bernneq2 - Metamath Proof Explorer

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Theorem bernneq2 14186

Description: Variation of Bernoulli's inequality bernneq 14185. (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 11455	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ)
2	1	3ad2ant1 1134	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (𝐴 − 1) ∈ ℝ)
3		simp2 1138	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → 𝑁 ∈ ℕ₀)
4		df-neg 11374	. . . . 5 ⊢ -1 = (0 − 1)
5		0re 11140	. . . . . . 7 ⊢ 0 ∈ ℝ
6		1re 11138	. . . . . . 7 ⊢ 1 ∈ ℝ
7		lesub1 11638	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1)))
8	5, 6, 7	mp3an13 1455	. . . . . 6 ⊢ (𝐴 ∈ ℝ → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1)))
9	8	biimpa 476	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → (0 − 1) ≤ (𝐴 − 1))
10	4, 9	eqbrtrid 5121	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1))
11	10	3adant2 1132	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1))
12		bernneq 14185	. . 3 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ -1 ≤ (𝐴 − 1)) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁))
13	2, 3, 11, 12	syl3anc 1374	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁))
14		ax-1cn 11090	. . . 4 ⊢ 1 ∈ ℂ
15	1	recnd 11167	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℂ)
16		nn0cn 12441	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
17		mulcl 11116	. . . . 5 ⊢ (((𝐴 − 1) ∈ ℂ ∧ 𝑁 ∈ ℂ) → ((𝐴 − 1) · 𝑁) ∈ ℂ)
18	15, 16, 17	syl2an 597	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀) → ((𝐴 − 1) · 𝑁) ∈ ℂ)
19		addcom 11326	. . . 4 ⊢ ((1 ∈ ℂ ∧ ((𝐴 − 1) · 𝑁) ∈ ℂ) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1))
20	14, 18, 19	sylancr 588	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1))
21	20	3adant3 1133	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1))
22		recn 11122	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
23		pncan3 11395	. . . . 5 ⊢ ((1 ∈ ℂ ∧ 𝐴 ∈ ℂ) → (1 + (𝐴 − 1)) = 𝐴)
24	14, 22, 23	sylancr 588	. . . 4 ⊢ (𝐴 ∈ ℝ → (1 + (𝐴 − 1)) = 𝐴)
25	24	oveq1d 7376	. . 3 ⊢ (𝐴 ∈ ℝ → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁))
26	25	3ad2ant1 1134	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁))
27	13, 21, 26	3brtr3d 5117	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (((𝐴 − 1) · 𝑁) + 1) ≤ (𝐴↑𝑁))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 class class class wbr 5086 (class class class)co 7361 ℂcc 11030 ℝcr 11031 0cc0 11032 1c1 11033 + caddc 11035 · cmul 11037 ≤ cle 11174 − cmin 11371 -cneg 11372 ℕ₀cn0 12431 ↑cexp 14017

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089 ax-1cn 11090 ax-icn 11091 ax-addcl 11092 ax-addrcl 11093 ax-mulcl 11094 ax-mulrcl 11095 ax-mulcom 11096 ax-addass 11097 ax-mulass 11098 ax-distr 11099 ax-i2m1 11100 ax-1ne0 11101 ax-1rid 11102 ax-rnegex 11103 ax-rrecex 11104 ax-cnre 11105 ax-pre-lttri 11106 ax-pre-lttrn 11107 ax-pre-ltadd 11108 ax-pre-mulgt0 11109

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3344 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-pss 3910 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-iun 4936 df-br 5087 df-opab 5149 df-mpt 5168 df-tr 5194 df-id 5520 df-eprel 5525 df-po 5533 df-so 5534 df-fr 5578 df-we 5580 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-pred 6260 df-ord 6321 df-on 6322 df-lim 6323 df-suc 6324 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-riota 7318 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7812 df-2nd 7937 df-frecs 8225 df-wrecs 8256 df-recs 8305 df-rdg 8343 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177 df-ltxr 11178 df-le 11179 df-sub 11373 df-neg 11374 df-nn 12169 df-n0 12432 df-z 12519 df-uz 12783 df-seq 13958 df-exp 14018

This theorem is referenced by: bernneq3 14187 expnbnd 14188 expmulnbnd 14191 expcnv 15823 ostth2lem1 27598

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