bernneq2 - Intuitionistic Logic Explorer

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Theorem bernneq2 10878

Description: Variation of Bernoulli's inequality bernneq 10877. (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 8409	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℝ)
2	1	3ad2ant1 1042	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (𝐴 − 1) ∈ ℝ)
3		simp2 1022	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → 𝑁 ∈ ℕ₀)
4		df-neg 8316	. . . . 5 ⊢ -1 = (0 − 1)
5		0re 8142	. . . . . . 7 ⊢ 0 ∈ ℝ
6		1re 8141	. . . . . . 7 ⊢ 1 ∈ ℝ
7		lesub1 8599	. . . . . . 7 ⊢ ((0 ∈ ℝ ∧ 𝐴 ∈ ℝ ∧ 1 ∈ ℝ) → (0 ≤ 𝐴 ↔ (0 − 1) ≤ (𝐴 − 1)))
8	5, 6, 7	mp3an13 1362	. . . . . 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 4117	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1))
11	10	3adant2 1040	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → -1 ≤ (𝐴 − 1))
12		bernneq 10877	. . 3 ⊢ (((𝐴 − 1) ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ -1 ≤ (𝐴 − 1)) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁))
13	2, 3, 11, 12	syl3anc 1271	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) ≤ ((1 + (𝐴 − 1))↑𝑁))
14		ax-1cn 8088	. . . 4 ⊢ 1 ∈ ℂ
15	1	recnd 8171	. . . . 5 ⊢ (𝐴 ∈ ℝ → (𝐴 − 1) ∈ ℂ)
16		nn0cn 9375	. . . . 5 ⊢ (𝑁 ∈ ℕ₀ → 𝑁 ∈ ℂ)
17		mulcl 8122	. . . . 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 8279	. . . 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 1041	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → (1 + ((𝐴 − 1) · 𝑁)) = (((𝐴 − 1) · 𝑁) + 1))
22		recn 8128	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ∈ ℂ)
23		pncan3 8350	. . . . 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 6015	. . 3 ⊢ (𝐴 ∈ ℝ → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁))
26	25	3ad2ant1 1042	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℕ₀ ∧ 0 ≤ 𝐴) → ((1 + (𝐴 − 1))↑𝑁) = (𝐴↑𝑁))
27	13, 21, 26	3brtr3d 4113	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 1002 = wceq 1395 ∈ wcel 2200 class class class wbr 4082 (class class class)co 6000 ℂcc 7993 ℝcr 7994 0cc0 7995 1c1 7996 + caddc 7998 · cmul 8000 ≤ cle 8178 − cmin 8313 -cneg 8314 ℕ₀cn0 9365 ↑cexp 10755

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 617 ax-in2 618 ax-io 714 ax-5 1493 ax-7 1494 ax-gen 1495 ax-ie1 1539 ax-ie2 1540 ax-8 1550 ax-10 1551 ax-11 1552 ax-i12 1553 ax-bndl 1555 ax-4 1556 ax-17 1572 ax-i9 1576 ax-ial 1580 ax-i5r 1581 ax-13 2202 ax-14 2203 ax-ext 2211 ax-coll 4198 ax-sep 4201 ax-nul 4209 ax-pow 4257 ax-pr 4292 ax-un 4523 ax-setind 4628 ax-iinf 4679 ax-cnex 8086 ax-resscn 8087 ax-1cn 8088 ax-1re 8089 ax-icn 8090 ax-addcl 8091 ax-addrcl 8092 ax-mulcl 8093 ax-mulrcl 8094 ax-addcom 8095 ax-mulcom 8096 ax-addass 8097 ax-mulass 8098 ax-distr 8099 ax-i2m1 8100 ax-0lt1 8101 ax-1rid 8102 ax-0id 8103 ax-rnegex 8104 ax-precex 8105 ax-cnre 8106 ax-pre-ltirr 8107 ax-pre-ltwlin 8108 ax-pre-lttrn 8109 ax-pre-apti 8110 ax-pre-ltadd 8111 ax-pre-mulgt0 8112 ax-pre-mulext 8113

This theorem depends on definitions: df-bi 117 df-dc 840 df-3or 1003 df-3an 1004 df-tru 1398 df-fal 1401 df-nf 1507 df-sb 1809 df-eu 2080 df-mo 2081 df-clab 2216 df-cleq 2222 df-clel 2225 df-nfc 2361 df-ne 2401 df-nel 2496 df-ral 2513 df-rex 2514 df-reu 2515 df-rmo 2516 df-rab 2517 df-v 2801 df-sbc 3029 df-csb 3125 df-dif 3199 df-un 3201 df-in 3203 df-ss 3210 df-nul 3492 df-if 3603 df-pw 3651 df-sn 3672 df-pr 3673 df-op 3675 df-uni 3888 df-int 3923 df-iun 3966 df-br 4083 df-opab 4145 df-mpt 4146 df-tr 4182 df-id 4383 df-po 4386 df-iso 4387 df-iord 4456 df-on 4458 df-ilim 4459 df-suc 4461 df-iom 4682 df-xp 4724 df-rel 4725 df-cnv 4726 df-co 4727 df-dm 4728 df-rn 4729 df-res 4730 df-ima 4731 df-iota 5277 df-fun 5319 df-fn 5320 df-f 5321 df-f1 5322 df-fo 5323 df-f1o 5324 df-fv 5325 df-riota 5953 df-ov 6003 df-oprab 6004 df-mpo 6005 df-1st 6284 df-2nd 6285 df-recs 6449 df-frec 6535 df-pnf 8179 df-mnf 8180 df-xr 8181 df-ltxr 8182 df-le 8183 df-sub 8315 df-neg 8316 df-reap 8718 df-ap 8725 df-div 8816 df-inn 9107 df-n0 9366 df-z 9443 df-uz 9719 df-seqfrec 10665 df-exp 10756

This theorem is referenced by: bernneq3 10879 expnbnd 10880 expcnvap0 12008 cvgratnnlembern 12029

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