Theorem expgt1 10759

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 8106	. . 3 |- 1 e. RR
2	1	a1i 9	. 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 e. RR )
3		simp1 1000	. 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A e. RR )
4		simp2 1001	. . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN )
5	4	nnnn0d 9383	. . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> N e. NN0 )
6		reexpcl 10738	. . 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 1002	. 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < A )
9		nnm1nn0 9371	. . . . . 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 8195	. . . . . . 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 10758	. . . . 5 |- ( ( A e. RR /\ ( N - 1 ) e. NN0 /\ 1 <_ A ) -> 1 <_ ( A ^ ( N - 1 ) ) )
15	3, 10, 13, 14	syl3anc 1250	. . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 <_ ( A ^ ( N - 1 ) ) )
16		reexpcl 10738	. . . . . 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 8108	. . . . . 6 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 e. RR )
19		0lt1 8234	. . . . . . 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 8233	. . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 0 < A )
22		lemul1 8701	. . . . 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 1254	. . . 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 8093	. . . . . 6 |- ( A e. RR -> A e. CC )
26	25	3ad2ant1 1021	. . . . 5 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A e. CC )
27	26	mulid2d 8126	. . . 4 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> ( 1 x. A ) = A )
28	27	eqcomd 2213	. . 3 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A = ( 1 x. A ) )
29		expm1t 10749	. . . 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 4091	. 2 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> A <_ ( A ^ N ) )
32	2, 3, 7, 8, 31	ltletrd 8531	1 |- ( ( A e. RR /\ N e. NN /\ 1 < A ) -> 1 < ( A ^ N ) )

Syntax hints:   -> wi 4   <-> wb 105   /\ w3a 981   = wceq 1373   e. wcel 2178   class class class wbr 4059  (class class class)co 5967   CCcc 7958   RRcr 7959   0cc0 7960   1c1 7961   x. cmul 7965   < clt 8142   <_ cle 8143   - cmin 8278   NNcn 9071   NN0cn0 9330   ^cexp 10720

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654  ax-cnex 8051  ax-resscn 8052  ax-1cn 8053  ax-1re 8054  ax-icn 8055  ax-addcl 8056  ax-addrcl 8057  ax-mulcl 8058  ax-mulrcl 8059  ax-addcom 8060  ax-mulcom 8061  ax-addass 8062  ax-mulass 8063  ax-distr 8064  ax-i2m1 8065  ax-0lt1 8066  ax-1rid 8067  ax-0id 8068  ax-rnegex 8069  ax-precex 8070  ax-cnre 8071  ax-pre-ltirr 8072  ax-pre-ltwlin 8073  ax-pre-lttrn 8074  ax-pre-apti 8075  ax-pre-ltadd 8076  ax-pre-mulgt0 8077  ax-pre-mulext 8078

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-reu 2493  df-rmo 2494  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-po 4361  df-iso 4362  df-iord 4431  df-on 4433  df-ilim 4434  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-riota 5922  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-recs 6414  df-frec 6500  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147  df-le 8148  df-sub 8280  df-neg 8281  df-reap 8683  df-ap 8690  df-div 8781  df-inn 9072  df-n0 9331  df-z 9408  df-uz 9684  df-seqfrec 10630  df-exp 10721

This theorem is referenced by:  ltexp2a  10773  dvdsprmpweqle  12775  logbgcd1irraplemexp  15555  perfectlem1  15586  perfectlem2  15587