cdiv 10684 ℕcn 11020 2c2 11070 3c3 11071 ℤcz 11377 ;cdc 11493 ℝ+crp 11832 (,)cioo 12175 ↑cexp 12860 ψcchp 24819 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-div 10685 df-nn 11021 df-2 11079 df-3 11080 df-4 11081 df-5 11082 df-6 11083 df-7 11084 df-8 11085 df-9 11086 df-n0 11293 df-z 11378 df-dec 11494 df-uz 11688 df-rp 11833 df-ioo 12179 df-seq 12802 df-exp 12861 |
This theorem is referenced by: pntlemc 25284 pntlema 25285 pntlemb 25286 pntlemq 25290 pntlemr 25291 pntlemj 25292 pntlemf 25294 pntlemo 25296 pntleml 25300 |
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