Theorem rpcxpmul2 15581

Description: Product of exponents law for complex exponentiation. Variation on cxpmul 15580 with more general conditions on A and B when C is a nonnegative integer. (Contributed by Mario Carneiro, 9-Aug-2014.)

Assertion

Ref	Expression
rpcxpmul2	|- ( ( A e. RR+ /\ B e. CC /\ C e. NN0 ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )

Proof of Theorem rpcxpmul2

Dummy variables x k are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		oveq2 6008	. . . . . . 7 |- ( x = 0 -> ( B x. x ) = ( B x. 0 ) )
2	1	oveq2d 6016	. . . . . 6 |- ( x = 0 -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. 0 ) ) )
3		oveq2 6008	. . . . . 6 |- ( x = 0 -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ 0 ) )
4	2, 3	eqeq12d 2244	. . . . 5 |- ( x = 0 -> ( ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) <-> ( A ^c ( B x. 0 ) ) = ( ( A ^c B ) ^ 0 ) ) )
5	4	imbi2d 230	. . . 4 |- ( x = 0 -> ( ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) ) <-> ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( ( A ^c B ) ^ 0 ) ) ) )
6		oveq2 6008	. . . . . . 7 |- ( x = k -> ( B x. x ) = ( B x. k ) )
7	6	oveq2d 6016	. . . . . 6 |- ( x = k -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. k ) ) )
8		oveq2 6008	. . . . . 6 |- ( x = k -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ k ) )
9	7, 8	eqeq12d 2244	. . . . 5 |- ( x = k -> ( ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) <-> ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) ) )
10	9	imbi2d 230	. . . 4 |- ( x = k -> ( ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) ) <-> ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) ) ) )
11		oveq2 6008	. . . . . . 7 |- ( x = ( k + 1 ) -> ( B x. x ) = ( B x. ( k + 1 ) ) )
12	11	oveq2d 6016	. . . . . 6 |- ( x = ( k + 1 ) -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. ( k + 1 ) ) ) )
13		oveq2 6008	. . . . . 6 |- ( x = ( k + 1 ) -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2244	. . . . 5 |- ( x = ( k + 1 ) -> ( ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) <-> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) ) )
15	14	imbi2d 230	. . . 4 |- ( x = ( k + 1 ) -> ( ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) ) <-> ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) ) ) )
16		oveq2 6008	. . . . . . 7 |- ( x = C -> ( B x. x ) = ( B x. C ) )
17	16	oveq2d 6016	. . . . . 6 |- ( x = C -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. C ) ) )
18		oveq2 6008	. . . . . 6 |- ( x = C -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ C ) )
19	17, 18	eqeq12d 2244	. . . . 5 |- ( x = C -> ( ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) <-> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) )
20	19	imbi2d 230	. . . 4 |- ( x = C -> ( ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. x ) ) = ( ( A ^c B ) ^ x ) ) <-> ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) ) )
21		rpcxp0 15566	. . . . . 6 |- ( A e. RR+ -> ( A ^c 0 ) = 1 )
22	21	adantr 276	. . . . 5 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c 0 ) = 1 )
23		mul01 8531	. . . . . . 7 |- ( B e. CC -> ( B x. 0 ) = 0 )
24	23	adantl 277	. . . . . 6 |- ( ( A e. RR+ /\ B e. CC ) -> ( B x. 0 ) = 0 )
25	24	oveq2d 6016	. . . . 5 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( A ^c 0 ) )
26		rpcncxpcl 15570	. . . . . 6 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) e. CC )
27	26	exp0d 10884	. . . . 5 |- ( ( A e. RR+ /\ B e. CC ) -> ( ( A ^c B ) ^ 0 ) = 1 )
28	22, 25, 27	3eqtr4d 2272	. . . 4 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( ( A ^c B ) ^ 0 ) )
29		oveq1 6007	. . . . . . 7 |- ( ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) -> ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) = ( ( ( A ^c B ) ^ k ) x. ( A ^c B ) ) )
30		simplr 528	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> B e. CC )
31		nn0cn 9375	. . . . . . . . . . . . 13 |- ( k e. NN0 -> k e. CC )
32	31	adantl 277	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> k e. CC )
33		1cnd 8158	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> 1 e. CC )
34	30, 32, 33	adddid 8167	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. ( k + 1 ) ) = ( ( B x. k ) + ( B x. 1 ) ) )
35	30	mulridd 8159	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. 1 ) = B )
36	35	oveq2d 6016	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( ( B x. k ) + ( B x. 1 ) ) = ( ( B x. k ) + B ) )
37	34, 36	eqtrd 2262	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. ( k + 1 ) ) = ( ( B x. k ) + B ) )
38	37	oveq2d 6016	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( A ^c ( ( B x. k ) + B ) ) )
39		simpll 527	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> A e. RR+ )
40	30, 32	mulcld 8163	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. k ) e. CC )
41		rpcxpadd 15573	. . . . . . . . . 10 |- ( ( A e. RR+ /\ ( B x. k ) e. CC /\ B e. CC ) -> ( A ^c ( ( B x. k ) + B ) ) = ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) )
42	39, 40, 30, 41	syl3anc 1271	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( A ^c ( ( B x. k ) + B ) ) = ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) )
43	38, 42	eqtrd 2262	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) )
44		expp1 10763	. . . . . . . . 9 |- ( ( ( A ^c B ) e. CC /\ k e. NN0 ) -> ( ( A ^c B ) ^ ( k + 1 ) ) = ( ( ( A ^c B ) ^ k ) x. ( A ^c B ) ) )
45	26, 44	sylan 283	. . . . . . . 8 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( ( A ^c B ) ^ ( k + 1 ) ) = ( ( ( A ^c B ) ^ k ) x. ( A ^c B ) ) )
46	43, 45	eqeq12d 2244	. . . . . . 7 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) <-> ( ( A ^c ( B x. k ) ) x. ( A ^c B ) ) = ( ( ( A ^c B ) ^ k ) x. ( A ^c B ) ) ) )
47	29, 46	imbitrrid 156	. . . . . 6 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) ) )
48	47	expcom 116	. . . . 5 |- ( k e. NN0 -> ( ( A e. RR+ /\ B e. CC ) -> ( ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) ) ) )
49	48	a2d 26	. . . 4 |- ( k e. NN0 -> ( ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. k ) ) = ( ( A ^c B ) ^ k ) ) -> ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( ( A ^c B ) ^ ( k + 1 ) ) ) ) )
50	5, 10, 15, 20, 28, 49	nn0ind 9557	. . 3 |- ( C e. NN0 -> ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) )
51	50	com12 30	. 2 |- ( ( A e. RR+ /\ B e. CC ) -> ( C e. NN0 -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) ) )
52	51	3impia 1224	1 |- ( ( A e. RR+ /\ B e. CC /\ C e. NN0 ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1002   = wceq 1395   e. wcel 2200  (class class class)co 6000   CCcc 7993   0cc0 7995   1c1 7996   + caddc 7998   x. cmul 8000   NN0cn0 9365   RR+crp 9845   ^cexp 10755   ^c ccxp 15525

