| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > eflegeo | Structured version Visualization version GIF version | ||
| Description: The exponential function on the reals between 0 and 1 lies below the comparable geometric series sum. (Contributed by Paul Chapman, 11-Sep-2007.) |
| Ref | Expression |
|---|---|
| eflegeo.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
| eflegeo.2 | ⊢ (𝜑 → 0 ≤ 𝐴) |
| eflegeo.3 | ⊢ (𝜑 → 𝐴 < 1) |
| Ref | Expression |
|---|---|
| eflegeo | ⊢ (𝜑 → (exp‘𝐴) ≤ (1 / (1 − 𝐴))) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nn0uz 12920 | . . 3 ⊢ ℕ0 = (ℤ≥‘0) | |
| 2 | 0zd 12625 | . . 3 ⊢ (𝜑 → 0 ∈ ℤ) | |
| 3 | eqid 2737 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) = (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛))) | |
| 4 | 3 | eftval 16112 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
| 5 | 4 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))‘𝑘) = ((𝐴↑𝑘) / (!‘𝑘))) |
| 6 | eflegeo.1 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
| 7 | reeftcl 16110 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) | |
| 8 | 6, 7 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ∈ ℝ) |
| 9 | oveq2 7439 | . . . . 5 ⊢ (𝑛 = 𝑘 → (𝐴↑𝑛) = (𝐴↑𝑘)) | |
| 10 | eqid 2737 | . . . . 5 ⊢ (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) = (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛)) | |
| 11 | ovex 7464 | . . . . 5 ⊢ (𝐴↑𝑘) ∈ V | |
| 12 | 9, 10, 11 | fvmpt 7016 | . . . 4 ⊢ (𝑘 ∈ ℕ0 → ((𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))‘𝑘) = (𝐴↑𝑘)) |
| 13 | 12 | adantl 481 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))‘𝑘) = (𝐴↑𝑘)) |
| 14 | reexpcl 14119 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ) | |
| 15 | 6, 14 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℝ) |
| 16 | faccl 14322 | . . . . . . 7 ⊢ (𝑘 ∈ ℕ0 → (!‘𝑘) ∈ ℕ) | |
| 17 | 16 | adantl 481 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℕ) |
| 18 | 17 | nnred 12281 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (!‘𝑘) ∈ ℝ) |
| 19 | 6 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝐴 ∈ ℝ) |
| 20 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 𝑘 ∈ ℕ0) | |
| 21 | eflegeo.2 | . . . . . . 7 ⊢ (𝜑 → 0 ≤ 𝐴) | |
| 22 | 21 | adantr 480 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 ≤ 𝐴) |
| 23 | 19, 20, 22 | expge0d 14204 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 ≤ (𝐴↑𝑘)) |
| 24 | 17 | nnge1d 12314 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 1 ≤ (!‘𝑘)) |
| 25 | 15, 18, 23, 24 | lemulge12d 12206 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ≤ ((!‘𝑘) · (𝐴↑𝑘))) |
| 26 | 17 | nngt0d 12315 | . . . . 5 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → 0 < (!‘𝑘)) |
| 27 | ledivmul 12144 | . . . . 5 ⊢ (((𝐴↑𝑘) ∈ ℝ ∧ (𝐴↑𝑘) ∈ ℝ ∧ ((!‘𝑘) ∈ ℝ ∧ 0 < (!‘𝑘))) → (((𝐴↑𝑘) / (!‘𝑘)) ≤ (𝐴↑𝑘) ↔ (𝐴↑𝑘) ≤ ((!‘𝑘) · (𝐴↑𝑘)))) | |
| 28 | 15, 15, 18, 26, 27 | syl112anc 1376 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (((𝐴↑𝑘) / (!‘𝑘)) ≤ (𝐴↑𝑘) ↔ (𝐴↑𝑘) ≤ ((!‘𝑘) · (𝐴↑𝑘)))) |
