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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  expmordi Structured version   Visualization version   Unicode version
Theorem expmordi 37512
Description: Mantissa ordering relationship for exponentiation. (Contributed by Stefan O'Rear, 16-Oct-2014.)
Assertion
Ref Expression
expmordi |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) /\ N e. NN ) -> ( A ^ N ) < ( B ^ N ) )
Proof of Theorem expmordi
Dummy variables a b are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 6658 . . . . . 6 |- ( a = 1 -> ( A ^ a ) = ( A ^ 1 ) )
2 oveq2 6658 . . . . . 6 |- ( a = 1 -> ( B ^ a ) = ( B ^ 1 ) )
31, 2breq12d 4666 . . . . 5 |- ( a = 1 -> ( ( A ^ a ) < ( B ^ a ) <-> ( A ^ 1 ) < ( B ^ 1 ) ) )
43imbi2d 330 . . . 4 |- ( a = 1 -> ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ a ) < ( B ^ a ) ) <-> ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ 1 ) < ( B ^ 1 ) ) ) )
5 oveq2 6658 . . . . . 6 |- ( a = b -> ( A ^ a ) = ( A ^ b ) )
6 oveq2 6658 . . . . . 6 |- ( a = b -> ( B ^ a ) = ( B ^ b ) )
75, 6breq12d 4666 . . . . 5 |- ( a = b -> ( ( A ^ a ) < ( B ^ a ) <-> ( A ^ b ) < ( B ^ b ) ) )
87imbi2d 330 . . . 4 |- ( a = b -> ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ a ) < ( B ^ a ) ) <-> ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ b ) < ( B ^ b ) ) ) )
9 oveq2 6658 . . . . . 6 |- ( a = ( b + 1 ) -> ( A ^ a ) = ( A ^ ( b + 1 ) ) )
10 oveq2 6658 . . . . . 6 |- ( a = ( b + 1 ) -> ( B ^ a ) = ( B ^ ( b + 1 ) ) )
119, 10breq12d 4666 . . . . 5 |- ( a = ( b + 1 ) -> ( ( A ^ a ) < ( B ^ a ) <-> ( A ^ ( b + 1 ) ) < ( B ^ ( b + 1 ) ) ) )
1211imbi2d 330 . . . 4 |- ( a = ( b + 1 ) -> ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ a ) < ( B ^ a ) ) <-> ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ ( b + 1 ) ) < ( B ^ ( b + 1 ) ) ) ) )
13 oveq2 6658 . . . . . 6 |- ( a = N -> ( A ^ a ) = ( A ^ N ) )
14 oveq2 6658 . . . . . 6 |- ( a = N -> ( B ^ a ) = ( B ^ N ) )
1513, 14breq12d 4666 . . . . 5 |- ( a = N -> ( ( A ^ a ) < ( B ^ a ) <-> ( A ^ N ) < ( B ^ N ) ) )
1615imbi2d 330 . . . 4 |- ( a = N -> ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ a ) < ( B ^ a ) ) <-> ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ N ) < ( B ^ N ) ) ) )
17 recn 10026 . . . . . . 7 |- ( A e. RR -> A e. CC )
18 recn 10026 . . . . . . 7 |- ( B e. RR -> B e. CC )
19 exp1 12866 . . . . . . . 8 |- ( A e. CC -> ( A ^ 1 ) = A )
20 exp1 12866 . . . . . . . 8 |- ( B e. CC -> ( B ^ 1 ) = B )
2119, 20breqan12d 4669 . . . . . . 7 |- ( ( A e. CC /\ B e. CC ) -> ( ( A ^ 1 ) < ( B ^ 1 ) <-> A < B ) )
2217, 18, 21syl2an 494 . . . . . 6 |- ( ( A e. RR /\ B e. RR ) -> ( ( A ^ 1 ) < ( B ^ 1 ) <-> A < B ) )
2322biimpar 502 . . . . 5 |- ( ( ( A e. RR /\ B e. RR ) /\ A < B ) -> ( A ^ 1 ) < ( B ^ 1 ) )
2423adantrl 752 . . . 4 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ 1 ) < ( B ^ 1 ) )
25 simp2ll 1128 . . . . . . . . . 10 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> A e. RR )
26 nnnn0 11299 . . . . . . . . . . 11 |- ( b e. NN -> b e. NN0 )
27263ad2ant1 1082 . . . . . . . . . 10 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> b e. NN0 )
2825, 27reexpcld 13025 . . . . . . . . 9 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( A ^ b ) e. RR )
29 simp2lr 1129 . . . . . . . . . 10 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> B e. RR )
3029, 27reexpcld 13025 . . . . . . . . 9 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( B ^ b ) e. RR )
