ltexp2r - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > ltexp2r		Structured version Visualization version GIF version

Theorem ltexp2r 13536

Description: The power of a positive number smaller than 1 decreases as its exponent increases. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)

**Assertion**
Ref	Expression
ltexp2r	⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ (𝐴↑𝑁) < (𝐴↑𝑀)))

**Proof of Theorem ltexp2r**
Step	Hyp	Ref	Expression
1		simpl1 1187	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ∈ ℝ⁺)
2	1	rpcnd 12432	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ∈ ℂ)
3	1	rpne0d 12435	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ≠ 0)
4		simpl2 1188	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝑀 ∈ ℤ)
5		exprec 13469	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑀 ∈ ℤ) → ((1 / 𝐴)↑𝑀) = (1 / (𝐴↑𝑀)))
6	2, 3, 4, 5	syl3anc 1367	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((1 / 𝐴)↑𝑀) = (1 / (𝐴↑𝑀)))
7		simpl3 1189	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝑁 ∈ ℤ)
8		exprec 13469	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
9	2, 3, 7, 8	syl3anc 1367	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
10	6, 9	breq12d 5078	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁) ↔ (1 / (𝐴↑𝑀)) < (1 / (𝐴↑𝑁))))
11	1	rprecred 12441	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (1 / 𝐴) ∈ ℝ)
12		simpr 487	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 < 1)
13	1	reclt1d 12443	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
14	12, 13	mpbid 234	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 1 < (1 / 𝐴))
15		ltexp2 13533	. . 3 ⊢ ((((1 / 𝐴) ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < (1 / 𝐴)) → (𝑀 < 𝑁 ↔ ((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁)))
16	11, 4, 7, 14, 15	syl31anc 1369	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ ((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁)))
17		rpexpcl 13447	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ⁺)
18	1, 7, 17	syl2anc 586	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴↑𝑁) ∈ ℝ⁺)
19		rpexpcl 13447	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ) → (𝐴↑𝑀) ∈ ℝ⁺)
20	1, 4, 19	syl2anc 586	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴↑𝑀) ∈ ℝ⁺)
21	18, 20	ltrecd 12448	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((𝐴↑𝑁) < (𝐴↑𝑀) ↔ (1 / (𝐴↑𝑀)) < (1 / (𝐴↑𝑁))))
22	10, 16, 21	3bitr4d 313	1 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ (𝐴↑𝑁) < (𝐴↑𝑀)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1083 = wceq 1533 ∈ wcel 2110 ≠ wne 3016 class class class wbr 5065 (class class class)co 7155 ℂcc 10534 ℝcr 10535 0cc0 10536 1c1 10537 < clt 10674 / cdiv 11296 ℤcz 11980 ℝ⁺crp 12388 ↑cexp 13428

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 ax-cnex 10592 ax-resscn 10593 ax-1cn 10594 ax-icn 10595 ax-addcl 10596 ax-addrcl 10597 ax-mulcl 10598 ax-mulrcl 10599 ax-mulcom 10600 ax-addass 10601 ax-mulass 10602 ax-distr 10603 ax-i2m1 10604 ax-1ne0 10605 ax-1rid 10606 ax-rnegex 10607 ax-rrecex 10608 ax-cnre 10609 ax-pre-lttri 10610 ax-pre-lttrn 10611 ax-pre-ltadd 10612 ax-pre-mulgt0 10613

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4567 df-pr 4569 df-tp 4571 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-tr 5172 df-id 5459 df-eprel 5464 df-po 5473 df-so 5474 df-fr 5513 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-ord 6193 df-on 6194 df-lim 6195 df-suc 6196 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-riota 7113 df-ov 7158 df-oprab 7159 df-mpo 7160 df-om 7580 df-2nd 7689 df-wrecs 7946 df-recs 8007 df-rdg 8045 df-er 8288 df-en 8509 df-dom 8510 df-sdom 8511 df-pnf 10676 df-mnf 10677 df-xr 10678 df-ltxr 10679 df-le 10680 df-sub 10871 df-neg 10872 df-div 11297 df-nn 11638 df-n0 11897 df-z 11981 df-uz 12243 df-rp 12389 df-seq 13369 df-exp 13429

This theorem is referenced by: ltexp2rd 13608

W3C validator