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Mirrors > Home > ILE Home > Th. List > faclbnd2 | GIF version |
Description: A lower bound for the factorial function. (Contributed by NM, 17-Dec-2005.) |
Ref | Expression |
---|---|
faclbnd2 | ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ≤ (!‘𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sq2 10656 | . . . . . 6 ⊢ (2↑2) = 4 | |
2 | 2t2e4 9108 | . . . . . 6 ⊢ (2 · 2) = 4 | |
3 | 1, 2 | eqtr4i 2213 | . . . . 5 ⊢ (2↑2) = (2 · 2) |
4 | 3 | oveq2i 5911 | . . . 4 ⊢ ((2↑(𝑁 + 1)) / (2↑2)) = ((2↑(𝑁 + 1)) / (2 · 2)) |
5 | 2cn 9025 | . . . . . 6 ⊢ 2 ∈ ℂ | |
6 | expp1 10567 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2)) | |
7 | 5, 6 | mpan 424 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2)) |
8 | 7 | oveq1d 5915 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((2↑(𝑁 + 1)) / (2 · 2)) = (((2↑𝑁) · 2) / (2 · 2))) |
9 | 4, 8 | eqtrid 2234 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((2↑(𝑁 + 1)) / (2↑2)) = (((2↑𝑁) · 2) / (2 · 2))) |
10 | expcl 10578 | . . . . 5 ⊢ ((2 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℂ) | |
11 | 5, 10 | mpan 424 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℂ) |
12 | 5 | a1i 9 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) |
13 | 2ap0 9047 | . . . . 5 ⊢ 2 # 0 | |
14 | 13 | a1i 9 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 2 # 0) |
15 | 11, 12, 12, 12, 14, 14 | divmuldivapd 8824 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) / 2) · (2 / 2)) = (((2↑𝑁) · 2) / (2 · 2))) |
16 | 2div2e1 9086 | . . . . 5 ⊢ (2 / 2) = 1 | |
17 | 16 | oveq2i 5911 | . . . 4 ⊢ (((2↑𝑁) / 2) · (2 / 2)) = (((2↑𝑁) / 2) · 1) |
18 | 11 | halfcld 9198 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ∈ ℂ) |
19 | 18 | mulridd 8009 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) / 2) · 1) = ((2↑𝑁) / 2)) |
20 | 17, 19 | eqtrid 2234 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) / 2) · (2 / 2)) = ((2↑𝑁) / 2)) |
21 | 9, 15, 20 | 3eqtr2rd 2229 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) = ((2↑(𝑁 + 1)) / (2↑2))) |
22 | 2nn0 9228 | . . . 4 ⊢ 2 ∈ ℕ0 | |
23 | faclbnd 10762 | . . . 4 ⊢ ((2 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2↑(𝑁 + 1)) ≤ ((2↑2) · (!‘𝑁))) | |
24 | 22, 23 | mpan 424 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (2↑(𝑁 + 1)) ≤ ((2↑2) · (!‘𝑁))) |
25 | 2re 9024 | . . . . 5 ⊢ 2 ∈ ℝ | |
26 | peano2nn0 9251 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
27 | reexpcl 10577 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ (𝑁 + 1) ∈ ℕ0) → (2↑(𝑁 + 1)) ∈ ℝ) | |
28 | 25, 26, 27 | sylancr 414 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2↑(𝑁 + 1)) ∈ ℝ) |
29 | faccl 10756 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
30 | 29 | nnred 8967 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℝ) |
31 | 4re 9031 | . . . . . . 7 ⊢ 4 ∈ ℝ | |
32 | 1, 31 | eqeltri 2262 | . . . . . 6 ⊢ (2↑2) ∈ ℝ |
