Theorem ltexp2a 10916

Description: Ordering relationship for exponentiation. (Contributed by NM, 2-Aug-2006.) (Revised by Mario Carneiro, 4-Jun-2014.)

Assertion

Ref	Expression
ltexp2a	|- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) < ( A ^ N ) )

Proof of Theorem ltexp2a

Step	Hyp	Ref	Expression
1		simpl1 1027	. . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. RR )
2		0red 8240	. . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 e. RR )
3		1red 8254	. . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 e. RR )
4		0lt1 8365	. . . . . . . . 9 |- 0 < 1
5	4	a1i 9	. . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 < 1 )
6		simprl 531	. . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < A )
7	2, 3, 1, 5, 6	lttrd 8364	. . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 < A )
8	1, 7	elrpd 9989	. . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. RR+ )
9		simpl2 1028	. . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> M e. ZZ )
10		rpexpcl 10883	. . . . . 6 |- ( ( A e. RR+ /\ M e. ZZ ) -> ( A ^ M ) e. RR+ )
11	8, 9, 10	syl2anc 411	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. RR+ )
12	11	rpred 9992	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. RR )
13	12	recnd 8267	. . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. CC )
14	13	mullidd 8257	. 2 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( 1 x. ( A ^ M ) ) = ( A ^ M ) )
15		simprr 533	. . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> M < N )
16		simpl3 1029	. . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> N e. ZZ )
17		znnsub 9592	. . . . . . 7 |- ( ( M e. ZZ /\ N e. ZZ ) -> ( M < N <-> ( N - M ) e. NN ) )
18	9, 16, 17	syl2anc 411	. . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( M < N <-> ( N - M ) e. NN ) )
19	15, 18	mpbid 147	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( N - M ) e. NN )
20		expgt1 10902	. . . . 5 |- ( ( A e. RR /\ ( N - M ) e. NN /\ 1 < A ) -> 1 < ( A ^ ( N - M ) ) )
21	1, 19, 6, 20	syl3anc 1274	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < ( A ^ ( N - M ) ) )
22	1	recnd 8267	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. CC )
23	1, 7	gt0ap0d 8868	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A # 0 )
24		expsubap 10912	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. ZZ /\ M e. ZZ ) ) -> ( A ^ ( N - M ) ) = ( ( A ^ N ) / ( A ^ M ) ) )
25	22, 23, 16, 9, 24	syl22anc 1275	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ ( N - M ) ) = ( ( A ^ N ) / ( A ^ M ) ) )
26	21, 25	breqtrd 4119	. . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < ( ( A ^ N ) / ( A ^ M ) ) )
27		rpexpcl 10883	. . . . . 6 |- ( ( A e. RR+ /\ N e. ZZ ) -> ( A ^ N ) e. RR+ )
28	8, 16, 27	syl2anc 411	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ N ) e. RR+ )
29	28	rpred 9992	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ N ) e. RR )
30	3, 29, 11	ltmuldivd 10040	. . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( ( 1 x. ( A ^ M ) ) < ( A ^ N ) <-> 1 < ( ( A ^ N ) / ( A ^ M ) ) ) )
31	26, 30	mpbird 167	. 2 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( 1 x. ( A ^ M ) ) < ( A ^ N ) )
32	14, 31	eqbrtrrd 4117	1 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) < ( A ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093  (class class class)co 6028   CCcc 8090   RRcr 8091   0cc0 8092   1c1 8093   x. cmul 8097   < clt 8273   - cmin 8409   # cap 8820   / cdiv 8911   NNcn 9202   ZZcz 9540   RR+crp 9949   ^cexp 10863

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 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-n0 9462  df-z 9541  df-uz 9817  df-rp 9950  df-seqfrec 10773  df-exp 10864

This theorem is referenced by:  expnass  10970  nn0ltexp2  11034  expcanlem  11040  perfectlem2  15814