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Theorem cnfldexp 17977
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 6210 . . . . 5  |-  ( x  =  0  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( 0 (.g `  (mulGrp ` fld ) ) A ) )
2 oveq2 6211 . . . . 5  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
31, 2eqeq12d 2476 . . . 4  |-  ( x  =  0  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( 0 (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
0 ) ) )
43imbi2d 316 . . 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 6210 . . . . 5  |-  ( x  =  y  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( y (.g `  (mulGrp ` fld ) ) A ) )
6 oveq2 6211 . . . . 5  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
75, 6eqeq12d 2476 . . . 4  |-  ( x  =  y  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
y ) ) )
87imbi2d 316 . . 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 6210 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A ) )
10 oveq2 6211 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A ^ x )  =  ( A ^ (
y  +  1 ) ) )
119, 10eqeq12d 2476 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
( y  +  1 ) ) ) )
1211imbi2d 316 . . 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 6210 . . . . 5  |-  ( x  =  B  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( B (.g `  (mulGrp ` fld ) ) A ) )
14 oveq2 6211 . . . . 5  |-  ( x  =  B  ->  ( A ^ x )  =  ( A ^ B
) )
1513, 14eqeq12d 2476 . . . 4  |-  ( x  =  B  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( B (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ B ) ) )
1615imbi2d 316 . . 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 2454 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
18 cnfldbas 17950 . . . . . 6  |-  CC  =  ( Base ` fld )
1917, 18mgpbas 16722 . . . . 5  |-  CC  =  ( Base `  (mulGrp ` fld ) )
20 cnfld1 17969 . . . . . 6  |-  1  =  ( 1r ` fld )
2117, 20rngidval 16730 . . . . 5  |-  1  =  ( 0g `  (mulGrp ` fld ) )
22 eqid 2454 . . . . 5  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
2319, 21, 22mulg0 15754 . . . 4  |-  ( A  e.  CC  ->  (
0 (.g `  (mulGrp ` fld ) ) A )  =  1 )
24 exp0 11989 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2523, 24eqtr4d 2498 . . 3  |-  ( A  e.  CC  ->  (
0 (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
0 ) )
26 oveq1 6210 . . . . . 6  |-  ( ( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
y )  ->  (
( y (.g `  (mulGrp ` fld ) ) A )  x.  A )  =  ( ( A ^ y
)  x.  A ) )
27 cnrng 17966 . . . . . . . . . 10  |-fld  e.  Ring
2817rngmgp 16777 . . . . . . . . . 10  |-  (fld  e.  Ring  -> 
(mulGrp ` fld )  e.  Mnd )
2927, 28ax-mp 5 . . . . . . . . 9  |-  (mulGrp ` fld )  e.  Mnd
30 cnfldmul 17952 . . . . . . . . . . 11  |-  x.  =  ( .r ` fld )
3117, 30mgpplusg 16720 . . . . . . . . . 10  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3219, 22, 31mulgnn0p1 15760 . . . . . . . . 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 1302 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  A  e.  CC )  ->  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( ( y (.g `  (mulGrp ` fld ) ) A )  x.  A ) )
3433ancoms 453 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( ( y (.g `  (mulGrp ` fld ) ) A )  x.  A ) )
35 expp1 11992 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ (
y  +  1 ) )  =  ( ( A ^ y )  x.  A ) )
3634, 35eqeq12d 2476 . . . . . 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 221 . . . . 5  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( ( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
y )  ->  (
( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
( y  +  1 ) ) ) )
3837expcom 435 . . . 4  |-  ( y  e.  NN0  ->  ( A  e.  CC  ->  (
( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ y
)  ->  ( (
y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
( y  +  1 ) ) ) ) )
3938a2d 26 . . 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 10852 . 2  |-  ( B  e.  NN0  ->  ( A  e.  CC  ->  ( B (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ B ) ) )
4140impcom 430 1  |-  ( ( A  e.  CC  /\  B  e.  NN0 )  -> 
( B (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   ` cfv 5529  (class class class)co 6203   CCcc 9394   0cc0 9396   1c1 9397    + caddc 9399    x. cmul 9401   NN0cn0 10693   ^cexp 11985   Mndcmnd 15531  .gcmg 15536  mulGrpcmgp 16716   Ringcrg 16771  ℂfldccnfld 17946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485  ax-inf2 7961  ax-cnex 9452  ax-resscn 9453  ax-1cn 9454  ax-icn 9455  ax-addcl 9456  ax-addrcl 9457  ax-mulcl 9458  ax-mulrcl 9459  ax-mulcom 9460  ax-addass 9461  ax-mulass 9462  ax-distr 9463  ax-i2m1 9464  ax-1ne0 9465  ax-1rid 9466  ax-rnegex 9467  ax-rrecex 9468  ax-cnre 9469  ax-pre-lttri 9470  ax-pre-lttrn 9471  ax-pre-ltadd 9472  ax-pre-mulgt0 9473  ax-addf 9475  ax-mulf 9476
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-pss 3455  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-tp 3993  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-br 4404  df-opab 4462  df-mpt 4463  df-tr 4497  df-eprel 4743  df-id 4747  df-po 4752  df-so 4753  df-fr 4790  df-we 4792  df-ord 4833  df-on 4834  df-lim 4835  df-suc 4836  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-riota 6164  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-om 6590  df-1st 6690  df-2nd 6691  df-recs 6945  df-rdg 6979  df-1o 7033  df-oadd 7037  df-er 7214  df-en 7424  df-dom 7425  df-sdom 7426  df-fin 7427  df-pnf 9534  df-mnf 9535  df-xr 9536  df-ltxr 9537  df-le 9538  df-sub 9711  df-neg 9712  df-nn 10437  df-2 10494  df-3 10495  df-4 10496  df-5 10497  df-6 10498  df-7 10499  df-8 10500  df-9 10501  df-10 10502  df-n0 10694  df-z 10761  df-dec 10870  df-uz 10976  df-fz 11558  df-seq 11927  df-exp 11986  df-struct 14297  df-ndx 14298  df-slot 14299  df-base 14300  df-sets 14301  df-plusg 14373  df-mulr 14374  df-starv 14375  df-tset 14379  df-ple 14380  df-ds 14382  df-unif 14383  df-0g 14502  df-mnd 15537  df-grp 15667  df-mulg 15670  df-cmn 16403  df-mgp 16717  df-ur 16729  df-rng 16773  df-cring 16774  df-cnfld 17947
This theorem is referenced by:  plypf1  21816  dchrfi  22730  dchrabs  22735  lgsqrlem1  22816  lgseisenlem4  22827  dchrisum0flblem1  22893  proot1ex  29737
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