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Theorem qabvexp 21320
Description: Induct the product rule abvmul 15917 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 6089 . . . . . . 7  |-  ( k  =  0  ->  ( M ^ k )  =  ( M ^ 0 ) )
21fveq2d 5732 . . . . . 6  |-  ( k  =  0  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ 0 ) ) )
3 oveq2 6089 . . . . . 6  |-  ( k  =  0  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
0 ) )
42, 3eqeq12d 2450 . . . . 5  |-  ( k  =  0  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ 0 ) )  =  ( ( F `  M
) ^ 0 ) ) )
54imbi2d 308 . . . 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 6089 . . . . . . 7  |-  ( k  =  n  ->  ( M ^ k )  =  ( M ^ n
) )
76fveq2d 5732 . . . . . 6  |-  ( k  =  n  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ n ) ) )
8 oveq2 6089 . . . . . 6  |-  ( k  =  n  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
n ) )
97, 8eqeq12d 2450 . . . . 5  |-  ( k  =  n  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ n
) )  =  ( ( F `  M
) ^ n ) ) )
109imbi2d 308 . . . 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 6089 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  ( M ^ k )  =  ( M ^ (
n  +  1 ) ) )
1211fveq2d 5732 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ ( n  +  1 ) ) ) )
13 oveq2 6089 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
( n  +  1 ) ) )
1412, 13eqeq12d 2450 . . . . 5  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ (
n  +  1 ) ) )  =  ( ( F `  M
) ^ ( n  +  1 ) ) ) )
1514imbi2d 308 . . . 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 6089 . . . . . . 7  |-  ( k  =  N  ->  ( M ^ k )  =  ( M ^ N
) )
1716fveq2d 5732 . . . . . 6  |-  ( k  =  N  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ N ) ) )
18 oveq2 6089 . . . . . 6  |-  ( k  =  N  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^ N ) )
1917, 18eqeq12d 2450 . . . . 5  |-  ( k  =  N  ->  (
( F `  ( M ^ k ) )  =  ( ( F `
 M ) ^
k )  <->  ( F `  ( M ^ N
) )  =  ( ( F `  M
) ^ N ) ) )
2019imbi2d 308 . . . 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 9059 . . . . . . 7  |-  1  =/=  0
22 qabsabv.a . . . . . . . 8  |-  A  =  (AbsVal `  Q )
23 qrng.q . . . . . . . . 9  |-  Q  =  (flds  QQ )
2423qrng1 21316 . . . . . . . 8  |-  1  =  ( 1r `  Q )
2523qrng0 21315 . . . . . . . 8  |-  0  =  ( 0g `  Q )
2622, 24, 25abv1z 15920 . . . . . . 7  |-  ( ( F  e.  A  /\  1  =/=  0 )  -> 
( F `  1
)  =  1 )
2721, 26mpan2 653 . . . . . 6  |-  ( F  e.  A  ->  ( F `  1 )  =  1 )
2827adantr 452 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  1
)  =  1 )
29 qcn 10588 . . . . . . . 8  |-  ( M  e.  QQ  ->  M  e.  CC )
3029adantl 453 . . . . . . 7  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  M  e.  CC )
3130exp0d 11517 . . . . . 6  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( M ^ 0 )  =  1 )
3231fveq2d 5732 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( F ` 
1 ) )
3323qrngbas 21313 . . . . . . . 8  |-  QQ  =  ( Base `  Q )
3422, 33abvcl 15912 . . . . . . 7  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  M
)  e.  RR )
3534recnd 9114 . . . . . 6  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  M
)  e.  CC )
3635exp0d 11517 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( ( F `  M ) ^ 0 )  =  1 )
3728, 32, 363eqtr4d 2478 . . . 4  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( ( F `
 M ) ^
0 ) )
38 oveq1 6088 . . . . . . 7  |-  ( ( F `  ( M ^ n ) )  =  ( ( F `
 M ) ^
n )  ->  (
( F `  ( M ^ n ) )  x.  ( F `  M ) )  =  ( ( ( F `
 M ) ^
n )  x.  ( F `  M )
) )
39 expp1 11388 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  n  e.  NN0 )  -> 