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679  ax-cnex 8086  ax-resscn 8087  ax-1cn 8088  ax-1re 8089  ax-icn 8090  ax-addcl 8091  ax-addrcl 8092  ax-mulcl 8093  ax-mulrcl 8094  ax-addcom 8095  ax-mulcom 8096  ax-addass 8097  ax-mulass 8098  ax-distr 8099  ax-i2m1 8100  ax-0lt1 8101  ax-1rid 8102  ax-0id 8103  ax-rnegex 8104  ax-precex 8105  ax-cnre 8106  ax-pre-ltirr 8107  ax-pre-ltwlin 8108  ax-pre-lttrn 8109  ax-pre-apti 8110  ax-pre-ltadd 8111  ax-pre-mulgt0 8112  ax-pre-mulext 8113  ax-arch 8114  ax-caucvg 8115  ax-pre-suploc 8116  ax-addf 8117  ax-mulf 8118

This theorem depends on definitions:  df-bi 117  df-stab 836  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-reu 2515  df-rmo 2516  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-iun 3966  df-disj 4059  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-po 4386  df-iso 4387  df-iord 4456  df-on 4458  df-ilim 4459  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326  df-riota 5953  df-ov 6003  df-oprab 6004  df-mpo 6005  df-of 6216  df-1st 6284  df-2nd 6285  df-recs 6449  df-irdg 6514  df-frec 6535  df-1o 6560  df-oadd 6564  df-er 6678  df-map 6795  df-pm 6796  df-en 6886  df-dom 6887  df-fin 6888  df-sup 7147  df-inf 7148  df-pnf 8179  df-mnf 8180  df-xr 8181  df-ltxr 8182  df-le 8183  df-sub 8315  df-neg 8316  df-reap 8718  df-ap 8725  df-div 8816  df-inn 9107  df-2 9165  df-3 9166  df-4 9167  df-n0 9366  df-z 9443  df-uz 9719  df-q 9811  df-rp 9846  df-xneg 9964  df-xadd 9965  df-ioo 10084  df-ico 10086  df-icc 10087  df-fz 10201  df-fzo 10335  df-seqfrec 10665  df-exp 10756  df-fac 10943  df-bc 10965  df-ihash 10993  df-shft 11321  df-cj 11348  df-re 11349  df-im 11350  df-rsqrt 11504  df-abs 11505  df-clim 11785  df-sumdc 11860  df-ef 12154  df-e 12155  df-rest 13269  df-topgen 13288  df-psmet 14501  df-xmet 14502  df-met 14503  df-bl 14504  df-mopn 14505  df-top 14666  df-topon 14679  df-bases 14711  df-ntr 14764  df-cn 14856  df-cnp 14857  df-tx 14921  df-cncf 15239  df-limced 15324  df-dvap 15325  df-relog 15526  df-rpcxp 15527

This theorem is referenced by:  rpcxpmul2d  15600