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Mirrors > Home > MPE Home > Th. List > pntlemd | Structured version Visualization version GIF version |
Description: Lemma for pnt 25872. 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 12648 | . . . 4 ⊢ (0(,)1) ⊆ ℝ | |
2 | pntlem1.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) | |
3 | 1, 2 | sseldi 3887 | . . 3 ⊢ (𝜑 → 𝐿 ∈ ℝ) |
4 | eliooord 12646 | . . . . 5 ⊢ (𝐿 ∈ (0(,)1) → (0 < 𝐿 ∧ 𝐿 < 1)) | |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝜑 → (0 < 𝐿 ∧ 𝐿 < 1)) |
6 | 5 | simpld 495 | . . 3 ⊢ (𝜑 → 0 < 𝐿) |
7 | 3, 6 | elrpd 12278 | . 2 ⊢ (𝜑 → 𝐿 ∈ ℝ+) |
8 | pntlem1.d | . . 3 ⊢ 𝐷 = (𝐴 + 1) | |
9 | pntlem1.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
10 | 1rp 12243 | . . . 4 ⊢ 1 ∈ ℝ+ | |
11 | rpaddcl 12261 | . . . 4 ⊢ ((𝐴 ∈ ℝ+ ∧ 1 ∈ ℝ+) → (𝐴 + 1) ∈ ℝ+) | |
12 | 9, 10, 11 | sylancl 586 | . . 3 ⊢ (𝜑 → (𝐴 + 1) ∈ ℝ+) |
13 | 8, 12 | syl5eqel 2887 | . 2 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
14 | pntlem1.f | . . 3 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) | |
15 | 1re 10487 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
16 | ltaddrp 12276 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
17 | 15, 9, 16 | sylancr 587 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
18 | 9 | rpcnd 12283 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
19 | ax-1cn 10441 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
20 | addcom 10673 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
21 | 18, 19, 20 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
22 | 8, 21 | syl5eq 2843 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
23 | 17, 22 | breqtrrd 4990 | . . . . . 6 ⊢ (𝜑 → 1 < 𝐷) |
24 | 13 | recgt1d 12295 | . . . . . 6 ⊢ (𝜑 → (1 < 𝐷 ↔ (1 / 𝐷) < 1)) |
25 | 23, 24 | mpbid 233 | . . . . 5 ⊢ (𝜑 → (1 / 𝐷) < 1) |
26 | 13 | rprecred 12292 | . . . . . 6 ⊢ (𝜑 → (1 / 𝐷) ∈ ℝ) |
27 | difrp 12277 | . . . . . 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 233 | . . . 4 ⊢ (𝜑 → (1 − (1 / 𝐷)) ∈ ℝ+) |
30 | 3nn0 11763 | . . . . . . . . 9 ⊢ 3 ∈ ℕ0 | |
31 | 2nn 11558 | . . . . . . . . 9 ⊢ 2 ∈ ℕ | |
32 | 30, 31 | decnncl 11967 | . . . . . . . 8 ⊢ ;32 ∈ ℕ |
33 | nnrp 12250 | . . . . . . . 8 ⊢ (;32 ∈ ℕ → ;32 ∈ ℝ+) | |
34 | 32, 33 | ax-mp 5 | . . . . . . 7 ⊢ ;32 ∈ ℝ+ |
35 | pntlem1.b | . . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
36 | rpmulcl 12262 | . . . . . . 7 ⊢ ((;32 ∈ ℝ+ ∧ 𝐵 ∈ ℝ+) → (;32 · 𝐵) ∈ ℝ+) | |
37 | 34, 35, 36 | sylancr 587 | . . . . . 6 ⊢ (𝜑 → (;32 · 𝐵) ∈ ℝ+) |
