Theorem rpcxpmul2 15938

Description: Product of exponents law for complex exponentiation. Variation on cxpmul 15937 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 6083	. . . . . . 7 |- ( x = 0 -> ( B x. x ) = ( B x. 0 ) )
2	1	oveq2d 6091	. . . . . 6 |- ( x = 0 -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. 0 ) ) )
3		oveq2 6083	. . . . . 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 6083	. . . . . . 7 |- ( x = k -> ( B x. x ) = ( B x. k ) )
7	6	oveq2d 6091	. . . . . 6 |- ( x = k -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. k ) ) )
8		oveq2 6083	. . . . . 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 6083	. . . . . . 7 |- ( x = ( k + 1 ) -> ( B x. x ) = ( B x. ( k + 1 ) ) )
12	11	oveq2d 6091	. . . . . 6 |- ( x = ( k + 1 ) -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. ( k + 1 ) ) ) )
13		oveq2 6083	. . . . . 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 6083	. . . . . . 7 |- ( x = C -> ( B x. x ) = ( B x. C ) )
17	16	oveq2d 6091	. . . . . 6 |- ( x = C -> ( A ^c ( B x. x ) ) = ( A ^c ( B x. C ) ) )
18		oveq2 6083	. . . . . 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 15923	. . . . . 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 8706	. . . . . . 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 6091	. . . . 5 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c ( B x. 0 ) ) = ( A ^c 0 ) )
26		rpcncxpcl 15927	. . . . . 6 |- ( ( A e. RR+ /\ B e. CC ) -> ( A ^c B ) e. CC )
27	26	exp0d 11083	. . . . 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 6082	. . . . . . 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 9552	. . . . . . . . . . . . 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 8332	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> 1 e. CC )
34	30, 32, 33	adddid 8340	. . . . . . . . . . 11 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. ( k + 1 ) ) = ( ( B x. k ) + ( B x. 1 ) ) )
35	30	mulridd 8333	. . . . . . . . . . . 12 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. 1 ) = B )
36	35	oveq2d 6091	. . . . . . . . . . 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 6091	. . . . . . . . 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 8336	. . . . . . . . . 10 |- ( ( ( A e. RR+ /\ B e. CC ) /\ k e. NN0 ) -> ( B x. k ) e. CC )
41		rpcxpadd 15930	. . . . . . . . . 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 10961	. . . . . . . . 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 9739	. . 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 ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1009   = wceq 1402   e. wcel 2209  (class class class)co 6075   CCcc 8167   0cc0 8169   1c1 8170   + caddc 8172   x. cmul 8174   NN0cn0 9542   RR+crp 10033   ^cexp 10953   ^c ccxp 15881

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 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 4241  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-setind 4679  ax-iinf 4730  ax-cnex 8260  ax-resscn 8261  ax-1cn 8262  ax-1re 8263  ax-icn 8264  ax-addcl 8265  ax-addrcl 8266  ax-mulcl 8267  ax-mulrcl 8268  ax-addcom 8269  ax-mulcom 8270  ax-addass 8271  ax-mulass 8272  ax-distr 8273  ax-i2m1 8274  ax-0lt1 8275  ax-1rid 8276  ax-0id 8277  ax-rnegex 8278  ax-precex 8279  ax-cnre 8280  ax-pre-ltirr 8281  ax-pre-ltwlin 8282  ax-pre-lttrn 8283  ax-pre-apti 8284  ax-pre-ltadd 8285  ax-pre-mulgt0 8286  ax-pre-mulext 8287  ax-arch 8288  ax-caucvg 8289  ax-pre-suploc 8290  ax-addf 8291  ax-mulf 8292

This theorem 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 3636  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-iun 4009  df-disj 4102  df-br 4126  df-opab 4188  df-mpt 4189  df-tr 4225  df-id 4433  df-po 4436  df-iso 4437  df-iord 4506  df-on 4508  df-ilim 4509  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-isom 5381  df-riota 6028  df-ov 6078  df-oprab 6079  df-mpo 6080  df-of 6292  df-1st 6364  df-2nd 6365  df-recs 6566  df-irdg 6631  df-frec 6652  df-1o 6677  df-oadd 6681  df-er 6797  df-map 6914  df-pm 6915  df-en 7013  df-dom 7014  df-fin 7015  df-sup 7314  df-inf 7315  df-pnf 8352  df-mnf 8353  df-xr 8354  df-ltxr 8355  df-le 8356  df-sub 8489  df-neg 8490  df-reap 8893  df-ap 8900  df-div 8993  df-inn 9284  df-2 9342  df-3 9343  df-4 9344  df-n0 9543  df-z 9624  df-uz 9901  df-q 9999  df-rp 10034  df-xneg 10153  df-xadd 10154  df-ioo 10273  df-ico 10275  df-icc 10276  df-fz 10391  df-fzo 10528  df-seqfrec 10863  df-exp 10954  df-fac 11142  df-bc 11164  df-ihash 11193  df-shft 11558  df-cj 11585  df-re 11586  df-im 11587  df-rsqrt 11742  df-abs 11743  df-clim 12023  df-sumdc 12098  df-ef 12393  df-e 12394  df-rest 13572  df-topgen 13591  df-psmet 14852  df-xmet 14853  df-met 14854  df-bl 14855  df-mopn 14856  df-top 15022  df-topon 15035  df-bases 15067  df-ntr 15120  df-cn 15212  df-cnp 15213  df-tx 15277  df-cncf 15595  df-limced 15680  df-dvap 15681  df-relog 15882  df-rpcxp 15883

This theorem is referenced by:  rpcxpmul2d  15957