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Mirrors > Home > MPE Home > Th. List > pntlemc | Structured version Visualization version GIF version |
Description: Lemma for pnt 26190. 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 26170 | . . . . 5 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
10 | 9 | simp2d 1139 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
11 | 2, 10 | rpdivcld 12449 | . . 3 ⊢ (𝜑 → (𝑈 / 𝐷) ∈ ℝ+) |
12 | 1, 11 | eqeltrid 2917 | . 2 ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
13 | pntlem1.k | . . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) | |
14 | 5, 12 | rpdivcld 12449 | . . . . 5 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ+) |
15 | 14 | rpred 12432 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ) |
16 | 15 | rpefcld 15458 | . . 3 ⊢ (𝜑 → (exp‘(𝐵 / 𝐸)) ∈ ℝ+) |
17 | 13, 16 | eqeltrid 2917 | . 2 ⊢ (𝜑 → 𝐾 ∈ ℝ+) |
18 | 12 | rpred 12432 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
19 | 12 | rpgt0d 12435 | . . . 4 ⊢ (𝜑 → 0 < 𝐸) |
20 | 2 | rpred 12432 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ) |
21 | 4 | rpred 12432 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
22 | 10 | rpred 12432 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
23 | pntlem1.u2 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝐴) | |
24 | 21 | ltp1d 11570 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 < (𝐴 + 1)) |
25 | 24, 7 | breqtrrdi 5108 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐷) |
26 | 20, 21, 22, 23, 25 | lelttrd 10798 | . . . . . . 7 ⊢ (𝜑 → 𝑈 < 𝐷) |
27 | 10 | rpcnd 12434 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
28 | 27 | mulid1d 10658 | . . . . . . 7 ⊢ (𝜑 → (𝐷 · 1) = 𝐷) |
29 | 26, 28 | breqtrrd 5094 | . . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐷 · 1)) |
30 | 1red 10642 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ) | |
31 | 20, 30, 10 | ltdivmuld 12483 | . . . . . 6 ⊢ (𝜑 → ((𝑈 / 𝐷) < 1 ↔ 𝑈 < (𝐷 · 1))) |
32 | 29, 31 | mpbird 259 | . . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 1) |
33 | 1, 32 | eqbrtrid 5101 | . . . 4 ⊢ (𝜑 → 𝐸 < 1) |
34 | 0xr 10688 | . . . . 5 ⊢ 0 ∈ ℝ* | |
35 | 1xr 10700 | . . . . 5 ⊢ 1 ∈ ℝ* | |
36 | elioo2 12780 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1))) | |
37 | 34, 35, 36 | mp2an 690 | . . . 4 ⊢ (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1)) |
38 | 18, 19, 33, 37 | syl3anbrc 1339 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) |
39 | efgt1 15469 | . . . . 5 ⊢ ((𝐵 / 𝐸) ∈ ℝ+ → 1 < (exp‘(𝐵 / 𝐸))) | |
40 | 14, 39 | syl 17 | . . . 4 ⊢ (𝜑 → 1 < (exp‘(𝐵 / 𝐸))) |
41 | 40, 13 | breqtrrdi 5108 | . . 3 ⊢ (𝜑 → 1 < 𝐾) |
42 | 1re 10641 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
43 | ltaddrp 12427 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
44 | 42, 4, 43 | sylancr 589 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
45 | 2 | rpcnne0d 12441 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑈 ≠ 0)) |
46 | divid 11327 | . . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ 𝑈 ≠ 0) → (𝑈 / 𝑈) = 1) | |
47 | 45, 46 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑈 / 𝑈) = 1) |
48 | 4 | rpcnd 12434 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
49 | ax-1cn 10595 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
50 | addcom 10826 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
51 | 48, 49, 50 | sylancl 588 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
52 | 7, 51 | syl5eq 2868 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
53 | 44, 47, 52 | 3brtr4d 5098 | . . . . . 6 ⊢ (𝜑 → (𝑈 / 𝑈) < 𝐷) |
54 | 20, 2, 10, 53 | ltdiv23d 12499 | . . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 𝑈) |
55 | 1, 54 | eqbrtrid 5101 | . . . 4 ⊢ (𝜑 → 𝐸 < 𝑈) |
56 | difrp 12428 | . . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ+)) | |
57 | 18, 20, 56 | syl2anc 586 | . . . 4 ⊢ (𝜑 → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ+)) |
58 | 55, 57 | mpbid 234 | . . 3 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ+) |
59 | 38, 41, 58 | 3jca 1124 | . 2 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+)) |
60 | 12, 17, 59 | 3jca 1124 | 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 208 ∧ wa 398 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ≠ wne 3016 class class class wbr 5066 ↦ cmpt 5146 ‘cfv 6355 (class class class)co 7156 ℂcc 10535 ℝcr 10536 0cc0 10537 1c1 10538 + caddc 10540 · cmul 10542 ℝ*cxr 10674 < clt 10675 ≤ cle 10676 − cmin 10870 / cdiv 11297 2c2 11693 3c3 11694 ;cdc 12099 ℝ+crp 12390 (,)cioo 12739 ↑cexp 13430 expce 15415 ψcchp 25670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-inf2 9104 ax-cnex 10593 ax-resscn 10594 ax-1cn 10595 ax-icn 10596 ax-addcl 10597 ax-addrcl 10598 ax-mulcl 10599 ax-mulrcl 10600 ax-mulcom 10601 ax-addass 10602 ax-mulass 10603 ax-distr 10604 ax-i2m1 10605 ax-1ne0 10606 ax-1rid 10607 ax-rnegex 10608 ax-rrecex 10609 ax-cnre 10610 ax-pre-lttri 10611 ax-pre-lttrn 10612 ax-pre-ltadd 10613 ax-pre-mulgt0 10614 ax-pre-sup 10615 ax-addf 10616 ax-mulf 10617 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-recs 8008 df-rdg 8046 df-1o 8102 df-oadd 8106 df-er 8289 df-pm 8409 df-en 8510 df-dom 8511 df-sdom 8512 df-fin 8513 df-sup 8906 df-inf 8907 df-oi 8974 df-card 9368 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-sub 10872 df-neg 10873 df-div 11298 df-nn 11639 df-2 11701 df-3 11702 df-4 11703 df-5 11704 df-6 11705 df-7 11706 df-8 11707 df-9 11708 df-n0 11899 df-z 11983 df-dec 12100 df-uz 12245 df-rp 12391 df-ioo 12743 df-ico 12745 df-fz 12894 df-fzo 13035 df-fl 13163 df-seq 13371 df-exp 13431 df-fac 13635 df-bc 13664 df-hash 13692 df-shft 14426 df-cj 14458 df-re 14459 df-im 14460 df-sqrt 14594 df-abs 14595 df-limsup 14828 df-clim 14845 df-rlim 14846 df-sum 15043 df-ef 15421 |
This theorem is referenced by: pntlema 26172 pntlemb 26173 pntlemg 26174 pntlemh 26175 pntlemq 26177 pntlemr 26178 pntlemj 26179 pntlemi 26180 pntlemf 26181 pntlemo 26183 pntleme 26184 pntlemp 26186 |
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