Theorem oexpneg 11814

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 9210	. . . . 5 |- ( N e. NN -> N e. ZZ )
2		odd2np1 11810	. . . . 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 294	. . 3 |- ( ( N e. NN /\ -. 2 || N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
5	4	3adant1 1005	. 2 |- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
6		simpl1 990	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> A e. CC )
7		simprr 522	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( 2 x. n ) + 1 ) = N )
8		simpl2 991	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> N e. NN )
9	8	nncnd 8871	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> N e. CC )
10		1cnd 7915	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 1 e. CC )
11		2z 9219	. . . . . . . . . . 11 |- 2 e. ZZ
12		simprl 521	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. ZZ )
13		zmulcl 9244	. . . . . . . . . . 11 |- ( ( 2 e. ZZ /\ n e. ZZ ) -> ( 2 x. n ) e. ZZ )
14	11, 12, 13	sylancr 411	. . . . . . . . . 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 9314	. . . . . . . . 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 8228	. . . . . . . 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 9155	. . . . . . . 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 2244	. . . . . 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 10588	. . . . 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 8310	. . . 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 10514	. . . . . . . . 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 5857	. . . . . . 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 8196	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> -u A e. CC )
27		2re 8927	. . . . . . . . . . 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 9313	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. RR )
30		2pos 8948	. . . . . . . . . . 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 9170	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 <_ ( 2 x. n ) )
33		prodge0 8749	. . . . . . . . . 10 |- ( ( ( 2 e. RR /\ n e. RR ) /\ ( 0 < 2 /\ 0 <_ ( 2 x. n ) ) ) -> 0 <_ n )
34	28, 29, 31, 32, 33	syl22anc 1229	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 <_ n )
35		elnn0z 9204	. . . . . . . . 9 |- ( n e. NN0 <-> ( n e. ZZ /\ 0 <_ n ) )
36	12, 34, 35	sylanbrc 414	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. NN0 )
37		2nn0 9131	. . . . . . . . 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 10591	. . . . . . 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 10591	. . . . . . 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 2208	. . . . . 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 5857	. . . . 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 10589	. . . . . 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 5858	. . . . . 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 2200	. . . . 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 2200	. . . 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 2200	. . 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 10589	. . . . 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 5858	. . . . 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 2200	. . . 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 8093	. . 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 2200	. 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 2588	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 968   = wceq 1343   e. wcel 2136   E.wrex 2445   class class class wbr 3982  (class class class)co 5842   CCcc 7751   RRcr 7752   0cc0 7753   1c1 7754   + caddc 7756   x. cmul 7758   < clt 7933   <_ cle 7934   - cmin 8069   -ucneg 8070   NNcn 8857   2c2 8908   NN0cn0 9114   ZZcz 9191   ^cexp 10454   || cdvds 11727

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-mulrcl 7852  ax-addcom 7853  ax-mulcom 7854  ax-addass 7855  ax-mulass 7856  ax-distr 7857  ax-i2m1 7858  ax-0lt1 7859  ax-1rid 7860  ax-0id 7861  ax-rnegex 7862  ax-precex 7863  ax-cnre 7864  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-apti 7868  ax-pre-ltadd 7869  ax-pre-mulgt0 7870  ax-pre-mulext 7871

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-xor 1366  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-reu 2451  df-rmo 2452  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-tr 4081  df-id 4271  df-po 4274  df-iso 4275  df-iord 4344  df-on 4346  df-ilim 4347  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-riota 5798  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-recs 6273  df-frec 6359  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-sub 8071  df-neg 8072  df-reap 8473  df-ap 8480  df-div 8569  df-inn 8858  df-2 8916  df-n0 9115  df-z 9192  df-uz 9467  df-seqfrec 10381  df-exp 10455  df-dvds 11728

This theorem is referenced by: (None)