Theorem rpmulcxp 13470

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 987	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> A e. RR+ )
2		simp2 988	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> B e. RR+ )
3	1, 2	relogmuld 13445	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` ( A x. B ) ) = ( ( log ` A ) + ( log ` B ) ) )
4	3	oveq2d 5858	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` ( A x. B ) ) ) = ( C x. ( ( log ` A ) + ( log ` B ) ) ) )
5		simp3 989	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> C e. CC )
6	1	relogcld 13443	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` A ) e. RR )
7	6	recnd 7927	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` A ) e. CC )
8	2	relogcld 13443	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` B ) e. RR )
9	8	recnd 7927	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` B ) e. CC )
10	5, 7, 9	adddid 7923	. . . . 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 2198	. . . 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 5490	. . 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 7919	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` A ) ) e. CC )
14	5, 9	mulcld 7919	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` B ) ) e. CC )
15		efadd 11616	. . . 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 409	. . 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 2198	. 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 9649	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( A x. B ) e. RR+ )
19		rpcxpef 13455	. . 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 409	. 2 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( ( A x. B ) ^c C ) = ( exp ` ( C x. ( log ` ( A x. B ) ) ) ) )
21		rpcxpef 13455	. . . 4 |- ( ( A e. RR+ /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
22	1, 5, 21	syl2anc 409	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( A ^c C ) = ( exp ` ( C x. ( log ` A ) ) ) )
23		rpcxpef 13455	. . . 4 |- ( ( B e. RR+ /\ C e. CC ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
24	2, 5, 23	syl2anc 409	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( B ^c C ) = ( exp ` ( C x. ( log ` B ) ) ) )
25	22, 24	oveq12d 5860	. 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 2208	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 968   = wceq 1343   e. wcel 2136   ` cfv 5188  (class class class)co 5842   CCcc 7751   + caddc 7756   x. cmul 7758   RR+crp 9589   expce 11583   logclog 13417   ^c ccxp 13418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871  ax-arch 7872  ax-caucvg 7873  ax-pre-suploc 7874  ax-addf 7875  ax-mulf 7876

This theorem depends on definitions:  df-bi 116  df-stab 821  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-disj 3960  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-isom 5197  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-of 6050  df-1st 6108  df-2nd 6109  df-recs 6273  df-irdg 6338  df-frec 6359  df-1o 6384  df-oadd 6388  df-er 6501  df-map 6616  df-pm 6617  df-en 6707  df-dom 6708  df-fin 6709  df-sup 6949  df-inf 6950  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-3 8917  df-4 8918  df-n0 9115  df-z 9192  df-uz 9467  df-q 9558  df-rp 9590  df-xneg 9708  df-xadd 9709  df-ioo 9828  df-ico 9830  df-icc 9831  df-fz 9945  df-fzo 10078  df-seqfrec 10381  df-exp 10455  df-fac 10639  df-bc 10661  df-ihash 10689  df-shft 10757  df-cj 10784  df-re 10785  df-im 10786  df-rsqrt 10940  df-abs 10941  df-clim 11220  df-sumdc 11295  df-ef 11589  df-e 11590  df-rest 12558  df-topgen 12577  df-psmet 12627  df-xmet 12628  df-met 12629  df-bl 12630  df-mopn 12631  df-top 12636  df-topon 12649  df-bases 12681  df-ntr 12736  df-cn 12828  df-cnp 12829  df-tx 12893  df-cncf 13198  df-limced 13265  df-dvap 13266  df-relog 13419  df-rpcxp 13420

This theorem is referenced by:  cxprec  13471  rpdivcxp  13472