Theorem rpmulcxp 14626

Description: Complex exponentiation of a product. Proposition 10-4.2(c) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Assertion

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

Proof of Theorem rpmulcxp

Step	Hyp	Ref	Expression
1		simp1 998	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> A e. RR+ )
2		simp2 999	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> B e. RR+ )
3	1, 2	relogmuld 14601	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` ( A x. B ) ) = ( ( log ` A ) + ( log ` B ) ) )
4	3	oveq2d 5904	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` ( A x. B ) ) ) = ( C x. ( ( log ` A ) + ( log ` B ) ) ) )
5		simp3 1000	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> C e. CC )
6	1	relogcld 14599	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` A ) e. RR )
7	6	recnd 8000	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` A ) e. CC )
8	2	relogcld 14599	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` B ) e. RR )
9	8	recnd 8000	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` B ) e. CC )
10	5, 7, 9	adddid 7996	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( ( log ` A ) + ( log ` B ) ) ) = ( ( C x. ( log ` A ) ) + ( C x. ( log ` B ) ) ) )
11	4, 10	eqtrd 2220	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` ( A x. B ) ) ) = ( ( C x. ( log ` A ) ) + ( C x. ( log ` B ) ) ) )
12	11	fveq2d 5531	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( exp ` ( C x. ( log ` ( A x. B ) ) ) ) = ( exp ` ( ( C x. ( log ` A ) ) + ( C x. ( log ` B ) ) ) ) )
13	5, 7	mulcld 7992	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` A ) ) e. CC )
14	5, 9	mulcld 7992	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` B ) ) e. CC )
15		efadd 11697	. . . 4 |- ( ( ( C x. ( log ` A ) ) e. CC /\ ( C x. ( log ` B ) ) e. CC ) -> ( exp ` ( ( C x. ( log ` A ) ) + ( C x. ( log ` B ) ) ) ) = ( ( exp ` ( C x. ( log ` A ) ) ) x. ( exp ` ( C x. ( log ` B ) ) ) ) )
16	13, 14, 15	syl2anc 411	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( exp ` ( ( C x. ( log ` A ) ) + ( C x. ( log ` B ) ) ) ) = ( ( exp ` ( C x. ( log ` A ) ) ) x. ( exp ` ( C x. ( log ` B ) ) ) ) )
17	12, 16	eqtrd 2220	. 2 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( exp ` ( C x. ( log ` ( A x. B ) ) ) ) = ( ( exp ` ( C x. ( log ` A ) ) ) x. ( exp ` ( C x. ( log ` B ) ) ) ) )
18	1, 2	rpmulcld 9727	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( A x. B ) e. RR+ )
19		rpcxpef 14611	. . 3 |- ( ( ( A x. B ) e. RR+ /\ C e. CC ) -> ( ( A x. B ) ^c C ) = ( exp ` ( C x. ( log ` ( A x. B ) ) ) ) )
20	18, 5, 19	syl2anc 411	. 2 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( ( A x. B ) ^c C ) = ( exp ` ( C x. ( log ` ( A x. B ) ) ) ) )
21		rpcxpef 14611	. . . 4 |- ( ( A e. RR+ /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
22	1, 5, 21	syl2anc 411	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
23		rpcxpef 14611	. . . 4 |- ( ( B e. RR+ /\ C e. CC ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
24	2, 5, 23	syl2anc 411	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
25	22, 24	oveq12d 5906	. 2 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( ( A ^c C ) x. ( B ^c C ) ) = ( ( exp ` ( C x. ( log ` A ) ) ) x. ( exp ` ( C x. ( log ` B ) ) ) ) )
26	17, 20, 25	3eqtr4d 2230	1 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( ( A x. B ) ^c C ) = ( ( A ^c C ) x. ( B ^c C ) ) )

Syntax hints:   -> wi 4   /\ w3a 979   = wceq 1363   e. wcel 2158   ` cfv 5228  (class class class)co 5888   CCcc 7823   + caddc 7828   x. cmul 7830   RR+crp 9667   expce 11664   logclog 14573   ^c ccxp 14574

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599  ax-cnex 7916  ax-resscn 7917  ax-1cn 7918  ax-1re 7919  ax-icn 7920  ax-addcl 7921  ax-addrcl 7922  ax-mulcl 7923  ax-mulrcl 7924  ax-addcom 7925  ax-mulcom 7926  ax-addass 7927  ax-mulass 7928  ax-distr 7929  ax-i2m1 7930  ax-0lt1 7931  ax-1rid 7932  ax-0id 7933  ax-rnegex 7934  ax-precex 7935  ax-cnre 7936  ax-pre-ltirr 7937  ax-pre-ltwlin 7938  ax-pre-lttrn 7939  ax-pre-apti 7940  ax-pre-ltadd 7941  ax-pre-mulgt0 7942  ax-pre-mulext 7943  ax-arch 7944  ax-caucvg 7945  ax-pre-suploc 7946  ax-addf 7947  ax-mulf 7948

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-reu 2472  df-rmo 2473  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-disj 3993  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-po 4308  df-iso 4309  df-iord 4378  df-on 4380  df-ilim 4381  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-isom 5237  df-riota 5844  df-ov 5891  df-oprab 5892  df-mpo 5893  df-of 6097  df-1st 6155  df-2nd 6156  df-recs 6320  df-irdg 6385  df-frec 6406  df-1o 6431  df-oadd 6435  df-er 6549  df-map 6664  df-pm 6665  df-en 6755  df-dom 6756  df-fin 6757  df-sup 6997  df-inf 6998  df-pnf 8008  df-mnf 8009  df-xr 8010  df-ltxr 8011  df-le 8012  df-sub 8144  df-neg 8145  df-reap 8546  df-ap 8553  df-div 8644  df-inn 8934  df-2 8992  df-3 8993  df-4 8994  df-n0 9191  df-z 9268  df-uz 9543  df-q 9634  df-rp 9668  df-xneg 9786  df-xadd 9787  df-ioo 9906  df-ico 9908  df-icc 9909  df-fz 10023  df-fzo 10157  df-seqfrec 10460  df-exp 10534  df-fac 10720  df-bc 10742  df-ihash 10770  df-shft 10838  df-cj 10865  df-re 10866  df-im 10867  df-rsqrt 11021  df-abs 11022  df-clim 11301  df-sumdc 11376  df-ef 11670  df-e 11671  df-rest 12708  df-topgen 12727  df-psmet 13729  df-xmet 13730  df-met 13731  df-bl 13732  df-mopn 13733  df-top 13794  df-topon 13807  df-bases 13839  df-ntr 13892  df-cn 13984  df-cnp 13985  df-tx 14049  df-cncf 14354  df-limced 14421  df-dvap 14422  df-relog 14575  df-rpcxp 14576

This theorem is referenced by:  cxprec  14627  rpdivcxp  14628