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Theorem ltexp2a 10665
Description: Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
ltexp2a |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) < ( A ^ N ) )
Proof of Theorem ltexp2a
StepHypRef Expression
1 simpl1 1002 . . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. RR )
2 0red 8022 . . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 e. RR )
3 1red 8036 . . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 e. RR )
4 0lt1 8148 . . . . . . . . 9 |- 0 < 1
54a1i 9 . . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 < 1 )
6 simprl 529 . . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < A )
72, 3, 1, 5, 6lttrd 8147 . . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 < A )
81, 7elrpd 9762 . . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. RR+ )
9 simpl2 1003 . . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> M e. ZZ )
10 rpexpcl 10632 . . . . . 6 |- ( ( A e. RR+ /\ M e. ZZ ) -> ( A ^ M ) e. RR+ )
118, 9, 10syl2anc 411 . . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. RR+ )
1211rpred 9765 . . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. RR )
1312recnd 8050 . . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. CC )
1413mulid2d 8040 . 2 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( 1 x. ( A ^ M ) ) = ( A ^ M ) )
15 simprr 531 . . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> M < N )
16 simpl3 1004 . . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> N e. ZZ )
17 znnsub 9371 . . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )
189, 16, 17syl2anc 411 . . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( M < N <-> ( N - M ) e. NN ) )
1915, 18mpbid 147 . . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( N - M ) e. NN )
20 expgt1 10651 . . . . 5 |- ( ( A e. RR /\ ( N - M ) e. NN /\ 1 < A ) -> 1 < ( A ^ ( N - M ) ) )
211, 19, 6, 20syl3anc 1249 . . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < ( A ^ ( N - M ) ) )
221recnd 8050 . . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. CC )
231, 7gt0ap0d 8650 . . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A # 0 )
24 expsubap 10661 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( A ^ ( N - M ) ) = ( ( A ^ N ) / ( A ^ M ) ) )
2522, 23, 16, 9, 24syl22anc 1250 . . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ ( N - M ) ) = ( ( A ^ N ) / ( A ^ M ) ) )
2621, 25breqtrd 4056 . . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < ( ( A ^ N ) / ( A ^ M ) ) )
27 rpexpcl 10632 . . . . . 6 |- ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) e. RR+ )
288, 16, 27syl2anc 411 . . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ N ) e. RR+ )
2928rpred 9765 . . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ N ) e. RR )
303, 29, 11ltmuldivd 9813 . . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( ( 1 x. ( A ^ M ) ) < ( A ^ N ) <-> 1 < ( ( A ^ N ) / ( A ^ M ) ) ) )
3126, 30mpbird 167 . 2 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( 1 x. ( A ^ M ) ) < ( A ^ N ) )
3214, 31eqbrtrrd 4054 1 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) < ( A ^ N ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   = wceq 1364   e. wcel 2164   class class class wbr 4030  (class class class)co 5919   CCcc 7872   RRcr 7873   0cc0 7874   1c1 7875   x. cmul 7879   < clt 8056   - cmin 8192   # cap 8602   / cdiv 8693   NNcn 8984   ZZcz 9320   RR+crp 9722   ^cexp 10612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-iinf 4621  ax-cnex 7965  ax-resscn 7966  ax-1cn 7967  ax-1re 7968  ax-icn 7969  ax-addcl 7970  ax-addrcl 7971  ax-mulcl 7972  ax-mulrcl 7973  ax-addcom 7974  ax-mulcom 7975  ax-addass 7976  ax-mulass 7977  ax-distr 7978  ax-i2m1 7979  ax-0lt1 7980  ax-1rid 7981  ax-0id 7982  ax-rnegex 7983  ax-precex 7984  ax-cnre 7985  ax-pre-ltirr 7986  ax-pre-ltwlin 7987  ax-pre-lttrn 7988  ax-pre-apti 7989  ax-pre-ltadd 7990  ax-pre-mulgt0 7991  ax-pre-mulext 7992
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-reu 2479  df-rmo 2480  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-if 3559  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-tr 4129  df-id 4325  df-po 4328  df-iso 4329  df-iord 4398  df-on 4400  df-ilim 4401  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-riota 5874  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-recs 6360  df-frec 6446  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061  df-le 8062  df-sub 8194  df-neg 8195  df-reap 8596  df-ap 8603  df-div 8694  df-inn 8985  df-n0 9244  df-z 9321  df-uz 9596  df-rp 9723  df-seqfrec 10522  df-exp 10613
This theorem is referenced by:  expnass  10719  nn0ltexp2  10783  expcanlem  10789
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