bernneq2 - Intuitionistic Logic Explorer

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Theorem bernneq2 10413

Description: Variation of Bernoulli's inequality bernneq 10412. (Contributed by NM, 18-Oct-2007.)

**Assertion**
Ref	Expression
bernneq2	⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁))

**Proof of Theorem bernneq2**
Step	Hyp	Ref	Expression
1		peano2rem 8029	. . . 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 7936	. . . . 5 ⊢ -1 = (0 − 1)
5		0re 7766	. . . . . . 7 ⊢ 0 ∈ ℝ
6		1re 7765	. . . . . . 7 ⊢ 1 ∈ ℝ
7		lesub1 8218	. . . . . . 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 3963	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1))
11	10	3adant2 1000	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1))
12		bernneq 10412	. . 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 7713	. . . 4 ⊢ 1 ∈ ℂ
15	1	recnd 7794	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℂ)
16		nn0cn 8987	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
17		mulcl 7747	. . . . 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 7899	. . . 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 7753	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
23		pncan3 7970	. . . . 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 5789	. . 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 3959	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 3929 (class class class)co 5774 ℂcc 7618 ℝcr 7619 0cc0 7620 1c1 7621 + caddc 7623 · cmul 7625 ≤ cle 7801 − cmin 7933 -cneg 7934 ℕ₀cn0 8977 ↑cexp 10292

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 2121 ax-coll 4043 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-iinf 4502 ax-cnex 7711 ax-resscn 7712 ax-1cn 7713 ax-1re 7714 ax-icn 7715 ax-addcl 7716 ax-addrcl 7717 ax-mulcl 7718 ax-mulrcl 7719 ax-addcom 7720 ax-mulcom 7721 ax-addass 7722 ax-mulass 7723 ax-distr 7724 ax-i2m1 7725 ax-0lt1 7726 ax-1rid 7727 ax-0id 7728 ax-rnegex 7729 ax-precex 7730 ax-cnre 7731 ax-pre-ltirr 7732 ax-pre-ltwlin 7733 ax-pre-lttrn 7734 ax-pre-apti 7735 ax-pre-ltadd 7736 ax-pre-mulgt0 7737 ax-pre-mulext 7738

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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-reu 2423 df-rmo 2424 df-rab 2425 df-v 2688 df-sbc 2910 df-csb 3004 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-if 3475 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-iun 3815 df-br 3930 df-opab 3990 df-mpt 3991 df-tr 4027 df-id 4215 df-po 4218 df-iso 4219 df-iord 4288 df-on 4290 df-ilim 4291 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-riota 5730 df-ov 5777 df-oprab 5778 df-mpo 5779 df-1st 6038 df-2nd 6039 df-recs 6202 df-frec 6288 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-sub 7935 df-neg 7936 df-reap 8337 df-ap 8344 df-div 8433 df-inn 8721 df-n0 8978 df-z 9055 df-uz 9327 df-seqfrec 10219 df-exp 10293

This theorem is referenced by: bernneq3 10414 expnbnd 10415 expcnvap0 11271 cvgratnnlembern 11292

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