ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  expgt1 Unicode version
Theorem expgt1 10997
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 8319 . . 3 |- 1 e. RR
21a1i 9 . 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 e. RR )
3 simp1 1028 . 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A e. RR )
4 simp2 1029 . . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN )
54nnnn0d 9603 . . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN0 )
6 reexpcl 10976 . . 3 |- ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR )
73, 5, 6syl2anc 415 . 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ N ) e. RR )
8 simp3 1030 . 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < A )
9 nnm1nn0 9587 . . . . . 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 8407 . . . . . . 7 |- ( ( 1 e. RR /\ A e. RR ) -> ( 1 < A -> 1 <_ A ) )
121, 3, 11sylancr 418 . . . . . 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 10996 . . . . 5 |- ( ( A e. RR /\ ( N - 1 ) e. NN0 /\ 1 <_ A ) -> 1 <_ ( A ^ ( N - 1 ) ) )
153, 10, 13, 14syl3anc 1278 . . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 <_ ( A ^ ( N - 1 ) ) )
16 reexpcl 10976 . . . . . 6 |- ( ( A e. RR /\ ( N - 1 ) e. NN0 ) -> ( A ^ ( N - 1 ) ) e. RR )
173, 10, 16syl2anc 415 . . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ ( N - 1 ) ) e. RR )
18 0red 8321 . . . . . 6 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 e. RR )
19 0lt1 8447 . . . . . . 7 |- 0 < 1
2019a1i 9 . . . . . 6 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 < 1 )
2118, 2, 3, 20, 8lttrd 8446 . . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 < A )
22 lemul1 8915 . . . . 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 1282 . . . 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 8306 . . . . . 6 |- ( A e. RR -> A e. CC )
26253ad2ant1 1049 . . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A e. CC )
2726mullidd 8338 . . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 x. A ) = A )
2827eqcomd 2244 . . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A = ( 1 x. A ) )
29 expm1t 10987 . . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( ( A ^ ( N - 1 ) ) x. A ) )
3026, 4, 29syl2anc 415 . . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ N ) = ( ( A ^ ( N - 1 ) ) x. A ) )
3124, 28, 303brtr4d 4160 . 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A <_ ( A ^ N ) )
322, 3, 7, 8, 31ltletrd 8745 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 1009   = wceq 1402   e. wcel 2209   class class class wbr 4128  (class class class)co 6079   CCcc 8171   RRcr 8172   0cc0 8173   1c1 8174   x. cmul 8178   < clt 8354   <_ cle 8355   - cmin 8491   NNcn 9287   NN0cn0 9546   ^cexp 10958
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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4244  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-setind 4682  ax-iinf 4733  ax-cnex 8264  ax-resscn 8265  ax-1cn 8266  ax-1re 8267  ax-icn 8268  ax-addcl 8269  ax-addrcl 8270  ax-mulcl 8271  ax-mulrcl 8272  ax-addcom 8273  ax-mulcom 8274  ax-addass 8275  ax-mulass 8276  ax-distr 8277  ax-i2m1 8278  ax-0lt1 8279  ax-1rid 8280  ax-0id 8281  ax-rnegex 8282  ax-precex 8283  ax-cnre 8284  ax-pre-ltirr 8285  ax-pre-ltwlin 8286  ax-pre-lttrn 8287  ax-pre-apti 8288  ax-pre-ltadd 8289  ax-pre-mulgt0 8290  ax-pre-mulext 8291
This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-iun 4012  df-br 4129  df-opab 4191  df-mpt 4192  df-tr 4228  df-id 4436  df-po 4439  df-iso 4440  df-iord 4509  df-on 4511  df-ilim 4512  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-riota 6032  df-ov 6082  df-oprab 6083  df-mpo 6084  df-1st 6368  df-2nd 6369  df-recs 6570  df-frec 6656  df-pnf 8356  df-mnf 8357  df-xr 8358  df-ltxr 8359  df-le 8360  df-sub 8493  df-neg 8494  df-reap 8897  df-ap 8904  df-div 8997  df-inn 9288  df-n0 9547  df-z 9628  df-uz 9905  df-seqfrec 10868  df-exp 10959
This theorem is referenced by:  ltexp2a  11011  dvdsprmpweqle  13099  logbgcd1irraplemexp  16053  perfectlem1  16096  perfectlem2  16097
  Copyright terms: Public domain W3C validator