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Theorem efgh 19735
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 8706 . . . 4  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( A  x.  ( B  +  C ) )  =  ( ( A  x.  B )  +  ( A  x.  C ) ) )
21fveq2d 5381 . . 3  |-  ( ( A  e.  CC  /\  B  e.  CC  /\  C  e.  CC )  ->  ( exp `  ( A  x.  ( B  +  C
) ) )  =  ( exp `  (
( A  x.  B
)  +  ( A  x.  C ) ) ) )
3 mulcl 8701 . . . . 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 8701 . . . . 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 12249 . . . 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 2285 . 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 8699 . . . 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 5718 . . . . 5  |-  ( x  =  ( B  +  C )  ->  ( A  x.  x )  =  ( A  x.  ( B  +  C
) ) )
1312fveq2d 5381 . . . 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 5391 . . . 4  |-  ( exp `  ( A  x.  ( B  +  C )
) )  e.  _V
1613, 14, 15fvmpt 5454 . . 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 5718 . . . . . 6  |-  ( x  =  B  ->  ( A  x.  x )  =  ( A  x.  B ) )
1918fveq2d 5381 . . . . 5  |-  ( x  =  B  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  B )
) )
20 fvex 5391 . . . . 5  |-  ( exp `  ( A  x.  B
) )  e.  _V
2119, 14, 20fvmpt 5454 . . . 4  |-  ( B  e.  CC  ->  ( F `  B )  =  ( exp `  ( A  x.  B )
) )
22 oveq2 5718 . . . . . 6  |-  ( x  =  C  ->  ( A  x.  x )  =  ( A  x.  C ) )
2322fveq2d 5381 . . . . 5  |-  ( x  =  C  ->  ( exp `  ( A  x.  x ) )  =  ( exp `  ( A  x.  C )
) )
24 fvex 5391 . . . . 5  |-  ( exp `  ( A  x.  C
) )  e.  _V
2523, 14, 24fvmpt 5454 . . . 4  |-  ( C  e.  CC  ->  ( F `  C )  =  ( exp `  ( A  x.  C )
) )
2621, 25oveqan12d 5729 . . 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 2295 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 3974   ` cfv 4592  (class class class)co 5710   CCcc 8615    + caddc 8620    x. cmul 8622   expce 12217
This theorem is referenced by:  efghgrp  20870
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 1926  ax-ext 2234  ax-rep 4028  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403  ax-inf2 7226  ax-cnex 8673  ax-resscn 8674  ax-1cn 8675  ax-icn 8676  ax-addcl 8677  ax-addrcl 8678  ax-mulcl 8679  ax-mulrcl 8680  ax-mulcom 8681  ax-addass 8682  ax-mulass 8683  ax-distr 8684  ax-i2m1 8685  ax-1ne0 8686  ax-1rid 8687  ax-rnegex 8688  ax-rrecex 8689  ax-cnre 8690  ax-pre-lttri 8691  ax-pre-lttrn 8692  ax-pre-ltadd 8693  ax-pre-mulgt0 8694  ax-pre-sup 8695  ax-addf 8696  ax-mulf 8697
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-iun 3805  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-se 4246  df-we 4247  df-ord 4288  df-on 4289  df-lim 4290  df-suc 4291  df-om 4548  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-isom 4609  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-1st 5974  df-2nd 5975  df-iota 6143  df-riota 6190  df-recs 6274  df-rdg 6309  df-1o 6365  df-oadd 6369  df-er 6546  df-pm 6661  df-en 6750  df-dom 6751  df-sdom 6752  df-fin 6753  df-sup 7078  df-oi 7109  df-card 7456  df-pnf 8749  df-mnf 8750  df-xr 8751  df-ltxr 8752  df-le 8753  df-sub 8919  df-neg 8920  df-div 9304  df-n 9627  df-2 9684  df-3 9685  df-n0 9845  df-z 9904  df-uz 10110  df-rp 10234  df-ico 10540  df-fz 10661  df-fzo 10749  df-fl 10803  df-seq 10925  df-exp 10983  df-fac 11167  df-bc 11194  df-hash 11216  df-shft 11439  df-cj 11461  df-re 11462  df-im 11463  df-sqr 11597  df-abs 11598  df-limsup 11822  df-clim 11839  df-rlim 11840  df-sum 12036  df-ef 12223
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