bernneq2 - Metamath Proof Explorer

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Theorem bernneq2 14183

Description: Variation of Bernoulli's inequality bernneq 14182. (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 11452	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ)
2	1	3ad2ant1 1139	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (𝐴 − 1) ∈ ℝ)
3		simp2 1143	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → 𝑁 ∈ ℕ₀)
4		df-neg 11371	. . . . 5 ⊢ -1 = (0 − 1)
5		0re 11137	. . . . . . 7 ⊢ 0 ∈ ℝ
6		1re 11135	. . . . . . 7 ⊢ 1 ∈ ℝ
7		lesub1 11635	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1)))
8	5, 6, 7	mp3an13 1460	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1)))
9	8	biimpa 477	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0 − 1) ≤ (𝐴 − 1))
10	4, 9	eqbrtrid 5107	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1))
11	10	3adant2 1137	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1))
12		bernneq 14182	. . 3 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ -1 ≤ (𝐴 − 1)) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁))
13	2, 3, 11, 12	syl3anc 1379	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁))
14		ax-1cn 11087	. . . 4 ⊢ 1 ∈ ℂ
15	1	recnd 11164	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℂ)
16		nn0cn 12438	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
17		mulcl 11113	. . . . 5 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐴 − 1) · 𝑁) ∈ ℂ)
18	15, 16, 17	syl2an 602	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 − 1) · 𝑁) ∈ ℂ)
19		addcom 11323	. . . 4 ⊢ ((1 ∈ ℂ ∧ ((𝐴 − 1) · 𝑁) ∈ ℂ) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1))
20	14, 18, 19	sylancr 593	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1))
21	20	3adant3 1138	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1))
22		recn 11119	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
23		pncan3 11392	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴)
24	14, 22, 23	sylancr 593	. . . 4 ⊢ (𝐴 ∈ ℝ → (1 + (𝐴 − 1)) = 𝐴)
25	24	oveq1d 7371	. . 3 ⊢ (𝐴 ∈ ℝ → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁))
26	25	3ad2ant1 1139	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁))
27	13, 21, 26	3brtr3d 5103	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1092 = wceq 1547 ∈ wcel 2119 class class class wbr 5072 (class class class)co 7356 ℂcc 11027 ℝcr 11028 0cc0 11029 1c1 11030 + caddc 11032 · cmul 11034 ≤ cle 11171 − cmin 11368 -cneg 11369 ℕ₀cn0 12428 ↑cexp 14014

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678 ax-cnex 11085 ax-resscn 11086 ax-1cn 11087 ax-icn 11088 ax-addcl 11089 ax-addrcl 11090 ax-mulcl 11091 ax-mulrcl 11092 ax-mulcom 11093 ax-addass 11094 ax-mulass 11095 ax-distr 11096 ax-i2m1 11097 ax-1ne0 11098 ax-1rid 11099 ax-rnegex 11100 ax-rrecex 11101 ax-cnre 11102 ax-pre-lttri 11103 ax-pre-lttrn 11104 ax-pre-ltadd 11105 ax-pre-mulgt0 11106

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3or 1093 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-nel 3039 df-ral 3054 df-rex 3064 df-reu 3345 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3903 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-tr 5180 df-id 5513 df-eprel 5518 df-po 5526 df-so 5527 df-fr 5571 df-we 5573 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-pred 6252 df-ord 6313 df-on 6314 df-lim 6315 df-suc 6316 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-riota 7313 df-ov 7359 df-oprab 7360 df-mpo 7361 df-om 7807 df-2nd 7932 df-frecs 8221 df-wrecs 8252 df-recs 8301 df-rdg 8339 df-er 8633 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11172 df-mnf 11173 df-xr 11174 df-ltxr 11175 df-le 11176 df-sub 11370 df-neg 11371 df-nn 12166 df-n0 12429 df-z 12516 df-uz 12780 df-seq 13955 df-exp 14015

This theorem is referenced by: bernneq3 14184 expnbnd 14185 expmulnbnd 14188 expcnv 15820 ostth2lem1 27599

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