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Theorem cnfldexp 18661
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 6239 . . . . 5  |-  ( x  =  0  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( 0 (.g `  (mulGrp ` fld ) ) A ) )
2 oveq2 6240 . . . . 5  |-  ( x  =  0  ->  ( A ^ x )  =  ( A ^ 0 ) )
31, 2eqeq12d 2422 . . . 4  |-  ( x  =  0  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( 0 (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
0 ) ) )
43imbi2d 314 . . 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 6239 . . . . 5  |-  ( x  =  y  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( y (.g `  (mulGrp ` fld ) ) A ) )
6 oveq2 6240 . . . . 5  |-  ( x  =  y  ->  ( A ^ x )  =  ( A ^ y
) )
75, 6eqeq12d 2422 . . . 4  |-  ( x  =  y  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
y ) ) )
87imbi2d 314 . . 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 6239 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A ) )
10 oveq2 6240 . . . . 5  |-  ( x  =  ( y  +  1 )  ->  ( A ^ x )  =  ( A ^ (
y  +  1 ) ) )
119, 10eqeq12d 2422 . . . 4  |-  ( x  =  ( y  +  1 )  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
( y  +  1 ) ) ) )
1211imbi2d 314 . . 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 6239 . . . . 5  |-  ( x  =  B  ->  (
x (.g `  (mulGrp ` fld ) ) A )  =  ( B (.g `  (mulGrp ` fld ) ) A ) )
14 oveq2 6240 . . . . 5  |-  ( x  =  B  ->  ( A ^ x )  =  ( A ^ B
) )
1513, 14eqeq12d 2422 . . . 4  |-  ( x  =  B  ->  (
( x (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ x
)  <->  ( B (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ B ) ) )
1615imbi2d 314 . . 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 2400 . . . . . 6  |-  (mulGrp ` fld )  =  (mulGrp ` fld )
18 cnfldbas 18634 . . . . . 6  |-  CC  =  ( Base ` fld )
1917, 18mgpbas 17357 . . . . 5  |-  CC  =  ( Base `  (mulGrp ` fld ) )
20 cnfld1 18653 . . . . . 6  |-  1  =  ( 1r ` fld )
2117, 20ringidval 17365 . . . . 5  |-  1  =  ( 0g `  (mulGrp ` fld ) )
22 eqid 2400 . . . . 5  |-  (.g `  (mulGrp ` fld ) )  =  (.g `  (mulGrp ` fld ) )
2319, 21, 22mulg0 16361 . . . 4  |-  ( A  e.  CC  ->  (
0 (.g `  (mulGrp ` fld ) ) A )  =  1 )
24 exp0 12122 . . . 4  |-  ( A  e.  CC  ->  ( A ^ 0 )  =  1 )
2523, 24eqtr4d 2444 . . 3  |-  ( A  e.  CC  ->  (
0 (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
0 ) )
26 oveq1 6239 . . . . . 6  |-  ( ( y (.g `  (mulGrp ` fld ) ) A )  =  ( A ^
y )  ->  (
( y (.g `  (mulGrp ` fld ) ) A )  x.  A )  =  ( ( A ^ y
)  x.  A ) )
27 cnring 18650 . . . . . . . . . 10  |-fld  e.  Ring
2817ringmgp 17414 . . . . . . . . . 10  |-  (fld  e.  Ring  -> 
(mulGrp ` fld )  e.  Mnd )
2927, 28ax-mp 5 . . . . . . . . 9  |-  (mulGrp ` fld )  e.  Mnd
30 cnfldmul 18636 . . . . . . . . . . 11  |-  x.  =  ( .r ` fld )
3117, 30mgpplusg 17355 . . . . . . . . . 10  |-  x.  =  ( +g  `  (mulGrp ` fld )
)
3219, 22, 31mulgnn0p1 16367 . . . . . . . . 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 1311 . . . . . . . 8  |-  ( ( y  e.  NN0  /\  A  e.  CC )  ->  ( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( ( y (.g `  (mulGrp ` fld ) ) A )  x.  A ) )
