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Mirrors > Home > MPE Home > Th. List > pntlemc | Structured version Visualization version GIF version |
Description: Lemma for pnt 27567. 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 27547 | . . . . 5 ⊢ (𝜑 → (𝐿 ∈ ℝ+ ∧ 𝐷 ∈ ℝ+ ∧ 𝐹 ∈ ℝ+)) |
10 | 9 | simp2d 1140 | . . . 4 ⊢ (𝜑 → 𝐷 ∈ ℝ+) |
11 | 2, 10 | rpdivcld 13073 | . . 3 ⊢ (𝜑 → (𝑈 / 𝐷) ∈ ℝ+) |
12 | 1, 11 | eqeltrid 2833 | . 2 ⊢ (𝜑 → 𝐸 ∈ ℝ+) |
13 | pntlem1.k | . . 3 ⊢ 𝐾 = (exp‘(𝐵 / 𝐸)) | |
14 | 5, 12 | rpdivcld 13073 | . . . . 5 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ+) |
15 | 14 | rpred 13056 | . . . 4 ⊢ (𝜑 → (𝐵 / 𝐸) ∈ ℝ) |
16 | 15 | rpefcld 16089 | . . 3 ⊢ (𝜑 → (exp‘(𝐵 / 𝐸)) ∈ ℝ+) |
17 | 13, 16 | eqeltrid 2833 | . 2 ⊢ (𝜑 → 𝐾 ∈ ℝ+) |
18 | 12 | rpred 13056 | . . . 4 ⊢ (𝜑 → 𝐸 ∈ ℝ) |
19 | 12 | rpgt0d 13059 | . . . 4 ⊢ (𝜑 → 0 < 𝐸) |
20 | 2 | rpred 13056 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ∈ ℝ) |
21 | 4 | rpred 13056 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 ∈ ℝ) |
22 | 10 | rpred 13056 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℝ) |
23 | pntlem1.u2 | . . . . . . . 8 ⊢ (𝜑 → 𝑈 ≤ 𝐴) | |
24 | 21 | ltp1d 12182 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 < (𝐴 + 1)) |
25 | 24, 7 | breqtrrdi 5194 | . . . . . . . 8 ⊢ (𝜑 → 𝐴 < 𝐷) |
26 | 20, 21, 22, 23, 25 | lelttrd 11410 | . . . . . . 7 ⊢ (𝜑 → 𝑈 < 𝐷) |
27 | 10 | rpcnd 13058 | . . . . . . . 8 ⊢ (𝜑 → 𝐷 ∈ ℂ) |
28 | 27 | mulridd 11269 | . . . . . . 7 ⊢ (𝜑 → (𝐷 · 1) = 𝐷) |
29 | 26, 28 | breqtrrd 5180 | . . . . . 6 ⊢ (𝜑 → 𝑈 < (𝐷 · 1)) |
30 | 1red 11253 | . . . . . . 7 ⊢ (𝜑 → 1 ∈ ℝ) | |
31 | 20, 30, 10 | ltdivmuld 13107 | . . . . . 6 ⊢ (𝜑 → ((𝑈 / 𝐷) < 1 ↔ 𝑈 < (𝐷 · 1))) |
32 | 29, 31 | mpbird 256 | . . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 1) |
33 | 1, 32 | eqbrtrid 5187 | . . . 4 ⊢ (𝜑 → 𝐸 < 1) |
34 | 0xr 11299 | . . . . 5 ⊢ 0 ∈ ℝ* | |
35 | 1xr 11311 | . . . . 5 ⊢ 1 ∈ ℝ* | |
36 | elioo2 13405 | . . . . 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 1340 | . . 3 ⊢ (𝜑 → 𝐸 ∈ (0(,)1)) |
39 | efgt1 16100 | . . . . 5 ⊢ ((𝐵 / 𝐸) ∈ ℝ+ → 1 < (exp‘(𝐵 / 𝐸))) | |
40 | 14, 39 | syl 17 | . . . 4 ⊢ (𝜑 → 1 < (exp‘(𝐵 / 𝐸))) |
41 | 40, 13 | breqtrrdi 5194 | . . 3 ⊢ (𝜑 → 1 < 𝐾) |
42 | 1re 11252 | . . . . . . . 8 ⊢ 1 ∈ ℝ | |
43 | ltaddrp 13051 | . . . . . . . 8 ⊢ ((1 ∈ ℝ ∧ 𝐴 ∈ ℝ+) → 1 < (1 + 𝐴)) | |
44 | 42, 4, 43 | sylancr 585 | . . . . . . 7 ⊢ (𝜑 → 1 < (1 + 𝐴)) |
45 | 2 | rpcnne0d 13065 | . . . . . . . 8 ⊢ (𝜑 → (𝑈 ∈ ℂ ∧ 𝑈 ≠ 0)) |
46 | divid 11939 | . . . . . . . 8 ⊢ ((𝑈 ∈ ℂ ∧ 𝑈 ≠ 0) → (𝑈 / 𝑈) = 1) | |
47 | 45, 46 | syl 17 | . . . . . . 7 ⊢ (𝜑 → (𝑈 / 𝑈) = 1) |
