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Theorem ltexp2r 14114

Description: The integer powers of a fixed positive real less 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 1192	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ∈ ℝ⁺)
2	1	rpcnd 12973	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ∈ ℂ)
3	1	rpne0d 12976	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ≠ 0)
4		simpl2 1193	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝑀 ∈ ℤ)
5		exprec 14044	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑀 ∈ ℤ) → ((1 / 𝐴)↑𝑀) = (1 / (𝐴↑𝑀)))
6	2, 3, 4, 5	syl3anc 1373	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((1 / 𝐴)↑𝑀) = (1 / (𝐴↑𝑀)))
7		simpl3 1194	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝑁 ∈ ℤ)
8		exprec 14044	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
9	2, 3, 7, 8	syl3anc 1373	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
10	6, 9	breq12d 5115	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁) ↔ (1 / (𝐴↑𝑀)) < (1 / (𝐴↑𝑁))))
11	1	rprecred 12982	. . 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 12984	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
14	12, 13	mpbid 232	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 1 < (1 / 𝐴))
15		ltexp2 14111	. . 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 1375	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ ((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁)))
17		rpexpcl 14021	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ⁺)
18	1, 7, 17	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴↑𝑁) ∈ ℝ⁺)
19		rpexpcl 14021	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ) → (𝐴↑𝑀) ∈ ℝ⁺)
20	1, 4, 19	syl2anc 584	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴↑𝑀) ∈ ℝ⁺)
21	18, 20	ltrecd 12989	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((𝐴↑𝑁) < (𝐴↑𝑀) ↔ (1 / (𝐴↑𝑀)) < (1 / (𝐴↑𝑁))))
22	10, 16, 21	3bitr4d 311	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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∈ wcel 2109 ≠ wne 2925 class class class wbr 5102 (class class class)co 7369 ℂcc 11042 ℝcr 11043 0cc0 11044 1c1 11045 < clt 11184 / cdiv 11811 ℤcz 12505 ℝ⁺crp 12927 ↑cexp 14002

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-sep 5246 ax-nul 5256 ax-pow 5315 ax-pr 5382 ax-un 7691 ax-cnex 11100 ax-resscn 11101 ax-1cn 11102 ax-icn 11103 ax-addcl 11104 ax-addrcl 11105 ax-mulcl 11106 ax-mulrcl 11107 ax-mulcom 11108 ax-addass 11109 ax-mulass 11110 ax-distr 11111 ax-i2m1 11112 ax-1ne0 11113 ax-1rid 11114 ax-rnegex 11115 ax-rrecex 11116 ax-cnre 11117 ax-pre-lttri 11118 ax-pre-lttrn 11119 ax-pre-ltadd 11120 ax-pre-mulgt0 11121

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3351 df-reu 3352 df-rab 3403 df-v 3446 df-sbc 3751 df-csb 3860 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-pss 3931 df-nul 4293 df-if 4485 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4868 df-iun 4953 df-br 5103 df-opab 5165 df-mpt 5184 df-tr 5210 df-id 5526 df-eprel 5531 df-po 5539 df-so 5540 df-fr 5584 df-we 5586 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-pred 6262 df-ord 6323 df-on 6324 df-lim 6325 df-suc 6326 df-iota 6452 df-fun 6501 df-fn 6502 df-f 6503 df-f1 6504 df-fo 6505 df-f1o 6506 df-fv 6507 df-riota 7326 df-ov 7372 df-oprab 7373 df-mpo 7374 df-om 7823 df-2nd 7948 df-frecs 8237 df-wrecs 8268 df-recs 8317 df-rdg 8355 df-er 8648 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11186 df-mnf 11187 df-xr 11188 df-ltxr 11189 df-le 11190 df-sub 11383 df-neg 11384 df-div 11812 df-nn 12163 df-n0 12419 df-z 12506 df-uz 12770 df-rp 12928 df-seq 13943 df-exp 14003

This theorem is referenced by: ltexp2rd 14189

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