Theorem absexpzap 9966

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 8365	. 2 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2		absexp 9965	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
3	2	ex 113	. . . . 5 |- ( A e. CC -> ( N e. NN0 -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
4	3	adantr 270	. . . 4 |- ( ( A e. CC /\ A # 0 ) -> ( N e. NN0 -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
5		1cnd 7135	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> 1 e. CC )
6		simpll 495	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
7		nnnn0 8295	. . . . . . . . . 10 |- ( -u N e. NN -> -u N e. NN0 )
8	7	ad2antll 474	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
9	6, 8	expcld 9605	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) e. CC )
10		simplr 496	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> A # 0 )
11		nnz 8370	. . . . . . . . . 10 |- ( -u N e. NN -> -u N e. ZZ )
12	11	ad2antll 474	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
13	6, 10, 12	expap0d 9611	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) # 0 )
14		absdivap 9956	. . . . . . . 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 1169	. . . . . . 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 9958	. . . . . . . . 9 |- ( abs ` 1 ) = 1
17	16	oveq1i 5542	. . . . . . . 8 |- ( ( abs ` 1 ) / ( abs ` ( A ^ -u N ) ) ) = ( 1 / ( abs ` ( A ^ -u N ) ) )
18		absexp 9965	. . . . . . . . . 10 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( abs ` ( A ^ -u N ) ) = ( ( abs ` A ) ^ -u N ) )
19	6, 8, 18	syl2anc 403	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ -u N ) ) = ( ( abs ` A ) ^ -u N ) )
20	19	oveq2d 5548	. . . . . . . 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 2125	. . . . . . 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 2113	. . . . . 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 497	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
24	23	recnd 7147	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
25		expineg2 9485	. . . . . . . 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 1170	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
27	26	fveq2d 5202	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ N ) ) = ( abs ` ( 1 / ( A ^ -u N ) ) ) )
28		abscl 9937	. . . . . . . . 9 |- ( A e. CC -> ( abs ` A ) e. RR )
29	28	ad2antrr 471	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` A ) e. RR )
30	29	recnd 7147	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` A ) e. CC )
31		abs00ap 9948	. . . . . . . . 9 |- ( A e. CC -> ( ( abs ` A ) # 0 <-> A # 0 ) )
32	31	ad2antrr 471	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( abs ` A ) # 0 <-> A # 0 ) )
33	10, 32	mpbird 165	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` A ) # 0 )
34		expineg2 9485	. . . . . . 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 1170	. . . . . 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 2123	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
37	36	ex 113	. . . 4 |- ( ( A e. CC /\ A # 0 ) -> ( ( N e. RR /\ -u N e. NN ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
38	4, 37	jaod 669	. . 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 1135	. 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 1207	1 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 661   /\ w3a 919   = wceq 1284   e. wcel 1433   class class class wbr 3785   ` cfv 4922  (class class class)co 5532   CCcc 6979   RRcr 6980   0cc0 6981   1c1 6982   -ucneg 7280   # cap 7681   / cdiv 7760   NNcn 8039   NN0cn0 8288   ZZcz 8351   ^cexp 9475   abscabs 9883

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-coll 3893  ax-sep 3896  ax-nul 3904  ax-pow 3948  ax-pr 3964  ax-un 4188  ax-setind 4280  ax-iinf 4329  ax-cnex 7067  ax-resscn 7068  ax-1cn 7069  ax-1re 7070  ax-icn 7071  ax-addcl 7072  ax-addrcl 7073  ax-mulcl 7074  ax-mulrcl 7075  ax-addcom 7076  ax-mulcom 7077  ax-addass 7078  ax-mulass 7079  ax-distr 7080  ax-i2m1 7081  ax-0lt1 7082  ax-1rid 7083  ax-0id 7084  ax-rnegex 7085  ax-precex 7086  ax-cnre 7087  ax-pre-ltirr 7088  ax-pre-ltwlin 7089  ax-pre-lttrn 7090  ax-pre-apti 7091  ax-pre-ltadd 7092  ax-pre-mulgt0 7093  ax-pre-mulext 7094  ax-arch 7095  ax-caucvg 7096

This theorem depends on definitions:  df-bi 115  df-dc 776  df-3or 920  df-3an 921  df-tru 1287  df-fal 1290  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ne 2246  df-nel 2340  df-ral 2353  df-rex 2354  df-reu 2355  df-rmo 2356  df-rab 2357  df-v 2603  df-sbc 2816  df-csb 2909  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-if 3352  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-iun 3680  df-br 3786  df-opab 3840  df-mpt 3841  df-tr 3876  df-id 4048  df-po 4051  df-iso 4052  df-iord 4121  df-on 4123  df-suc 4126  df-iom 4332  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-res 4375  df-ima 4376  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-riota 5488  df-ov 5535  df-oprab 5536  df-mpt2 5537  df-1st 5787  df-2nd 5788  df-recs 5943  df-frec 6001  df-pnf 7155  df-mnf 7156  df-xr 7157  df-ltxr 7158  df-le 7159  df-sub 7281  df-neg 7282  df-reap 7675  df-ap 7682  df-div 7761  df-inn 8040  df-2 8098  df-3 8099  df-4 8100  df-n0 8289  df-z 8352  df-uz 8620  df-rp 8735  df-iseq 9432  df-iexp 9476  df-cj 9729  df-re 9730  df-im 9731  df-rsqrt 9884  df-abs 9885

This theorem is referenced by: (None)