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Theorem qabvexp 22759
Description: Induct the product rule abvmul 16837 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 6088 . . . . . . 7  |-  ( k  =  0  ->  ( M ^ k )  =  ( M ^ 0 ) )
21fveq2d 5683 . . . . . 6  |-  ( k  =  0  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ 0 ) ) )
3 oveq2 6088 . . . . . 6  |-  ( k  =  0  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
0 ) )
42, 3eqeq12d 2447 . . . . 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 6088 . . . . . . 7  |-  ( k  =  n  ->  ( M ^ k )  =  ( M ^ n
) )
76fveq2d 5683 . . . . . 6  |-  ( k  =  n  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ n ) ) )
8 oveq2 6088 . . . . . 6  |-  ( k  =  n  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
n ) )
97, 8eqeq12d 2447 . . . . 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 6088 . . . . . . 7  |-  ( k  =  ( n  + 
1 )  ->  ( M ^ k )  =  ( M ^ (
n  +  1 ) ) )
1211fveq2d 5683 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ ( n  +  1 ) ) ) )
13 oveq2 6088 . . . . . 6  |-  ( k  =  ( n  + 
1 )  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^
( n  +  1 ) ) )
1412, 13eqeq12d 2447 . . . . 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 6088 . . . . . . 7  |-  ( k  =  N  ->  ( M ^ k )  =  ( M ^ N
) )
1716fveq2d 5683 . . . . . 6  |-  ( k  =  N  ->  ( F `  ( M ^ k ) )  =  ( F `  ( M ^ N ) ) )
18 oveq2 6088 . . . . . 6  |-  ( k  =  N  ->  (
( F `  M
) ^ k )  =  ( ( F `
 M ) ^ N ) )
1917, 18eqeq12d 2447 . . . . 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 9338 . . . . . . 7  |-  1  =/=  0
22 qabsabv.a . . . . . . . 8  |-  A  =  (AbsVal `  Q )
23 qrng.q . . . . . . . . 9  |-  Q  =  (flds  QQ )
2423qrng1 22755 . . . . . . . 8  |-  1  =  ( 1r `  Q )
2523qrng0 22754 . . . . . . . 8  |-  0  =  ( 0g `  Q )
2622, 24, 25abv1z 16840 . . . . . . 7  |-  ( ( F  e.  A  /\  1  =/=  0 )  -> 
( F `  1
)  =  1 )
2721, 26mpan2 664 . . . . . 6  |-  ( F  e.  A  ->  ( F `  1 )  =  1 )
2827adantr 462 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  1
)  =  1 )
29 qcn 10954 . . . . . . . 8  |-  ( M  e.  QQ  ->  M  e.  CC )
3029adantl 463 . . . . . . 7  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  M  e.  CC )
3130exp0d 11985 . . . . . 6  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( M ^ 0 )  =  1 )
3231fveq2d 5683 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( F ` 
1 ) )
3323qrngbas 22752 . . . . . . . 8  |-  QQ  =  ( Base `  Q )
3422, 33abvcl 16832 . . . . . . 7  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  M
)  e.  RR )
3534recnd 9399 . . . . . 6  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  M
)  e.  CC )
3635exp0d 11985 . . . . 5  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( ( F `  M ) ^ 0 )  =  1 )
3728, 32, 363eqtr4d 2475 . . . 4  |-  ( ( F  e.  A  /\  M  e.  QQ )  ->  ( F `  ( M ^ 0 ) )  =  ( ( F `
 M ) ^
0 ) )
38 oveq1 6087 . . . . . . 7  |-  ( ( F `  ( M ^ n ) )  =  ( ( F `
 M ) ^
n )  ->  (
( F `  ( M ^ n ) )  x.  ( F `  M ) )  =  ( ( ( F `
 M ) ^
n )  x.  ( F `  M )
) )
39 expp1 11855 . . . . . . . . . . 11  |-  ( ( M  e.  CC  /\  n  e.  NN0 )  -> 
( M ^ (
n  +  1 ) )  =  ( ( M ^ n )  x.  M ) )
4030, 39sylan 468 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( M ^
( n  +  1 ) )  =  ( ( M ^ n
)  x.  M ) )
4140fveq2d 5683 . . . . . . . . 9  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( M ^ ( n  +  1 ) ) )  =  ( F `
 ( ( M ^ n )  x.  M ) ) )
42 simpll 746 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  F  e.  A
)
43 qexpcl 11864 . . . . . . . . . . 11  |-  ( ( M  e.  QQ  /\  n  e.  NN0 )  -> 
( M ^ n
)  e.  QQ )
4443adantll 706 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( M ^
