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Theorem efgh 20396
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 9035 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
21fveq2d 5691 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( exp `  ( A  x.  ( B  +  C
) ) )  =  ( exp `  (
( A  x.  B
)  +  ( A  x.  C ) ) ) )
3 mulcl 9030 . . . . 5  |-  ( ( A  e.  CC  /\  B  e.  CC )  ->  ( A  x.  B
)  e.  CC )
433adant3 977 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  B )  e.  CC )
5 mulcl 9030 . . . . 5  |-  ( ( A  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C
)  e.  CC )
653adant2 976 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  C )  e.  CC )
7 efadd 12651 . . . 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 643 . . 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 2436 . 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 9028 . . . 4  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C
)  e.  CC )
11103adant1 975 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( B  +  C )  e.  CC )
12 oveq2 6048 . . . . 5  |-  ( x  =  ( B  +  C )  ->  ( A  x.  x )  =  ( A  x.  ( B  +  C
) ) )
1312fveq2d 5691 . . . 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 5701 . . . 4  |-  ( exp `  ( A  x.  ( B  +  C )
) )  e.  _V
1613, 14, 15fvmpt 5765 . . 3  |-  ( ( B  +  C )  e.  CC  ->  ( F `  ( B  +  C ) )  =  ( exp `  ( A  x.  ( B  +  C ) ) ) )
1711, 16syl 16 . 2  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( F `  ( B  +  C ) )  =  ( exp `  ( A  x.  ( B  +  C ) ) ) )
18 oveq2 6048 . . . . . 6  |-  ( x  =  B  ->  ( A  x.  x )  =  ( A  x.  B ) )
1918fveq2d 5691 . . . . 5  |-  ( x  =  B  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  B )
) )
20 fvex 5701 . . . . 5  |-  ( exp `  ( A  x.  B
) )  e.  _V
2119, 14, 20fvmpt 5765 . . . 4  |-  ( B  e.  CC  ->  ( F `  B )  =  ( exp `  ( A  x.  B )
) )
22 oveq2 6048 . . . . . 6  |-  ( x  =  C  ->  ( A  x.  x )  =  ( A  x.  C ) )
2322fveq2d 5691 . . . . 5  |-  ( x  =  C  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  C )
) )
24 fvex 5701 . . . . 5  |-  ( exp `  ( A  x.  C
) )  e.  _V
2523, 14, 24fvmpt 5765 . . . 4  |-  ( C  e.  CC  ->  ( F `  C )  =  ( exp `  ( A  x.  C )
) )
2621, 25oveqan12d 6059 . . 3  |-  ( ( B  e.  CC  /\  C  e.  CC )  ->  ( ( F `  B )  x.  ( F `  C )
)  =  ( ( exp `  ( A  x.  B ) )  x.  ( exp `  ( A  x.  C )
) ) )
27263adant1 975 . 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 2446 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 936    = wceq 1649    e. wcel 1721    e. cmpt 4226   ` cfv 5413  (class class class)co 6040   CCcc 8944    + caddc 8949    x. cmul 8951   expce 12619
This theorem is referenced by:  efghgrp  21914
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660  ax-inf2 7552  ax-cnex 9002  ax-resscn 9003  ax-1cn 9004  ax-icn 9005  ax-addcl 9006  ax-addrcl 9007  ax-mulcl 9008  ax-mulrcl 9009  ax-mulcom 9010  ax-addass 9011  ax-mulass 9012  ax-distr 9013  ax-i2m1 9014  ax-1ne0 9015  ax-1rid 9016  ax-rnegex 9017  ax-rrecex 9018  ax-cnre 9019  ax-pre-lttri 9020  ax-pre-lttrn 9021  ax-pre-ltadd 9022  ax-pre-mulgt0 9023  ax-pre-sup 9024  ax-addf 9025  ax-mulf 9026
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-nel 2570  df-ral 2671  df-rex 2672  df-reu 2673  df-rmo 2674  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-br 4173  df-opab 4227  df-mpt 4228  df-tr 4263  df-eprel 4454  df-id 4458  df-po 4463  df-so 4464  df-fr 4501  df-se 4502  df-we 4503  df-ord 4544  df-on 4545  df-lim 4546  df-suc 4547  df-om 4805  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-isom 5422  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-1st 6308  df-2nd 6309  df-riota 6508  df-recs 6592  df-rdg 6627  df-1o 6683  df-oadd 6687  df-er 6864  df-pm 6980  df-en 7069  df-dom 7070  df-sdom 7071  df-fin 7072  df-sup 7404  df-oi 7435  df-card 7782  df-pnf 9078  df-mnf 9079  df-xr 9080  df-ltxr 9081  df-le 9082  df-sub 9249  df-neg 9250  df-div 9634  df-nn 9957  df-2 10014  df-3 10015  df-n0 10178  df-z 10239  df-uz 10445  df-rp 10569  df-ico 10878  df-fz 11000  df-fzo 11091  df-fl 11157  df-seq 11279  df-exp 11338  df-fac 11522  df-bc 11549  df-hash 11574  df-shft 11837  df-cj 11859  df-re 11860  df-im 11861  df-sqr 11995  df-abs 11996  df-limsup 12220  df-clim 12237  df-rlim 12238  df-sum 12435  df-ef 12625
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