Theorem rpmulcxp 15414

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 1000	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> A e. RR+ )
2		simp2 1001	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> B e. RR+ )
3	1, 2	relogmuld 15389	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` ( A x. B ) ) = ( ( log ` A ) + ( log ` B ) ) )
4	3	oveq2d 5962	. . . . 5 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` ( A x. B ) ) ) = ( C x. ( ( log ` A ) + ( log ` B ) ) ) )
5		simp3 1002	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> C e. CC )
6	1	relogcld 15387	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` A ) e. RR )
7	6	recnd 8103	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` A ) e. CC )
8	2	relogcld 15387	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` B ) e. RR )
9	8	recnd 8103	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( log ` B ) e. CC )
10	5, 7, 9	adddid 8099	. . . . 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 2238	. . . 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 5582	. . 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 8095	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` A ) ) e. CC )
14	5, 9	mulcld 8095	. . . 4 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( C x. ( log ` B ) ) e. CC )
15		efadd 12019	. . . 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 2238	. 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 9837	. . 3 |- ( ( A e. RR+ /\ B e. RR+ /\ C e. CC ) -> ( A x. B ) e. RR+ )
19		rpcxpef 15399	. . 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 15399	. . . 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 15399	. . . 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 5964	. 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 2248	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 981   = wceq 1373   e. wcel 2176   ` cfv 5272  (class class class)co 5946   CCcc 7925   + caddc 7930   x. cmul 7932   RR+crp 9777   expce 11986   logclog 15361   ^c ccxp 15362

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-coll 4160  ax-sep 4163  ax-nul 4171  ax-pow 4219  ax-pr 4254  ax-un 4481  ax-setind 4586  ax-iinf 4637  ax-cnex 8018  ax-resscn 8019  ax-1cn 8020  ax-1re 8021  ax-icn 8022  ax-addcl 8023  ax-addrcl 8024  ax-mulcl 8025  ax-mulrcl 8026  ax-addcom 8027  ax-mulcom 8028  ax-addass 8029  ax-mulass 8030  ax-distr 8031  ax-i2m1 8032  ax-0lt1 8033  ax-1rid 8034  ax-0id 8035  ax-rnegex 8036  ax-precex 8037  ax-cnre 8038  ax-pre-ltirr 8039  ax-pre-ltwlin 8040  ax-pre-lttrn 8041  ax-pre-apti 8042  ax-pre-ltadd 8043  ax-pre-mulgt0 8044  ax-pre-mulext 8045  ax-arch 8046  ax-caucvg 8047  ax-pre-suploc 8048  ax-addf 8049  ax-mulf 8050

This theorem depends on definitions:  df-bi 117  df-stab 833  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-nel 2472  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-csb 3094  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-if 3572  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-iun 3929  df-disj 4022  df-br 4046  df-opab 4107  df-mpt 4108  df-tr 4144  df-id 4341  df-po 4344  df-iso 4345  df-iord 4414  df-on 4416  df-ilim 4417  df-suc 4419  df-iom 4640  df-xp 4682  df-rel 4683  df-cnv 4684  df-co 4685  df-dm 4686  df-rn 4687  df-res 4688  df-ima 4689  df-iota 5233  df-fun 5274  df-fn 5275  df-f 5276  df-f1 5277  df-fo 5278  df-f1o 5279  df-fv 5280  df-isom 5281  df-riota 5901  df-ov 5949  df-oprab 5950  df-mpo 5951  df-of 6160  df-1st 6228  df-2nd 6229  df-recs 6393  df-irdg 6458  df-frec 6479  df-1o 6504  df-oadd 6508  df-er 6622  df-map 6739  df-pm 6740  df-en 6830  df-dom 6831  df-fin 6832  df-sup 7088  df-inf 7089  df-pnf 8111  df-mnf 8112  df-xr 8113  df-ltxr 8114  df-le 8115  df-sub 8247  df-neg 8248  df-reap 8650  df-ap 8657  df-div 8748  df-inn 9039  df-2 9097  df-3 9098  df-4 9099  df-n0 9298  df-z 9375  df-uz 9651  df-q 9743  df-rp 9778  df-xneg 9896  df-xadd 9897  df-ioo 10016  df-ico 10018  df-icc 10019  df-fz 10133  df-fzo 10267  df-seqfrec 10595  df-exp 10686  df-fac 10873  df-bc 10895  df-ihash 10923  df-shft 11159  df-cj 11186  df-re 11187  df-im 11188  df-rsqrt 11342  df-abs 11343  df-clim 11623  df-sumdc 11698  df-ef 11992  df-e 11993  df-rest 13106  df-topgen 13125  df-psmet 14338  df-xmet 14339  df-met 14340  df-bl 14341  df-mopn 14342  df-top 14503  df-topon 14516  df-bases 14548  df-ntr 14601  df-cn 14693  df-cnp 14694  df-tx 14758  df-cncf 15076  df-limced 15161  df-dvap 15162  df-relog 15363  df-rpcxp 15364

This theorem is referenced by:  cxprec  15415  rpdivcxp  15416  sgmmul  15501