n )  e.  QQ )
45 simplr 747 . . . . . . . . . 10  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  M  e.  QQ )
46 qex 10952 . . . . . . . . . . . 12  |-  QQ  e.  _V
47 cnfldmul 17667 . . . . . . . . . . . . 13  |-  x.  =  ( .r ` fld )
4823, 47ressmulr 14273 . . . . . . . . . . . 12  |-  ( QQ  e.  _V  ->  x.  =  ( .r `  Q ) )
4946, 48ax-mp 5 . . . . . . . . . . 11  |-  x.  =  ( .r `  Q )
5022, 33, 49abvmul 16837 . . . . . . . . . 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 1211 . . . . . . . . 9  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( ( M ^
n )  x.  M
) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `  M ) ) )
5241, 51eqtrd 2465 . . . . . . . 8  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( F `  ( M ^ ( n  +  1 ) ) )  =  ( ( F `  ( M ^ n ) )  x.  ( F `  M ) ) )
53 expp1 11855 . . . . . . . . 9  |-  ( ( ( F `  M
)  e.  CC  /\  n  e.  NN0 )  -> 
( ( F `  M ) ^ (
n  +  1 ) )  =  ( ( ( F `  M
) ^ n )  x.  ( F `  M ) ) )
5435, 53sylan 468 . . . . . . . 8  |-  ( ( ( F  e.  A  /\  M  e.  QQ )  /\  n  e.  NN0 )  ->  ( ( F `
 M ) ^
( n  +  1 ) )  =  ( ( ( F `  M ) ^ n
)  x.  ( F `
 M ) ) )
5552, 54eqeq12d 2447 . . . . . . 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 10725 . . 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 1177 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 958    = wceq 1362    e. wcel 1755    =/= wne 2596   _Vcvv 2962   ` cfv 5406  (class class class)co 6080   CCcc 9267   0cc0 9269   1c1 9270    + caddc 9272    x. cmul 9274   NN0cn0 10566   QQcq 10940   ^cexp 11848   ↾s cress 14157   .rcmulr 14221  AbsValcabv 16824  ℂfldccnfld 17661
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361  ax-cnex 9325  ax-resscn 9326  ax-1cn 9327  ax-icn 9328  ax-addcl 9329  ax-addrcl 9330  ax-mulcl 9331  ax-mulrcl 9332  ax-mulcom 9333  ax-addass 9334  ax-mulass 9335  ax-distr 9336  ax-i2m1 9337  ax-1ne0 9338  ax-1rid 9339  ax-rnegex 9340  ax-rrecex 9341  ax-cnre 9342  ax-pre-lttri 9343  ax-pre-lttrn 9344  ax-pre-ltadd 9345  ax-pre-mulgt0 9346  ax-addf 9348  ax-mulf 9349
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 959  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-pss 3332  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-tp 3870  df-op 3872  df-uni 4080  df-int 4117  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-tr 4374  df-eprel 4619  df-id 4623  df-po 4628  df-so 4629  df-fr 4666  df-we 4668  df-ord 4709  df-on 4710  df-lim 4711  df-suc 4712  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-riota 6039  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-om 6466  df-1st 6566  df-2nd 6567  df-tpos 6734  df-recs 6818  df-rdg 6852  df-1o 6908  df-oadd 6912  df-er 7089  df-map 7204  df-en 7299  df-dom 7300  df-sdom 7301  df-fin 7302  df-pnf 9407  df-mnf 9408  df-xr 9409  df-ltxr 9410  df-le 9411  df-sub 9584  df-neg 9585  df-div 9981  df-nn 10310  df-2 10367  df-3 10368  df-4 10369  df-5 10370  df-6 10371  df-7 10372  df-8 10373  df-9 10374  df-10 10375  df-n0 10567  df-z 10634  df-dec 10743  df-uz 10849  df-q 10941  df-ico 11293  df-fz 11424  df-seq 11790  df-exp 11849  df-struct 14158  df-ndx 14159  df-slot 14160  df-base 14161  df-sets 14162  df-ress 14163  df-plusg 14233  df-mulr 14234  df-starv 14235  df-tset 14239  df-ple 14240  df-ds 14242  df-unif 14243  df-0g 14362  df-mnd 15397  df-grp 15524  df-minusg 15525  df-subg 15657  df-cmn 16258  df-mgp 16565  df-rng 16579  df-cring 16580  df-ur 16581  df-oppr 16648  df-dvdsr 16666  df-unit 16667  df-invr 16697  df-dvr 16708  df-drng 16757  df-subrg 16786  df-abv 16825  df-cnfld 17662
This theorem is referenced by:  ostth2lem2  22767  ostth2lem3  22768  ostth3  22771
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