Theorem absexpzap 11245

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 9340	. 2 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2		absexp 11244	. . . . . 6 |- ( ( A e. CC /\ N e. NN0 ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
3	2	ex 115	. . . . 5 |- ( A e. CC -> ( N e. NN0 -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
4	3	adantr 276	. . . 4 |- ( ( A e. CC /\ A # 0 ) -> ( N e. NN0 -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
5		1cnd 8042	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> 1 e. CC )
6		simpll 527	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> A e. CC )
7		nnnn0 9256	. . . . . . . . . 10 |- ( -u N e. NN -> -u N e. NN0 )
8	7	ad2antll 491	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. NN0 )
9	6, 8	expcld 10765	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) e. CC )
10		simplr 528	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> A # 0 )
11		nnz 9345	. . . . . . . . . 10 |- ( -u N e. NN -> -u N e. ZZ )
12	11	ad2antll 491	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> -u N e. ZZ )
13	6, 10, 12	expap0d 10771	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) # 0 )
14		absdivap 11235	. . . . . . . 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 1249	. . . . . . 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 11237	. . . . . . . . 9 |- ( abs ` 1 ) = 1
17	16	oveq1i 5932	. . . . . . . 8 |- ( ( abs ` 1 ) / ( abs ` ( A ^ -u N ) ) ) = ( 1 / ( abs ` ( A ^ -u N ) ) )
18		absexp 11244	. . . . . . . . . 10 |- ( ( A e. CC /\ -u N e. NN0 ) -> ( abs ` ( A ^ -u N ) ) = ( ( abs ` A ) ^ -u N ) )
19	6, 8, 18	syl2anc 411	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ -u N ) ) = ( ( abs ` A ) ^ -u N ) )
20	19	oveq2d 5938	. . . . . . . 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	eqtrid 2241	. . . . . . 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 2229	. . . . . 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 529	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
24	23	recnd 8055	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
25		expineg2 10640	. . . . . . . 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 1250	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
27	26	fveq2d 5562	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ N ) ) = ( abs ` ( 1 / ( A ^ -u N ) ) ) )
28		abscl 11216	. . . . . . . . 9 |- ( A e. CC -> ( abs ` A ) e. RR )
29	28	ad2antrr 488	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` A ) e. RR )
30	29	recnd 8055	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` A ) e. CC )
31		abs00ap 11227	. . . . . . . . 9 |- ( A e. CC -> ( ( abs ` A ) # 0 <-> A # 0 ) )
32	31	ad2antrr 488	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( ( abs ` A ) # 0 <-> A # 0 ) )
33	10, 32	mpbird 167	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` A ) # 0 )
34		expineg2 10640	. . . . . . 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 1250	. . . . . 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 2239	. . . . 5 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )
37	36	ex 115	. . . 4 |- ( ( A e. CC /\ A # 0 ) -> ( ( N e. RR /\ -u N e. NN ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) ) )
38	4, 37	jaod 718	. . 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 1202	. 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 1287	1 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   /\ w3a 980   = wceq 1364   e. wcel 2167   class class class wbr 4033   ` cfv 5258  (class class class)co 5922   CCcc 7877   RRcr 7878   0cc0 7879   1c1 7880   -ucneg 8198   # cap 8608   / cdiv 8699   NNcn 8990   NN0cn0 9249   ZZcz 9326   ^cexp 10630   abscabs 11162

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 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4148  ax-sep 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-setind 4573  ax-iinf 4624  ax-cnex 7970  ax-resscn 7971  ax-1cn 7972  ax-1re 7973  ax-icn 7974  ax-addcl 7975  ax-addrcl 7976  ax-mulcl 7977  ax-mulrcl 7978  ax-addcom 7979  ax-mulcom 7980  ax-addass 7981  ax-mulass 7982  ax-distr 7983  ax-i2m1 7984  ax-0lt1 7985  ax-1rid 7986  ax-0id 7987  ax-rnegex 7988  ax-precex 7989  ax-cnre 7990  ax-pre-ltirr 7991  ax-pre-ltwlin 7992  ax-pre-lttrn 7993  ax-pre-apti 7994  ax-pre-ltadd 7995  ax-pre-mulgt0 7996  ax-pre-mulext 7997  ax-arch 7998  ax-caucvg 7999

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-nel 2463  df-ral 2480  df-rex 2481  df-reu 2482  df-rmo 2483  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-if 3562  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-iun 3918  df-br 4034  df-opab 4095  df-mpt 4096  df-tr 4132  df-id 4328  df-po 4331  df-iso 4332  df-iord 4401  df-on 4403  df-ilim 4404  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-riota 5877  df-ov 5925  df-oprab 5926  df-mpo 5927  df-1st 6198  df-2nd 6199  df-recs 6363  df-frec 6449  df-pnf 8063  df-mnf 8064  df-xr 8065  df-ltxr 8066  df-le 8067  df-sub 8199  df-neg 8200  df-reap 8602  df-ap 8609  df-div 8700  df-inn 8991  df-2 9049  df-3 9050  df-4 9051  df-n0 9250  df-z 9327  df-uz 9602  df-rp 9729  df-seqfrec 10540  df-exp 10631  df-cj 11007  df-re 11008  df-im 11009  df-rsqrt 11163  df-abs 11164

This theorem is referenced by:  lgseisen  15315