Theorem cxpmul 13180

Description: Product of exponents law for complex exponentiation. Proposition 10-4.2(b) of [Gleason] p. 135. (Contributed by Mario Carneiro, 2-Aug-2014.)

Assertion

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

Proof of Theorem cxpmul

Step	Hyp	Ref	Expression
1		simp3 984	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> C e. CC )
2		simp2 983	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> B e. RR )
3	2	recnd 7885	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> B e. CC )
4		relogcl 13130	. . . . . . 7 |- ( A e. RR+ -> ( log ` A ) e. RR )
5	4	3ad2ant1 1003	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( log ` A ) e. RR )
6	5	recnd 7885	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( log ` A ) e. CC )
7	1, 3, 6	mulassd 7880	. . . 4 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( ( C x. B ) x. ( log ` A ) ) = ( C x. ( B x. ( log ` A ) ) ) )
8	3, 1	mulcomd 7878	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
9	8	oveq1d 5829	. . . 4 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( ( B x. C ) x. ( log ` A ) ) = ( ( C x. B ) x. ( log ` A ) ) )
10		simp1 982	. . . . . . . 8 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> A e. RR+ )
11		rpcxpef 13162	. . . . . . . 8 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
12	10, 3, 11	syl2anc 409	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
13	12	fveq2d 5465	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( log ` ( A ^c B ) ) = ( log ` ( exp ` ( B x. ( log ` A ) ) ) ) )
14	2, 5	remulcld 7887	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( B x. ( log ` A ) ) e. RR )
15	14	relogefd 13155	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( log ` ( exp ` ( B x. ( log ` A ) ) ) ) = ( B x. ( log ` A ) ) )
16	13, 15	eqtrd 2187	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( log ` ( A ^c B ) ) = ( B x. ( log ` A ) ) )
17	16	oveq2d 5830	. . . 4 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( C x. ( log ` ( A ^c B ) ) ) = ( C x. ( B x. ( log ` A ) ) ) )
18	7, 9, 17	3eqtr4d 2197	. . 3 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( ( B x. C ) x. ( log ` A ) ) = ( C x. ( log ` ( A ^c B ) ) ) )
19	18	fveq2d 5465	. 2 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( exp ` ( ( B x. C ) x. ( log ` A ) ) ) = ( exp ` ( C x. ( log ` ( A ^c B ) ) ) ) )
20	3, 1	mulcld 7877	. . 3 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( B x. C ) e. CC )
21		rpcxpef 13162	. . 3 |- ( ( A e. RR+ /\ ( B x. C ) e. CC ) -> ( A ^c ( B x. C ) ) = ( exp ` ( ( B x. C ) x. ( log ` A ) ) ) )
22	10, 20, 21	syl2anc 409	. 2 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( A ^c ( B x. C ) ) = ( exp ` ( ( B x. C ) x. ( log ` A ) ) ) )
23		rpcxpcl 13171	. . 3 |- ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) e. RR+ )
24		rpcxpef 13162	. . 3 |- ( ( ( A ^c B ) e. RR+ /\ C e. CC ) -> ( ( A ^c B ) ^c C ) = ( exp ` ( C x. ( log ` ( A ^c B ) ) ) ) )
25	23, 24	stoic3 1408	. 2 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( ( A ^c B ) ^c C ) = ( exp ` ( C x. ( log ` ( A ^c B ) ) ) ) )
26	19, 22, 25	3eqtr4d 2197	1 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^c C ) )

Syntax hints:   -> wi 4   /\ w3a 963   = wceq 1332   e. wcel 2125   ` cfv 5163  (class class class)co 5814   CCcc 7709   RRcr 7710   x. cmul 7716   RR+crp 9538   expce 11516   logclog 13124   ^c ccxp 13125

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-coll 4075  ax-sep 4078  ax-nul 4086  ax-pow 4130  ax-pr 4164  ax-un 4388  ax-setind 4490  ax-iinf 4541  ax-cnex 7802  ax-resscn 7803  ax-1cn 7804  ax-1re 7805  ax-icn 7806  ax-addcl 7807  ax-addrcl 7808  ax-mulcl 7809  ax-mulrcl 7810  ax-addcom 7811  ax-mulcom 7812  ax-addass 7813  ax-mulass 7814  ax-distr 7815  ax-i2m1 7816  ax-0lt1 7817  ax-1rid 7818  ax-0id 7819  ax-rnegex 7820  ax-precex 7821  ax-cnre 7822  ax-pre-ltirr 7823  ax-pre-ltwlin 7824  ax-pre-lttrn 7825  ax-pre-apti 7826  ax-pre-ltadd 7827  ax-pre-mulgt0 7828  ax-pre-mulext 7829  ax-arch 7830  ax-caucvg 7831  ax-pre-suploc 7832  ax-addf 7833  ax-mulf 7834

This theorem depends on definitions:  df-bi 116  df-stab 817  df-dc 821  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ne 2325  df-nel 2420  df-ral 2437  df-rex 2438  df-reu 2439  df-rmo 2440  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-dif 3100  df-un 3102  df-in 3104  df-ss 3111  df-nul 3391  df-if 3502  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-int 3804  df-iun 3847  df-disj 3939  df-br 3962  df-opab 4022  df-mpt 4023  df-tr 4059  df-id 4248  df-po 4251  df-iso 4252  df-iord 4321  df-on 4323  df-ilim 4324  df-suc 4326  df-iom 4544  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-f1 5168  df-fo 5169  df-f1o 5170  df-fv 5171  df-isom 5172  df-riota 5770  df-ov 5817  df-oprab 5818  df-mpo 5819  df-of 6022  df-1st 6078  df-2nd 6079  df-recs 6242  df-irdg 6307  df-frec 6328  df-1o 6353  df-oadd 6357  df-er 6469  df-map 6584  df-pm 6585  df-en 6675  df-dom 6676  df-fin 6677  df-sup 6916  df-inf 6917  df-pnf 7893  df-mnf 7894  df-xr 7895  df-ltxr 7896  df-le 7897  df-sub 8027  df-neg 8028  df-reap 8429  df-ap 8436  df-div 8525  df-inn 8813  df-2 8871  df-3 8872  df-4 8873  df-n0 9070  df-z 9147  df-uz 9419  df-q 9507  df-rp 9539  df-xneg 9657  df-xadd 9658  df-ioo 9774  df-ico 9776  df-icc 9777  df-fz 9891  df-fzo 10020  df-seqfrec 10323  df-exp 10397  df-fac 10577  df-bc 10599  df-ihash 10627  df-shft 10692  df-cj 10719  df-re 10720  df-im 10721  df-rsqrt 10875  df-abs 10876  df-clim 11153  df-sumdc 11228  df-ef 11522  df-e 11523  df-rest 12300  df-topgen 12319  df-psmet 12334  df-xmet 12335  df-met 12336  df-bl 12337  df-mopn 12338  df-top 12343  df-topon 12356  df-bases 12388  df-ntr 12443  df-cn 12535  df-cnp 12536  df-tx 12600  df-cncf 12905  df-limced 12972  df-dvap 12973  df-relog 13126  df-rpcxp 13127

This theorem is referenced by:  rpcxproot  13181  cxpmuld  13203  cxpcom  13204