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Theorem qabvexp 22855
Description: Induct the product rule abvmul 16894 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 6094 . . . . . . 7  |-  ( k  =  0  ->  ( M ^ k )  =  ( M ^ 0 ) )
21fveq2d 5690 . . . . . 6  |-  ( k  =  0  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ 0 ) ) )
3 oveq2 6094 . . . . . 6  |-  ( k  =  0  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
0 ) )
42, 3eqeq12d 2452 . . . . 5  |-  ( k  =  0  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ 0 ) )  =  ( ( F `  M
) ^ 0 ) ) )
54imbi2d 316 . . . 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 6094 . . . . . . 7  |-  ( k  =  n  ->  ( M ^ k )  =  ( M ^ n
) )
76fveq2d 5690 . . . . . 6  |-  ( k  =  n  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ n ) ) )
8 oveq2 6094 . . . . . 6  |-  ( k  =  n  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
n ) )
97, 8eqeq12d 2452 . . . . 5  |-  ( k  =  n  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ n
) )  =  ( ( F `  M
) ^ n ) ) )
109imbi2d 316 . . . 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 6094 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  ( M ^ k )  =  ( M ^ (
n  +  1 ) ) )
1211fveq2d 5690 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ ( n  +  1 ) ) ) )
13 oveq2 6094 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
( n  +  1 ) ) )
1412, 13eqeq12d 2452 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ (
n  +  1 ) ) )  =  ( ( F `  M
) ^ ( n  +  1 ) ) ) )
1514imbi2d 316 . . . 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 6094 . . . . . . 7  |-  ( k  =  N  ->  ( M ^ k )  =  ( M ^ N
) )
1716fveq2d 5690 . . . . . 6  |-  ( k  =  N  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ N ) ) )
18 oveq2 6094 . . . . . 6  |-  ( k  =  N  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^ N ) )
1917, 18eqeq12d 2452 . . . . 5  |-  ( k  =  N  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ N
) )  =  ( ( F `  M
) ^ N ) ) )
2019imbi2d 316 . . . 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 9343 . . . . . . 7  |-  1  =/=  0
22 qabsabv.a . . . . . . . 8  |-  A  =  (AbsVal `  Q )
23 qrng.q . . . . . . . . 9  |-  Q  =  (flds  QQ )
2423qrng1 22851 . . . . . . . 8  |-  1  =  ( 1r `  Q )
2523qrng0 22850 . . . . . . . 8  |-  0  =  ( 0g `  Q )
2622, 24, 25abv1z 16897 . . . . . . 7  |-  ( ( F  e.  A  /\  1  =/=  0 )  -> 
( F `  1
)  =  1 )
2721, 26mpan2 671 . . . . . 6  |-  ( F  e.  A  ->  ( F `  1 )  =  1 )
2827adantr 465 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  1
)  =  1 )
29 qcn 10959 . . . . . . . 8  |-  ( M  e.  QQ  ->  M  e.  CC )
3029adantl 466 . . . . . . 7  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  M  e.  CC )
3130exp0d 11994 . . . . . 6  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( M ^ 0 )  =  1 )
3231fveq2d 5690 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( F ` 
1 ) )
3323qrngbas 22848 . . . . . . . 8  |-  QQ  =  ( Base `  Q )
3422, 33abvcl 16889 . . . . . . 7  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  M
)  e.  RR )
3534recnd 9404 . . . . . 6  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  M
)  e.  CC )
3635exp0d 11994 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( ( F `  M ) ^ 0 )  =  1 )
3728, 32, 363eqtr4d 2480 . . . 4  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( ( F `
 M ) ^
0 ) )
38 oveq1 6093 . . . . . . 7  |-  ( ( F `  ( M ^ n ) )  =  ( ( F `
 M ) ^
n )  ->  (
( F `  ( M ^ n ) )  x.  ( F `  M ) )  =  ( ( ( F `
 M ) ^
n )  x.  ( F `  M )
) )
39 expp1 11864 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  n  e.  NN0 )  -> 
( M ^ (
n  +  1 ) )  =  ( ( M ^ n )  x.  M ) )
4030, 39sylan 471 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( M ^
( n  +  1 ) )  =  ( ( M ^ n
)  x.  M ) )
4140fveq2d 5690 . . . . . . . . 9  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( M ^ ( n  +  1 ) ) )  =  ( F `
 ( ( M ^ n )  x.  M ) ) )
