Theorem oexpneg 12588

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 9613	. . . . 5 |- ( N e. NN -> N e. ZZ )
2		odd2np1 12584	. . . . 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 296	. . 3 |- ( ( N e. NN /\ -. 2 || N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
5	4	3adant1 1042	. 2 |- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
6		simpl1 1027	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> A e. CC )
7		simprr 533	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( 2 x. n ) + 1 ) = N )
8		simpl2 1028	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> N e. NN )
9	8	nncnd 9268	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> N e. CC )
10		1cnd 8306	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 1 e. CC )
11		2z 9622	. . . . . . . . . . 11 |- 2 e. ZZ
12		simprl 531	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. ZZ )
13		zmulcl 9648	. . . . . . . . . . 11 |- ( ( 2 e. ZZ /\ n e. ZZ ) -> ( 2 x. n ) e. ZZ )
14	11, 12, 13	sylancr 414	. . . . . . . . . 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 9719	. . . . . . . . 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 8619	. . . . . . . 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 167	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( N - 1 ) = ( 2 x. n ) )
18		nnm1nn0 9554	. . . . . . . 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 2312	. . . . . 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 11060	. . . . 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 8702	. . . 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 10984	. . . . . . . . 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 6073	. . . . . . 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 8587	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> -u A e. CC )
27		2re 9324	. . . . . . . . . . 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 9718	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. RR )
30		2pos 9345	. . . . . . . . . . 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 9573	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 <_ ( 2 x. n ) )
33		prodge0 9145	. . . . . . . . . 10 |- ( ( ( 2 e. RR /\ n e. RR ) /\ ( 0 < 2 /\ 0 <_ ( 2 x. n ) ) ) -> 0 <_ n )
34	28, 29, 31, 32, 33	syl22anc 1275	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 <_ n )
35		elnn0z 9607	. . . . . . . . 9 |- ( n e. NN0 <-> ( n e. ZZ /\ 0 <_ n ) )
36	12, 34, 35	sylanbrc 417	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. NN0 )
37		2nn0 9530	. . . . . . . . 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 11063	. . . . . . 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 11063	. . . . . . 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 2277	. . . . . 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 6073	. . . . 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 11061	. . . . . 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 6074	. . . . . 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 2269	. . . . 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 2269	. . . 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 2269	. . 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 11061	. . . . 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 6074	. . . . 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 2269	. . . 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 8484	. . 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 2269	. 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 2667	1 |- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> ( -u A ^ N ) = -u ( A ^ N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1005   = wceq 1398   e. wcel 2205   E.wrex 2523   class class class wbr 4114  (class class class)co 6058   CCcc 8141   RRcr 8142   0cc0 8143   1c1 8144   + caddc 8146   x. cmul 8148   < clt 8324   <_ cle 8325   - cmin 8460   -ucneg 8461   NNcn 9254   2c2 9305   NN0cn0 9513   ZZcz 9594   ^cexp 10924   || cdvds 12498

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-mulrcl 8242  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-0lt1 8249  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-precex 8253  ax-cnre 8254  ax-pre-ltirr 8255  ax-pre-ltwlin 8256  ax-pre-lttrn 8257  ax-pre-apti 8258  ax-pre-ltadd 8259  ax-pre-mulgt0 8260  ax-pre-mulext 8261

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-xor 1421  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-reu 2529  df-rmo 2530  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-po 4422  df-iso 4423  df-iord 4492  df-on 4494  df-ilim 4495  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-recs 6549  df-frec 6635  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329  df-le 8330  df-sub 8462  df-neg 8463  df-reap 8866  df-ap 8873  df-div 8964  df-inn 9255  df-2 9313  df-n0 9514  df-z 9595  df-uz 9872  df-seqfrec 10834  df-exp 10925  df-dvds 12499

This theorem is referenced by:  lgseisenlem1  16069  lgseisenlem4  16072  m1lgs  16084