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Theorem efgh 19851
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 8780 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
21fveq2d 5448 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( exp `  ( A  x.  ( B  +  C
) ) )  =  ( exp `  (
( A  x.  B
)  +  ( A  x.  C ) ) ) )
3 mulcl 8775 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
433adant3 980 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  e.  CC )
5 mulcl 8775 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
653adant2 979 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  e.  CC )
7 efadd 12323 . . . 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 645 . . 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 2288 . 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 8773 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C
)  e.  CC )
11103adant1 978 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  e.  CC )
12 oveq2 5786 . . . . 5  |-  ( x  =  ( B  +  C )  ->  ( A  x.  x )  =  ( A  x.  ( B  +  C
) ) )
1312fveq2d 5448 . . . 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 5458 . . . 4  |-  ( exp `  ( A  x.  ( B  +  C )
) )  e.  _V
1613, 14, 15fvmpt 5522 . . 3  |-  ( ( B  +  C )  e.  CC  ->  ( F `  ( B  +  C ) )  =  ( exp `  ( A  x.  ( B  +  C ) ) ) )
1711, 16syl 17 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( F `  ( B  +  C ) )  =  ( exp `  ( A  x.  ( B  +  C ) ) ) )
18 oveq2 5786 . . . . . 6  |-  ( x  =  B  ->  ( A  x.  x )  =  ( A  x.  B ) )
1918fveq2d 5448 . . . . 5  |-  ( x  =  B  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  B )
) )
20 fvex 5458 . . . . 5  |-  ( exp `  ( A  x.  B
) )  e.  _V
2119, 14, 20fvmpt 5522 . . . 4  |-  ( B  e.  CC  ->  ( F `  B )  =  ( exp `  ( A  x.  B )
) )
22 oveq2 5786 . . . . . 6  |-  ( x  =  C  ->  ( A  x.  x )  =  ( A  x.  C ) )
2322fveq2d 5448 . . . . 5  |-  ( x  =  C  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  C )
) )
24 fvex 5458 . . . . 5  |-  ( exp `  ( A  x.  C
) )  e.  _V
2523, 14, 24fvmpt 5522 . . . 4  |-  ( C  e.  CC  ->  ( F `  C )  =  ( exp `  ( A  x.  C )
) )
2621, 25oveqan12d 5797 . . 3  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( F `  B )  x.  ( F `  C )
)  =  ( ( exp `  ( A  x.  B ) )  x.  ( exp `  ( A  x.  C )
) ) )
27263adant1 978 . 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 2298 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 6    /\ w3a 939    = wceq 1619    e. wcel 1621    e. cmpt 4037   ` cfv 4659  (class class class)co 5778   CCcc 8689    + caddc 8694    x. cmul 8696   expce 12291
This theorem is referenced by:  efghgrp  20986
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-rep 4091  ax-sep 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470  ax-inf2 7296  ax-cnex 8747  ax-resscn 8748  ax-1cn 8749  ax-icn 8750  ax-addcl 8751  ax-addrcl 8752  ax-mulcl 8753  ax-mulrcl 8754  ax-mulcom 8755  ax-addass 8756  ax-mulass 8757  ax-distr 8758  ax-i2m1 8759  ax-1ne0 8760  ax-1rid 8761  ax-rnegex 8762  ax-rrecex 8763  ax-cnre 8764  ax-pre-lttri 8765  ax-pre-lttrn 8766  ax-pre-ltadd 8767  ax-pre-mulgt0 8768  ax-pre-sup 8769  ax-addf 8770  ax-mulf 8771
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-nel 2422  df-ral 2521  df-rex 2522  df-reu 2523  df-rmo 2524  df-rab 2525  df-v 2759  df-sbc 2953  df-csb 3043  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-iun 3867  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-se 4311  df-we 4312  df-ord 4353  df-on 4354  df-lim 4355  df-suc 4356  df-om 4615  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-isom 4676  df-ov 5781  df-oprab 5782  df-mpt2 5783  df-1st 6042  df-2nd 6043  df-iota 6211  df-riota 6258  df-recs 6342  df-rdg 6377  df-1o 6433  df-oadd 6437  df-er 6614  df-pm 6729  df-en 6818  df-dom 6819  df-sdom 6820  df-fin 6821  df-sup 7148  df-oi 7179  df-card 7526  df-pnf 8823  df-mnf 8824  df-xr 8825  df-ltxr 8826  df-le 8827  df-sub 8993  df-neg 8994  df-div 9378  df-n 9701  df-2 9758  df-3 9759  df-n0 9919  df-z 9978  df-uz 10184  df-rp 10308  df-ico 10614  df-fz 10735  df-fzo 10823  df-fl 10877  df-seq 10999  df-exp 11057  df-fac 11241  df-bc 11268  df-hash 11290  df-shft 11513  df-cj 11535  df-re 11536  df-im 11537  df-sqr 11671  df-abs 11672  df-limsup 11896  df-clim 11913  df-rlim 11914  df-sum 12110  df-ef 12297
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