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Theorem qabvexp 20775
Description: Induct the product rule abvmul 15594 to find the absolute value of a power. (Contributed by Mario Carneiro, 10-Sep-2014.)
Hypotheses
Ref Expression
qrng.q  |-  Q  =  (flds  QQ )
qabsabv.a  |-  A  =  (AbsVal `  Q )
Assertion
Ref Expression
qabvexp  |-  ( ( F  e.  A  /\  M  e.  QQ  /\  N  e.  NN0 )  ->  ( F `  ( M ^ N ) )  =  ( ( F `  M ) ^ N
) )

Proof of Theorem qabvexp
Dummy variables  k  n are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq2 5866 . . . . . . 7  |-  ( k  =  0  ->  ( M ^ k )  =  ( M ^ 0 ) )
21fveq2d 5529 . . . . . 6  |-  ( k  =  0  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ 0 ) ) )
3 oveq2 5866 . . . . . 6  |-  ( k  =  0  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
0 ) )
42, 3eqeq12d 2297 . . . . 5  |-  ( k  =  0  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ 0 ) )  =  ( ( F `  M
) ^ 0 ) ) )
54imbi2d 307 . . . 4  |-  ( k  =  0  ->  (
( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k ) )  <->  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( ( F `
 M ) ^
0 ) ) ) )
6 oveq2 5866 . . . . . . 7  |-  ( k  =  n  ->  ( M ^ k )  =  ( M ^ n
) )
76fveq2d 5529 . . . . . 6  |-  ( k  =  n  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ n ) ) )
8 oveq2 5866 . . . . . 6  |-  ( k  =  n  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
n ) )
97, 8eqeq12d 2297 . . . . 5  |-  ( k  =  n  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ n
) )  =  ( ( F `  M
) ^ n ) ) )
109imbi2d 307 . . . 4  |-  ( k  =  n  ->  (
( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k ) )  <->  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ n ) )  =  ( ( F `
 M ) ^
n ) ) ) )
11 oveq2 5866 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  ( M ^ k )  =  ( M ^ (
n  +  1 ) ) )
1211fveq2d 5529 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ ( n  +  1 ) ) ) )
13 oveq2 5866 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
( n  +  1 ) ) )
1412, 13eqeq12d 2297 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ (
n  +  1 ) ) )  =  ( ( F `  M
) ^ ( n  +  1 ) ) ) )
1514imbi2d 307 . . . 4  |-  ( k  =  ( n  + 
1 )  ->  (
( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k ) )  <->  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ ( n  + 
1 ) ) )  =  ( ( F `
 M ) ^
( n  +  1 ) ) ) ) )
16 oveq2 5866 . . . . . . 7  |-  ( k  =  N  ->  ( M ^ k )  =  ( M ^ N
) )
1716fveq2d 5529 . . . . . 6  |-  ( k  =  N  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ N ) ) )
18 oveq2 5866 . . . . . 6  |-  ( k  =  N  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^ N ) )
1917, 18eqeq12d 2297 . . . . 5  |-  ( k  =  N  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ N
) )  =  ( ( F `  M
) ^ N ) ) )
2019imbi2d 307 . . . 4  |-  ( k  =  N  ->  (
( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k ) )  <->  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ N ) )  =  ( ( F `
 M ) ^ N ) ) ) )
21 ax-1ne0 8806 . . . . . . 7  |-  1  =/=  0
22 qabsabv.a . . . . . . . 8  |-  A  =  (AbsVal `  Q )
23 qrng.q . . . . . . . . 9  |-  Q  =  (flds  QQ )
2423qrng1 20771 . . . . . . . 8  |-  1  =  ( 1r `  Q )
2523qrng0 20770 . . . . . . . 8  |-  0  =  ( 0g `  Q )
2622, 24, 25abv1z 15597 . . . . . . 7  |-  ( ( F  e.  A  /\  1  =/=  0 )  -> 
( F `  1
)  =  1 )
2721, 26mpan2 652 . . . . . 6  |-  ( F  e.  A  ->  ( F `  1 )  =  1 )
2827adantr 451 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  1
)  =  1 )
29 qcn 10330 . . . . . . . 8  |-  ( M  e.  QQ  ->  M  e.  CC )
3029adantl 452 . . . . . . 7  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  M  e.  CC )
3130exp0d 11239 . . . . . 6  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( M ^ 0 )  =  1 )
3231fveq2d 5529 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( F ` 
1 ) )
3323qrngbas 20768 . . . . . . . 8  |-  QQ  =  ( Base `  Q )
3422, 33abvcl 15589 . . . . . . 7  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  M
)  e.  RR )
3534recnd 8861 . . . . . 6  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  M
)  e.  CC )
3635exp0d 11239 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( ( F `  M ) ^ 0 )  =  1 )
3728, 32, 363eqtr4d 2325 . . . 4  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( ( F `
 M ) ^
0 ) )
38 oveq1 5865 . . . . . . 7  |-  ( ( F `  ( M ^ n ) )  =  ( ( F `
 M ) ^
n )  ->  (
( F `  ( M ^ n ) )  x.  ( F `  M ) )  =  ( ( ( F `
 M ) ^
n )  x.  ( F `  M )
) )
39 expp1 11110 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  n  e.  NN0 )  -> 
( M ^ (
