Theorem rpmulcxp 13709

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 993	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> A e. RR+ )
2		simp2 994	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> B e. RR+ )
3	1, 2	relogmuld 13684	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` ( A x. B ) ) = ( ( log ` A ) + ( log ` B ) ) )
4	3	oveq2d 5873	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` ( A x. B ) ) ) = ( C x. ( ( log ` A ) + ( log ` B ) ) ) )
5		simp3 995	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> C e. CC )
6	1	relogcld 13682	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` A ) e. RR )
7	6	recnd 7952	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` A ) e. CC )
8	2	relogcld 13682	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` B ) e. RR )
9	8	recnd 7952	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` B ) e. CC )
10	5, 7, 9	adddid 7948	. . . . 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 2204	. . . 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 5503	. . 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 7944	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` A ) ) e. CC )
14	5, 9	mulcld 7944	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` B ) ) e. CC )
15		efadd 11642	. . . 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 2204	. 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 9674	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( A x. B ) e. RR+ )
19		rpcxpef 13694	. . 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 13694	. . . 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 13694	. . . 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 5875	. 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 2214	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 974   = wceq 1349   e. wcel 2142   ` cfv 5200  (class class class)co 5857   CCcc 7776   + caddc 7781   x. cmul 7783   RR+crp 9614   expce 11609   logclog 13656   ^c ccxp 13657

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 610  ax-in2 611  ax-io 705  ax-5 1441  ax-7 1442  ax-gen 1443  ax-ie1 1487  ax-ie2 1488  ax-8 1498  ax-10 1499  ax-11 1500  ax-i12 1501  ax-bndl 1503  ax-4 1504  ax-17 1520  ax-i9 1524  ax-ial 1528  ax-i5r 1529  ax-13 2144  ax-14 2145  ax-ext 2153  ax-coll 4105  ax-sep 4108  ax-nul 4116  ax-pow 4161  ax-pr 4195  ax-un 4419  ax-setind 4522  ax-iinf 4573  ax-cnex 7869  ax-resscn 7870  ax-1cn 7871  ax-1re 7872  ax-icn 7873  ax-addcl 7874  ax-addrcl 7875  ax-mulcl 7876  ax-mulrcl 7877  ax-addcom 7878  ax-mulcom 7879  ax-addass 7880  ax-mulass 7881  ax-distr 7882  ax-i2m1 7883  ax-0lt1 7884  ax-1rid 7885  ax-0id 7886  ax-rnegex 7887  ax-precex 7888  ax-cnre 7889  ax-pre-ltirr 7890  ax-pre-ltwlin 7891  ax-pre-lttrn 7892  ax-pre-apti 7893  ax-pre-ltadd 7894  ax-pre-mulgt0 7895  ax-pre-mulext 7896  ax-arch 7897  ax-caucvg 7898  ax-pre-suploc 7899  ax-addf 7900  ax-mulf 7901

This theorem depends on definitions:  df-bi 116  df-stab 827  df-dc 831  df-3or 975  df-3an 976  df-tru 1352  df-fal 1355  df-nf 1455  df-sb 1757  df-eu 2023  df-mo 2024  df-clab 2158  df-cleq 2164  df-clel 2167  df-nfc 2302  df-ne 2342  df-nel 2437  df-ral 2454  df-rex 2455  df-reu 2456  df-rmo 2457  df-rab 2458  df-v 2733  df-sbc 2957  df-csb 3051  df-dif 3124  df-un 3126  df-in 3128  df-ss 3135  df-nul 3416  df-if 3528  df-pw 3569  df-sn 3590  df-pr 3591  df-op 3593  df-uni 3798  df-int 3833  df-iun 3876  df-disj 3968  df-br 3991  df-opab 4052  df-mpt 4053  df-tr 4089  df-id 4279  df-po 4282  df-iso 4283  df-iord 4352  df-on 4354  df-ilim 4355  df-suc 4357  df-iom 4576  df-xp 4618  df-rel 4619  df-cnv 4620  df-co 4621  df-dm 4622  df-rn 4623  df-res 4624  df-ima 4625  df-iota 5162  df-fun 5202  df-fn 5203  df-f 5204  df-f1 5205  df-fo 5206  df-f1o 5207  df-fv 5208  df-isom 5209  df-riota 5813  df-ov 5860  df-oprab 5861  df-mpo 5862  df-of 6065  df-1st 6123  df-2nd 6124  df-recs 6288  df-irdg 6353  df-frec 6374  df-1o 6399  df-oadd 6403  df-er 6517  df-map 6632  df-pm 6633  df-en 6723  df-dom 6724  df-fin 6725  df-sup 6965  df-inf 6966  df-pnf 7960  df-mnf 7961  df-xr 7962  df-ltxr 7963  df-le 7964  df-sub 8096  df-neg 8097  df-reap 8498  df-ap 8505  df-div 8594  df-inn 8883  df-2 8941  df-3 8942  df-4 8943  df-n0 9140  df-z 9217  df-uz 9492  df-q 9583  df-rp 9615  df-xneg 9733  df-xadd 9734  df-ioo 9853  df-ico 9855  df-icc 9856  df-fz 9970  df-fzo 10103  df-seqfrec 10406  df-exp 10480  df-fac 10664  df-bc 10686  df-ihash 10714  df-shft 10783  df-cj 10810  df-re 10811  df-im 10812  df-rsqrt 10966  df-abs 10967  df-clim 11246  df-sumdc 11321  df-ef 11615  df-e 11616  df-rest 12585  df-topgen 12604  df-psmet 12866  df-xmet 12867  df-met 12868  df-bl 12869  df-mopn 12870  df-top 12875  df-topon 12888  df-bases 12920  df-ntr 12975  df-cn 13067  df-cnp 13068  df-tx 13132  df-cncf 13437  df-limced 13504  df-dvap 13505  df-relog 13658  df-rpcxp 13659

This theorem is referenced by:  cxprec  13710  rpdivcxp  13711