Theorem oexpneg 11422

Description: The exponential of the negative of a number, when the exponent is odd. (Contributed by Mario Carneiro, 25-Apr-2015.)

Assertion

Ref	Expression
oexpneg	|- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> ( -u A ^ N ) = -u ( A ^ N ) )

Proof of Theorem oexpneg

Dummy variable n is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nnz 8977	. . . . 5 |- ( N e. NN -> N e. ZZ )
2		odd2np1 11418	. . . . 5 |- ( N e. ZZ -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
3	1, 2	syl 14	. . . 4 |- ( N e. NN -> ( -. 2 || N <-> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N ) )
4	3	biimpa 292	. . 3 |- ( ( N e. NN /\ -. 2 || N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
5	4	3adant1 982	. 2 |- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
6		simpl1 967	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> A e. CC )
7		simprr 504	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( 2 x. n ) + 1 ) = N )
8		simpl2 968	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> N e. NN )
9	8	nncnd 8644	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> N e. CC )
10		1cnd 7706	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 1 e. CC )
11		2z 8986	. . . . . . . . . . 11 |- 2 e. ZZ
12		simprl 503	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. ZZ )
13		zmulcl 9011	. . . . . . . . . . 11 |- ( ( 2 e. ZZ /\ n e. ZZ ) -> ( 2 x. n ) e. ZZ )
14	11, 12, 13	sylancr 408	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( 2 x. n ) e. ZZ )
15	14	zcnd 9078	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( 2 x. n ) e. CC )
16	9, 10, 15	subadd2d 8015	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( N - 1 ) = ( 2 x. n ) <-> ( ( 2 x. n ) + 1 ) = N ) )
17	7, 16	mpbird 166	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( N - 1 ) = ( 2 x. n ) )
18		nnm1nn0 8922	. . . . . . . 8 |- ( N e. NN -> ( N - 1 ) e. NN0 )
19	8, 18	syl 14	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( N - 1 ) e. NN0 )
20	17, 19	eqeltrrd 2192	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( 2 x. n ) e. NN0 )
21	6, 20	expcld 10317	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( A ^ ( 2 x. n ) ) e. CC )
22	21, 6	mulneg2d 8093	. . . 4 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( A ^ ( 2 x. n ) ) x. -u A ) = -u ( ( A ^ ( 2 x. n ) ) x. A ) )
23		sqneg 10245	. . . . . . . . 9 |- ( A e. CC -> ( -u A ^ 2 ) = ( A ^ 2 ) )
24	6, 23	syl 14	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( -u A ^ 2 ) = ( A ^ 2 ) )
25	24	oveq1d 5743	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( -u A ^ 2 ) ^ n ) = ( ( A ^ 2 ) ^ n ) )
26	6	negcld 7983	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> -u A e. CC )
27		2re 8700	. . . . . . . . . . 11 |- 2 e. RR
28	27	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 2 e. RR )
29	12	zred 9077	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. RR )
30		2pos 8721	. . . . . . . . . . 11 |- 0 < 2
31	30	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 < 2 )
32	20	nn0ge0d 8937	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 <_ ( 2 x. n ) )
33		prodge0 8522	. . . . . . . . . 10 |- ( ( ( 2 e. RR /\ n e. RR ) /\ ( 0 < 2 /\ 0 <_ ( 2 x. n ) ) ) -> 0 <_ n )
34	28, 29, 31, 32, 33	syl22anc 1200	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 <_ n )
35		elnn0z 8971	. . . . . . . . 9 |- ( n e. NN0 <-> ( n e. ZZ /\ 0 <_ n ) )
36	12, 34, 35	sylanbrc 411	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. NN0 )
37		2nn0 8898	. . . . . . . . 9 |- 2 e. NN0
38	37	a1i 9	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 2 e. NN0 )
39	26, 36, 38	expmuld 10320	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( -u A ^ ( 2 x. n ) ) = ( ( -u A ^ 2 ) ^ n ) )
40	6, 36, 38	expmuld 10320	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( A ^ ( 2 x. n ) ) = ( ( A ^ 2 ) ^ n ) )
41	25, 39, 40	3eqtr4d 2157	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( -u A ^ ( 2 x. n ) ) = ( A ^ ( 2 x. n ) ) )
42	41	oveq1d 5743	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( -u A ^ ( 2 x. n ) ) x. -u A ) = ( ( A ^ ( 2 x. n ) ) x. -u A ) )
43	26, 20	expp1d 10318	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( -u A ^ ( ( 2 x. n ) + 1 ) ) = ( ( -u A ^ ( 2 x. n ) ) x. -u A ) )
44	7	oveq2d 5744	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( -u A ^ ( ( 2 x. n ) + 1 ) ) = ( -u A ^ N ) )
45	43, 44	eqtr3d 2149	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( -u A ^ ( 2 x. n ) ) x. -u A ) = ( -u A ^ N ) )
46	42, 45	eqtr3d 2149	. . . 4 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( A ^ ( 2 x. n ) ) x. -u A ) = ( -u A ^ N ) )
47	22, 46	eqtr3d 2149	. . 3 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> -u ( ( A ^ ( 2 x. n ) ) x. A ) = ( -u A ^ N ) )
48	6, 20	expp1d 10318	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( A ^ ( ( 2 x. n ) + 1 ) ) = ( ( A ^ ( 2 x. n ) ) x. A ) )
49	7	oveq2d 5744	. . . . 5 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( A ^ ( ( 2 x. n ) + 1 ) ) = ( A ^ N ) )
50	48, 49	eqtr3d 2149	. . . 4 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( A ^ ( 2 x. n ) ) x. A ) = ( A ^ N ) )
51	50	negeqd 7880	. . 3 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> -u ( ( A ^ ( 2 x. n ) ) x. A ) = -u ( A ^ N ) )
52	47, 51	eqtr3d 2149	. 2 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( -u A ^ N ) = -u ( A ^ N ) )
53	5, 52	rexlimddv 2528	1 |- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> ( -u A ^ N ) = -u ( A ^ N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   /\ w3a 945   = wceq 1314   e. wcel 1463   E.wrex 2391   class class class wbr 3895  (class class class)co 5728   CCcc 7545   RRcr 7546   0cc0 7547   1c1 7548   + caddc 7550   x. cmul 7552   < clt 7724   <_ cle 7725   - cmin 7856   -ucneg 7857   NNcn 8630   2c2 8681   NN0cn0 8881   ZZcz 8958   ^cexp 10185   || cdvds 11341