n  +  1 ) )  =  ( ( M ^ n )  x.  M ) )
4030, 39sylan 457 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( M ^
( n  +  1 ) )  =  ( ( M ^ n
)  x.  M ) )
4140fveq2d 5529 . . . . . . . . 9  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( M ^ ( n  +  1 ) ) )  =  ( F `
 ( ( M ^ n )  x.  M ) ) )
42 simpll 730 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  F  e.  A
)
43 qexpcl 11119 . . . . . . . . . . 11  |-  ( ( M  e.  QQ  /\  n  e.  NN0 )  -> 
( M ^ n
)  e.  QQ )
4443adantll 694 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( M ^
n )  e.  QQ )
45 simplr 731 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  M  e.  QQ )
46 qex 10328 . . . . . . . . . . . 12  |-  QQ  e.  _V
47 cnfldmul 16385 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
4823, 47ressmulr 13261 . . . . . . . . . . . 12  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
4946, 48ax-mp 8 . . . . . . . . . . 11  |-  x.  =  ( .r `  Q )
5022, 33, 49abvmul 15594 . . . . . . . . . 10  |-  ( ( F  e.  A  /\  ( M ^ n )  e.  QQ  /\  M  e.  QQ )  ->  ( F `  ( ( M ^ n )  x.  M ) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `
 M ) ) )
5142, 44, 45, 50syl3anc 1182 . . . . . . . . 9  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( ( M ^
n )  x.  M
) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `  M ) ) )
5241, 51eqtrd 2315 . . . . . . . 8  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( M ^ ( n  +  1 ) ) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `  M ) ) )
53 expp1 11110 . . . . . . . . 9  |-  ( ( ( F `  M
)  e.  CC  /\  n  e.  NN0 )  -> 
( ( F `  M ) ^ (
n  +  1 ) )  =  ( ( ( F `  M
) ^ n )  x.  ( F `  M ) ) )
5435, 53sylan 457 . . . . . . . 8  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( ( F `
 M ) ^
( n  +  1 ) )  =  ( ( ( F `  M ) ^ n
)  x.  ( F `
 M ) ) )
5552, 54eqeq12d 2297 . . . . . . 7  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( ( F `
 ( M ^
( n  +  1 ) ) )  =  ( ( F `  M ) ^ (
n  +  1 ) )  <->  ( ( F `
 ( M ^
n ) )  x.  ( F `  M
) )  =  ( ( ( F `  M ) ^ n
)  x.  ( F `
 M ) ) ) )
5638, 55syl5ibr 212 . . . . . 6  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( ( F `
 ( M ^
n ) )  =  ( ( F `  M ) ^ n
)  ->  ( F `  ( M ^ (
n  +  1 ) ) )  =  ( ( F `  M
) ^ ( n  +  1 ) ) ) )
5756expcom 424 . . . . 5  |-  ( n  e.  NN0  ->  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( ( F `  ( M ^ n ) )  =  ( ( F `  M ) ^ n )  -> 
( F `  ( M ^ ( n  + 
1 ) ) )  =  ( ( F `
 M ) ^
( n  +  1 ) ) ) ) )
5857a2d 23 . . . 4  |-  ( n  e.  NN0  ->  ( ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ n ) )  =  ( ( F `  M ) ^ n ) )  ->  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ ( n  + 
1 ) ) )  =  ( ( F `
 M ) ^
( n  +  1 ) ) ) ) )
595, 10, 15, 20, 37, 58nn0ind 10108 . . 3  |-  ( N  e.  NN0  ->  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ N ) )  =  ( ( F `
 M ) ^ N ) ) )
6059com12 27 . 2  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( N  e.  NN0  ->  ( F `  ( M ^ N ) )  =  ( ( F `
 M ) ^ N ) ) )
61603impia 1148 1  |-  ( ( F  e.  A  /\  M  e.  QQ  /\  N  e.  NN0 )  ->  ( F `  ( M ^ N ) )  =  ( ( F `  M ) ^ N
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   _Vcvv 2788   ` cfv 5255  (class class class)co 5858   CCcc 8735   0cc0 8737   1c1 8738    + caddc 8740    x. cmul 8742   NN0cn0 9965   QQcq 10316   ^cexp 11104   ↾s cress 13149   .rcmulr 13209  AbsValcabv 15581  ℂfldccnfld 16377
This theorem is referenced by:  ostth2lem2  20783  ostth2lem3  20784  ostth3  20787
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814  ax-addf 8816  ax-mulf 8817
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-tpos 6234  df-riota 6304  df-recs 6388  df-rdg 6423  df-1o 6479  df-oadd 6483  df-er 6660  df-map 6774  df-en 6864  df-dom 6865  df-sdom 6866  df-fin 6867  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-div 9424  df-nn 9747  df-2 9804  df-3 9805  df-4 9806  df-5 9807  df-6 9808  df-7 9809  df-8 9810  df-9 9811  df-10 9812  df-n0 9966  df-z 10025  df-dec 10125  df-uz 10231  df-q 10317  df-ico 10662  df-fz 10783  df-seq 11047  df-exp 11105  df-struct 13150  df-ndx 13151  df-slot 13152  df-base 13153  df-sets 13154  df-ress 13155  df-plusg 13221  df-mulr 13222  df-starv 13223  df-tset 13227  df-ple 13228  df-ds 13230  df-0g 13404  df-mnd 14367  df-grp 14489  df-minusg 14490  df-subg 14618  df-cmn 15091  df-mgp 15326  df-rng 15340  df-cring 15341  df-ur 15342  df-oppr 15405  df-dvdsr 15423  df-unit 15424  df-invr 15454  df-dvr 15465  df-drng 15514  df-subrg 15543  df-abv 15582  df-cnfld 16378
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