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Theorem expmordi 35766
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 6314 . . . . . 6  |-  ( a  =  1  ->  ( A ^ a )  =  ( A ^ 1 ) )
2 oveq2 6314 . . . . . 6  |-  ( a  =  1  ->  ( B ^ a )  =  ( B ^ 1 ) )
31, 2breq12d 4436 . . . . 5  |-  ( a  =  1  ->  (
( A ^ a
)  <  ( B ^ a )  <->  ( A ^ 1 )  < 
( B ^ 1 ) ) )
43imbi2d 317 . . . 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 6314 . . . . . 6  |-  ( a  =  b  ->  ( A ^ a )  =  ( A ^ b
) )
6 oveq2 6314 . . . . . 6  |-  ( a  =  b  ->  ( B ^ a )  =  ( B ^ b
) )
75, 6breq12d 4436 . . . . 5  |-  ( a  =  b  ->  (
( A ^ a
)  <  ( B ^ a )  <->  ( A ^ b )  < 
( B ^ b
) ) )
87imbi2d 317 . . . 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 6314 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( A ^ a )  =  ( A ^ (
b  +  1 ) ) )
10 oveq2 6314 . . . . . 6  |-  ( a  =  ( b  +  1 )  ->  ( B ^ a )  =  ( B ^ (
b  +  1 ) ) )
119, 10breq12d 4436 . . . . 5  |-  ( a  =  ( b  +  1 )  ->  (
( A ^ a
)  <  ( B ^ a )  <->  ( A ^ ( b  +  1 ) )  < 
( B ^ (
b  +  1 ) ) ) )
1211imbi2d 317 . . . 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 6314 . . . . . 6  |-  ( a  =  N  ->  ( A ^ a )  =  ( A ^ N
) )
14 oveq2 6314 . . . . . 6  |-  ( a  =  N  ->  ( B ^ a )  =  ( B ^ N
) )
1513, 14breq12d 4436 . . . . 5  |-  ( a  =  N  ->  (
( A ^ a
)  <  ( B ^ a )  <->  ( A ^ N )  <  ( B ^ N ) ) )
1615imbi2d 317 . . . 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 9637 . . . . . . 7  |-  ( A  e.  RR  ->  A  e.  CC )
18 recn 9637 . . . . . . 7  |-  ( B  e.  RR  ->  B  e.  CC )
19 exp1 12285 . . . . . . . 8  |-  ( A  e.  CC  ->  ( A ^ 1 )  =  A )
20 exp1 12285 . . . . . . . 8  |-  ( B  e.  CC  ->  ( B ^ 1 )  =  B )
2119, 20breqan12d 4439 . . . . . . 7  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( ( A ^
1 )  <  ( B ^ 1 )  <->  A  <  B ) )
2217, 18, 21syl2an 479 . . . . . 6  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( ( A ^
1 )  <  ( B ^ 1 )  <->  A  <  B ) )
2322biimpar 487 . . . . 5  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  A  <  B
)  ->  ( A ^ 1 )  < 
( B ^ 1 ) )
2423adantrl 720 . . . 4  |-  ( ( ( A  e.  RR  /\  B  e.  RR )  /\  ( 0  <_  A  /\  A  <  B
) )  ->  ( A ^ 1 )  < 
( B ^ 1 ) )
25 simp2ll 1072 . . . . . . . . . 10  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  A  e.  RR )
26 nnnn0 10884 . . . . . . . . . . 11  |-  ( b  e.  NN  ->  b  e.  NN0 )
27263ad2ant1 1026 . . . . . . . . . 10  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  b  e.  NN0 )
2825, 27reexpcld 12440 . . . . . . . . 9  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( A ^
b )  e.  RR )
29 simp2lr 1073 . . . . . . . . . 10  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  B  e.  RR )
3029, 27reexpcld 12440 . . . . . . . . 9  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( B ^
b )  e.  RR )
3128, 30jca 534 . . . . . . . 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 1074 . . . . . . . . . 10  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  0  <_  A
)
3325, 27, 32expge0d 12441 . . . . . . . . 9  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  0  <_  ( A ^ b ) )
34 simp3 1007 . . . . . . . . 9  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( A ^
b )  <  ( B ^ b ) )
3533, 34jca 534 . . . . . . . 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 1031 . . . . . . . 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 1032 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  ( 0  <_  A  /\  A  <  B
) )
38 ltmul12a 10469 . . . . . . . 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 1265 . . . . . . 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 9677 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  A  e.  CC )
4140, 27expp1d 12424 . . . . . . 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 9677 . . . . . . . 8  |-  ( ( b  e.  NN  /\  ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  ( A ^
b )  <  ( B ^ b ) )  ->  B  e.  CC )
4342, 27expp1d 12424 . . . . . . 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 4454 . . . . . 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 1204 . . . . 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 10635 . . 3  |-  ( N  e.  NN  ->  (
( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  ->  ( A ^ N )  <  ( B ^ N ) ) )
4847impcom 431 . 2  |-  ( ( ( ( A  e.  RR  /\  B  e.  RR )  /\  (
0  <_  A  /\  A  <  B ) )  /\  N  e.  NN )  ->  ( A ^ N )  <  ( B ^ N ) )
49483impa 1200 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 187    /\ wa 370    /\ w3a 982    = wceq 1437    e. wcel 1872   class class class wbr 4423  (class class class)co 6306   CCcc 9545   RRcr 9546   0cc0 9547   1c1 9548    + caddc 9550    x. cmul 9552    < clt 9683    <_ cle 9684   NNcn 10617   NN0cn0 10877   ^cexp 12279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pow 4602  ax-pr 4660  ax-un 6598  ax-cnex 9603  ax-resscn 9604  ax-1cn 9605  ax-icn 9606  ax-addcl 9607  ax-addrcl 9608  ax-mulcl 9609  ax-mulrcl 9610  ax-mulcom 9611  ax-addass 9612  ax-mulass 9613  ax-distr 9614  ax-i2m1 9615  ax-1ne0 9616  ax-1rid 9617  ax-rnegex 9618  ax-rrecex 9619  ax-cnre 9620  ax-pre-lttri 9621  ax-pre-lttrn 9622  ax-pre-ltadd 9623  ax-pre-mulgt0 9624
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-nel 2617  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-csb 3396  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-pss 3452  df-nul 3762  df-if 3912  df-pw 3983  df-sn 3999  df-pr 4001  df-tp 4003  df-op 4005  df-uni 4220  df-iun 4301  df-br 4424  df-opab 4483  df-mpt 4484  df-tr 4519  df-eprel 4764  df-id 4768  df-po 4774  df-so 4775  df-fr 4812  df-we 4814  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6268  df-ov 6309  df-oprab 6310  df-mpt2 6311  df-om 6708  df-2nd 6809  df-wrecs 7040  df-recs 7102  df-rdg 7140  df-er 7375  df-en 7582  df-dom 7583  df-sdom 7584  df-pnf 9685  df-mnf 9686  df-xr 9687  df-ltxr 9688  df-le 9689  df-sub 9870  df-neg 9871  df-nn 10618  df-n0 10878  df-z 10946  df-uz 11168  df-seq 12221  df-exp 12280
This theorem is referenced by:  rpexpmord  35767
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