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Mirrors > Home > MPE Home > Th. List > pntlemc | Structured version Visualization version GIF version |
Description: Lemma for pnt 27672. 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 27652 | . . . . 5 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
10 | 9 | simp2d 1142 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
11 | 2, 10 | rpdivcld 13091 | . . 3 ⊢ (𝜑 → (𝑈 / 𝐷) ∈ ℝ+) |
12 | 1, 11 | eqeltrid 2842 | . 2 ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
13 | pntlem1.k | . . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) | |
14 | 5, 12 | rpdivcld 13091 | . . . . 5 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ+) |
15 | 14 | rpred 13074 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ) |
16 | 15 | rpefcld 16137 | . . 3 ⊢ (𝜑 → (exp‘(𝐵 / 𝐸)) ∈ ℝ+) |
17 | 13, 16 | eqeltrid 2842 | . 2 ⊢ (𝜑 → 𝐾 ∈ ℝ+) |
18 | 12 | rpred 13074 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
19 | 12 | rpgt0d 13077 | . . . 4 ⊢ (𝜑 → 0 < 𝐸) |
20 | 2 | rpred 13074 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ) |
21 | 4 | rpred 13074 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
22 | 10 | rpred 13074 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
23 | pntlem1.u2 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝐴) | |
24 | 21 | ltp1d 12195 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 < (𝐴 + 1)) |
25 | 24, 7 | breqtrrdi 5189 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐷) |
26 | 20, 21, 22, 23, 25 | lelttrd 11416 | . . . . . . 7 ⊢ (𝜑 → 𝑈 < 𝐷) |
27 | 10 | rpcnd 13076 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
28 | 27 | mulridd 11275 | . . . . . . 7 ⊢ (𝜑 → (𝐷 · 1) = 𝐷) |
29 | 26, 28 | breqtrrd 5175 | . . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐷 · 1)) |
30 | 1red 11259 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ) | |
31 | 20, 30, 10 | ltdivmuld 13125 | . . . . . 6 ⊢ (𝜑 → ((𝑈 / 𝐷) < 1 ↔ 𝑈 < (𝐷 · 1))) |
32 | 29, 31 | mpbird 257 | . . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 1) |
33 | 1, 32 | eqbrtrid 5182 | . . . 4 ⊢ (𝜑 → 𝐸 < 1) |
34 | 0xr 11305 | . . . . 5 ⊢ 0 ∈ ℝ* | |
35 | 1xr 11317 | . . . . 5 ⊢ 1 ∈ ℝ* | |
36 | elioo2 13424 | . . . . 5 ⊢ ((0 ∈ ℝ* ∧ 1 ∈ ℝ*) → (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1))) | |
37 | 34, 35, 36 | mp2an 692 | . . . 4 ⊢ (𝐸 ∈ (0(,)1) ↔ (𝐸 ∈ ℝ ∧ 0 < 𝐸 ∧ 𝐸 < 1)) |
38 | 18, 19, 33, 37 | syl3anbrc 1342 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) |
39 | efgt1 16148 | . . . . 5 ⊢ ((𝐵 / 𝐸) ∈ ℝ+ → 1 < (exp‘(𝐵 / 𝐸))) | |
40 | 14, 39 | syl 17 | . . . 4 ⊢ (𝜑 → 1 < (exp‘(𝐵 / 𝐸))) |
41 | 40, 13 | breqtrrdi 5189 | . . 3 ⊢ (𝜑 → 1 < 𝐾) |
42 | 1re 11258 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
43 | ltaddrp 13069 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
44 | 42, 4, 43 | sylancr 587 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
45 | 2 | rpcnne0d 13083 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑈 ≠ 0)) |
46 | divid 11950 | . . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ 𝑈 ≠ 0) → (𝑈 / 𝑈) = 1) | |
47 | 45, 46 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑈 / 𝑈) = 1) |
48 | 4 | rpcnd 13076 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
49 | ax-1cn 11210 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
50 | addcom 11444 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
51 | 48, 49, 50 | sylancl 586 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
52 | 7, 51 | eqtrid 2786 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
53 | 44, 47, 52 | 3brtr4d 5179 | . . . . . 6 ⊢ (𝜑 → (𝑈 / 𝑈) < 𝐷) |
54 | 20, 2, 10, 53 | ltdiv23d 13141 | . . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 𝑈) |
55 | 1, 54 | eqbrtrid 5182 | . . . 4 ⊢ (𝜑 → 𝐸 < 𝑈) |
56 | difrp 13070 | . . . . 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 232 | . . 3 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ+) |
59 | 38, 41, 58 | 3jca 1127 | . 2 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+)) |
60 | 12, 17, 59 | 3jca 1127 | 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 ≠ wne 2937 class class class wbr 5147 ↦ cmpt 5230 ‘cfv 6562 (class class class)co 7430 ℂcc 11150 ℝcr 11151 0cc0 11152 1c1 11153 + caddc 11155 · cmul 11157 ℝ*cxr 11291 < clt 11292 ≤ cle 11293 − cmin 11489 / cdiv 11917 2c2 12318 3c3 12319 ;cdc 12730 ℝ+crp 13031 (,)cioo 13383 ↑cexp 14098 expce 16093 ψcchp 27150 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 ax-inf2 9678 ax-cnex 11208 ax-resscn 11209 ax-1cn 11210 ax-icn 11211 ax-addcl 11212 ax-addrcl 11213 ax-mulcl 11214 ax-mulrcl 11215 ax-mulcom 11216 ax-addass 11217 ax-mulass 11218 ax-distr 11219 ax-i2m1 11220 ax-1ne0 11221 ax-1rid 11222 ax-rnegex 11223 ax-rrecex 11224 ax-cnre 11225 ax-pre-lttri 11226 ax-pre-lttrn 11227 ax-pre-ltadd 11228 ax-pre-mulgt0 11229 ax-pre-sup 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3377 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-int 4951 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5582 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-se 5641 df-we 5642 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-pred 6322 df-ord 6388 df-on 6389 df-lim 6390 df-suc 6391 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 df-isom 6571 df-riota 7387 df-ov 7433 df-oprab 7434 df-mpo 7435 df-om 7887 df-1st 8012 df-2nd 8013 df-frecs 8304 df-wrecs 8335 df-recs 8409 df-rdg 8448 df-1o 8504 df-er 8743 df-pm 8867 df-en 8984 df-dom 8985 df-sdom 8986 df-fin 8987 df-sup 9479 df-inf 9480 df-oi 9547 df-card 9976 df-pnf 11294 df-mnf 11295 df-xr 11296 df-ltxr 11297 df-le 11298 df-sub 11491 df-neg 11492 df-div 11918 df-nn 12264 df-2 12326 df-3 12327 df-4 12328 df-5 12329 df-6 12330 df-7 12331 df-8 12332 df-9 12333 df-n0 12524 df-z 12611 df-dec 12731 df-uz 12876 df-rp 13032 df-ioo 13387 df-ico 13389 df-fz 13544 df-fzo 13691 df-fl 13828 df-seq 14039 df-exp 14099 df-fac 14309 df-bc 14338 df-hash 14366 df-shft 15102 df-cj 15134 df-re 15135 df-im 15136 df-sqrt 15270 df-abs 15271 df-limsup 15503 df-clim 15520 df-rlim 15521 df-sum 15719 df-ef 16099 |
This theorem is referenced by: pntlema 27654 pntlemb 27655 pntlemg 27656 pntlemh 27657 pntlemq 27659 pntlemr 27660 pntlemj 27661 pntlemi 27662 pntlemf 27663 pntlemo 27665 pntleme 27666 pntlemp 27668 |
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