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Mirrors > Home > MPE Home > Th. List > pntlemd | Structured version Visualization version GIF version |
Description: Lemma for pnt 25348. 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 12273 | . . . 4 ⊢ (0(,)1) ⊆ ℝ | |
2 | pntlem1.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) | |
3 | 1, 2 | sseldi 3634 | . . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
4 | eliooord 12271 | . . . . 5 ⊢ (𝐿 ∈ (0(,)1) → (0 < 𝐿 ∧ 𝐿 < 1)) | |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (0 < 𝐿 ∧ 𝐿 < 1)) |
6 | 5 | simpld 474 | . . 3 ⊢ (𝜑 → 0 < 𝐿) |
7 | 3, 6 | elrpd 11907 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℝ+) |
8 | pntlem1.d | . . 3 ⊢ 𝐷 = (𝐴 + 1) | |
9 | pntlem1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
10 | 1rp 11874 | . . . 4 ⊢ 1 ∈ ℝ+ | |
11 | rpaddcl 11892 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝐴 + 1) ∈ ℝ+) | |
12 | 9, 10, 11 | sylancl 695 | . . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
13 | 8, 12 | syl5eqel 2734 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
14 | pntlem1.f | . . 3 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) | |
15 | 1re 10077 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
16 | ltaddrp 11905 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
17 | 15, 9, 16 | sylancr 696 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
18 | 9 | rpcnd 11912 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | ax-1cn 10032 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
20 | addcom 10260 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
21 | 18, 19, 20 | sylancl 695 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
22 | 8, 21 | syl5eq 2697 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
23 | 17, 22 | breqtrrd 4713 | . . . . . 6 ⊢ (𝜑 → 1 < 𝐷) |
24 | 13 | recgt1d 11924 | . . . . . 6 ⊢ (𝜑 → (1 < 𝐷 ↔ (1 / 𝐷) < 1)) |
25 | 23, 24 | mpbid 222 | . . . . 5 ⊢ (𝜑 → (1 / 𝐷) < 1) |
26 | 13 | rprecred 11921 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐷) ∈ ℝ) |
27 | difrp 11906 | . . . . . 6 ⊢ (((1 / 𝐷) ∈ ℝ ∧ 1 ∈ ℝ) → ((1 / 𝐷) < 1 ↔ (1 − (1 / 𝐷)) ∈ ℝ+)) | |
28 | 26, 15, 27 | sylancl 695 | . . . . 5 ⊢ (𝜑 → ((1 / 𝐷) < 1 ↔ (1 − (1 / 𝐷)) ∈ ℝ+)) |
29 | 25, 28 | mpbid 222 | . . . 4 ⊢ (𝜑 → (1 − (1 / 𝐷)) ∈ ℝ+) |
30 | 3nn0 11348 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
31 | 2nn 11223 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
32 | 30, 31 | decnncl 11556 | . . . . . . . 8 ⊢ ;32 ∈ ℕ |
33 | nnrp 11880 | . . . . . . . 8 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
34 | 32, 33 | ax-mp 5 | . . . . . . 7 ⊢ ;32 ∈ ℝ+ |
35 | pntlem1.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
36 | rpmulcl 11893 | . . . . . . 7 ⊢ ((;32 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (;32 · 𝐵) ∈ ℝ+) | |
37 | 34, 35, 36 | sylancr 696 | . . . . . 6 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ+) |
38 | 7, 37 | rpdivcld 11927 | . . . . 5 ⊢ (𝜑 → (𝐿 / (;32 · 𝐵)) ∈ ℝ+) |
39 | 2z 11447 | . . . . . 6 ⊢ 2 ∈ ℤ | |
40 | rpexpcl 12919 | . . . . . 6 ⊢ ((𝐷 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐷↑2) ∈ ℝ+) | |
41 | 13, 39, 40 | sylancl 695 | . . . . 5 ⊢ (𝜑 → (𝐷↑2) ∈ ℝ+) |
42 | 38, 41 | rpdivcld 11927 | . . . 4 ⊢ (𝜑 → ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) ∈ ℝ+) |
43 | 29, 42 | rpmulcld 11926 | . . 3 ⊢ (𝜑 → ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) ∈ ℝ+) |
44 | 14, 43 | syl5eqel 2734 | . 2 ⊢ (𝜑 → 𝐹 ∈ ℝ+) |
45 | 7, 13, 44 | 3jca 1261 | 1 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∧ w3a 1054 = wceq 1523 ∈ wcel 2030 class class class wbr 4685 ↦ cmpt 4762 ‘cfv 5926 (class class class)co 6690 ℂcc 9972 ℝcr 9973 0cc0 9974 1c1 9975 + caddc 9977 · cmul 9979 < clt 10112 − cmin 10304 / cdiv 10722 ℕcn 11058 2c2 11108 3c3 11109 ℤcz 11415 ;cdc 11531 ℝ+crp 11870 (,)cioo 12213 ↑cexp 12900 ψcchp 24864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-cnex 10030 ax-resscn 10031 ax-1cn 10032 ax-icn 10033 ax-addcl 10034 ax-addrcl 10035 ax-mulcl 10036 ax-mulrcl 10037 ax-mulcom 10038 ax-addass 10039 ax-mulass 10040 ax-distr 10041 ax-i2m1 10042 ax-1ne0 10043 ax-1rid 10044 ax-rnegex 10045 ax-rrecex 10046 ax-cnre 10047 ax-pre-lttri 10048 ax-pre-lttrn 10049 ax-pre-ltadd 10050 ax-pre-mulgt0 10051 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-nel 2927 df-ral 2946 df-rex 2947 df-reu 2948 df-rmo 2949 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-riota 6651 df-ov 6693 df-oprab 6694 df-mpt2 6695 df-om 7108 df-1st 7210 df-2nd 7211 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-er 7787 df-en 7998 df-dom 7999 df-sdom 8000 df-pnf 10114 df-mnf 10115 df-xr 10116 df-ltxr 10117 df-le 10118 df-sub 10306 df-neg 10307 df-div 10723 df-nn 11059 df-2 11117 df-3 11118 df-4 11119 df-5 11120 df-6 11121 df-7 11122 df-8 11123 df-9 11124 df-n0 11331 df-z 11416 df-dec 11532 df-uz 11726 df-rp 11871 df-ioo 12217 df-seq 12842 df-exp 12901 |
This theorem is referenced by: pntlemc 25329 pntlema 25330 pntlemb 25331 pntlemq 25335 pntlemr 25336 pntlemj 25337 pntlemf 25339 pntlemo 25341 pntleml 25345 |
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