Theorem cxpmul 14521

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 999	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> C e. CC )
2		simp2 998	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> B e. RR )
3	2	recnd 7989	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> B e. CC )
4		relogcl 14471	. . . . . . 7 |- ( A e. RR+ -> ( log ` A ) e. RR )
5	4	3ad2ant1 1018	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( log ` A ) e. RR )
6	5	recnd 7989	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( log ` A ) e. CC )
7	1, 3, 6	mulassd 7984	. . . 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 7982	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( B x. C ) = ( C x. B ) )
9	8	oveq1d 5893	. . . 4 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( ( B x. C ) x. ( log ` A ) ) = ( ( C x. B ) x. ( log ` A ) ) )
10		simp1 997	. . . . . . . 8 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> A e. RR+ )
11		rpcxpef 14503	. . . . . . . 8 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
12	10, 3, 11	syl2anc 411	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( A ^c B ) = ( exp ` ( B x. ( log ` A ) ) ) )
13	12	fveq2d 5521	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( log ` ( A ^c B ) ) = ( log ` ( exp ` ( B x. ( log ` A ) ) ) ) )
14	2, 5	remulcld 7991	. . . . . . 7 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( B x. ( log ` A ) ) e. RR )
15	14	relogefd 14496	. . . . . 6 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( log ` ( exp ` ( B x. ( log ` A ) ) ) ) = ( B x. ( log ` A ) ) )
16	13, 15	eqtrd 2210	. . . . 5 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( log ` ( A ^c B ) ) = ( B x. ( log ` A ) ) )
17	16	oveq2d 5894	. . . 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 2220	. . 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 5521	. 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 7981	. . 3 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( B x. C ) e. CC )
21		rpcxpef 14503	. . 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 411	. 2 |- ( ( A e. RR+ /\ B e. RR /\ C e. CC ) -> ( A ^c ( B x. C ) ) = ( exp ` ( ( B x. C ) x. ( log ` A ) ) ) )
23		rpcxpcl 14512	. . 3 |- ( ( A e. RR+ /\ B e. RR ) -> ( A ^c B ) e. RR+ )
24		rpcxpef 14503	. . 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 1431	. 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 2220	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 978   = wceq 1353   e. wcel 2148   ` cfv 5218  (class class class)co 5878   CCcc 7812   RRcr 7813   x. cmul 7819   RR+crp 9656   expce 11653   logclog 14465   ^c ccxp 14466

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4120  ax-sep 4123  ax-nul 4131  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-iinf 4589  ax-cnex 7905  ax-resscn 7906  ax-1cn 7907  ax-1re 7908  ax-icn 7909  ax-addcl 7910  ax-addrcl 7911  ax-mulcl 7912  ax-mulrcl 7913  ax-addcom 7914  ax-mulcom 7915  ax-addass 7916  ax-mulass 7917  ax-distr 7918  ax-i2m1 7919  ax-0lt1 7920  ax-1rid 7921  ax-0id 7922  ax-rnegex 7923  ax-precex 7924  ax-cnre 7925  ax-pre-ltirr 7926  ax-pre-ltwlin 7927  ax-pre-lttrn 7928  ax-pre-apti 7929  ax-pre-ltadd 7930  ax-pre-mulgt0 7931  ax-pre-mulext 7932  ax-arch 7933  ax-caucvg 7934  ax-pre-suploc 7935  ax-addf 7936  ax-mulf 7937

This theorem depends on definitions:  df-bi 117  df-stab 831  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-if 3537  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-int 3847  df-iun 3890  df-disj 3983  df-br 4006  df-opab 4067  df-mpt 4068  df-tr 4104  df-id 4295  df-po 4298  df-iso 4299  df-iord 4368  df-on 4370  df-ilim 4371  df-suc 4373  df-iom 4592  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-isom 5227  df-riota 5834  df-ov 5881  df-oprab 5882  df-mpo 5883  df-of 6086  df-1st 6144  df-2nd 6145  df-recs 6309  df-irdg 6374  df-frec 6395  df-1o 6420  df-oadd 6424  df-er 6538  df-map 6653  df-pm 6654  df-en 6744  df-dom 6745  df-fin 6746  df-sup 6986  df-inf 6987  df-pnf 7997  df-mnf 7998  df-xr 7999  df-ltxr 8000  df-le 8001  df-sub 8133  df-neg 8134  df-reap 8535  df-ap 8542  df-div 8633  df-inn 8923  df-2 8981  df-3 8982  df-4 8983  df-n0 9180  df-z 9257  df-uz 9532  df-q 9623  df-rp 9657  df-xneg 9775  df-xadd 9776  df-ioo 9895  df-ico 9897  df-icc 9898  df-fz 10012  df-fzo 10146  df-seqfrec 10449  df-exp 10523  df-fac 10709  df-bc 10731  df-ihash 10759  df-shft 10827  df-cj 10854  df-re 10855  df-im 10856  df-rsqrt 11010  df-abs 11011  df-clim 11290  df-sumdc 11365  df-ef 11659  df-e 11660  df-rest 12696  df-topgen 12715  df-psmet 13621  df-xmet 13622  df-met 13623  df-bl 13624  df-mopn 13625  df-top 13686  df-topon 13699  df-bases 13731  df-ntr 13784  df-cn 13876  df-cnp 13877  df-tx 13941  df-cncf 14246  df-limced 14313  df-dvap 14314  df-relog 14467  df-rpcxp 14468

This theorem is referenced by:  rpcxproot  14522  cxpmuld  14544  cxpcom  14545