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Theorem leexp2 14085

Description: Ordering law for exponentiation of a fixed real base greater than 1 to integer exponents. (Contributed by Mario Carneiro, 26-Apr-2016.)

**Assertion**
Ref	Expression
leexp2	⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 ≤ 𝑁 ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑁)))

**Proof of Theorem leexp2**
Step	Hyp	Ref	Expression
1		3ancomb 1098	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ↔ (𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ))
2		ltexp2 14084	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ) ∧ 1 < 𝐴) → (𝑁 < 𝑀 ↔ (𝐴↑𝑁) < (𝐴↑𝑀)))
3	1, 2	sylanb 581	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑁 < 𝑀 ↔ (𝐴↑𝑁) < (𝐴↑𝑀)))
4	3	notbid 318	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (¬ 𝑁 < 𝑀 ↔ ¬ (𝐴↑𝑁) < (𝐴↑𝑀)))
5		simpl2 1193	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑀 ∈ ℤ)
6		simpl3 1194	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑁 ∈ ℤ)
7		zre 12483	. . . 4 ⊢ (𝑀 ∈ ℤ → 𝑀 ∈ ℝ)
8		zre 12483	. . . 4 ⊢ (𝑁 ∈ ℤ → 𝑁 ∈ ℝ)
9		lenlt 11202	. . . 4 ⊢ ((𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
10	7, 8, 9	syl2an 596	. . 3 ⊢ ((𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
11	5, 6, 10	syl2anc 584	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 ≤ 𝑁 ↔ ¬ 𝑁 < 𝑀))
12		simpl1 1192	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝐴 ∈ ℝ)
13		0red 11126	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 ∈ ℝ)
14		1red 11124	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 1 ∈ ℝ)
15		0lt1 11650	. . . . . . 7 ⊢ 0 < 1
16	15	a1i 11	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 < 1)
17		simpr 484	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 1 < 𝐴)
18	13, 14, 12, 16, 17	lttrd 11285	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 < 𝐴)
19	18	gt0ne0d 11692	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝐴 ≠ 0)
20		reexpclz 13996	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ∧ 𝑀 ∈ ℤ) → (𝐴↑𝑀) ∈ ℝ)
21	12, 19, 5, 20	syl3anc 1373	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝐴↑𝑀) ∈ ℝ)
22		reexpclz 13996	. . . 4 ⊢ ((𝐴 ∈ ℝ ∧ 𝐴 ≠ 0 ∧ 𝑁 ∈ ℤ) → (𝐴↑𝑁) ∈ ℝ)
23	12, 19, 6, 22	syl3anc 1373	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝐴↑𝑁) ∈ ℝ)
24	21, 23	lenltd 11270	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑𝑀) ≤ (𝐴↑𝑁) ↔ ¬ (𝐴↑𝑁) < (𝐴↑𝑀)))
25	4, 11, 24	3bitr4d 311	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 ≤ 𝑁 ↔ (𝐴↑𝑀) ≤ (𝐴↑𝑁)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2113 ≠ wne 2929 class class class wbr 5095 (class class class)co 7355 ℝcr 11016 0cc0 11017 1c1 11018 < clt 11157 ≤ cle 11158 ℤcz 12479 ↑cexp 13975

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2705 ax-sep 5238 ax-nul 5248 ax-pow 5307 ax-pr 5374 ax-un 7677 ax-cnex 11073 ax-resscn 11074 ax-1cn 11075 ax-icn 11076 ax-addcl 11077 ax-addrcl 11078 ax-mulcl 11079 ax-mulrcl 11080 ax-mulcom 11081 ax-addass 11082 ax-mulass 11083 ax-distr 11084 ax-i2m1 11085 ax-1ne0 11086 ax-1rid 11087 ax-rnegex 11088 ax-rrecex 11089 ax-cnre 11090 ax-pre-lttri 11091 ax-pre-lttrn 11092 ax-pre-ltadd 11093 ax-pre-mulgt0 11094

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2068 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2725 df-clel 2808 df-nfc 2882 df-ne 2930 df-nel 3034 df-ral 3049 df-rex 3058 df-rmo 3347 df-reu 3348 df-rab 3397 df-v 3439 df-sbc 3738 df-csb 3847 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3918 df-nul 4283 df-if 4477 df-pw 4553 df-sn 4578 df-pr 4580 df-op 4584 df-uni 4861 df-iun 4945 df-br 5096 df-opab 5158 df-mpt 5177 df-tr 5203 df-id 5516 df-eprel 5521 df-po 5529 df-so 5530 df-fr 5574 df-we 5576 df-xp 5627 df-rel 5628 df-cnv 5629 df-co 5630 df-dm 5631 df-rn 5632 df-res 5633 df-ima 5634 df-pred 6256 df-ord 6317 df-on 6318 df-lim 6319 df-suc 6320 df-iota 6445 df-fun 6491 df-fn 6492 df-f 6493 df-f1 6494 df-fo 6495 df-f1o 6496 df-fv 6497 df-riota 7312 df-ov 7358 df-oprab 7359 df-mpo 7360 df-om 7806 df-2nd 7931 df-frecs 8220 df-wrecs 8251 df-recs 8300 df-rdg 8338 df-er 8631 df-en 8880 df-dom 8881 df-sdom 8882 df-pnf 11159 df-mnf 11160 df-xr 11161 df-ltxr 11162 df-le 11163 df-sub 11357 df-neg 11358 df-div 11786 df-nn 12137 df-n0 12393 df-z 12480 df-uz 12743 df-rp 12897 df-seq 13916 df-exp 13976

This theorem is referenced by: leexp2d 14166 hgt750leme 34743

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