Theorem oexpneg 10410

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 8440	. . . . 5 |- ( N e. NN -> N e. ZZ )
2		odd2np1 10406	. . . . 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 290	. . 3 |- ( ( N e. NN /\ -. 2 || N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
5	4	3adant1 957	. 2 |- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> E. n e. ZZ ( ( 2 x. n ) + 1 ) = N )
6		simpl1 942	. . . . . 6 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> A e. CC )
7		simprr 499	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( ( 2 x. n ) + 1 ) = N )
8		simpl2 943	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> N e. NN )
9	8	nncnd 8109	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> N e. CC )
10		1cnd 7186	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 1 e. CC )
11		2z 8449	. . . . . . . . . . 11 |- 2 e. ZZ
12		simprl 498	. . . . . . . . . . 11 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. ZZ )
13		zmulcl 8474	. . . . . . . . . . 11 |- ( ( 2 e. ZZ /\ n e. ZZ ) -> ( 2 x. n ) e. ZZ )
14	11, 12, 13	sylancr 405	. . . . . . . . . 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 8540	. . . . . . . . 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 7494	. . . . . . . 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 165	. . . . . . 7 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> ( N - 1 ) = ( 2 x. n ) )
18		nnm1nn0 8385	. . . . . . . 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 2157	. . . . . 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 9691	. . . . 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 7572	. . . 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 9621	. . . . . . . . 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 5552	. . . . . . 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 7462	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> -u A e. CC )
27		2re 8165	. . . . . . . . . . 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 8539	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. RR )
30		2pos 8186	. . . . . . . . . . 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 8400	. . . . . . . . . 10 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 <_ ( 2 x. n ) )
33		prodge0 7988	. . . . . . . . . 10 |- ( ( ( 2 e. RR /\ n e. RR ) /\ ( 0 < 2 /\ 0 <_ ( 2 x. n ) ) ) -> 0 <_ n )
34	28, 29, 31, 32, 33	syl22anc 1171	. . . . . . . . 9 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> 0 <_ n )
35		elnn0z 8434	. . . . . . . . 9 |- ( n e. NN0 <-> ( n e. ZZ /\ 0 <_ n ) )
36	12, 34, 35	sylanbrc 408	. . . . . . . 8 |- ( ( ( A e. CC /\ N e. NN /\ -. 2 || N ) /\ ( n e. ZZ /\ ( ( 2 x. n ) + 1 ) = N ) ) -> n e. NN0 )
37		2nn0 8361	. . . . . . . . 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 9694	. . . . . . 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 9694	. . . . . . 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 2124	. . . . . 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 5552	. . . . 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 9692	. . . . . 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 5553	. . . . . 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 2116	. . . . 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 2116	. . . 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 2116	. . 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 9692	. . . . 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 5553	. . . . 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 2116	. . . 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 7359	. . 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 2116	. 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 2482	1 |- ( ( A e. CC /\ N e. NN /\ -. 2 || N ) -> ( -u A ^ N ) = -u ( A ^ N ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 102   <-> wb 103   /\ w3a 920   = wceq 1285   e. wcel 1434   E.wrex 2350   class class class wbr 3787  (class class class)co 5537   CCcc 7030   RRcr 7031   0cc0 7032   1c1 7033   + caddc 7035   x. cmul 7037   < clt 7204   <_ cle 7205   - cmin 7335   -ucneg 7336   NNcn 8095   2c2 8145   NN0cn0 8344   ZZcz 8421   ^cexp 9561   || cdvds 10329

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3895  ax-sep 3898  ax-nul 3906  ax-pow 3950  ax-pr 3966  ax-un 4190  ax-setind 4282  ax-iinf 4331  ax-cnex 7118  ax-resscn 7119  ax-1cn 7120  ax-1re 7121  ax-icn 7122  ax-addcl 7123  ax-addrcl 7124  ax-mulcl 7125  ax-mulrcl 7126  ax-addcom 7127  ax-mulcom 7128  ax-addass 7129  ax-mulass 7130  ax-distr 7131  ax-i2m1 7132  ax-0lt1 7133  ax-1rid 7134  ax-0id 7135  ax-rnegex 7136  ax-precex 7137  ax-cnre 7138  ax-pre-ltirr 7139  ax-pre-ltwlin 7140  ax-pre-lttrn 7141  ax-pre-apti 7142  ax-pre-ltadd 7143  ax-pre-mulgt0 7144  ax-pre-mulext 7145

This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-xor 1308  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-nel 2341  df-ral 2354  df-rex 2355  df-reu 2356  df-rmo 2357  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3253  df-if 3354  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3604  df-int 3639  df-iun 3682  df-br 3788  df-opab 3842  df-mpt 3843  df-tr 3878  df-id 4050  df-po 4053  df-iso 4054  df-iord 4123  df-on 4125  df-ilim 4126  df-suc 4128  df-iom 4334  df-xp 4371  df-rel 4372  df-cnv 4373  df-co 4374  df-dm 4375  df-rn 4376  df-res 4377  df-ima 4378  df-iota 4891  df-fun 4928  df-fn 4929  df-f 4930  df-f1 4931  df-fo 4932  df-f1o 4933  df-fv 4934  df-riota 5493  df-ov 5540  df-oprab 5541  df-mpt2 5542  df-1st 5792  df-2nd 5793  df-recs 5948  df-frec 6034  df-pnf 7206  df-mnf 7207  df-xr 7208  df-ltxr 7209  df-le 7210  df-sub 7337  df-neg 7338  df-reap 7731  df-ap 7738  df-div 7817  df-inn 8096  df-2 8154  df-n0 8345  df-z 8422  df-uz 8690  df-iseq 9511  df-iexp 9562  df-dvds 10330

This theorem is referenced by: (None)