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 13819

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 1189	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ∈ ℝ⁺)
2	1	rpcnd 12703	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ∈ ℂ)
3	1	rpne0d 12706	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ≠ 0)
4		simpl2 1190	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝑀 ∈ ℤ)
5		exprec 13752	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑀 ∈ ℤ) → ((1 / 𝐴)↑𝑀) = (1 / (𝐴↑𝑀)))
6	2, 3, 4, 5	syl3anc 1369	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((1 / 𝐴)↑𝑀) = (1 / (𝐴↑𝑀)))
7		simpl3 1191	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝑁 ∈ ℤ)
8		exprec 13752	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
9	2, 3, 7, 8	syl3anc 1369	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
10	6, 9	breq12d 5083	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁) ↔ (1 / (𝐴↑𝑀)) < (1 / (𝐴↑𝑁))))
11	1	rprecred 12712	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (1 / 𝐴) ∈ ℝ)
12		simpr 484	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 < 1)
13	1	reclt1d 12714	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
14	12, 13	mpbid 231	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 1 < (1 / 𝐴))
15		ltexp2 13816	. . 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 1371	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ ((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁)))
17		rpexpcl 13729	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ⁺)
18	1, 7, 17	syl2anc 583	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴↑𝑁) ∈ ℝ⁺)
19		rpexpcl 13729	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ) → (𝐴↑𝑀) ∈ ℝ⁺)
20	1, 4, 19	syl2anc 583	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴↑𝑀) ∈ ℝ⁺)
21	18, 20	ltrecd 12719	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((𝐴↑𝑁) < (𝐴↑𝑀) ↔ (1 / (𝐴↑𝑀)) < (1 / (𝐴↑𝑁))))
22	10, 16, 21	3bitr4d 310	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 205 ∧ wa 395 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 ≠ wne 2942 class class class wbr 5070 (class class class)co 7255 ℂcc 10800 ℝcr 10801 0cc0 10802 1c1 10803 < clt 10940 / cdiv 11562 ℤcz 12249 ℝ⁺crp 12659 ↑cexp 13710

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-1cn 10860 ax-icn 10861 ax-addcl 10862 ax-addrcl 10863 ax-mulcl 10864 ax-mulrcl 10865 ax-mulcom 10866 ax-addass 10867 ax-mulass 10868 ax-distr 10869 ax-i2m1 10870 ax-1ne0 10871 ax-1rid 10872 ax-rnegex 10873 ax-rrecex 10874 ax-cnre 10875 ax-pre-lttri 10876 ax-pre-lttrn 10877 ax-pre-ltadd 10878 ax-pre-mulgt0 10879

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-reu 3070 df-rmo 3071 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-pss 3902 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-tp 4563 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-tr 5188 df-id 5480 df-eprel 5486 df-po 5494 df-so 5495 df-fr 5535 df-we 5537 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-pred 6191 df-ord 6254 df-on 6255 df-lim 6256 df-suc 6257 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-riota 7212 df-ov 7258 df-oprab 7259 df-mpo 7260 df-om 7688 df-2nd 7805 df-frecs 8068 df-wrecs 8099 df-recs 8173 df-rdg 8212 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-sub 11137 df-neg 11138 df-div 11563 df-nn 11904 df-n0 12164 df-z 12250 df-uz 12512 df-rp 12660 df-seq 13650 df-exp 13711

This theorem is referenced by: ltexp2rd 13891

W3C validator