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Theorem efgh 19919
Description: The exponential function of a scaled complex number is a group homomorphism from the group of complex numbers under addition to the set of complex numbers under multiplication. (Contributed by Paul Chapman, 25-Apr-2008.) (Revised by Mario Carneiro, 11-May-2014.)
Hypothesis
Ref Expression
efgh.1  |-  F  =  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )
Assertion
Ref Expression
efgh  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( F `  ( B  +  C ) )  =  ( ( F `  B )  x.  ( F `  C )
) )
Distinct variable groups:    x, A    x, B    x, C
Allowed substitution hint:    F( x)

Proof of Theorem efgh
StepHypRef Expression
1 adddi 8842 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
21fveq2d 5545 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( exp `  ( A  x.  ( B  +  C
) ) )  =  ( exp `  (
( A  x.  B
)  +  ( A  x.  C ) ) ) )
3 mulcl 8837 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
433adant3 975 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  e.  CC )
5 mulcl 8837 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
653adant2 974 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  e.  CC )
7 efadd 12391 . . . 4  |-  ( ( ( A  x.  B
)  e.  CC  /\  ( A  x.  C
)  e.  CC )  ->  ( exp `  (
( A  x.  B
)  +  ( A  x.  C ) ) )  =  ( ( exp `  ( A  x.  B ) )  x.  ( exp `  ( A  x.  C )
) ) )
84, 6, 7syl2anc 642 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( exp `  ( ( A  x.  B )  +  ( A  x.  C
) ) )  =  ( ( exp `  ( A  x.  B )
)  x.  ( exp `  ( A  x.  C
) ) ) )
92, 8eqtrd 2328 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( exp `  ( A  x.  ( B  +  C
) ) )  =  ( ( exp `  ( A  x.  B )
)  x.  ( exp `  ( A  x.  C
) ) ) )
10 addcl 8835 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C
)  e.  CC )
11103adant1 973 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  e.  CC )
12 oveq2 5882 . . . . 5  |-  ( x  =  ( B  +  C )  ->  ( A  x.  x )  =  ( A  x.  ( B  +  C
) ) )
1312fveq2d 5545 . . . 4  |-  ( x  =  ( B  +  C )  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  ( B  +  C ) ) ) )
14 efgh.1 . . . 4  |-  F  =  ( x  e.  CC  |->  ( exp `  ( A  x.  x ) ) )
15 fvex 5555 . . . 4  |-  ( exp `  ( A  x.  ( B  +  C )
) )  e.  _V
1613, 14, 15fvmpt 5618 . . 3  |-  ( ( B  +  C )  e.  CC  ->  ( F `  ( B  +  C ) )  =  ( exp `  ( A  x.  ( B  +  C ) ) ) )
1711, 16syl 15 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( F `  ( B  +  C ) )  =  ( exp `  ( A  x.  ( B  +  C ) ) ) )
18 oveq2 5882 . . . . . 6  |-  ( x  =  B  ->  ( A  x.  x )  =  ( A  x.  B ) )
1918fveq2d 5545 . . . . 5  |-  ( x  =  B  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  B )
) )
20 fvex 5555 . . . . 5  |-  ( exp `  ( A  x.  B
) )  e.  _V
2119, 14, 20fvmpt 5618 . . . 4  |-  ( B  e.  CC  ->  ( F `  B )  =  ( exp `  ( A  x.  B )
) )
22 oveq2 5882 . . . . . 6  |-  ( x  =  C  ->  ( A  x.  x )  =  ( A  x.  C ) )
2322fveq2d 5545 . . . . 5  |-  ( x  =  C  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  C )
) )
24 fvex 5555 . . . . 5  |-  ( exp `  ( A  x.  C
) )  e.  _V
2523, 14, 24fvmpt 5618 . . . 4  |-  ( C  e.  CC  ->  ( F `  C )  =  ( exp `  ( A  x.  C )
) )
2621, 25oveqan12d 5893 . . 3  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( F `  B )  x.  ( F `  C )
)  =  ( ( exp `  ( A  x.  B ) )  x.  ( exp `  ( A  x.  C )
) ) )
27263adant1 973 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  (
( F `  B
)  x.  ( F `
 C ) )  =  ( ( exp `  ( A  x.  B
) )  x.  ( exp `  ( A  x.  C ) ) ) )
289, 17, 273eqtr4d 2338 1  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( F `  ( B  +  C ) )  =  ( ( F `  B )  x.  ( F `  C )
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ w3a 934    = wceq 1632    e. wcel 1696    e. cmpt 4093   ` cfv 5271  (class class class)co 5874   CCcc 8751    + caddc 8756    x. cmul 8758   expce 12359
This theorem is referenced by:  efghgrp  21056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831  ax-addf 8832  ax-mulf 8833
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-se 4369  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-isom 5280  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-pm 6791  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-sup 7210  df-oi 7241  df-card 7588  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-ico 10678  df-fz 10799  df-fzo 10887  df-fl 10941  df-seq 11063  df-exp 11121  df-fac 11305  df-bc 11332  df-hash 11354  df-shft 11578  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-limsup 11961  df-clim 11978  df-rlim 11979  df-sum 12175  df-ef 12365
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