38 | 7, 37 | rpdivcld 12298 | . . . . 5 ⊢ (𝜑 → (𝐿 / (;32 · 𝐵)) ∈ ℝ+) |
39 | 2z 11863 | . . . . . 6 ⊢ 2 ∈ ℤ | |
40 | rpexpcl 13298 | . . . . . 6 ⊢ ((𝐷 ∈ ℝ+ ∧ 2 ∈ ℤ) → (𝐷↑2) ∈ ℝ+) | |
41 | 13, 39, 40 | sylancl 586 | . . . . 5 ⊢ (𝜑 → (𝐷↑2) ∈ ℝ+) |
42 | 38, 41 | rpdivcld 12298 | . . . 4 ⊢ (𝜑 → ((𝐿 / (;32 · 𝐵)) / (𝐷↑2)) ∈ ℝ+) |
43 | 29, 42 | rpmulcld 12297 | . . 3 ⊢ (𝜑 → ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) ∈ ℝ+) |
44 | 14, 43 | syl5eqel 2887 | . 2 ⊢ (𝜑 → 𝐹 ∈ ℝ+) |
45 | 7, 13, 44 | 3jca 1121 | 1 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 class class class wbr 4962 ↦ cmpt 5041 ‘cfv 6225 (class class class)co 7016 ℂcc 10381 ℝcr 10382 0cc0 10383 1c1 10384 + caddc 10386 · cmul 10388 < clt 10521 − cmin 10717 / cdiv 11145 ℕcn 11486 2c2 11540 3c3 11541 ℤcz 11829 ;cdc 11947 ℝ+crp 12239 (,)cioo 12588 ↑cexp 13279 ψcchp 25352 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-cnex 10439 ax-resscn 10440 ax-1cn 10441 ax-icn 10442 ax-addcl 10443 ax-addrcl 10444 ax-mulcl 10445 ax-mulrcl 10446 ax-mulcom 10447 ax-addass 10448 ax-mulass 10449 ax-distr 10450 ax-i2m1 10451 ax-1ne0 10452 ax-1rid 10453 ax-rnegex 10454 ax-rrecex 10455 ax-cnre 10456 ax-pre-lttri 10457 ax-pre-lttrn 10458 ax-pre-ltadd 10459 ax-pre-mulgt0 10460 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rmo 3113 df-rab 3114 df-v 3439 df-sbc 3707 df-csb 3812 df-dif 3862 df-un 3864 df-in 3866 df-ss 3874 df-pss 3876 df-nul 4212 df-if 4382 df-pw 4455 df-sn 4473 df-pr 4475 df-tp 4477 df-op 4479 df-uni 4746 df-iun 4827 df-br 4963 df-opab 5025 df-mpt 5042 df-tr 5064 df-id 5348 df-eprel 5353 df-po 5362 df-so 5363 df-fr 5402 df-we 5404 df-xp 5449 df-rel 5450 df-cnv 5451 df-co 5452 df-dm 5453 df-rn 5454 df-res 5455 df-ima 5456 df-pred 6023 df-ord 6069 df-on 6070 df-lim 6071 df-suc 6072 df-iota 6189 df-fun 6227 df-fn 6228 df-f 6229 df-f1 6230 df-fo 6231 df-f1o 6232 df-fv 6233 df-riota 6977 df-ov 7019 df-oprab 7020 df-mpo 7021 df-om 7437 df-1st 7545 df-2nd 7546 df-wrecs 7798 df-recs 7860 df-rdg 7898 df-er 8139 df-en 8358 df-dom 8359 df-sdom 8360 df-pnf 10523 df-mnf 10524 df-xr 10525 df-ltxr 10526 df-le 10527 df-sub 10719 df-neg 10720 df-div 11146 df-nn 11487 df-2 11548 df-3 11549 df-4 11550 df-5 11551 df-6 11552 df-7 11553 df-8 11554 df-9 11555 df-n0 11746 df-z 11830 df-dec 11948 df-uz 12094 df-rp 12240 df-ioo 12592 df-seq 13220 df-exp 13280 |
This theorem is referenced by: pntlemc 25853 pntlema 25854 pntlemb 25855 pntlemq 25859 pntlemr 25860 pntlemj 25861 pntlemf 25863 pntlemo 25865 pntleml 25869 |
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