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Mirrors > Home > ILE Home > Th. List > bernneq2 | GIF version |
Description: Variation of Bernoulli's inequality bernneq 10405. (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 8022 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ) | |
2 | 1 | 3ad2ant1 1002 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (𝐴 − 1) ∈ ℝ) |
3 | simp2 982 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 𝑁 ∈ ℕ0) | |
4 | df-neg 7929 | . . . . 5 ⊢ -1 = (0 − 1) | |
5 | 0re 7759 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
6 | 1re 7758 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
7 | lesub1 8211 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1))) | |
8 | 5, 6, 7 | mp3an13 1306 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1))) |
9 | 8 | biimpa 294 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0 − 1) ≤ (𝐴 − 1)) |
10 | 4, 9 | eqbrtrid 3958 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1)) |
11 | 10 | 3adant2 1000 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1)) |
12 | bernneq 10405 | . . 3 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ -1 ≤ (𝐴 − 1)) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁)) | |
13 | 2, 3, 11, 12 | syl3anc 1216 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁)) |
14 | ax-1cn 7706 | . . . 4 ⊢ 1 ∈ ℂ | |
15 | 1 | recnd 7787 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℂ) |
16 | nn0cn 8980 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
17 | mulcl 7740 | . . . . 5 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐴 − 1) · 𝑁) ∈ ℂ) | |
18 | 15, 16, 17 | syl2an 287 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → ((𝐴 − 1) · 𝑁) ∈ ℂ) |
19 | addcom 7892 | . . . 4 ⊢ ((1 ∈ ℂ ∧ ((𝐴 − 1) · 𝑁) ∈ ℂ) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1)) | |
20 | 14, 18, 19 | sylancr 410 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1)) |
21 | 20 | 3adant3 1001 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1)) |
22 | recn 7746 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
23 | pncan3 7963 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴) | |
24 | 14, 22, 23 | sylancr 410 | . . . 4 ⊢ (𝐴 ∈ ℝ → (1 + (𝐴 − 1)) = 𝐴) |
25 | 24 | oveq1d 5782 | . . 3 ⊢ (𝐴 ∈ ℝ → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁)) |
26 | 25 | 3ad2ant1 1002 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁)) |
27 | 13, 21, 26 | 3brtr3d 3954 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 962 = wceq 1331 ∈ wcel 1480 class class class wbr 3924 (class class class)co 5767 ℂcc 7611 ℝcr 7612 0cc0 7613 1c1 7614 + caddc 7616 · cmul 7618 ≤ cle 7794 − cmin 7926 -cneg 7927 ℕ0cn0 8970 ↑cexp 10285 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-coll 4038 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 ax-cnex 7704 ax-resscn 7705 ax-1cn 7706 ax-1re 7707 ax-icn 7708 ax-addcl 7709 ax-addrcl 7710 ax-mulcl 7711 ax-mulrcl 7712 ax-addcom 7713 ax-mulcom 7714 ax-addass 7715 ax-mulass 7716 ax-distr 7717 ax-i2m1 7718 ax-0lt1 7719 ax-1rid 7720 ax-0id 7721 ax-rnegex 7722 ax-precex 7723 ax-cnre 7724 ax-pre-ltirr 7725 ax-pre-ltwlin 7726 ax-pre-lttrn 7727 ax-pre-apti 7728 ax-pre-ltadd 7729 ax-pre-mulgt0 7730 ax-pre-mulext 7731 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-reu 2421 df-rmo 2422 df-rab 2423 df-v 2683 df-sbc 2905 df-csb 2999 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-if 3470 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-iun 3810 df-br 3925 df-opab 3985 df-mpt 3986 df-tr 4022 df-id 4210 df-po 4213 df-iso 4214 df-iord 4283 df-on 4285 df-ilim 4286 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-riota 5723 df-ov 5770 df-oprab 5771 df-mpo 5772 df-1st 6031 df-2nd 6032 df-recs 6195 df-frec 6281 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-sub 7928 df-neg 7929 df-reap 8330 df-ap 8337 df-div 8426 df-inn 8714 df-n0 8971 df-z 9048 df-uz 9320 df-seqfrec 10212 df-exp 10286 |
This theorem is referenced by: bernneq3 10407 expnbnd 10408 expcnvap0 11264 cvgratnnlembern 11285 |
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