Theorem rpcxpadd 16060

Description: Sum of exponents law for complex exponentiation. (Contributed by Mario Carneiro, 2-Aug-2014.) (Revised by Jim Kingdon, 13-Jun-2024.)

Assertion

Ref	Expression
rpcxpadd	|- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A ^c ( B + C ) ) = ( ( A ^c B ) x. ( A ^c C ) ) )

Proof of Theorem rpcxpadd

Step	Hyp	Ref	Expression
1		simp2 1029	. . . . 5 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> B e. CC )
2		simp3 1030	. . . . 5 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> C e. CC )
3		relogcl 16013	. . . . . . 7 |- ( A e. RR+ -> ( log ` A ) e. RR )
4	3	3ad2ant1 1049	. . . . . 6 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( log ` A ) e. RR )
5	4	recnd 8354	. . . . 5 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( log ` A ) e. CC )
6	1, 2, 5	adddird 8351	. . . 4 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( ( B + C ) x. ( log ` A ) ) = ( ( B x. ( log ` A ) ) + ( C x. ( log ` A ) ) ) )
7	6	fveq2d 5699	. . 3 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( exp ` ( ( B + C ) x. ( log ` A ) ) ) = ( exp ` ( ( B x. ( log ` A ) ) + ( C x. ( log ` A ) ) ) ) )
8	1, 5	mulcld 8346	. . . 4 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( B x. ( log ` A ) ) e. CC )
9	2, 5	mulcld 8346	. . . 4 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( C x. ( log ` A ) ) e. CC )
10		efadd 12458	. . . 4 |- ( ( ( B x. ( log ` A ) ) e. CC /\ ( C x. ( log ` A ) ) e. CC ) -> ( exp ` ( ( B x. ( log ` A ) ) + ( C x. ( log ` A ) ) ) ) = ( ( exp ` ( B x. ( log ` A ) ) ) x. ( exp ` ( C x. ( log ` A ) ) ) ) )
11	8, 9, 10	syl2anc 415	. . 3 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( exp ` ( ( B x. ( log ` A ) ) + ( C x. ( log ` A ) ) ) ) = ( ( exp ` ( B x. ( log ` A ) ) ) x. ( exp ` ( C x. ( log ` A ) ) ) ) )
12	7, 11	eqtrd 2271	. 2 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( exp ` ( ( B + C ) x. ( log ` A ) ) ) = ( ( exp ` ( B x. ( log ` A ) ) ) x. ( exp ` ( C x. ( log ` A ) ) ) ) )
13		simp1 1028	. . 3 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> A e. RR+ )
14	1, 2	addcld 8345	. . 3 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( B + C ) e. CC )
15		rpcxpef 16049	. . 3 |- ( ( A e. RR+ /\ ( B + C ) e. CC ) -> ( A ^c ( B + C ) ) = ( exp ` ( ( B + C ) x. ( log ` A ) ) ) )
16	13, 14, 15	syl2anc 415	. 2 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A ^c ( B + C ) ) = ( exp ` ( ( B + C ) x. ( log ` A ) ) ) )
17		rpcxpef 16049	. . . 4 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
18	13, 1, 17	syl2anc 415	. . 3 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
19		rpcxpef 16049	. . . 4 |- ( ( A e. RR+ /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
20	13, 2, 19	syl2anc 415	. . 3 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
21	18, 20	oveq12d 6103	. 2 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( ( A ^c B ) x. ( A ^c C ) ) = ( ( exp ` ( B x. ( log ` A ) ) ) x. ( exp ` ( C x. ( log ` A ) ) ) ) )
22	12, 16, 21	3eqtr4d 2281	1 |- ( ( A e. RR+ /\ B e. CC /\ C e. CC ) -> ( A ^c ( B + C ) ) = ( ( A ^c B ) x. ( A ^c C ) ) )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   ` cfv 5377  (class class class)co 6085   CCcc 8177   RRcr 8178   + caddc 8182   x. cmul 8184   RR+crp 10064   expce 12425   logclog 16007   ^c ccxp 16008

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8500  df-neg 8501  df-reap 8905  df-ap 8912  df-div 9005  df-inn 9307  df-2 9365  df-3 9366  df-4 9367  df-n0 9568  df-z 9649  df-uz 9931  df-q 10029  df-rp 10065  df-xneg 10184  df-xadd 10185  df-ioo 10304  df-ico 10306  df-icc 10307  df-fz 10422  df-fzo 10560  df-seqfrec 10898  df-exp 10989  df-fac 11178  df-bc 11200  df-ihash 11229  df-shft 11594  df-cj 11621  df-re 11622  df-im 11623  df-rsqrt 11778  df-abs 11779  df-clim 12061  df-sumdc 12136  df-ef 12431  df-e 12432  df-rest 13644  df-topgen 13663  df-psmet 14929  df-xmet 14930  df-met 14931  df-bl 14932  df-mopn 14933  df-top 15148  df-topon 15161  df-bases 15193  df-ntr 15246  df-cn 15338  df-cnp 15339  df-tx 15403  df-cncf 15721  df-limced 15806  df-dvap 15807  df-relog 16009  df-rpcxp 16010

This theorem is used by:  rpcxpp1  16061  rpcxpneg  16062  rpcxpsub  16063  rpcxpmul2  16068  rpcxpsqrt  16077