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Mirrors > Home > MPE Home > Th. List > pntlemc | 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, 𝑈 is α, 𝐸 is ε, and 𝐾 is K. (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)) |
Ref | Expression |
---|---|
pntlemc | ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pntlem1.e | . . 3 ⊢ 𝐸 = (𝑈 / 𝐷) | |
2 | pntlem1.u | . . . 4 ⊢ (𝜑 → 𝑈 ∈ ℝ+) | |
3 | pntlem1.r | . . . . . 6 ⊢ 𝑅 = (𝑎 ∈ ℝ+ ↦ ((ψ‘𝑎) − 𝑎)) | |
4 | pntlem1.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ ℝ+) | |
5 | pntlem1.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ ℝ+) | |
6 | pntlem1.l | . . . . . 6 ⊢ (𝜑 → 𝐿 ∈ (0(,)1)) | |
7 | pntlem1.d | . . . . . 6 ⊢ 𝐷 = (𝐴 + 1) | |
8 | pntlem1.f | . . . . . 6 ⊢ 𝐹 = ((1 − (1 / 𝐷)) · ((𝐿 / (;32 · 𝐵)) / (𝐷↑2))) | |
9 | 3, 4, 5, 6, 7, 8 | pntlemd 25852 | . . . . 5 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
10 | 9 | simp2d 1136 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
11 | 2, 10 | rpdivcld 12298 | . . 3 ⊢ (𝜑 → (𝑈 / 𝐷) ∈ ℝ+) |
12 | 1, 11 | syl5eqel 2887 | . 2 ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
13 | pntlem1.k | . . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) | |
14 | 5, 12 | rpdivcld 12298 | . . . . 5 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ+) |
15 | 14 | rpred 12281 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ) |
16 | 15 | rpefcld 15291 | . . 3 ⊢ (𝜑 → (exp‘(𝐵 / 𝐸)) ∈ ℝ+) |
17 | 13, 16 | syl5eqel 2887 | . 2 ⊢ (𝜑 → 𝐾 ∈ ℝ+) |
18 | 12 | rpred 12281 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
19 | 12 | rpgt0d 12284 | . . . 4 ⊢ (𝜑 → 0 < 𝐸) |
20 | 2 | rpred 12281 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ) |
21 | 4 | rpred 12281 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
22 | 10 | rpred 12281 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
23 | pntlem1.u2 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝐴) | |
24 | 21 | ltp1d 11418 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 < (𝐴 + 1)) |
25 | 24, 7 | syl6breqr 5004 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐷) |
26 | 20, 21, 22, 23, 25 | lelttrd 10645 | . . . . . . 7 ⊢ (𝜑 → 𝑈 < 𝐷) |
27 | 10 | rpcnd 12283 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
28 | 27 | mulid1d 10504 | . . . . . . 7 ⊢ (𝜑 → (𝐷 · 1) = 𝐷) |
29 | 26, 28 | breqtrrd 4990 | . . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐷 · 1)) |
30 | 1red 10488 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ) | |
31 | 20, 30, 10 | ltdivmuld 12332 | . . . . . 6 ⊢ (𝜑 → ((𝑈 / 𝐷) < 1 ↔ 𝑈 < (𝐷 · 1))) |
32 | 29, 31 | mpbird 258 | . . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 1) |
33 | 1, 32 | eqbrtrid 4997 | . . . 4 ⊢ (𝜑 → 𝐸 < 1) |
34 | 0xr 10534 | . . . . 5 ⊢ 0 ∈ ℝ* | |
35 | 1xr 10547 | . . . . 5 ⊢ 1 ∈ ℝ* | |
36 | elioo2 12629 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1))) | |
37 | 34, 35, 36 | mp2an 688 | . . . 4 ⊢ (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1)) |
38 | 18, 19, 33, 37 | syl3anbrc 1336 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) |
39 | efgt1 15302 | . . . . 5 ⊢ ((𝐵 / 𝐸) ∈ ℝ+ → 1 < (exp‘(𝐵 / 𝐸))) | |
40 | 14, 39 | syl 17 | . . . 4 ⊢ (𝜑 → 1 < (exp‘(𝐵 / 𝐸))) |
41 | 40, 13 | syl6breqr 5004 | . . 3 ⊢ (𝜑 → 1 < 𝐾) |
42 | 1re 10487 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
43 | ltaddrp 12276 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
44 | 42, 4, 43 | sylancr 587 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
45 | 2 | rpcnne0d 12290 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑈 ≠ 0)) |
46 | divid 11175 | . . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ 𝑈 ≠ 0) → (𝑈 / 𝑈) = 1) | |
47 | 45, 46 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑈 / 𝑈) = 1) |
48 | 4 | rpcnd 12283 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
49 | ax-1cn 10441 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
50 | addcom 10673 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
51 | 48, 49, 50 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
52 | 7, 51 | syl5eq 2843 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
53 | 44, 47, 52 | 3brtr4d 4994 | . . . . . 6 ⊢ (𝜑 → (𝑈 / 𝑈) < 𝐷) |
54 | 20, 2, 10, 53 | ltdiv23d 12348 | . . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 𝑈) |
55 | 1, 54 | eqbrtrid 4997 | . . . 4 ⊢ (𝜑 → 𝐸 < 𝑈) |
56 | difrp 12277 | . . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ+)) | |
57 | 18, 20, 56 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ+)) |
58 | 55, 57 | mpbid 233 | . . 3 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ+) |
59 | 38, 41, 58 | 3jca 1121 | . 2 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+)) |
60 | 12, 17, 59 | 3jca 1121 | 1 ⊢ (𝜑 → (𝐸 ∈ ℝ+ ∧ 𝐾 ∈ ℝ+ ∧ (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1080 = wceq 1522 ∈ wcel 2081 ≠ wne 2984 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 ℝ*cxr 10520 < clt 10521 ≤ cle 10522 − cmin 10717 / cdiv 11145 2c2 11540 3c3 11541 ;cdc 11947 ℝ+crp 12239 (,)cioo 12588 ↑cexp 13279 expce 15248 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-rep 5081 ax-sep 5094 ax-nul 5101 ax-pow 5157 ax-pr 5221 ax-un 7319 ax-inf2 8950 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 ax-pre-sup 10461 ax-addf 10462 ax-mulf 10463 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-fal 1535 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-int 4783 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-se 5403 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-isom 6234 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-1o 7953 df-oadd 7957 df-er 8139 df-pm 8259 df-en 8358 df-dom 8359 df-sdom 8360 df-fin 8361 df-sup 8752 df-inf 8753 df-oi 8820 df-card 9214 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-ico 12594 df-fz 12743 df-fzo 12884 df-fl 13012 df-seq 13220 df-exp 13280 df-fac 13484 df-bc 13513 df-hash 13541 df-shft 14260 df-cj 14292 df-re 14293 df-im 14294 df-sqrt 14428 df-abs 14429 df-limsup 14662 df-clim 14679 df-rlim 14680 df-sum 14877 df-ef 15254 |
This theorem is referenced by: pntlema 25854 pntlemb 25855 pntlemg 25856 pntlemh 25857 pntlemq 25859 pntlemr 25860 pntlemj 25861 pntlemi 25862 pntlemf 25863 pntlemo 25865 pntleme 25866 pntlemp 25868 |
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