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Theorem bernneq2 14158

Description: Variation of Bernoulli's inequality bernneq 14157. (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 11492	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ)
2	1	3ad2ant1 1133	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (𝐴 − 1) ∈ ℝ)
3		simp2 1137	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → 𝑁 ∈ ℕ₀)
4		df-neg 11412	. . . . 5 ⊢ -1 = (0 − 1)
5		0re 11181	. . . . . . 7 ⊢ 0 ∈ ℝ
6		1re 11179	. . . . . . 7 ⊢ 1 ∈ ℝ
7		lesub1 11673	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1)))
8	5, 6, 7	mp3an13 1452	. . . . . 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 5160	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1))
11	10	3adant2 1131	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1))
12		bernneq 14157	. . 3 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ -1 ≤ (𝐴 − 1)) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁))
13	2, 3, 11, 12	syl3anc 1371	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁))
14		ax-1cn 11133	. . . 4 ⊢ 1 ∈ ℂ
15	1	recnd 11207	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℂ)
16		nn0cn 12447	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
17		mulcl 11159	. . . . 5 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐴 − 1) · 𝑁) ∈ ℂ)
18	15, 16, 17	syl2an 596	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 − 1) · 𝑁) ∈ ℂ)
19		addcom 11365	. . . 4 ⊢ ((1 ∈ ℂ ∧ ((𝐴 − 1) · 𝑁) ∈ ℂ) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1))
20	14, 18, 19	sylancr 587	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1))
21	20	3adant3 1132	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1))
22		recn 11165	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
23		pncan3 11433	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴)
24	14, 22, 23	sylancr 587	. . . 4 ⊢ (𝐴 ∈ ℝ → (1 + (𝐴 − 1)) = 𝐴)
25	24	oveq1d 7392	. . 3 ⊢ (𝐴 ∈ ℝ → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁))
26	25	3ad2ant1 1133	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁))
27	13, 21, 26	3brtr3d 5156	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5125 (class class class)co 7377 ℂcc 11073 ℝcr 11074 0cc0 11075 1c1 11076 + caddc 11078 · cmul 11080 ≤ cle 11214 − cmin 11409 -cneg 11410 ℕ₀cn0 12437 ↑cexp 13992

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 ax-cnex 11131 ax-resscn 11132 ax-1cn 11133 ax-icn 11134 ax-addcl 11135 ax-addrcl 11136 ax-mulcl 11137 ax-mulrcl 11138 ax-mulcom 11139 ax-addass 11140 ax-mulass 11141 ax-distr 11142 ax-i2m1 11143 ax-1ne0 11144 ax-1rid 11145 ax-rnegex 11146 ax-rrecex 11147 ax-cnre 11148 ax-pre-lttri 11149 ax-pre-lttrn 11150 ax-pre-ltadd 11151 ax-pre-mulgt0 11152

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-rdg 8376 df-er 8670 df-en 8906 df-dom 8907 df-sdom 8908 df-pnf 11215 df-mnf 11216 df-xr 11217 df-ltxr 11218 df-le 11219 df-sub 11411 df-neg 11412 df-nn 12178 df-n0 12438 df-z 12524 df-uz 12788 df-seq 13932 df-exp 13993

This theorem is referenced by: bernneq3 14159 expnbnd 14160 expmulnbnd 14163 expcnv 15775 ostth2lem1 27018

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