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Theorem ltexp2r 14177

Description: The integer powers of a fixed positive real smaller than 1 decrease as the 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 1188	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ∈ ℝ⁺)
2	1	rpcnd 13058	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ∈ ℂ)
3	1	rpne0d 13061	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ≠ 0)
4		simpl2 1189	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝑀 ∈ ℤ)
5		exprec 14108	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑀 ∈ ℤ) → ((1 / 𝐴)↑𝑀) = (1 / (𝐴↑𝑀)))
6	2, 3, 4, 5	syl3anc 1368	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((1 / 𝐴)↑𝑀) = (1 / (𝐴↑𝑀)))
7		simpl3 1190	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝑁 ∈ ℤ)
8		exprec 14108	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
9	2, 3, 7, 8	syl3anc 1368	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
10	6, 9	breq12d 5165	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁) ↔ (1 / (𝐴↑𝑀)) < (1 / (𝐴↑𝑁))))
11	1	rprecred 13067	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (1 / 𝐴) ∈ ℝ)
12		simpr 483	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 < 1)
13	1	reclt1d 13069	. . . 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 14174	. . 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 1370	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ ((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁)))
17		rpexpcl 14085	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ⁺)
18	1, 7, 17	syl2anc 582	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴↑𝑁) ∈ ℝ⁺)
19		rpexpcl 14085	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ) → (𝐴↑𝑀) ∈ ℝ⁺)
20	1, 4, 19	syl2anc 582	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴↑𝑀) ∈ ℝ⁺)
21	18, 20	ltrecd 13074	. 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ≠ wne 2937 class class class wbr 5152 (class class class)co 7426 ℂcc 11144 ℝcr 11145 0cc0 11146 1c1 11147 < clt 11286 / cdiv 11909 ℤcz 12596 ℝ⁺crp 13014 ↑cexp 14066

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7746 ax-cnex 11202 ax-resscn 11203 ax-1cn 11204 ax-icn 11205 ax-addcl 11206 ax-addrcl 11207 ax-mulcl 11208 ax-mulrcl 11209 ax-mulcom 11210 ax-addass 11211 ax-mulass 11212 ax-distr 11213 ax-i2m1 11214 ax-1ne0 11215 ax-1rid 11216 ax-rnegex 11217 ax-rrecex 11218 ax-cnre 11219 ax-pre-lttri 11220 ax-pre-lttrn 11221 ax-pre-ltadd 11222 ax-pre-mulgt0 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rmo 3374 df-reu 3375 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-tr 5270 df-id 5580 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-pred 6310 df-ord 6377 df-on 6378 df-lim 6379 df-suc 6380 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-riota 7382 df-ov 7429 df-oprab 7430 df-mpo 7431 df-om 7877 df-2nd 8000 df-frecs 8293 df-wrecs 8324 df-recs 8398 df-rdg 8437 df-er 8731 df-en 8971 df-dom 8972 df-sdom 8973 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-sub 11484 df-neg 11485 df-div 11910 df-nn 12251 df-n0 12511 df-z 12597 df-uz 12861 df-rp 13015 df-seq 14007 df-exp 14067

This theorem is referenced by: ltexp2rd 14250

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