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Theorem expgt1 10935
Description: A real greater than 1 raised to a positive integer is greater than 1. (Contributed by NM, 13-Feb-2005.) (Revised by Mario Carneiro, 4-Jun-2014.)
Assertion
Ref Expression
expgt1 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < ( A ^ N ) )
Proof of Theorem expgt1
StepHypRef Expression
1 1re 8269 . . 3 |- 1 e. RR
21a1i 9 . 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 e. RR )
3 simp1 1024 . 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A e. RR )
4 simp2 1025 . . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN )
54nnnn0d 9549 . . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN0 )
6 reexpcl 10914 . . 3 |- ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR )
73, 5, 6syl2anc 411 . 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ N ) e. RR )
8 simp3 1026 . 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < A )
9 nnm1nn0 9533 . . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
104, 9syl 14 . . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( N - 1 ) e. NN0 )
11 ltle 8357 . . . . . . 7 |- ( ( 1 e. RR /\ A e. RR ) -> ( 1 < A -> 1 <_ A ) )
121, 3, 11sylancr 414 . . . . . 6 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 < A -> 1 <_ A ) )
138, 12mpd 13 . . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 <_ A )
14 expge1 10934 . . . . 5 |- ( ( A e. RR /\ ( N - 1 ) e. NN0 /\ 1 <_ A ) -> 1 <_ ( A ^ ( N - 1 ) ) )
153, 10, 13, 14syl3anc 1274 . . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 <_ ( A ^ ( N - 1 ) ) )
16 reexpcl 10914 . . . . . 6 |- ( ( A e. RR /\ ( N - 1 ) e. NN0 ) -> ( A ^ ( N - 1 ) ) e. RR )
173, 10, 16syl2anc 411 . . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ ( N - 1 ) ) e. RR )
18 0red 8271 . . . . . 6 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 e. RR )
19 0lt1 8396 . . . . . . 7 |- 0 < 1
2019a1i 9 . . . . . 6 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 < 1 )
2118, 2, 3, 20, 8lttrd 8395 . . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 < A )
22 lemul1 8863 . . . . 5 |- ( ( 1 e. RR /\ ( A ^ ( N - 1 ) ) e. RR /\ ( A e. RR /\ 0 < A ) ) -> ( 1 <_ ( A ^ ( N - 1 ) ) <-> ( 1 x. A ) <_ ( ( A ^ ( N - 1 ) ) x. A ) ) )
232, 17, 3, 21, 22syl112anc 1278 . . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 <_ ( A ^ ( N - 1 ) ) <-> ( 1 x. A ) <_ ( ( A ^ ( N - 1 ) ) x. A ) ) )
2415, 23mpbid 147 . . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 x. A ) <_ ( ( A ^ ( N - 1 ) ) x. A ) )
25 recn 8256 . . . . . 6 |- ( A e. RR -> A e. CC )
26253ad2ant1 1045 . . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A e. CC )
2726mullidd 8288 . . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 x. A ) = A )
2827eqcomd 2238 . . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A = ( 1 x. A ) )
29 expm1t 10925 . . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( ( A ^ ( N - 1 ) ) x. A ) )
3026, 4, 29syl2anc 411 . . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ N ) = ( ( A ^ ( N - 1 ) ) x. A ) )
3124, 28, 303brtr4d 4140 . 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A <_ ( A ^ N ) )
322, 3, 7, 8, 31ltletrd 8693 1 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < ( A ^ N ) )
Colors of variables: wff set class
Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2203   class class class wbr 4108  (class class class)co 6049   CCcc 8121   RRcr 8122   0cc0 8123   1c1 8124   x. cmul 8128   < clt 8304   <_ cle 8305   - cmin 8440   NNcn 9233   NN0cn0 9492   ^cexp 10896
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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-coll 4224  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-setind 4658  ax-iinf 4709  ax-cnex 8214  ax-resscn 8215  ax-1cn 8216  ax-1re 8217  ax-icn 8218  ax-addcl 8219  ax-addrcl 8220  ax-mulcl 8221  ax-mulrcl 8222  ax-addcom 8223  ax-mulcom 8224  ax-addass 8225  ax-mulass 8226  ax-distr 8227  ax-i2m1 8228  ax-0lt1 8229  ax-1rid 8230  ax-0id 8231  ax-rnegex 8232  ax-precex 8233  ax-cnre 8234  ax-pre-ltirr 8235  ax-pre-ltwlin 8236  ax-pre-lttrn 8237  ax-pre-apti 8238  ax-pre-ltadd 8239  ax-pre-mulgt0 8240  ax-pre-mulext 8241
This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-nel 2508  df-ral 2525  df-rex 2526  df-reu 2527  df-rmo 2528  df-rab 2529  df-v 2814  df-sbc 3042  df-csb 3138  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-if 3620  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-iun 3992  df-br 4109  df-opab 4171  df-mpt 4172  df-tr 4208  df-id 4413  df-po 4416  df-iso 4417  df-iord 4486  df-on 4488  df-ilim 4489  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-riota 6002  df-ov 6052  df-oprab 6053  df-mpo 6054  df-1st 6333  df-2nd 6334  df-recs 6535  df-frec 6621  df-pnf 8306  df-mnf 8307  df-xr 8308  df-ltxr 8309  df-le 8310  df-sub 8442  df-neg 8443  df-reap 8845  df-ap 8852  df-div 8943  df-inn 9234  df-n0 9493  df-z 9574  df-uz 9850  df-seqfrec 10806  df-exp 10897
This theorem is referenced by:  ltexp2a  10949  dvdsprmpweqle  13028  logbgcd1irraplemexp  15820  perfectlem1  15854  perfectlem2  15855
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