Theorem ltexp2a 10736

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 1003	. . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. RR )
2		0red 8073	. . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 e. RR )
3		1red 8087	. . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 e. RR )
4		0lt1 8199	. . . . . . . . 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 529	. . . . . . . 8 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < A )
7	2, 3, 1, 5, 6	lttrd 8198	. . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 0 < A )
8	1, 7	elrpd 9815	. . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. RR+ )
9		simpl2 1004	. . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> M e. ZZ )
10		rpexpcl 10703	. . . . . 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 9818	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. RR )
13	12	recnd 8101	. . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ M ) e. CC )
14	13	mulid2d 8091	. 2 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( 1 x. ( A ^ M ) ) = ( A ^ M ) )
15		simprr 531	. . . . . 6 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> M < N )
16		simpl3 1005	. . . . . . 7 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> N e. ZZ )
17		znnsub 9424	. . . . . . 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 10722	. . . . 5 |- ( ( A e. RR /\ ( N - M ) e. NN /\ 1 < A ) -> 1 < ( A ^ ( N - M ) ) )
21	1, 19, 6, 20	syl3anc 1250	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < ( A ^ ( N - M ) ) )
22	1	recnd 8101	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A e. CC )
23	1, 7	gt0ap0d 8702	. . . . 5 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> A # 0 )
24		expsubap 10732	. . . . 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 1251	. . . 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 4070	. . 3 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> 1 < ( ( A ^ N ) / ( A ^ M ) ) )
27		rpexpcl 10703	. . . . . 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 9818	. . . 4 |- ( ( ( A e. RR /\ M e. ZZ /\ N e. ZZ ) /\ ( 1 < A /\ M < N ) ) -> ( A ^ N ) e. RR )
30	3, 29, 11	ltmuldivd 9866	. . 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 4068	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 981   = wceq 1373   e. wcel 2176   class class class wbr 4044  (class class class)co 5944   CCcc 7923   RRcr 7924   0cc0 7925   1c1 7926   x. cmul 7930   < clt 8107   - cmin 8243   # cap 8654   / cdiv 8745   NNcn 9036   ZZcz 9372   RR+crp 9775   ^cexp 10683

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 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4159  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-setind 4585  ax-iinf 4636  ax-cnex 8016  ax-resscn 8017  ax-1cn 8018  ax-1re 8019  ax-icn 8020  ax-addcl 8021  ax-addrcl 8022  ax-mulcl 8023  ax-mulrcl 8024  ax-addcom 8025  ax-mulcom 8026  ax-addass 8027  ax-mulass 8028  ax-distr 8029  ax-i2m1 8030  ax-0lt1 8031  ax-1rid 8032  ax-0id 8033  ax-rnegex 8034  ax-precex 8035  ax-cnre 8036  ax-pre-ltirr 8037  ax-pre-ltwlin 8038  ax-pre-lttrn 8039  ax-pre-apti 8040  ax-pre-ltadd 8041  ax-pre-mulgt0 8042  ax-pre-mulext 8043

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 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-br 4045  df-opab 4106  df-mpt 4107  df-tr 4143  df-id 4340  df-po 4343  df-iso 4344  df-iord 4413  df-on 4415  df-ilim 4416  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-riota 5899  df-ov 5947  df-oprab 5948  df-mpo 5949  df-1st 6226  df-2nd 6227  df-recs 6391  df-frec 6477  df-pnf 8109  df-mnf 8110  df-xr 8111  df-ltxr 8112  df-le 8113  df-sub 8245  df-neg 8246  df-reap 8648  df-ap 8655  df-div 8746  df-inn 9037  df-n0 9296  df-z 9373  df-uz 9649  df-rp 9776  df-seqfrec 10593  df-exp 10684

This theorem is referenced by:  expnass  10790  nn0ltexp2  10854  expcanlem  10860  perfectlem2  15472