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Theorem cnfldexp 16734
Description: The exponentiation operator in the field of complex numbers (for nonnegative exponents). (Contributed by Mario Carneiro, 15-Jun-2015.)
Assertion
Ref Expression
cnfldexp  |-  ( ( A  e.  CC  /\  B  e.  NN0 )  -> 
( B (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ B
) )

Proof of Theorem cnfldexp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 oveq1 6088 . . . . 5  |-  ( x  =  0  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( 0 (.g `  (mulGrp ` fld ) ) A ) )
2 oveq2 6089 . . . . 5  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
31, 2eqeq12d 2450 . . . 4  |-  ( x  =  0  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( 0 (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
0 ) ) )
43imbi2d 308 . . 3  |-  ( x  =  0  ->  (
( A  e.  CC  ->  ( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
) )  <->  ( A  e.  CC  ->  ( 0 (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
0 ) ) ) )
5 oveq1 6088 . . . . 5  |-  ( x  =  y  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( y (.g `  (mulGrp ` fld ) ) A ) )
6 oveq2 6089 . . . . 5  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
75, 6eqeq12d 2450 . . . 4  |-  ( x  =  y  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
y ) ) )
87imbi2d 308 . . 3  |-  ( x  =  y  ->  (
( A  e.  CC  ->  ( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
) )  <->  ( A  e.  CC  ->  ( y
(.g `  (mulGrp ` fld ) ) A )  =  ( A ^
y ) ) ) )
9 oveq1 6088 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A ) )
10 oveq2 6089 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A ^ x )  =  ( A ^ (
y  +  1 ) ) )
119, 10eqeq12d 2450 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
( y  +  1 ) ) ) )
1211imbi2d 308 . . 3  |-  ( x  =  ( y  +  1 )  ->  (
( A  e.  CC  ->  ( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
) )  <->  ( A  e.  CC  ->  ( (
y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
( y  +  1 ) ) ) ) )
13 oveq1 6088 . . . . 5  |-  ( x  =  B  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( B (.g `  (mulGrp ` fld ) ) A ) )
14 oveq2 6089 . . . . 5  |-  ( x  =  B  ->  ( A ^ x )  =  ( A ^ B
) )
1513, 14eqeq12d 2450 . . . 4  |-  ( x  =  B  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( B (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ B ) ) )
1615imbi2d 308 . . 3  |-  ( x  =  B  ->  (
( A  e.  CC  ->  ( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
) )  <->  ( A  e.  CC  ->  ( B
(.g `  (mulGrp ` fld ) ) A )  =  ( A ^ B ) ) ) )
17 eqid 2436 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
18 cnfldbas 16707 . . . . . 6  |-  CC  =  ( Base ` fld )
1917, 18mgpbas 15654 . . . . 5  |-  CC  =  ( Base `  (mulGrp ` fld ) )
20 cnfld1 16726 . . . . . 6  |-  1  =  ( 1r ` fld )
2117, 20rngidval 15666 . . . . 5  |-  1  =  ( 0g `  (mulGrp ` fld ) )
22 eqid 2436 . . . . 5  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
2319, 21, 22mulg0 14895 . . . 4  |-  ( A  e.  CC  ->  (
0 (.g `  (mulGrp ` fld ) ) A )  =  1 )
24 exp0 11386 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2523, 24eqtr4d 2471 . . 3  |-  ( A  e.  CC  ->  (
0 (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
0 ) )
26 oveq1 6088 . . . . . 6  |-  ( ( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
y )  ->  (
( y (.g `  (mulGrp ` fld ) ) A )  x.  A )  =  ( ( A ^ y
)  x.  A ) )
27 cnrng 16723 . . . . . . . . . 10  |-fld  e.  Ring
2817rngmgp 15670 . . . . . . . . . 10  |-  (fld  e.  Ring  -> 
(mulGrp ` fld )  e.  Mnd )
2927, 28ax-mp 8 . . . . . . . . 9  |-  (mulGrp ` fld )  e.  Mnd
30 cnfldmul 16709 . . . . . . . . . . 11  |-  x.  =  ( .r ` fld )
3117, 30mgpplusg 15652 . . . . . . . . . 10  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3219, 22, 31mulgnn0p1 14901 . . . . . . . . 9  |-  ( ( (mulGrp ` fld )  e.  Mnd  /\  y  e.  NN0  /\  A  e.  CC )  ->  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( ( y (.g `  (mulGrp ` fld ) ) A )  x.  A ) )
3329, 32mp3an1 1266 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  A  e.  CC )  ->  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( ( y (.g `  (mulGrp ` fld ) ) A )  x.  A ) )
3433ancoms 440 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( ( y (.g `  (mulGrp ` fld ) ) A )  x.  A ) )
35 expp1 11388 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ (
y  +  1 ) )  =  ( ( A ^ y )  x.  A ) )
3634, 35eqeq12d 2450 . . . . . 6  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
( y  +  1 ) )  <->  ( (
y (.g `  (mulGrp ` fld ) ) A )  x.  A )  =  ( ( A ^
y )  x.  A
) ) )
3726, 36syl5ibr 213 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( ( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
y )  ->  (
( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
( y  +  1 ) ) ) )
3837expcom 425 . . . 4  |-  ( y  e.  NN0  ->  ( A  e.  CC  ->  (
( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ y
)  ->  ( (
y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
( y  +  1 ) ) ) ) )
3938a2d 24 . . 3  |-  ( y  e.  NN0  ->  ( ( A  e.  CC  ->  ( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
y ) )  -> 
( A  e.  CC  ->  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ (
y  +  1 ) ) ) ) )
404, 8, 12, 16, 25, 39nn0ind 10366 . 2  |-  ( B  e.  NN0  ->  ( A  e.  CC  ->  ( B (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ B ) ) )
4140impcom 420 1  |-  ( ( A  e.  CC  /\  B  e.  NN0 )  -> 
( B (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ B
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1652    e. wcel 1725   ` cfv 5454  (class class class)co 6081   CCcc 8988   0cc0 8990   1c1 8991    + caddc 8993    x. cmul 8995   NN0cn0 10221   ^cexp 11382   Mndcmnd 14684  .gcmg 14689  mulGrpcmgp 15648   Ringcrg 15660  ℂfldccnfld 16703
This theorem is referenced by:  plypf1  20131  dchrfi  21039  dchrabs  21044  lgsqrlem1  21125  lgseisenlem4  21136  dchrisum0flblem1  21202  proot1ex  27497
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-inf2 7596  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-riota 6549  df-recs 6633  df-rdg 6668  df-1o 6724  df-oadd 6728  df-er 6905  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-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-fz 11044  df-seq 11324  df-exp 11383  df-struct 13471  df-ndx 13472  df-slot 13473  df-base 13474  df-sets 13475  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-mulg 14815  df-cmn 15414  df-mgp 15649  df-rng 15663  df-cring 15664  df-ur 15665  df-cnfld 16704
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