ltexp2 - Intuitionistic Logic Explorer

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Theorem ltexp2 13356

Description: Ordering law for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 5-Jun-2014.)

**Assertion**
Ref	Expression
ltexp2	⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))

**Proof of Theorem ltexp2**
Step	Hyp	Ref	Expression
1		simpl1 985	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝐴 ∈ ℝ)
2		simpr 109	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 1 < 𝐴)
3		simpl2 986	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑀 ∈ ℤ)
4	3	zred 9292	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑀 ∈ ℝ)
5		simpl3 987	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑁 ∈ ℤ)
6	5	zred 9292	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝑁 ∈ ℝ)
7	1, 2, 4, 6	cxpltd 13344	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑_𝑐𝑀) < (𝐴↑_𝑐𝑁)))
8		0red 7882	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 ∈ ℝ)
9		1red 7896	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 1 ∈ ℝ)
10		0lt1 8007	. . . . . . 7 ⊢ 0 < 1
11	10	a1i 9	. . . . . 6 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 < 1)
12	8, 9, 1, 11, 2	lttrd 8006	. . . . 5 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 0 < 𝐴)
13	1, 12	elrpd 9607	. . . 4 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → 𝐴 ∈ ℝ⁺)
14		cxpexprp 13312	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑀 ∈ ℤ) → (𝐴↑_𝑐𝑀) = (𝐴↑𝑀))
15	13, 3, 14	syl2anc 409	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝐴↑_𝑐𝑀) = (𝐴↑𝑀))
16		cxpexprp 13312	. . . 4 ⊢ ((𝐴 ∈ ℝ⁺ ∧ 𝑁 ∈ ℤ) → (𝐴↑_𝑐𝑁) = (𝐴↑𝑁))
17	13, 5, 16	syl2anc 409	. . 3 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝐴↑_𝑐𝑁) = (𝐴↑𝑁))
18	15, 17	breq12d 3980	. 2 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → ((𝐴↑_𝑐𝑀) < (𝐴↑_𝑐𝑁) ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))
19	7, 18	bitrd 187	1 ⊢ (((𝐴 ∈ ℝ ∧ 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ) ∧ 1 < 𝐴) → (𝑀 < 𝑁 ↔ (𝐴↑𝑀) < (𝐴↑𝑁)))

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 103 ↔ wb 104 ∧ w3a 963 = wceq 1335 ∈ wcel 2128 class class class wbr 3967 (class class class)co 5827 ℝcr 7734 0cc0 7735 1c1 7736 < clt 7915 ℤcz 9173 ℝ⁺crp 9567 ↑cexp 10428 ↑_𝑐ccxp 13274

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-coll 4082 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 ax-cnex 7826 ax-resscn 7827 ax-1cn 7828 ax-1re 7829 ax-icn 7830 ax-addcl 7831 ax-addrcl 7832 ax-mulcl 7833 ax-mulrcl 7834 ax-addcom 7835 ax-mulcom 7836 ax-addass 7837 ax-mulass 7838 ax-distr 7839 ax-i2m1 7840 ax-0lt1 7841 ax-1rid 7842 ax-0id 7843 ax-rnegex 7844 ax-precex 7845 ax-cnre 7846 ax-pre-ltirr 7847 ax-pre-ltwlin 7848 ax-pre-lttrn 7849 ax-pre-apti 7850 ax-pre-ltadd 7851 ax-pre-mulgt0 7852 ax-pre-mulext 7853 ax-arch 7854 ax-caucvg 7855 ax-pre-suploc 7856 ax-addf 7857 ax-mulf 7858

This theorem depends on definitions: df-bi 116 df-stab 817 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-nel 2423 df-ral 2440 df-rex 2441 df-reu 2442 df-rmo 2443 df-rab 2444 df-v 2714 df-sbc 2938 df-csb 3032 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-iun 3853 df-disj 3945 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-id 4256 df-po 4259 df-iso 4260 df-iord 4329 df-on 4331 df-ilim 4332 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-f1 5178 df-fo 5179 df-f1o 5180 df-fv 5181 df-isom 5182 df-riota 5783 df-ov 5830 df-oprab 5831 df-mpo 5832 df-of 6035 df-1st 6091 df-2nd 6092 df-recs 6255 df-irdg 6320 df-frec 6341 df-1o 6366 df-oadd 6370 df-er 6483 df-map 6598 df-pm 6599 df-en 6689 df-dom 6690 df-fin 6691 df-sup 6931 df-inf 6932 df-pnf 7917 df-mnf 7918 df-xr 7919 df-ltxr 7920 df-le 7921 df-sub 8053 df-neg 8054 df-reap 8455 df-ap 8462 df-div 8551 df-inn 8840 df-2 8898 df-3 8899 df-4 8900 df-n0 9097 df-z 9174 df-uz 9446 df-q 9536 df-rp 9568 df-xneg 9686 df-xadd 9687 df-ioo 9803 df-ico 9805 df-icc 9806 df-fz 9920 df-fzo 10052 df-seqfrec 10355 df-exp 10429 df-fac 10612 df-bc 10634 df-ihash 10662 df-shft 10727 df-cj 10754 df-re 10755 df-im 10756 df-rsqrt 10910 df-abs 10911 df-clim 11188 df-sumdc 11263 df-ef 11557 df-e 11558 df-rest 12449 df-topgen 12468 df-psmet 12483 df-xmet 12484 df-met 12485 df-bl 12486 df-mopn 12487 df-top 12492 df-topon 12505 df-bases 12537 df-ntr 12592 df-cn 12684 df-cnp 12685 df-tx 12749 df-cncf 13054 df-limced 13121 df-dvap 13122 df-relog 13275 df-rpcxp 13276

This theorem is referenced by: (None)

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