Theorem rpcxpmul2 16015

Description: Product of exponents law for complex exponentiation. Variation on cxpmul 16014 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 6093	. . . . . . 7 |- ( x = 0 -> ( B x. x ) = ( B x. 0 ) )
2	1	oveq2d 6101	. . . . . 6 |- ( x = 0 -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. 0 ) ) )
3		oveq2 6093	. . . . . 6 |- ( x = 0 -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ 0 ) )
4	2, 3	eqeq12d 2253	. . . . 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 6093	. . . . . . 7 |- ( x = k -> ( B x. x ) = ( B x. k ) )
7	6	oveq2d 6101	. . . . . 6 |- ( x = k -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. k ) ) )
8		oveq2 6093	. . . . . 6 |- ( x = k -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ k ) )
9	7, 8	eqeq12d 2253	. . . . 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 6093	. . . . . . 7 |- ( x = ( k + 1 ) -> ( B x. x ) = ( B x. ( k + 1 ) ) )
12	11	oveq2d 6101	. . . . . 6 |- ( x = ( k + 1 ) -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. ( k + 1 ) ) ) )
13		oveq2 6093	. . . . . 6 |- ( x = ( k + 1 ) -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2253	. . . . 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 6093	. . . . . . 7 |- ( x = C -> ( B x. x ) = ( B x. C ) )
17	16	oveq2d 6101	. . . . . 6 |- ( x = C -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. C ) ) )
18		oveq2 6093	. . . . . 6 |- ( x = C -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ C ) )
19	17, 18	eqeq12d 2253	. . . . 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 16000	. . . . . 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 8716	. . . . . . 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 6101	. . . . 5 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( A ^c 0 ) )
26		rpcncxpcl 16004	. . . . . 6 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) e. CC )
27	26	exp0d 11105	. . . . 5 |- ( ( A e. RR+ /\ B e. CC ) -> ( ( A ^c B ) ^ 0 ) = 1 )
28	22, 25, 27	3eqtr4d 2281	. . . 4 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( ( A ^c B ) ^ 0 ) )
29		oveq1 6092	. . . . . . 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 533	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> B e. CC )
31		nn0cn 9573	. . . . . . . . . . . . 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 8342	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> 1 e. CC )
34	30, 32, 33	adddid 8350	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. ( k + 1 ) ) = ( ( B x. k ) + ( B x. 1 ) ) )
35	30	mulridd 8343	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. 1 ) = B )
36	35	oveq2d 6101	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( ( B x. k ) + ( B x. 1 ) ) = ( ( B x. k ) + B ) )
37	34, 36	eqtrd 2271	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. ( k + 1 ) ) = ( ( B x. k ) + B ) )
38	37	oveq2d 6101	. . . . . . . . 9 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( A ^c ( B x. ( k + 1 ) ) ) = ( A ^c ( ( B x. k ) + B ) ) )
39		simpll 531	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> A e. RR+ )
40	30, 32	mulcld 8346	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. k ) e. CC )
41		rpcxpadd 16007	. . . . . . . . . 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 1278	. . . . . . . . 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 2271	. . . . . . . 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 10983	. . . . . . . . 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 2253	. . . . . . 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 9760	. . 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 1231	1 |- ( ( A e. RR+ /\ B e. CC /\ C e. NN0 ) -> ( A ^c ( B x. C ) ) = ( ( A ^c B ) ^ C ) )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209  (class class class)co 6085   CCcc 8177   0cc0 8179   1c1 8180   + caddc 8182   x. cmul 8184   NN0cn0 9563   RR+crp 10054   ^cexp 10975   ^c ccxp 15958

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4246  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-mulrcl 8278  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-0lt1 8285  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-precex 8289  ax-cnre 8290  ax-pre-ltirr 8291  ax-pre-ltwlin 8292  ax-pre-lttrn 8293  ax-pre-apti 8294  ax-pre-ltadd 8295  ax-pre-mulgt0 8296  ax-pre-mulext 8297  ax-arch 8298  ax-caucvg 8299  ax-pre-suploc 8300  ax-addf 8301  ax-mulf 8302

This proof depends on definitions:  df-bi 117  df-stab 843  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-reu 2535  df-rmo 2536  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-iun 4014  df-disj 4107  df-br 4131  df-opab 4193  df-mpt 4194  df-tr 4230  df-id 4438  df-po 4441  df-iso 4442  df-iord 4511  df-on 4513  df-ilim 4514  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-isom 5386  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-of 6302  df-1st 6374  df-2nd 6375  df-recs 6576  df-irdg 6641  df-frec 6662  df-1o 6687  df-oadd 6691  df-er 6807  df-map 6924  df-pm 6925  df-en 7023  df-dom 7024  df-fin 7025  df-sup 7324  df-inf 7325  df-pnf 8362  df-mnf 8363  df-xr 8364  df-ltxr 8365  df-le 8366  df-sub 8499  df-neg 8500  df-reap 8903  df-ap 8910  df-div 9003  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-n0 9564  df-z 9645  df-uz 9922  df-q 10020  df-rp 10055  df-xneg 10174  df-xadd 10175  df-ioo 10294  df-ico 10296  df-icc 10297  df-fz 10412  df-fzo 10550  df-seqfrec 10885  df-exp 10976  df-fac 11164  df-bc 11186  df-ihash 11215  df-shft 11580  df-cj 11607  df-re 11608  df-im 11609  df-rsqrt 11764  df-abs 11765  df-clim 12045  df-sumdc 12120  df-ef 12415  df-e 12416  df-rest 13595  df-topgen 13614  df-psmet 14880  df-xmet 14881  df-met 14882  df-bl 14883  df-mopn 14884  df-top 15099  df-topon 15112  df-bases 15144  df-ntr 15197  df-cn 15289  df-cnp 15290  df-tx 15354  df-cncf 15672  df-limced 15757  df-dvap 15758  df-relog 15959  df-rpcxp 15960

This theorem is used by:  rpcxpmul2d  16034