![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > bernneq2 | GIF version |
Description: Variation of Bernoulli's inequality bernneq 10659. (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 8242 | . . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ) | |
2 | 1 | 3ad2ant1 1020 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (𝐴 − 1) ∈ ℝ) |
3 | simp2 1000 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → 𝑁 ∈ ℕ0) | |
4 | df-neg 8149 | . . . . 5 ⊢ -1 = (0 − 1) | |
5 | 0re 7975 | . . . . . . 7 ⊢ 0 ∈ ℝ | |
6 | 1re 7974 | . . . . . . 7 ⊢ 1 ∈ ℝ | |
7 | lesub1 8431 | . . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1))) | |
8 | 5, 6, 7 | mp3an13 1339 | . . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1))) |
9 | 8 | biimpa 296 | . . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0 − 1) ≤ (𝐴 − 1)) |
10 | 4, 9 | eqbrtrid 4053 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1)) |
11 | 10 | 3adant2 1018 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1)) |
12 | bernneq 10659 | . . 3 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ -1 ≤ (𝐴 − 1)) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁)) | |
13 | 2, 3, 11, 12 | syl3anc 1249 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁)) |
14 | ax-1cn 7922 | . . . 4 ⊢ 1 ∈ ℂ | |
15 | 1 | recnd 8004 | . . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℂ) |
16 | nn0cn 9204 | . . . . 5 ⊢ (𝑁 ∈ ℕ0 → 𝑁 ∈ ℂ) | |
17 | mulcl 7956 | . . . . 5 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐴 − 1) · 𝑁) ∈ ℂ) | |
18 | 15, 16, 17 | syl2an 289 | . . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → ((𝐴 − 1) · 𝑁) ∈ ℂ) |
19 | addcom 8112 | . . . 4 ⊢ ((1 ∈ ℂ ∧ ((𝐴 − 1) · 𝑁) ∈ ℂ) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1)) | |
20 | 14, 18, 19 | sylancr 414 | . . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1)) |
21 | 20 | 3adant3 1019 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1)) |
22 | recn 7962 | . . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ) | |
23 | pncan3 8183 | . . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴) | |
24 | 14, 22, 23 | sylancr 414 | . . . 4 ⊢ (𝐴 ∈ ℝ → (1 + (𝐴 − 1)) = 𝐴) |
25 | 24 | oveq1d 5906 | . . 3 ⊢ (𝐴 ∈ ℝ → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁)) |
26 | 25 | 3ad2ant1 1020 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁)) |
27 | 13, 21, 26 | 3brtr3d 4049 | 1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ0 ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∧ w3a 980 = wceq 1364 ∈ wcel 2160 class class class wbr 4018 (class class class)co 5891 ℂcc 7827 ℝcr 7828 0cc0 7829 1c1 7830 + caddc 7832 · cmul 7834 ≤ cle 8011 − cmin 8146 -cneg 8147 ℕ0cn0 9194 ↑cexp 10537 |
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 4133 ax-sep 4136 ax-nul 4144 ax-pow 4189 ax-pr 4224 ax-un 4448 ax-setind 4551 ax-iinf 4602 ax-cnex 7920 ax-resscn 7921 ax-1cn 7922 ax-1re 7923 ax-icn 7924 ax-addcl 7925 ax-addrcl 7926 ax-mulcl 7927 ax-mulrcl 7928 ax-addcom 7929 ax-mulcom 7930 ax-addass 7931 ax-mulass 7932 ax-distr 7933 ax-i2m1 7934 ax-0lt1 7935 ax-1rid 7936 ax-0id 7937 ax-rnegex 7938 ax-precex 7939 ax-cnre 7940 ax-pre-ltirr 7941 ax-pre-ltwlin 7942 ax-pre-lttrn 7943 ax-pre-apti 7944 ax-pre-ltadd 7945 ax-pre-mulgt0 7946 ax-pre-mulext 7947 |
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 3592 df-sn 3613 df-pr 3614 df-op 3616 df-uni 3825 df-int 3860 df-iun 3903 df-br 4019 df-opab 4080 df-mpt 4081 df-tr 4117 df-id 4308 df-po 4311 df-iso 4312 df-iord 4381 df-on 4383 df-ilim 4384 df-suc 4386 df-iom 4605 df-xp 4647 df-rel 4648 df-cnv 4649 df-co 4650 df-dm 4651 df-rn 4652 df-res 4653 df-ima 4654 df-iota 5193 df-fun 5233 df-fn 5234 df-f 5235 df-f1 5236 df-fo 5237 df-f1o 5238 df-fv 5239 df-riota 5847 df-ov 5894 df-oprab 5895 df-mpo 5896 df-1st 6159 df-2nd 6160 df-recs 6324 df-frec 6410 df-pnf 8012 df-mnf 8013 df-xr 8014 df-ltxr 8015 df-le 8016 df-sub 8148 df-neg 8149 df-reap 8550 df-ap 8557 df-div 8648 df-inn 8938 df-n0 9195 df-z 9272 df-uz 9547 df-seqfrec 10464 df-exp 10538 |
This theorem is referenced by: bernneq3 10661 expnbnd 10662 expcnvap0 11528 cvgratnnlembern 11549 |
Copyright terms: Public domain | W3C validator |