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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 10564 | . . . . . 6 ⊢ (2↑2) = 4 | |
2 | 2t2e4 9025 | . . . . . 6 ⊢ (2 · 2) = 4 | |
3 | 1, 2 | eqtr4i 2194 | . . . . 5 ⊢ (2↑2) = (2 · 2) |
4 | 3 | oveq2i 5862 | . . . 4 ⊢ ((2↑(𝑁 + 1)) / (2↑2)) = ((2↑(𝑁 + 1)) / (2 · 2)) |
5 | 2cn 8942 | . . . . . 6 ⊢ 2 ∈ ℂ | |
6 | expp1 10476 | . . . . . 6 ⊢ ((2 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2)) | |
7 | 5, 6 | mpan 422 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (2↑(𝑁 + 1)) = ((2↑𝑁) · 2)) |
8 | 7 | oveq1d 5866 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → ((2↑(𝑁 + 1)) / (2 · 2)) = (((2↑𝑁) · 2) / (2 · 2))) |
9 | 4, 8 | eqtrid 2215 | . . 3 ⊢ (𝑁 ∈ ℕ0 → ((2↑(𝑁 + 1)) / (2↑2)) = (((2↑𝑁) · 2) / (2 · 2))) |
10 | expcl 10487 | . . . . 5 ⊢ ((2 ∈ ℂ ∧ 𝑁 ∈ ℕ0) → (2↑𝑁) ∈ ℂ) | |
11 | 5, 10 | mpan 422 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2↑𝑁) ∈ ℂ) |
12 | 5 | a1i 9 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 2 ∈ ℂ) |
13 | 2ap0 8964 | . . . . 5 ⊢ 2 # 0 | |
14 | 13 | a1i 9 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → 2 # 0) |
15 | 11, 12, 12, 12, 14, 14 | divmuldivapd 8742 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) / 2) · (2 / 2)) = (((2↑𝑁) · 2) / (2 · 2))) |
16 | 2div2e1 9003 | . . . . 5 ⊢ (2 / 2) = 1 | |
17 | 16 | oveq2i 5862 | . . . 4 ⊢ (((2↑𝑁) / 2) · (2 / 2)) = (((2↑𝑁) / 2) · 1) |
18 | 11 | halfcld 9115 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ∈ ℂ) |
19 | 18 | mulid1d 7930 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) / 2) · 1) = ((2↑𝑁) / 2)) |
20 | 17, 19 | eqtrid 2215 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑𝑁) / 2) · (2 / 2)) = ((2↑𝑁) / 2)) |
21 | 9, 15, 20 | 3eqtr2rd 2210 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) = ((2↑(𝑁 + 1)) / (2↑2))) |
22 | 2nn0 9145 | . . . 4 ⊢ 2 ∈ ℕ0 | |
23 | faclbnd 10668 | . . . 4 ⊢ ((2 ∈ ℕ0 ∧ 𝑁 ∈ ℕ0) → (2↑(𝑁 + 1)) ≤ ((2↑2) · (!‘𝑁))) | |
24 | 22, 23 | mpan 422 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (2↑(𝑁 + 1)) ≤ ((2↑2) · (!‘𝑁))) |
25 | 2re 8941 | . . . . 5 ⊢ 2 ∈ ℝ | |
26 | peano2nn0 9168 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (𝑁 + 1) ∈ ℕ0) | |
27 | reexpcl 10486 | . . . . 5 ⊢ ((2 ∈ ℝ ∧ (𝑁 + 1) ∈ ℕ0) → (2↑(𝑁 + 1)) ∈ ℝ) | |
28 | 25, 26, 27 | sylancr 412 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (2↑(𝑁 + 1)) ∈ ℝ) |
29 | faccl 10662 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℕ) | |
30 | 29 | nnred 8884 | . . . 4 ⊢ (𝑁 ∈ ℕ0 → (!‘𝑁) ∈ ℝ) |
31 | 4re 8948 | . . . . . . 7 ⊢ 4 ∈ ℝ | |
