Theorem expgt1 10839

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

Step	Hyp	Ref	Expression
1		1re 8178	. . 3 |- 1 e. RR
2	1	a1i 9	. 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 e. RR )
3		simp1 1023	. 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A e. RR )
4		simp2 1024	. . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN )
5	4	nnnn0d 9455	. . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN0 )
6		reexpcl 10818	. . 3 |- ( ( A e. RR /\ N e. NN0 ) -> ( A ^ N ) e. RR )
7	3, 5, 6	syl2anc 411	. 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ N ) e. RR )
8		simp3 1025	. 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < A )
9		nnm1nn0 9443	. . . . . 6 |- ( N e. NN -> ( N - 1 ) e. NN0 )
10	4, 9	syl 14	. . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( N - 1 ) e. NN0 )
11		ltle 8267	. . . . . . 7 |- ( ( 1 e. RR /\ A e. RR ) -> ( 1 < A -> 1 <_ A ) )
12	1, 3, 11	sylancr 414	. . . . . 6 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 < A -> 1 <_ A ) )
13	8, 12	mpd 13	. . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 <_ A )
14		expge1 10838	. . . . 5 |- ( ( A e. RR /\ ( N - 1 ) e. NN0 /\ 1 <_ A ) -> 1 <_ ( A ^ ( N - 1 ) ) )
15	3, 10, 13, 14	syl3anc 1273	. . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 <_ ( A ^ ( N - 1 ) ) )
16		reexpcl 10818	. . . . . 6 |- ( ( A e. RR /\ ( N - 1 ) e. NN0 ) -> ( A ^ ( N - 1 ) ) e. RR )
17	3, 10, 16	syl2anc 411	. . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ ( N - 1 ) ) e. RR )
18		0red 8180	. . . . . 6 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 e. RR )
19		0lt1 8306	. . . . . . 7 |- 0 < 1
20	19	a1i 9	. . . . . 6 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 < 1 )
21	18, 2, 3, 20, 8	lttrd 8305	. . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 < A )
22		lemul1 8773	. . . . 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 ) ) )
23	2, 17, 3, 21, 22	syl112anc 1277	. . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 <_ ( A ^ ( N - 1 ) ) <-> ( 1 x. A ) <_ ( ( A ^ ( N - 1 ) ) x. A ) ) )
24	15, 23	mpbid 147	. . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 x. A ) <_ ( ( A ^ ( N - 1 ) ) x. A ) )
25		recn 8165	. . . . . 6 |- ( A e. RR -> A e. CC )
26	25	3ad2ant1 1044	. . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A e. CC )
27	26	mulid2d 8198	. . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 x. A ) = A )
28	27	eqcomd 2237	. . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A = ( 1 x. A ) )
29		expm1t 10829	. . . 4 |- ( ( A e. CC /\ N e. NN ) -> ( A ^ N ) = ( ( A ^ ( N - 1 ) ) x. A ) )
30	26, 4, 29	syl2anc 411	. . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( A ^ N ) = ( ( A ^ ( N - 1 ) ) x. A ) )
31	24, 28, 30	3brtr4d 4120	. 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A <_ ( A ^ N ) )
32	2, 3, 7, 8, 31	ltletrd 8603	1 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < ( A ^ N ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2202   class class class wbr 4088  (class class class)co 6018   CCcc 8030   RRcr 8031   0cc0 8032   1c1 8033   x. cmul 8037   < clt 8214   <_ cle 8215   - cmin 8350   NNcn 9143   NN0cn0 9402   ^cexp 10800

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-mulrcl 8131  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-0lt1 8138  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-precex 8142  ax-cnre 8143  ax-pre-ltirr 8144  ax-pre-ltwlin 8145  ax-pre-lttrn 8146  ax-pre-apti 8147  ax-pre-ltadd 8148  ax-pre-mulgt0 8149  ax-pre-mulext 8150

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-nel 2498  df-ral 2515  df-rex 2516  df-reu 2517  df-rmo 2518  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-iun 3972  df-br 4089  df-opab 4151  df-mpt 4152  df-tr 4188  df-id 4390  df-po 4393  df-iso 4394  df-iord 4463  df-on 4465  df-ilim 4466  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-recs 6471  df-frec 6557  df-pnf 8216  df-mnf 8217  df-xr 8218  df-ltxr 8219  df-le 8220  df-sub 8352  df-neg 8353  df-reap 8755  df-ap 8762  df-div 8853  df-inn 9144  df-n0 9403  df-z 9480  df-uz 9756  df-seqfrec 10710  df-exp 10801

This theorem is referenced by:  ltexp2a  10853  dvdsprmpweqle  12911  logbgcd1irraplemexp  15694  perfectlem1  15725  perfectlem2  15726