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Theorem efgh 19898
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 8822 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
21fveq2d 5490 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( exp `  ( A  x.  ( B  +  C
) ) )  =  ( exp `  (
( A  x.  B
)  +  ( A  x.  C ) ) ) )
3 mulcl 8817 . . . . 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 8817 . . . . 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 12370 . . . 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 644 . . 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 2317 . 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 8815 . . . 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 5828 . . . . 5  |-  ( x  =  ( B  +  C )  ->  ( A  x.  x )  =  ( A  x.  ( B  +  C
) ) )
1312fveq2d 5490 . . . 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 5500 . . . 4  |-  ( exp `  ( A  x.  ( B  +  C )
) )  e.  _V
1613, 14, 15fvmpt 5564 . . 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 5828 . . . . . 6  |-  ( x  =  B  ->  ( A  x.  x )  =  ( A  x.  B ) )
1918fveq2d 5490 . . . . 5  |-  ( x  =  B  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  B )
) )
20 fvex 5500 . . . . 5  |-  ( exp `  ( A  x.  B
) )  e.  _V
2119, 14, 20fvmpt 5564 . . . 4  |-  ( B  e.  CC  ->  ( F `  B )  =  ( exp `  ( A  x.  B )
) )
22 oveq2 5828 . . . . . 6  |-  ( x  =  C  ->  ( A  x.  x )  =  ( A  x.  C ) )
2322fveq2d 5490 . . . . 5  |-  ( x  =  C  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  C )
) )
24 fvex 5500 . . . . 5  |-  ( exp `  ( A  x.  C
) )  e.  _V
2523, 14, 24fvmpt 5564 . . . 4  |-  ( C  e.  CC  ->  ( F `  C )  =  ( exp `  ( A  x.  C )
) )
2621, 25oveqan12d 5839 . . 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 2327 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 936    = wceq 1624    e. wcel 1685    e. cmpt 4079   ` cfv 5222  (class class class)co 5820   CCcc 8731    + caddc 8736    x. cmul 8738   expce 12338
This theorem is referenced by:  efghgrp  21033
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7338  ax-cnex 8789  ax-resscn 8790  ax-1cn 8791  ax-icn 8792  ax-addcl 8793  ax-addrcl 8794  ax-mulcl 8795  ax-mulrcl 8796  ax-mulcom 8797  ax-addass 8798  ax-mulass 8799  ax-distr 8800  ax-i2m1 8801  ax-1ne0 8802  ax-1rid 8803  ax-rnegex 8804  ax-rrecex 8805  ax-cnre 8806  ax-pre-lttri 8807  ax-pre-lttrn 8808  ax-pre-ltadd 8809  ax-pre-mulgt0 8810  ax-pre-sup 8811  ax-addf 8812  ax-mulf 8813
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 937  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-nel 2451  df-ral 2550  df-rex 2551  df-reu 2552  df-rmo 2553  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-pss 3170  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-tp 3650  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-br 4026  df-opab 4080  df-mpt 4081  df-tr 4116  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-se 4353  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-isom 5231  df-ov 5823  df-oprab 5824  df-mpt2 5825  df-1st 6084  df-2nd 6085  df-iota 6253  df-riota 6300  df-recs 6384  df-rdg 6419  df-1o 6475  df-oadd 6479  df-er 6656  df-pm 6771  df-en 6860  df-dom 6861  df-sdom 6862  df-fin 6863  df-sup 7190  df-oi 7221  df-card 7568  df-pnf 8865  df-mnf 8866  df-xr 8867  df-ltxr 8868  df-le 8869  df-sub 9035  df-neg 9036  df-div 9420  df-nn 9743  df-2 9800  df-3 9801  df-n0 9962  df-z 10021  df-uz 10227  df-rp 10351  df-ico 10657  df-fz 10778  df-fzo 10866  df-fl 10920  df-seq 11042  df-exp 11100  df-fac 11284  df-bc 11311  df-hash 11333  df-shft 11557  df-cj 11579  df-re 11580  df-im 11581  df-sqr 11715  df-abs 11716  df-limsup 11940  df-clim 11957  df-rlim 11958  df-sum 12154  df-ef 12344
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