Theorem rpcxpmul2 15122

Description: Product of exponents law for complex exponentiation. Variation on cxpmul 15121 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 5930	. . . . . . 7 |- ( x = 0 -> ( B x. x ) = ( B x. 0 ) )
2	1	oveq2d 5938	. . . . . 6 |- ( x = 0 -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. 0 ) ) )
3		oveq2 5930	. . . . . 6 |- ( x = 0 -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ 0 ) )
4	2, 3	eqeq12d 2211	. . . . 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 5930	. . . . . . 7 |- ( x = k -> ( B x. x ) = ( B x. k ) )
7	6	oveq2d 5938	. . . . . 6 |- ( x = k -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. k ) ) )
8		oveq2 5930	. . . . . 6 |- ( x = k -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ k ) )
9	7, 8	eqeq12d 2211	. . . . 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 5930	. . . . . . 7 |- ( x = ( k + 1 ) -> ( B x. x ) = ( B x. ( k + 1 ) ) )
12	11	oveq2d 5938	. . . . . 6 |- ( x = ( k + 1 ) -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. ( k + 1 ) ) ) )
13		oveq2 5930	. . . . . 6 |- ( x = ( k + 1 ) -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ ( k + 1 ) ) )
14	12, 13	eqeq12d 2211	. . . . 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 5930	. . . . . . 7 |- ( x = C -> ( B x. x ) = ( B x. C ) )
17	16	oveq2d 5938	. . . . . 6 |- ( x = C -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. C ) ) )
18		oveq2 5930	. . . . . 6 |- ( x = C -> ( ( A ^c B ) ^ x ) = ( ( A ^c B ) ^ C ) )
19	17, 18	eqeq12d 2211	. . . . 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 15107	. . . . . 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 8413	. . . . . . 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 5938	. . . . 5 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( A ^c 0 ) )
26		rpcncxpcl 15111	. . . . . 6 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) e. CC )
27	26	exp0d 10744	. . . . 5 |- ( ( A e. RR+ /\ B e. CC ) -> ( ( A ^c B ) ^ 0 ) = 1 )
28	22, 25, 27	3eqtr4d 2239	. . . 4 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( ( A ^c B ) ^ 0 ) )
29		oveq1 5929	. . . . . . 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 9256	. . . . . . . . . . . . 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 8040	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> 1 e. CC )
34	30, 32, 33	adddid 8049	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. ( k + 1 ) ) = ( ( B x. k ) + ( B x. 1 ) ) )
35	30	mulridd 8041	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. 1 ) = B )
36	35	oveq2d 5938	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( ( B x. k ) + ( B x. 1 ) ) = ( ( B x. k ) + B ) )
37	34, 36	eqtrd 2229	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. ( k + 1 ) ) = ( ( B x. k ) + B ) )
38	37	oveq2d 5938	. . . . . . . . 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 8045	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. k ) e. CC )
41		rpcxpadd 15114	. . . . . . . . . 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 1249	. . . . . . . . 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 2229	. . . . . . . 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 10623	. . . . . . . . 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 2211	. . . . . . 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 9437	. . 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 1202	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 980   = wceq 1364   e. wcel 2167  (class class class)co 5922   CCcc 7875   0cc0 7877   1c1 7878   + caddc 7880   x. cmul 7882   NN0cn0 9246   RR+crp 9725   ^cexp 10615   ^c ccxp 15066

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7968  ax-resscn 7969  ax-1cn 7970  ax-1re 7971  ax-icn 7972  ax-addcl 7973  ax-addrcl 7974  ax-mulcl 7975  ax-mulrcl 7976  ax-addcom 7977  ax-mulcom 7978  ax-addass 7979  ax-mulass 7980  ax-distr 7981  ax-i2m1 7982  ax-0lt1 7983  ax-1rid 7984  ax-0id 7985  ax-rnegex 7986  ax-precex 7987  ax-cnre 7988  ax-pre-ltirr 7989  ax-pre-ltwlin 7990  ax-pre-lttrn 7991  ax-pre-apti 7992  ax-pre-ltadd 7993  ax-pre-mulgt0 7994  ax-pre-mulext 7995  ax-arch 7996  ax-caucvg 7997  ax-pre-suploc 7998  ax-addf 7999  ax-mulf 8000

This theorem depends on definitions:  df-bi 117  df-stab 832  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-disj 4011  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-isom 5267  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-of 6135  df-1st 6198  df-2nd 6199  df-recs 6363  df-irdg 6428  df-frec 6449  df-1o 6474  df-oadd 6478  df-er 6592  df-map 6709  df-pm 6710  df-en 6800  df-dom 6801  df-fin 6802  df-sup 7048  df-inf 7049  df-pnf 8061  df-mnf 8062  df-xr 8063  df-ltxr 8064  df-le 8065  df-sub 8197  df-neg 8198  df-reap 8599  df-ap 8606  df-div 8697  df-inn 8988  df-2 9046  df-3 9047  df-4 9048  df-n0 9247  df-z 9324  df-uz 9599  df-q 9691  df-rp 9726  df-xneg 9844  df-xadd 9845  df-ioo 9964  df-ico 9966  df-icc 9967  df-fz 10081  df-fzo 10215  df-seqfrec 10525  df-exp 10616  df-fac 10803  df-bc 10825  df-ihash 10853  df-shft 10965  df-cj 10992  df-re 10993  df-im 10994  df-rsqrt 11148  df-abs 11149  df-clim 11428  df-sumdc 11503  df-ef 11797  df-e 11798  df-rest 12888  df-topgen 12907  df-psmet 14075  df-xmet 14076  df-met 14077  df-bl 14078  df-mopn 14079  df-top 14210  df-topon 14223  df-bases 14255  df-ntr 14308  df-cn 14400  df-cnp 14401  df-tx 14465  df-cncf 14783  df-limced 14868  df-dvap 14869  df-relog 15067  df-rpcxp 15068

This theorem is referenced by:  rpcxpmul2d  15141