3128, 30jca 554 . . . . . . . 8 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( ( A ^ b ) e. RR /\ ( B ^ b ) e. RR ) )
32 simp2rl 1130 . . . . . . . . . 10 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> 0 <_ A )
3325, 27, 32expge0d 13026 . . . . . . . . 9 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> 0 <_ ( A ^ b ) )
34 simp3 1063 . . . . . . . . 9 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( A ^ b ) < ( B ^ b ) )
3533, 34jca 554 . . . . . . . 8 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( 0 <_ ( A ^ b ) /\ ( A ^ b ) < ( B ^ b ) ) )
36 simp2l 1087 . . . . . . . 8 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( A e. RR /\ B e. RR ) )
37 simp2r 1088 . . . . . . . 8 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( 0 <_ A /\ A < B ) )
38 ltmul12a 10879 . . . . . . . 8 |- ( ( ( ( ( A ^ b ) e. RR /\ ( B ^ b ) e. RR ) /\ ( 0 <_ ( A ^ b ) /\ ( A ^ b ) < ( B ^ b ) ) ) /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) ) -> ( ( A ^ b ) x. A ) < ( ( B ^ b ) x. B ) )
3931, 35, 36, 37, 38syl22anc 1327 . . . . . . 7 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( ( A ^ b ) x. A ) < ( ( B ^ b ) x. B ) )
4025recnd 10068 . . . . . . . 8 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> A e. CC )
4140, 27expp1d 13009 . . . . . . 7 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( A ^ ( b + 1 ) ) = ( ( A ^ b ) x. A ) )
4229recnd 10068 . . . . . . . 8 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> B e. CC )
4342, 27expp1d 13009 . . . . . . 7 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( B ^ ( b + 1 ) ) = ( ( B ^ b ) x. B ) )
4439, 41, 433brtr4d 4685 . . . . . 6 |- ( ( b e. NN /\ ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ ( A ^ b ) < ( B ^ b ) ) -> ( A ^ ( b + 1 ) ) < ( B ^ ( b + 1 ) ) )
45443exp 1264 . . . . 5 |- ( b e. NN -> ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( ( A ^ b ) < ( B ^ b ) -> ( A ^ ( b + 1 ) ) < ( B ^ ( b + 1 ) ) ) ) )
4645a2d 29 . . . 4 |- ( b e. NN -> ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ b ) < ( B ^ b ) ) -> ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ ( b + 1 ) ) < ( B ^ ( b + 1 ) ) ) ) )
474, 8, 12, 16, 24, 46nnind 11038 . . 3 |- ( N e. NN -> ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) -> ( A ^ N ) < ( B ^ N ) ) )
4847impcom 446 . 2 |- ( ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) ) /\ N e. NN ) -> ( A ^ N ) < ( B ^ N ) )
49483impa 1259 1 |- ( ( ( A e. RR /\ B e. RR ) /\ ( 0 <_ A /\ A < B ) /\ N e. NN ) -> ( A ^ N ) < ( B ^ N ) )
Colors of variables: wff setvar class
Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   class class class wbr 4653  (class class class)co 6650   CCcc 9934   RRcr 9935   0cc0 9936   1c1 9937   + caddc 9939   x. cmul 9941   < clt 10074   <_ cle 10075   NNcn 11020   NN0cn0 11292   ^cexp 12860
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-1cn 9994  ax-icn 9995  ax-addcl 9996  ax-addrcl 9997  ax-mulcl 9998  ax-mulrcl 9999  ax-mulcom 10000  ax-addass 10001  ax-mulass 10002  ax-distr 10003  ax-i2m1 10004  ax-1ne0 10005  ax-1rid 10006  ax-rnegex 10007  ax-rrecex 10008  ax-cnre 10009  ax-pre-lttri 10010  ax-pre-lttrn 10011  ax-pre-ltadd 10012  ax-pre-mulgt0 10013
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-nel 2898  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-2nd 7169  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-sub 10268  df-neg 10269  df-nn 11021  df-n0 11293  df-z 11378  df-uz 11688  df-seq 12802  df-exp 12861
This theorem is referenced by:  rpexpmord  37513
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