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-nul 4014  ax-pow 4058  ax-pr 4091  ax-un 4315  ax-setind 4412  ax-iinf 4462  ax-cnex 7636  ax-resscn 7637  ax-1cn 7638  ax-1re 7639  ax-icn 7640  ax-addcl 7641  ax-addrcl 7642  ax-mulcl 7643  ax-mulrcl 7644  ax-addcom 7645  ax-mulcom 7646  ax-addass 7647  ax-mulass 7648  ax-distr 7649  ax-i2m1 7650  ax-0lt1 7651  ax-1rid 7652  ax-0id 7653  ax-rnegex 7654  ax-precex 7655  ax-cnre 7656  ax-pre-ltirr 7657  ax-pre-ltwlin 7658  ax-pre-lttrn 7659  ax-pre-apti 7660  ax-pre-ltadd 7661  ax-pre-mulgt0 7662  ax-pre-mulext 7663

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-xor 1337  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ne 2283  df-nel 2378  df-ral 2395  df-rex 2396  df-reu 2397  df-rmo 2398  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-dif 3039  df-un 3041  df-in 3043  df-ss 3050  df-nul 3330  df-if 3441  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-int 3738  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-tr 3987  df-id 4175  df-po 4178  df-iso 4179  df-iord 4248  df-on 4250  df-ilim 4251  df-suc 4253  df-iom 4465  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-riota 5684  df-ov 5731  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-recs 6156  df-frec 6242  df-pnf 7726  df-mnf 7727  df-xr 7728  df-ltxr 7729  df-le 7730  df-sub 7858  df-neg 7859  df-reap 8255  df-ap 8262  df-div 8346  df-inn 8631  df-2 8689  df-n0 8882  df-z 8959  df-uz 9229  df-seqfrec 10112  df-exp 10186  df-dvds 11342

This theorem is referenced by: (None)