| 29 | 25, 28 | mpbird 257 | . . 3 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → ((𝐴↑𝑘) / (!‘𝑘)) ≤ (𝐴↑𝑘)) |
| 30 | 6 | recnd 11289 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ ℂ) |
| 31 | 3 | efcllem 16113 | . . . 4 ⊢ (𝐴 ∈ ℂ → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) ∈ dom ⇝ ) |
| 32 | 30, 31 | syl 17 | . . 3 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ ((𝐴↑𝑛) / (!‘𝑛)))) ∈ dom ⇝ ) |
| 33 | 6, 21 | absidd 15461 | . . . . . 6 ⊢ (𝜑 → (abs‘𝐴) = 𝐴) |
| 34 | eflegeo.3 | . . . . . 6 ⊢ (𝜑 → 𝐴 < 1) | |
| 35 | 33, 34 | eqbrtrd 5165 | . . . . 5 ⊢ (𝜑 → (abs‘𝐴) < 1) |
| 36 | 30, 35, 13 | geolim 15906 | . . . 4 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ⇝ (1 / (1 − 𝐴))) |
| 37 | seqex 14044 | . . . . 5 ⊢ seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ∈ V | |
| 38 | ovex 7464 | . . . . 5 ⊢ (1 / (1 − 𝐴)) ∈ V | |
| 39 | 37, 38 | breldm 5919 | . . . 4 ⊢ (seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ⇝ (1 / (1 − 𝐴)) → seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ∈ dom ⇝ ) |
| 40 | 36, 39 | syl 17 | . . 3 ⊢ (𝜑 → seq0( + , (𝑛 ∈ ℕ0 ↦ (𝐴↑𝑛))) ∈ dom ⇝ ) |
| 41 | 1, 2, 5, 8, 13, 15, 29, 32, 40 | isumle 15880 | . 2 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 ((𝐴↑𝑘) / (!‘𝑘)) ≤ Σ𝑘 ∈ ℕ0 (𝐴↑𝑘)) |
| 42 | efval 16115 | . . 3 ⊢ (𝐴 ∈ ℂ → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴↑𝑘) / (!‘𝑘))) | |
| 43 | 30, 42 | syl 17 | . 2 ⊢ (𝜑 → (exp‘𝐴) = Σ𝑘 ∈ ℕ0 ((𝐴↑𝑘) / (!‘𝑘))) |
| 44 | expcl 14120 | . . . . 5 ⊢ ((𝐴 ∈ ℂ ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ) | |
| 45 | 30, 44 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ ℕ0) → (𝐴↑𝑘) ∈ ℂ) |
| 46 | 1, 2, 13, 45, 36 | isumclim 15793 | . . 3 ⊢ (𝜑 → Σ𝑘 ∈ ℕ0 (𝐴↑𝑘) = (1 / (1 − 𝐴))) |
| 47 | 46 | eqcomd 2743 | . 2 ⊢ (𝜑 → (1 / (1 − 𝐴)) = Σ𝑘 ∈ ℕ0 (𝐴↑𝑘)) |
| 48 | 41, 43, 47 | 3brtr4d 5175 | 1 ⊢ (𝜑 → (exp‘𝐴) ≤ (1 / (1 − 𝐴))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ↦ cmpt 5225 dom cdm 5685 ‘cfv 6561 (class class class)co 7431 ℂcc 11153 ℝcr 11154 0cc0 11155 1c1 11156 + caddc 11158 · cmul 11160 < clt 11295 ≤ cle 11296 − cmin 11492 / cdiv 11920 ℕcn 12266 ℕ0cn0 12526 seqcseq 14042 ↑cexp 14102 !cfa 14312 abscabs 15273 ⇝ cli 15520 Σcsu 15722 expce 16097 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 ax-pre-sup 11233 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-sup 9482 df-inf 9483 df-oi 9550 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-div 11921 df-nn 12267 df-2 12329 df-3 12330 df-n0 12527 df-z 12614 df-uz 12879 df-rp 13035 df-ico 13393 df-fz 13548 df-fzo 13695 df-fl 13832 df-seq 14043 df-exp 14103 df-fac 14313 df-hash 14370 df-shft 15106 df-cj 15138 df-re 15139 df-im 15140 df-sqrt 15274 df-abs 15275 df-limsup 15507 df-clim 15524 df-rlim 15525 df-sum 15723 df-ef 16103 |
| This theorem is referenced by: birthdaylem3 26996 logdiflbnd 27038 emcllem2 27040 |
| Copyright terms: Public domain | W3C validator |