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 14229

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 1210	. . . . 5 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ∈ ℝ⁺)
2	1	rpcnd 13080	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ∈ ℂ)
3	1	rpne0d 13083	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 ≠ 0)
4		simpl2 1211	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝑀 ∈ ℤ)
5		exprec 14159	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑀 ∈ ℤ) → ((1 / 𝐴)↑𝑀) = (1 / (𝐴↑𝑀)))
6	2, 3, 4, 5	syl3anc 1398	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((1 / 𝐴)↑𝑀) = (1 / (𝐴↑𝑀)))
7		simpl3 1212	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝑁 ∈ ℤ)
8		exprec 14159	. . . 4 ⊢ ((𝐴 ∈ ℂ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
9	2, 3, 7, 8	syl3anc 1398	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((1 / 𝐴)↑𝑁) = (1 / (𝐴↑𝑁)))
10	6, 9	breq12d 5124	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁) ↔ (1 / (𝐴↑𝑀)) < (1 / (𝐴↑𝑁))))
11	1	rprecred 13089	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (1 / 𝐴) ∈ ℝ)
12		simpr 490	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 𝐴 < 1)
13	1	reclt1d 13091	. . . 4 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴 < 1 ↔ 1 < (1 / 𝐴)))
14	12, 13	mpbid 235	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → 1 < (1 / 𝐴))
15		ltexp2 14226	. . 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 1400	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ ((1 / 𝐴)↑𝑀) < ((1 / 𝐴)↑𝑁)))
17		rpexpcl 14136	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ⁺)
18	1, 7, 17	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴↑𝑁) ∈ ℝ⁺)
19		rpexpcl 14136	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ) → (𝐴↑𝑀) ∈ ℝ⁺)
20	1, 4, 19	syl2anc 596	. . 3 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝐴↑𝑀) ∈ ℝ⁺)
21	18, 20	ltrecd 13096	. 2 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → ((𝐴↑𝑁) < (𝐴↑𝑀) ↔ (1 / (𝐴↑𝑀)) < (1 / (𝐴↑𝑁))))
22	10, 16, 21	3bitr4d 314	1 ⊢ (((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 𝐴 < 1) → (𝑀 < 𝑁 ↔ (𝐴↑𝑁) < (𝐴↑𝑀)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∧ w3a 1103 = wceq 1570 ∈ wcel 2146 ≠ wne 2960 class class class wbr 5111 (class class class)co 7419 ℂcc 11115 ℝcr 11116 0cc0 11117 1c1 11118 < clt 11260 / cdiv 11888 ℤcz 12608 ℝ⁺crp 13034 ↑cexp 14117

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2148 ax-9 2156 ax-10 2179 ax-11 2195 ax-12 2216 ax-ext 2737 ax-sep 5259 ax-nul 5271 ax-pow 5338 ax-pr 5406 ax-un 7742 ax-cnex 11173 ax-resscn 11174 ax-1cn 11175 ax-icn 11176 ax-addcl 11177 ax-addrcl 11178 ax-mulcl 11179 ax-mulrcl 11180 ax-mulcom 11181 ax-addass 11182 ax-mulass 11183 ax-distr 11184 ax-i2m1 11185 ax-1ne0 11186 ax-1rid 11187 ax-rnegex 11188 ax-rrecex 11189 ax-cnre 11190 ax-pre-lttri 11191 ax-pre-lttrn 11192 ax-pre-ltadd 11193 ax-pre-mulgt0 11194

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2569 df-eu 2599 df-clab 2744 df-cleq 2757 df-clel 2840 df-nfc 2914 df-ne 2961 df-nel 3067 df-ral 3082 df-rex 3092 df-rmo 3371 df-reu 3372 df-rab 3419 df-v 3459 df-sbc 3747 df-csb 3855 df-dif 3909 df-un 3911 df-in 3913 df-ss 3923 df-pss 3926 df-nul 4287 df-if 4490 df-pw 4566 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4875 df-iun 4960 df-br 5112 df-opab 5176 df-mpt 5195 df-tr 5221 df-id 5558 df-eprel 5563 df-po 5571 df-so 5572 df-fr 5616 df-we 5618 df-xp 5669 df-rel 5670 df-cnv 5671 df-co 5672 df-dm 5673 df-rn 5674 df-res 5675 df-ima 5676 df-pred 6306 df-ord 6367 df-on 6368 df-lim 6369 df-suc 6370 df-iota 6496 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-riota 7376 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7869 df-2nd 7993 df-frecs 8284 df-wrecs 8315 df-recs 8364 df-rdg 8403 df-er 8700 df-en 8950 df-dom 8951 df-sdom 8952 df-pnf 11262 df-mnf 11263 df-xr 11264 df-ltxr 11265 df-le 11266 df-sub 11460 df-neg 11461 df-div 11889 df-nn 12251 df-n0 12522 df-z 12609 df-uz 12881 df-rp 13035 df-seq 14058 df-exp 14118

This theorem is used by: ltexp2rd 14304

W3C validator