48 | 4 | rpcnd 13058 | . . . . . . . . 9 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
49 | ax-1cn 11204 | . . . . . . . . 9 ⊢ 1 ∈ ℂ | |
50 | addcom 11438 | . . . . . . . . 9 ⊢ ((𝐴 ∈ ℂ ∧ 1 ∈ ℂ) → (𝐴 + 1) = (1 + 𝐴)) | |
51 | 48, 49, 50 | sylancl 584 | . . . . . . . 8 ⊢ (𝜑 → (𝐴 + 1) = (1 + 𝐴)) |
52 | 7, 51 | eqtrid 2780 | . . . . . . 7 ⊢ (𝜑 → 𝐷 = (1 + 𝐴)) |
53 | 44, 47, 52 | 3brtr4d 5184 | . . . . . 6 ⊢ (𝜑 → (𝑈 / 𝑈) < 𝐷) |
54 | 20, 2, 10, 53 | ltdiv23d 13123 | . . . . 5 ⊢ (𝜑 → (𝑈 / 𝐷) < 𝑈) |
55 | 1, 54 | eqbrtrid 5187 | . . . 4 ⊢ (𝜑 → 𝐸 < 𝑈) |
56 | difrp 13052 | . . . . 5 ⊢ ((𝐸 ∈ ℝ ∧ 𝑈 ∈ ℝ) → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ+)) | |
57 | 18, 20, 56 | syl2anc 582 | . . . 4 ⊢ (𝜑 → (𝐸 < 𝑈 ↔ (𝑈 − 𝐸) ∈ ℝ+)) |
58 | 55, 57 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝑈 − 𝐸) ∈ ℝ+) |
59 | 38, 41, 58 | 3jca 1125 | . 2 ⊢ (𝜑 → (𝐸 ∈ (0(,)1) ∧ 1 < 𝐾 ∧ (𝑈 − 𝐸) ∈ ℝ+)) |
60 | 12, 17, 59 | 3jca 1125 | 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 205 ∧ wa 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 ↦ cmpt 5235 ‘cfv 6553 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 + caddc 11149 · cmul 11151 ℝ*cxr 11285 < clt 11286 ≤ cle 11287 − cmin 11482 / cdiv 11909 2c2 12305 3c3 12306 ;cdc 12715 ℝ+crp 13014 (,)cioo 13364 ↑cexp 14066 expce 16045 ψcchp 27045 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-rep 5289 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-inf2 9672 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223 ax-pre-sup 11224 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-int 4954 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-se 5638 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-isom 6562 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-1st 7999 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-1o 8493 df-er 8731 df-pm 8854 df-en 8971 df-dom 8972 df-sdom 8973 df-fin 8974 df-sup 9473 df-inf 9474 df-oi 9541 df-card 9970 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-2 12313 df-3 12314 df-4 12315 df-5 12316 df-6 12317 df-7 12318 df-8 12319 df-9 12320 df-n0 12511 df-z 12597 df-dec 12716 df-uz 12861 df-rp 13015 df-ioo 13368 df-ico 13370 df-fz 13525 df-fzo 13668 df-fl 13797 df-seq 14007 df-exp 14067 df-fac 14273 df-bc 14302 df-hash 14330 df-shft 15054 df-cj 15086 df-re 15087 df-im 15088 df-sqrt 15222 df-abs 15223 df-limsup 15455 df-clim 15472 df-rlim 15473 df-sum 15673 df-ef 16051 |
This theorem is referenced by: pntlema 27549 pntlemb 27550 pntlemg 27551 pntlemh 27552 pntlemq 27554 pntlemr 27555 pntlemj 27556 pntlemi 27557 pntlemf 27558 pntlemo 27560 pntleme 27561 pntlemp 27563 |
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