42 simpll 753 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  F  e.  A
)
43 qexpcl 11873 . . . . . . . . . . 11  |-  ( ( M  e.  QQ  /\  n  e.  NN0 )  -> 
( M ^ n
)  e.  QQ )
4443adantll 713 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( M ^
n )  e.  QQ )
45 simplr 754 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  M  e.  QQ )
46 qex 10957 . . . . . . . . . . . 12  |-  QQ  e.  _V
47 cnfldmul 17804 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
4823, 47ressmulr 14283 . . . . . . . . . . . 12  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
4946, 48ax-mp 5 . . . . . . . . . . 11  |-  x.  =  ( .r `  Q )
5022, 33, 49abvmul 16894 . . . . . . . . . 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 1218 . . . . . . . . 9  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( ( M ^
n )  x.  M
) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `  M ) ) )
5241, 51eqtrd 2470 . . . . . . . 8  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( M ^ ( n  +  1 ) ) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `  M ) ) )
53 expp1 11864 . . . . . . . . 9  |-  ( ( ( F `  M
)  e.  CC  /\  n  e.  NN0 )  -> 
( ( F `  M ) ^ (
n  +  1 ) )  =  ( ( ( F `  M
) ^ n )  x.  ( F `  M ) ) )
5435, 53sylan 471 . . . . . . . 8  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( ( F `
 M ) ^
( n  +  1 ) )  =  ( ( ( F `  M ) ^ n
)  x.  ( F `
 M ) ) )
5552, 54eqeq12d 2452 . . . . . . 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 221 . . . . . 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 435 . . . . 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 26 . . . 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 10730 . . 3  |-  ( N  e.  NN0  ->  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ N ) )  =  ( ( F `
 M ) ^ N ) ) )
6059com12 31 . 2  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( N  e.  NN0  ->  ( F `  ( M ^ N ) )  =  ( ( F `
 M ) ^ N ) ) )
61603impia 1184 1  |-  ( ( F  e.  A  /\  M  e.  QQ  /\  N  e.  NN0 )  ->  ( F `  ( M ^ N ) )  =  ( ( F `  M ) ^ N
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   _Vcvv 2967   ` cfv 5413  (class class class)co 6086   CCcc 9272   0cc0 9274   1c1 9275    + caddc 9277    x. cmul 9279   NN0cn0 10571   QQcq 10945   ^cexp 11857   ↾s cress 14167   .rcmulr 14231  AbsValcabv 16881  ℂfldccnfld 17798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-rep 4398  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-addf 9353  ax-mulf 9354
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-pss 3339  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-tp 3877  df-op 3879  df-uni 4087  df-int 4124  df-iun 4168  df-br 4288  df-opab 4346  df-mpt 4347  df-tr 4381  df-eprel 4627  df-id 4631  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718  df-lim 4719  df-suc 4720  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-om 6472  df-1st 6572  df-2nd 6573  df-tpos 6740  df-recs 6824  df-rdg 6858  df-1o 6912  df-oadd 6916  df-er 7093  df-map 7208  df-en 7303  df-dom 7304  df-sdom 7305  df-fin 7306  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590  df-div 9986  df-nn 10315  df-2 10372  df-3 10373  df-4 10374  df-5 10375  df-6 10376  df-7 10377  df-8 10378  df-9 10379  df-10 10380  df-n0 10572  df-z 10639  df-dec 10748  df-uz 10854  df-q 10946  df-ico 11298  df-fz 11430  df-seq 11799  df-exp 11858  df-struct 14168  df-ndx 14169  df-slot 14170  df-base 14171  df-sets 14172  df-ress 14173  df-plusg 14243  df-mulr 14244  df-starv 14245  df-tset 14249  df-ple 14250  df-ds 14252  df-unif 14253  df-0g 14372  df-mnd 15407  df-grp 15536  df-minusg 15537  df-subg 15669  df-cmn 16270  df-mgp 16582  df-ur 16594  df-rng 16637  df-cring 16638  df-oppr 16705  df-dvdsr 16723  df-unit 16724  df-invr 16754  df-dvr 16765  df-drng 16814  df-subrg 16843  df-abv 16882  df-cnfld 17799
This theorem is referenced by:  ostth2lem2  22863  ostth2lem3  22864  ostth3  22867
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