( M ^ (
n  +  1 ) )  =  ( ( M ^ n )  x.  M ) )
4030, 39sylan 458 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( M ^
( n  +  1 ) )  =  ( ( M ^ n
)  x.  M ) )
4140fveq2d 5732 . . . . . . . . 9  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( M ^ ( n  +  1 ) ) )  =  ( F `
 ( ( M ^ n )  x.  M ) ) )
42 simpll 731 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  F  e.  A
)
43 qexpcl 11397 . . . . . . . . . . 11  |-  ( ( M  e.  QQ  /\  n  e.  NN0 )  -> 
( M ^ n
)  e.  QQ )
4443adantll 695 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( M ^
n )  e.  QQ )
45 simplr 732 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  M  e.  QQ )
46 qex 10586 . . . . . . . . . . . 12  |-  QQ  e.  _V
47 cnfldmul 16709 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
4823, 47ressmulr 13582 . . . . . . . . . . . 12  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
4946, 48ax-mp 8 . . . . . . . . . . 11  |-  x.  =  ( .r `  Q )
5022, 33, 49abvmul 15917 . . . . . . . . . 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 1184 . . . . . . . . 9  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( ( M ^
n )  x.  M
) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `  M ) ) )
5241, 51eqtrd 2468 . . . . . . . 8  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( M ^ ( n  +  1 ) ) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `  M ) ) )
53 expp1 11388 . . . . . . . . 9  |-  ( ( ( F `  M
)  e.  CC  /\  n  e.  NN0 )  -> 
( ( F `  M ) ^ (
n  +  1 ) )  =  ( ( ( F `  M
) ^ n )  x.  ( F `  M ) ) )
5435, 53sylan 458 . . . . . . . 8  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( ( F `
 M ) ^
( n  +  1 ) )  =  ( ( ( F `  M ) ^ n
)  x.  ( F `
 M ) ) )
5552, 54eqeq12d 2450 . . . . . . 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 213 . . . . . 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 425 . . . . 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 24 . . . 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 10366 . . 3  |-  ( N  e.  NN0  ->  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ N ) )  =  ( ( F `
 M ) ^ N ) ) )
6059com12 29 . 2  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( N  e.  NN0  ->  ( F `  ( M ^ N ) )  =  ( ( F `
 M ) ^ N ) ) )
61603impia 1150 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   _Vcvv 2956   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995   NN0cn0 10221   QQcq 10574   ^cexp 11382   ↾s cress 13470   .rcmulr 13530  AbsValcabv 15904  ℂfldccnfld 16703
This theorem is referenced by:  ostth2lem2  21328  ostth2lem3  21329  ostth3  21332
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067  ax-addf 9069  ax-mulf 9070
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-tpos 6479  df-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  df-map 7020  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-div 9678  df-nn 10001  df-2 10058  df-3 10059  df-4 10060  df-5 10061  df-6 10062  df-7 10063  df-8 10064  df-9 10065  df-10 10066  df-n0 10222  df-z 10283  df-dec 10383  df-uz 10489  df-q 10575  df-ico 10922  df-fz 11044  df-seq 11324  df-exp 11383  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  df-ress 13476  df-plusg 13542  df-mulr 13543  df-starv 13544  df-tset 13548  df-ple 13549  df-ds 13551  df-unif 13552  df-0g 13727  df-mnd 14690  df-grp 14812  df-minusg 14813  df-subg 14941  df-cmn 15414  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-oppr 15728  df-dvdsr 15746  df-unit 15747  df-invr 15777  df-dvr 15788  df-drng 15837  df-subrg 15866  df-abv 15905  df-cnfld 16704
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