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Mirrors > Home > MPE Home > Th. List > bernneq2 | Structured version Visualization version GIF version |
Description: Variation of Bernoulli's inequality bernneq 13586. (Contributed by NM, 18-Oct-2007.) |
Ref | Expression |
---|---|
bernneq2 | ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | peano2rem 10942 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ) | |
2 | 1 | 3ad2ant1 1130 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (𝐴 − 1) ∈ ℝ) |
3 | simp2 1134 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 𝑁 ∈ ℕ0) | |
4 | df-neg 10862 | . . . . 5 ⊢ -1 = (0 − 1) | |
5 | 0re 10632 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
6 | 1re 10630 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
7 | lesub1 11123 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1))) | |
8 | 5, 6, 7 | mp3an13 1449 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1))) |
9 | 8 | biimpa 480 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0 − 1) ≤ (𝐴 − 1)) |
10 | 4, 9 | eqbrtrid 5065 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1)) |
11 | 10 | 3adant2 1128 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1)) |
12 | bernneq 13586 | . . 3 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ -1 ≤ (𝐴 − 1)) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁)) | |
13 | 2, 3, 11, 12 | syl3anc 1368 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁)) |
14 | ax-1cn 10584 | . . . 4 ⊢ 1 ∈ ℂ | |
15 | 1 | recnd 10658 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℂ) |
16 | nn0cn 11895 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
17 | mulcl 10610 | . . . . 5 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐴 − 1) · 𝑁) ∈ ℂ) | |
18 | 15, 16, 17 | syl2an 598 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → ((𝐴 − 1) · 𝑁) ∈ ℂ) |
19 | addcom 10815 | . . . 4 ⊢ ((1 ∈ ℂ ∧ ((𝐴 − 1) · 𝑁) ∈ ℂ) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1)) | |
20 | 14, 18, 19 | sylancr 590 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1)) |
21 | 20 | 3adant3 1129 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1)) |
22 | recn 10616 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
23 | pncan3 10883 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴) | |
24 | 14, 22, 23 | sylancr 590 | . . . 4 ⊢ (𝐴 ∈ ℝ → (1 + (𝐴 − 1)) = 𝐴) |
25 | 24 | oveq1d 7150 | . . 3 ⊢ (𝐴 ∈ ℝ → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁)) |
26 | 25 | 3ad2ant1 1130 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁)) |
27 | 13, 21, 26 | 3brtr3d 5061 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 (class class class)co 7135 ℂcc 10524 ℝcr 10525 0cc0 10526 1c1 10527 + caddc 10529 · cmul 10531 ≤ cle 10665 − cmin 10859 -cneg 10860 ℕ0cn0 11885 ↑cexp 13425 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-riota 7093 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-er 8272 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11626 df-n0 11886 df-z 11970 df-uz 12232 df-seq 13365 df-exp 13426 |
This theorem is referenced by: bernneq3 13588 expnbnd 13589 expmulnbnd 13592 expcnv 15211 ostth2lem1 26202 |
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