Theorem absexpzap 10744

Description: Absolute value of integer exponentiation. (Contributed by Jim Kingdon, 11-Aug-2021.)

Assertion

Ref	Expression
absexpzap	|- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )

Proof of Theorem absexpzap

Step	Hyp	Ref	Expression
1		elznn0nn 8972	. 2 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2		absexp 10743	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
3	2	ex 114	. . . . 5 |- ( A e. CC -> ( N e. NN0 -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
4	3	adantr 272	. . . 4 |- ( ( A e. CC /\ A # 0 ) -> ( N e. NN0 -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
5		1cnd 7706	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> 1 e. CC )
6		simpll 501	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
7		nnnn0 8888	. . . . . . . . . 10 |- ( -u N e. NN -> -u N e. NN0 )
8	7	ad2antll 480	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
9	6, 8	expcld 10317	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) e. CC )
10		simplr 502	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> A # 0 )
11		nnz 8977	. . . . . . . . . 10 |- ( -u N e. NN -> -u N e. ZZ )
12	11	ad2antll 480	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
13	6, 10, 12	expap0d 10323	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) # 0 )
14		absdivap 10734	. . . . . . . 8 |- ( ( 1 e. CC /\ ( A ^ -u N ) e. CC /\ ( A ^ -u N ) # 0 ) -> ( abs ` ( 1 / ( A ^ -u N ) ) ) = ( ( abs ` 1 ) / ( abs ` ( A ^ -u N ) ) ) )
15	5, 9, 13, 14	syl3anc 1199	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( 1 / ( A ^ -u N ) ) ) = ( ( abs ` 1 ) / ( abs ` ( A ^ -u N ) ) ) )
16		abs1 10736	. . . . . . . . 9 |- ( abs ` 1 ) = 1
17	16	oveq1i 5738	. . . . . . . 8 |- ( ( abs ` 1 ) / ( abs ` ( A ^ -u N ) ) ) = ( 1 / ( abs ` ( A ^ -u N ) ) )
18		absexp 10743	. . . . . . . . . 10 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( abs ` ( A ^ -u N ) ) = ( ( abs ` A ) ^ -u N ) )
19	6, 8, 18	syl2anc 406	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ -u N ) ) = ( ( abs ` A ) ^ -u N ) )
20	19	oveq2d 5744	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( 1 / ( abs ` ( A ^ -u N ) ) ) = ( 1 / ( ( abs ` A ) ^ -u N ) ) )
21	17, 20	syl5eq 2159	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( abs ` 1 ) / ( abs ` ( A ^ -u N ) ) ) = ( 1 / ( ( abs ` A ) ^ -u N ) ) )
22	15, 21	eqtrd 2147	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( 1 / ( A ^ -u N ) ) ) = ( 1 / ( ( abs ` A ) ^ -u N ) ) )
23		simprl 503	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
24	23	recnd 7718	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
25		expineg2 10195	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
26	6, 10, 24, 8, 25	syl22anc 1200	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
27	26	fveq2d 5379	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ N ) ) = ( abs ` ( 1 / ( A ^ -u N ) ) ) )
28		abscl 10715	. . . . . . . . 9 |- ( A e. CC -> ( abs ` A ) e. RR )
29	28	ad2antrr 477	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` A ) e. RR )
30	29	recnd 7718	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` A ) e. CC )
31		abs00ap 10726	. . . . . . . . 9 |- ( A e. CC -> ( ( abs ` A ) # 0 <-> A # 0 ) )
32	31	ad2antrr 477	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( abs ` A ) # 0 <-> A # 0 ) )
33	10, 32	mpbird 166	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` A ) # 0 )
34		expineg2 10195	. . . . . . 7 |- ( ( ( ( abs ` A ) e. CC /\ ( abs ` A ) # 0 ) /\ ( N e. CC /\ -u N e. NN0 ) ) -> ( ( abs ` A ) ^ N ) = ( 1 / ( ( abs ` A ) ^ -u N ) ) )
35	30, 33, 24, 8, 34	syl22anc 1200	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( abs ` A ) ^ N ) = ( 1 / ( ( abs ` A ) ^ -u N ) ) )
36	22, 27, 35	3eqtr4d 2157	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
37	36	ex 114	. . . 4 |- ( ( A e. CC /\ A # 0 ) -> ( ( N e. RR /\ -u N e. NN ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
38	4, 37	jaod 689	. . 3 |- ( ( A e. CC /\ A # 0 ) -> ( ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
39	38	3impia 1161	. 2 |- ( ( A e. CC /\ A # 0 /\ ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
40	1, 39	syl3an3b 1237	1 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 680   /\ w3a 945   = wceq 1314   e. wcel 1463   class class class wbr 3895   ` cfv 5081  (class class class)co 5728   CCcc 7545   RRcr 7546   0cc0 7547   1c1 7548   -ucneg 7857   # cap 8261   / cdiv 8345   NNcn 8630   NN0cn0 8881   ZZcz 8958   ^cexp 10185   abscabs 10661

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  ax-arch 7664  ax-caucvg 7665

This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  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-3 8690  df-4 8691  df-n0 8882  df-z 8959  df-uz 9229  df-rp 9344  df-seqfrec 10112  df-exp 10186  df-cj 10507  df-re 10508  df-im 10509  df-rsqrt 10662  df-abs 10663

This theorem is referenced by: (None)