MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  qabvexp Structured version   Unicode version

Theorem qabvexp 23789
Description: Induct the product rule abvmul 17457 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 6289 . . . . . . 7  |-  ( k  =  0  ->  ( M ^ k )  =  ( M ^ 0 ) )
21fveq2d 5860 . . . . . 6  |-  ( k  =  0  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ 0 ) ) )
3 oveq2 6289 . . . . . 6  |-  ( k  =  0  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
0 ) )
42, 3eqeq12d 2465 . . . . 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 6289 . . . . . . 7  |-  ( k  =  n  ->  ( M ^ k )  =  ( M ^ n
) )
76fveq2d 5860 . . . . . 6  |-  ( k  =  n  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ n ) ) )
8 oveq2 6289 . . . . . 6  |-  ( k  =  n  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
n ) )
97, 8eqeq12d 2465 . . . . 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 6289 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  ( M ^ k )  =  ( M ^ (
n  +  1 ) ) )
1211fveq2d 5860 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ ( n  +  1 ) ) ) )
13 oveq2 6289 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
( n  +  1 ) ) )
1412, 13eqeq12d 2465 . . . . 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 6289 . . . . . . 7  |-  ( k  =  N  ->  ( M ^ k )  =  ( M ^ N
) )
1716fveq2d 5860 . . . . . 6  |-  ( k  =  N  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ N ) ) )
18 oveq2 6289 . . . . . 6  |-  ( k  =  N  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^ N ) )
1917, 18eqeq12d 2465 . . . . 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 9564 . . . . . . 7  |-  1  =/=  0
22 qabsabv.a . . . . . . . 8  |-  A  =  (AbsVal `  Q )
23 qrng.q . . . . . . . . 9  |-  Q  =  (flds  QQ )
2423qrng1 23785 . . . . . . . 8  |-  1  =  ( 1r `  Q )
2523qrng0 23784 . . . . . . . 8  |-  0  =  ( 0g `  Q )
2622, 24, 25abv1z 17460 . . . . . . 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 11207 . . . . . . . 8  |-  ( M  e.  QQ  ->  M  e.  CC )
3029adantl 466 . . . . . . 7  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  M  e.  CC )
3130exp0d 12286 . . . . . 6  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( M ^ 0 )  =  1 )
3231fveq2d 5860 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( F ` 
1 ) )
3323qrngbas 23782 . . . . . . . 8  |-  QQ  =  ( Base `  Q )
3422, 33abvcl 17452 . . . . . . 7  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  M
)  e.  RR )
3534recnd 9625 . . . . . 6  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  M
)  e.  CC )
3635exp0d 12286 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( ( F `  M ) ^ 0 )  =  1 )
3728, 32, 363eqtr4d 2494 . . . 4  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( ( F `
 M ) ^
0 ) )
38 oveq1 6288 . . . . . . 7  |-  ( ( F `  ( M ^ n ) )  =  ( ( F `
 M ) ^
n )  ->  (
( F `  ( M ^ n ) )  x.  ( F `  M ) )  =  ( ( ( F `
 M ) ^
n )  x.  ( F `  M )
) )
39 expp1 12155 . . . . . . . . . . 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 5860 . . . . . . . . 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 12164 . . . . . . . . . . 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 755 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  M  e.  QQ )
46 qex 11205 . . . . . . . . . . . 12  |-  QQ  e.  _V
47 cnfldmul 18405 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
4823, 47ressmulr 14732 . . . . . . . . . . . 12  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
4946, 48ax-mp 5 . . . . . . . . . . 11  |-  x.  =  ( .r `  Q )
5022, 33, 49abvmul 17457 . . . . . . . . . 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 1229 . . . . . . . . 9  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( ( M ^
n )  x.  M
) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `  M ) ) )
5241, 51eqtrd 2484 . . . . . . . 8  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( M ^ ( n  +  1 ) ) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `  M ) ) )
53 expp1 12155 . . . . . . . . 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 2465 . . . . . . 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 10966 . . 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 1194 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 974    = wceq 1383    e. wcel 1804    =/= wne 2638   _Vcvv 3095   ` cfv 5578  (class class class)co 6281   CCcc 9493   0cc0 9495   1c1 9496    + caddc 9498    x. cmul 9500   NN0cn0 10802   QQcq 11193   ^cexp 12148   ↾s cress 14615   .rcmulr 14680  AbsValcabv 17444  ℂfldccnfld 18399
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577  ax-cnex 9551  ax-resscn 9552  ax-1cn 9553  ax-icn 9554  ax-addcl 9555  ax-addrcl 9556  ax-mulcl 9557  ax-mulrcl 9558  ax-mulcom 9559  ax-addass 9560  ax-mulass 9561  ax-distr 9562  ax-i2m1 9563  ax-1ne0 9564  ax-1rid 9565  ax-rnegex 9566  ax-rrecex 9567  ax-cnre 9568  ax-pre-lttri 9569  ax-pre-lttrn 9570  ax-pre-ltadd 9571  ax-pre-mulgt0 9572  ax-addf 9574  ax-mulf 9575
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 975  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-nel 2641  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-tp 4019  df-op 4021  df-uni 4235  df-int 4272  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-tr 4531  df-eprel 4781  df-id 4785  df-po 4790  df-so 4791  df-fr 4828  df-we 4830  df-ord 4871  df-on 4872  df-lim 4873  df-suc 4874  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-om 6686  df-1st 6785  df-2nd 6786  df-tpos 6957  df-recs 7044  df-rdg 7078  df-1o 7132  df-oadd 7136  df-er 7313  df-map 7424  df-en 7519  df-dom 7520  df-sdom 7521  df-fin 7522  df-pnf 9633  df-mnf 9634  df-xr 9635  df-ltxr 9636  df-le 9637  df-sub 9812  df-neg 9813  df-div 10214  df-nn 10544  df-2 10601  df-3 10602  df-4 10603  df-5 10604  df-6 10605  df-7 10606  df-8 10607  df-9 10608  df-10 10609  df-n0 10803  df-z 10872  df-dec 10987  df-uz 11093  df-q 11194  df-ico 11546  df-fz 11684  df-seq 12090  df-exp 12149  df-struct 14616  df-ndx 14617  df-slot 14618  df-base 14619  df-sets 14620  df-ress 14621  df-plusg 14692  df-mulr 14693  df-starv 14694  df-tset 14698  df-ple 14699  df-ds 14701  df-unif 14702  df-0g 14821  df-mgm 15851  df-sgrp 15890  df-mnd 15900  df-grp 16036  df-minusg 16037  df-subg 16177  df-cmn 16779  df-mgp 17121  df-ur 17133  df-ring 17179  df-cring 17180  df-oppr 17251  df-dvdsr 17269  df-unit 17270  df-invr 17300  df-dvr 17311  df-drng 17377  df-subrg 17406  df-abv 17445  df-cnfld 18400
This theorem is referenced by:  ostth2lem2  23797  ostth2lem3  23798  ostth3  23801
  Copyright terms: Public domain W3C validator