Theorem expgt1 10845

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 8183	. . 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 9460	. . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN0 )
6		reexpcl 10824	. . 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 9448	. . . . . 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 8272	. . . . . . 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 10844	. . . . 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 10824	. . . . . 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 8185	. . . . . 6 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 e. RR )
19		0lt1 8311	. . . . . . 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 8310	. . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 < A )
22		lemul1 8778	. . . . 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 8170	. . . . . 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 8203	. . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 x. A ) = A )
28	27	eqcomd 2236	. . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A = ( 1 x. A ) )
29		expm1t 10835	. . . 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 4121	. 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A <_ ( A ^ N ) )
32	2, 3, 7, 8, 31	ltletrd 8608	1 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < ( A ^ N ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 1004   = wceq 1397   e. wcel 2201   class class class wbr 4089  (class class class)co 6023   CCcc 8035   RRcr 8036   0cc0 8037   1c1 8038   x. cmul 8042   < clt 8219   <_ cle 8220   - cmin 8355   NNcn 9148   NN0cn0 9407   ^cexp 10806

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 2203  ax-14 2204  ax-ext 2212  ax-coll 4205  ax-sep 4208  ax-nul 4216  ax-pow 4266  ax-pr 4301  ax-un 4532  ax-setind 4637  ax-iinf 4688  ax-cnex 8128  ax-resscn 8129  ax-1cn 8130  ax-1re 8131  ax-icn 8132  ax-addcl 8133  ax-addrcl 8134  ax-mulcl 8135  ax-mulrcl 8136  ax-addcom 8137  ax-mulcom 8138  ax-addass 8139  ax-mulass 8140  ax-distr 8141  ax-i2m1 8142  ax-0lt1 8143  ax-1rid 8144  ax-0id 8145  ax-rnegex 8146  ax-precex 8147  ax-cnre 8148  ax-pre-ltirr 8149  ax-pre-ltwlin 8150  ax-pre-lttrn 8151  ax-pre-apti 8152  ax-pre-ltadd 8153  ax-pre-mulgt0 8154  ax-pre-mulext 8155

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 1810  df-eu 2081  df-mo 2082  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-ral 2514  df-rex 2515  df-reu 2516  df-rmo 2517  df-rab 2518  df-v 2803  df-sbc 3031  df-csb 3127  df-dif 3201  df-un 3203  df-in 3205  df-ss 3212  df-nul 3494  df-if 3605  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-uni 3895  df-int 3930  df-iun 3973  df-br 4090  df-opab 4152  df-mpt 4153  df-tr 4189  df-id 4392  df-po 4395  df-iso 4396  df-iord 4465  df-on 4467  df-ilim 4468  df-suc 4470  df-iom 4691  df-xp 4733  df-rel 4734  df-cnv 4735  df-co 4736  df-dm 4737  df-rn 4738  df-res 4739  df-ima 4740  df-iota 5288  df-fun 5330  df-fn 5331  df-f 5332  df-f1 5333  df-fo 5334  df-f1o 5335  df-fv 5336  df-riota 5976  df-ov 6026  df-oprab 6027  df-mpo 6028  df-1st 6308  df-2nd 6309  df-recs 6476  df-frec 6562  df-pnf 8221  df-mnf 8222  df-xr 8223  df-ltxr 8224  df-le 8225  df-sub 8357  df-neg 8358  df-reap 8760  df-ap 8767  df-div 8858  df-inn 9149  df-n0 9408  df-z 9485  df-uz 9761  df-seqfrec 10716  df-exp 10807

This theorem is referenced by:  ltexp2a  10859  dvdsprmpweqle  12933  logbgcd1irraplemexp  15721  perfectlem1  15752  perfectlem2  15753