32 | 1, 31 | eqeltri 2243 | . . . . . 6 ⊢ (2↑2) ∈ ℝ |
33 | 4pos 8968 | . . . . . . 7 ⊢ 0 < 4 | |
34 | 33, 1 | breqtrri 4014 | . . . . . 6 ⊢ 0 < (2↑2) |
35 | 32, 34 | pm3.2i 270 | . . . . 5 ⊢ ((2↑2) ∈ ℝ ∧ 0 < (2↑2)) |
36 | ledivmul 8786 | . . . . 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 1321 | . . . 4 ⊢ (((2↑(𝑁 + 1)) ∈ ℝ ∧ (!‘𝑁) ∈ ℝ) → (((2↑(𝑁 + 1)) / (2↑2)) ≤ (!‘𝑁) ↔ (2↑(𝑁 + 1)) ≤ ((2↑2) · (!‘𝑁)))) |
38 | 28, 30, 37 | syl2anc 409 | . . 3 ⊢ (𝑁 ∈ ℕ0 → (((2↑(𝑁 + 1)) / (2↑2)) ≤ (!‘𝑁) ↔ (2↑(𝑁 + 1)) ≤ ((2↑2) · (!‘𝑁)))) |
39 | 24, 38 | mpbird 166 | . 2 ⊢ (𝑁 ∈ ℕ0 → ((2↑(𝑁 + 1)) / (2↑2)) ≤ (!‘𝑁)) |
40 | 21, 39 | eqbrtrd 4009 | 1 ⊢ (𝑁 ∈ ℕ0 → ((2↑𝑁) / 2) ≤ (!‘𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 = wceq 1348 ∈ wcel 2141 class class class wbr 3987 ‘cfv 5196 (class class class)co 5851 ℂcc 7765 ℝcr 7766 0cc0 7767 1c1 7768 + caddc 7770 · cmul 7772 < clt 7947 ≤ cle 7948 # cap 8493 / cdiv 8582 2c2 8922 4c4 8924 ℕ0cn0 9128 ↑cexp 10468 !cfa 10652 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-coll 4102 ax-sep 4105 ax-nul 4113 ax-pow 4158 ax-pr 4192 ax-un 4416 ax-setind 4519 ax-iinf 4570 ax-cnex 7858 ax-resscn 7859 ax-1cn 7860 ax-1re 7861 ax-icn 7862 ax-addcl 7863 ax-addrcl 7864 ax-mulcl 7865 ax-mulrcl 7866 ax-addcom 7867 ax-mulcom 7868 ax-addass 7869 ax-mulass 7870 ax-distr 7871 ax-i2m1 7872 ax-0lt1 7873 ax-1rid 7874 ax-0id 7875 ax-rnegex 7876 ax-precex 7877 ax-cnre 7878 ax-pre-ltirr 7879 ax-pre-ltwlin 7880 ax-pre-lttrn 7881 ax-pre-apti 7882 ax-pre-ltadd 7883 ax-pre-mulgt0 7884 ax-pre-mulext 7885 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-ral 2453 df-rex 2454 df-reu 2455 df-rmo 2456 df-rab 2457 df-v 2732 df-sbc 2956 df-csb 3050 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3526 df-pw 3566 df-sn 3587 df-pr 3588 df-op 3590 df-uni 3795 df-int 3830 df-iun 3873 df-br 3988 df-opab 4049 df-mpt 4050 df-tr 4086 df-id 4276 df-po 4279 df-iso 4280 df-iord 4349 df-on 4351 df-ilim 4352 df-suc 4354 df-iom 4573 df-xp 4615 df-rel 4616 df-cnv 4617 df-co 4618 df-dm 4619 df-rn 4620 df-res 4621 df-ima 4622 df-iota 5158 df-fun 5198 df-fn 5199 df-f 5200 df-f1 5201 df-fo 5202 df-f1o 5203 df-fv 5204 df-riota 5807 df-ov 5854 df-oprab 5855 df-mpo 5856 df-1st 6117 df-2nd 6118 df-recs 6282 df-frec 6368 df-pnf 7949 df-mnf 7950 df-xr 7951 df-ltxr 7952 df-le 7953 df-sub 8085 df-neg 8086 df-reap 8487 df-ap 8494 df-div 8583 df-inn 8872 df-2 8930 df-3 8931 df-4 8932 df-n0 9129 df-z 9206 df-uz 9481 df-rp 9604 df-seqfrec 10395 df-exp 10469 df-fac 10653 |
This theorem is referenced by: ege2le3 11627 |
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