Theorem oexpneg 11865

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 9261	. . . . 5 |- ( N e. NN -> N e. ZZ )
2		odd2np1 11861	. . . . 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 1015	. 2 |- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
6		simpl1 1000	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> A e. CC )
7		simprr 531	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( 2 x. n ) + 1 ) = N )
8		simpl2 1001	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> N e. NN )
9	8	nncnd 8922	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> N e. CC )
10		1cnd 7964	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 1 e. CC )
11		2z 9270	. . . . . . . . . . 11 |- 2 e. ZZ
12		simprl 529	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. ZZ )
13		zmulcl 9295	. . . . . . . . . . 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 9365	. . . . . . . . 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 8277	. . . . . . . 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 9206	. . . . . . . 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 2255	. . . . . 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 10639	. . . . 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 8359	. . . 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 10565	. . . . . . . . 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 5884	. . . . . . 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 8245	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> -u A e. CC )
27		2re 8978	. . . . . . . . . . 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 9364	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. RR )
30		2pos 8999	. . . . . . . . . . 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 9221	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 <_ ( 2 x. n ) )
33		prodge0 8800	. . . . . . . . . 10 |- ( ( ( 2 e. RR /\ n e. RR ) /\ ( 0 < 2 /\ 0 <_ ( 2 x. n ) ) ) -> 0 <_ n )
34	28, 29, 31, 32, 33	syl22anc 1239	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 <_ n )
35		elnn0z 9255	. . . . . . . . 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 9182	. . . . . . . . 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 10642	. . . . . . 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 10642	. . . . . . 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 2220	. . . . . 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 5884	. . . . 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 10640	. . . . . 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 5885	. . . . . 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 2212	. . . . 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 2212	. . . 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 2212	. . 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 10640	. . . . 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 5885	. . . . 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 2212	. . . 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 8142	. . 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 2212	. 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 2599	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 978   = wceq 1353   e. wcel 2148   E.wrex 2456   class class class wbr 4000  (class class class)co 5869   CCcc 7800   RRcr 7801   0cc0 7802   1c1 7803   + caddc 7805   x. cmul 7807   < clt 7982   <_ cle 7983   - cmin 8118   -ucneg 8119   NNcn 8908   2c2 8959   NN0cn0 9165   ZZcz 9242   ^cexp 10505   || cdvds 11778

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584  ax-cnex 7893  ax-resscn 7894  ax-1cn 7895  ax-1re 7896  ax-icn 7897  ax-addcl 7898  ax-addrcl 7899  ax-mulcl 7900  ax-mulrcl 7901  ax-addcom 7902  ax-mulcom 7903  ax-addass 7904  ax-mulass 7905  ax-distr 7906  ax-i2m1 7907  ax-0lt1 7908  ax-1rid 7909  ax-0id 7910  ax-rnegex 7911  ax-precex 7912  ax-cnre 7913  ax-pre-ltirr 7914  ax-pre-ltwlin 7915  ax-pre-lttrn 7916  ax-pre-apti 7917  ax-pre-ltadd 7918  ax-pre-mulgt0 7919  ax-pre-mulext 7920

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-xor 1376  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-reu 2462  df-rmo 2463  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-if 3535  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-ilim 4366  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-riota 5825  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-frec 6386  df-pnf 7984  df-mnf 7985  df-xr 7986  df-ltxr 7987  df-le 7988  df-sub 8120  df-neg 8121  df-reap 8522  df-ap 8529  df-div 8619  df-inn 8909  df-2 8967  df-n0 9166  df-z 9243  df-uz 9518  df-seqfrec 10432  df-exp 10506  df-dvds 11779

This theorem is referenced by: (None)