3433ancoms 451 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( ( y  +  1 ) (.g `  (mulGrp ` fld ) ) A )  =  ( ( y (.g `  (mulGrp ` fld ) ) A )  x.  A ) )
35 expp1 12125 . . . . . . 7  |-  ( ( A  e.  CC  /\  y  e.  NN0 )  -> 
( A ^ (
y  +  1 ) )  =  ( ( A ^ y )  x.  A ) )
3634, 35eqeq12d 2422 . . . . . 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 433 . . . 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 10916 . 2  |-  ( B  e.  NN0  ->  ( A  e.  CC  ->  ( B (.g `  (mulGrp ` fld ) ) A )  =  ( A ^ B ) ) )
4140impcom 428 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 367    = wceq 1403    e. wcel 1840   ` cfv 5523  (class class class)co 6232   CCcc 9438   0cc0 9440   1c1 9441    + caddc 9443    x. cmul 9445   NN0cn0 10754   ^cexp 12118   Mndcmnd 16133  .gcmg 16270  mulGrpcmgp 17351   Ringcrg 17408  ℂfldccnfld 18630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-rep 4504  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528  ax-inf2 8009  ax-cnex 9496  ax-resscn 9497  ax-1cn 9498  ax-icn 9499  ax-addcl 9500  ax-addrcl 9501  ax-mulcl 9502  ax-mulrcl 9503  ax-mulcom 9504  ax-addass 9505  ax-mulass 9506  ax-distr 9507  ax-i2m1 9508  ax-1ne0 9509  ax-1rid 9510  ax-rnegex 9511  ax-rrecex 9512  ax-cnre 9513  ax-pre-lttri 9514  ax-pre-lttrn 9515  ax-pre-ltadd 9516  ax-pre-mulgt0 9517  ax-addf 9519  ax-mulf 9520
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 973  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-nel 2599  df-ral 2756  df-rex 2757  df-reu 2758  df-rmo 2759  df-rab 2760  df-v 3058  df-sbc 3275  df-csb 3371  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-pss 3427  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-tp 3974  df-op 3976  df-uni 4189  df-int 4225  df-iun 4270  df-br 4393  df-opab 4451  df-mpt 4452  df-tr 4487  df-eprel 4731  df-id 4735  df-po 4741  df-so 4742  df-fr 4779  df-we 4781  df-ord 4822  df-on 4823  df-lim 4824  df-suc 4825  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-f1 5528  df-fo 5529  df-f1o 5530  df-fv 5531  df-riota 6194  df-ov 6235  df-oprab 6236  df-mpt2 6237  df-om 6637  df-1st 6736  df-2nd 6737  df-recs 6997  df-rdg 7031  df-1o 7085  df-oadd 7089  df-er 7266  df-en 7473  df-dom 7474  df-sdom 7475  df-fin 7476  df-pnf 9578  df-mnf 9579  df-xr 9580  df-ltxr 9581  df-le 9582  df-sub 9761  df-neg 9762  df-nn 10495  df-2 10553  df-3 10554  df-4 10555  df-5 10556  df-6 10557  df-7 10558  df-8 10559  df-9 10560  df-10 10561  df-n0 10755  df-z 10824  df-dec 10938  df-uz 11044  df-fz 11642  df-seq 12060  df-exp 12119  df-struct 14733  df-ndx 14734  df-slot 14735  df-base 14736  df-sets 14737  df-plusg 14812  df-mulr 14813  df-starv 14814  df-tset 14818  df-ple 14819  df-ds 14821  df-unif 14822  df-0g 14946  df-mgm 16086  df-sgrp 16125  df-mnd 16135  df-grp 16271  df-mulg 16274  df-cmn 17014  df-mgp 17352  df-ur 17364  df-ring 17410  df-cring 17411  df-cnfld 18631
This theorem is referenced by:  plypf1  22791  dchrfi  23801  dchrabs  23806  lgsqrlem1  23887  lgseisenlem4  23898  dchrisum0flblem1  23964  proot1ex  35489
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