33 | 4pos 9051 | . . . . . . 7 ⊢ 0 < 4 | |
34 | 33, 1 | breqtrri 4048 | . . . . . 6 ⊢ 0 < (2↑2) |
35 | 32, 34 | pm3.2i 272 | . . . . 5 ⊢ ((2↑2) ∈ ℝ ∧ 0 < (2↑2)) |
36 | ledivmul 8869 | . . . . 5 ⊢ (((2↑(𝑁 + 1)) ∈ ℝ ∧ (!‘𝑁) ∈ ℝ ∧ ((2↑2) ∈ ℝ ∧ 0 < (2↑2))) → (((2↑(𝑁 + 1)) / (2↑2)) ≤ (!‘𝑁) ↔ (2↑(𝑁 + 1)) ≤ ((2↑2) · (!‘𝑁)))) | |
37 | 35, 36 | mp3an3 1337 | . . . 4 ⊢ (((2↑(𝑁 + 1)) ∈ ℝ ∧ (!‘𝑁) ∈ ℝ) → (((2↑(𝑁 + 1)) / (2↑2)) ≤ (!‘𝑁) ↔ (2↑(𝑁 + 1)) ≤ ((2↑2) · (!‘𝑁)))) |
38 | 28, 30, 37 | syl2anc 411 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑(𝑁 + 1)) / (2↑2)) ≤ (!‘𝑁) ↔ (2↑(𝑁 + 1)) ≤ ((2↑2) · (!‘𝑁)))) |
39 | 24, 38 | mpbird 167 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑(𝑁 + 1)) / (2↑2)) ≤ (!‘𝑁)) |
40 | 21, 39 | eqbrtrd 4043 | 1 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ≤ (!‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 = wceq 1364 ∈ wcel 2160 class class class wbr 4021 ‘cfv 5238 (class class class)co 5900 ℂcc 7844 ℝcr 7845 0cc0 7846 1c1 7847 + caddc 7849 · cmul 7851 < clt 8027 ≤ cle 8028 # cap 8573 / cdiv 8664 2c2 9005 4c4 9007 ℕ0cn0 9211 ↑cexp 10559 !cfa 10746 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2162 ax-14 2163 ax-ext 2171 ax-coll 4136 ax-sep 4139 ax-nul 4147 ax-pow 4195 ax-pr 4230 ax-un 4454 ax-setind 4557 ax-iinf 4608 ax-cnex 7937 ax-resscn 7938 ax-1cn 7939 ax-1re 7940 ax-icn 7941 ax-addcl 7942 ax-addrcl 7943 ax-mulcl 7944 ax-mulrcl 7945 ax-addcom 7946 ax-mulcom 7947 ax-addass 7948 ax-mulass 7949 ax-distr 7950 ax-i2m1 7951 ax-0lt1 7952 ax-1rid 7953 ax-0id 7954 ax-rnegex 7955 ax-precex 7956 ax-cnre 7957 ax-pre-ltirr 7958 ax-pre-ltwlin 7959 ax-pre-lttrn 7960 ax-pre-apti 7961 ax-pre-ltadd 7962 ax-pre-mulgt0 7963 ax-pre-mulext 7964 |
This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-eu 2041 df-mo 2042 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ne 2361 df-nel 2456 df-ral 2473 df-rex 2474 df-reu 2475 df-rmo 2476 df-rab 2477 df-v 2754 df-sbc 2978 df-csb 3073 df-dif 3146 df-un 3148 df-in 3150 df-ss 3157 df-nul 3438 df-if 3550 df-pw 3595 df-sn 3616 df-pr 3617 df-op 3619 df-uni 3828 df-int 3863 df-iun 3906 df-br 4022 df-opab 4083 df-mpt 4084 df-tr 4120 df-id 4314 df-po 4317 df-iso 4318 df-iord 4387 df-on 4389 df-ilim 4390 df-suc 4392 df-iom 4611 df-xp 4653 df-rel 4654 df-cnv 4655 df-co 4656 df-dm 4657 df-rn 4658 df-res 4659 df-ima 4660 df-iota 5199 df-fun 5240 df-fn 5241 df-f 5242 df-f1 5243 df-fo 5244 df-f1o 5245 df-fv 5246 df-riota 5855 df-ov 5903 df-oprab 5904 df-mpo 5905 df-1st 6169 df-2nd 6170 df-recs 6334 df-frec 6420 df-pnf 8029 df-mnf 8030 df-xr 8031 df-ltxr 8032 df-le 8033 df-sub 8165 df-neg 8166 df-reap 8567 df-ap 8574 df-div 8665 df-inn 8955 df-2 9013 df-3 9014 df-4 9015 df-n0 9212 df-z 9289 df-uz 9564 df-rp 9690 df-seqfrec 10485 df-exp 10560 df-fac 10747 |
This theorem is referenced by: ege2le3 11720 |
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