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Theorem ltexp2a 10503
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 990 . . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. RR )
2 0red 7896 . . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 e. RR )
3 1red 7910 . . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 e. RR )
4 0lt1 8021 . . . . . . . . 9 |- 0 < 1
54a1i 9 . . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 < 1 )
6 simprl 521 . . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < A )
72, 3, 1, 5, 6lttrd 8020 . . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 < A )
81, 7elrpd 9625 . . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. RR+ )
9 simpl2 991 . . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> M e. ZZ )
10 rpexpcl 10470 . . . . . 6 |- ( ( A e. RR+ /\ M e. ZZ ) -> ( A ^ M ) e. RR+ )
118, 9, 10syl2anc 409 . . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. RR+ )
1211rpred 9628 . . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. RR )
1312recnd 7923 . . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. CC )
1413mulid2d 7913 . 2 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( 1 x. ( A ^ M ) ) = ( A ^ M ) )
15 simprr 522 . . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> M < N )
16 simpl3 992 . . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> N e. ZZ )
17 znnsub 9238 . . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )
189, 16, 17syl2anc 409 . . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( M < N <-> ( N - M ) e. NN ) )
1915, 18mpbid 146 . . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( N - M ) e. NN )
20 expgt1 10489 . . . . 5 |- ( ( A e. RR /\ ( N - M ) e. NN /\ 1 < A ) -> 1 < ( A ^ ( N - M ) ) )
211, 19, 6, 20syl3anc 1228 . . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < ( A ^ ( N - M ) ) )
221recnd 7923 . . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. CC )
231, 7gt0ap0d 8523 . . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A # 0 )
24 expsubap 10499 . . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( A ^ ( N - M ) ) = ( ( A ^ N ) / ( A ^ M ) ) )
2522, 23, 16, 9, 24syl22anc 1229 . . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ ( N - M ) ) = ( ( A ^ N ) / ( A ^ M ) ) )
2621, 25breqtrd 4007 . . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < ( ( A ^ N ) / ( A ^ M ) ) )
27 rpexpcl 10470 . . . . . 6 |- ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) e. RR+ )
288, 16, 27syl2anc 409 . . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ N ) e. RR+ )
2928rpred 9628 . . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ N ) e. RR )
303, 29, 11ltmuldivd 9676 . . 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 166 . 2 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( 1 x. ( A ^ M ) ) < ( A ^ N ) )
3214, 31eqbrtrrd 4005 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 103   <-> wb 104   /\ w3a 968   = wceq 1343   e. wcel 2136   class class class wbr 3981  (class class class)co 5841   CCcc 7747   RRcr 7748   0cc0 7749   1c1 7750   x. cmul 7754   < clt 7929   - cmin 8065   # cap 8475   / cdiv 8564   NNcn 8853   ZZcz 9187   RR+crp 9585   ^cexp 10450
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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-nul 4107  ax-pow 4152  ax-pr 4186  ax-un 4410  ax-setind 4513  ax-iinf 4564  ax-cnex 7840  ax-resscn 7841  ax-1cn 7842  ax-1re 7843  ax-icn 7844  ax-addcl 7845  ax-addrcl 7846  ax-mulcl 7847  ax-mulrcl 7848  ax-addcom 7849  ax-mulcom 7850  ax-addass 7851  ax-mulass 7852  ax-distr 7853  ax-i2m1 7854  ax-0lt1 7855  ax-1rid 7856  ax-0id 7857  ax-rnegex 7858  ax-precex 7859  ax-cnre 7860  ax-pre-ltirr 7861  ax-pre-ltwlin 7862  ax-pre-lttrn 7863  ax-pre-apti 7864  ax-pre-ltadd 7865  ax-pre-mulgt0 7866  ax-pre-mulext 7867
This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ne 2336  df-nel 2431  df-ral 2448  df-rex 2449  df-reu 2450  df-rmo 2451  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-dif 3117  df-un 3119  df-in 3121  df-ss 3128  df-nul 3409  df-if 3520  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-int 3824  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-tr 4080  df-id 4270  df-po 4273  df-iso 4274  df-iord 4343  df-on 4345  df-ilim 4346  df-suc 4348  df-iom 4567  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195  df-riota 5797  df-ov 5844  df-oprab 5845  df-mpo 5846  df-1st 6105  df-2nd 6106  df-recs 6269  df-frec 6355  df-pnf 7931  df-mnf 7932  df-xr 7933  df-ltxr 7934  df-le 7935  df-sub 8067  df-neg 8068  df-reap 8469  df-ap 8476  df-div 8565  df-inn 8854  df-n0 9111  df-z 9188  df-uz 9463  df-rp 9586  df-seqfrec 10377  df-exp 10451
This theorem is referenced by:  expnass  10556  nn0ltexp2  10619  expcanlem  10624
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