Theorem absexpzap 11720

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 9554	. 2 |- ( N e. ZZ <-> ( N e. NN0 \/ ( N e. RR /\ -u N e. NN ) ) )
2		absexp 11719	. . . . . 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 8255	. . . . . . . 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 9468	. . . . . . . . . 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 10998	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) e. CC )
10		simplr 529	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> A # 0 )
11		nnz 9559	. . . . . . . . . 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 11004	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ -u N ) # 0 )
14		absdivap 11710	. . . . . . . 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 1274	. . . . . . 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 11712	. . . . . . . . 9 |- ( abs ` 1 ) = 1
17	16	oveq1i 6038	. . . . . . . 8 |- ( ( abs ` 1 ) / ( abs ` ( A ^ -u N ) ) ) = ( 1 / ( abs ` ( A ^ -u N ) ) )
18		absexp 11719	. . . . . . . . . 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 6044	. . . . . . . 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 2276	. . . . . . 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 2264	. . . . . 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 531	. . . . . . . . 9 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. RR )
24	23	recnd 8267	. . . . . . . 8 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> N e. CC )
25		expineg2 10873	. . . . . . . 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 1275	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( A ^ N ) = ( 1 / ( A ^ -u N ) ) )
27	26	fveq2d 5652	. . . . . 6 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` ( A ^ N ) ) = ( abs ` ( 1 / ( A ^ -u N ) ) ) )
28		abscl 11691	. . . . . . . . 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 8267	. . . . . . 7 |- ( ( ( A e. CC /\ A # 0 ) /\ ( N e. RR /\ -u N e. NN ) ) -> ( abs ` A ) e. CC )
31		abs00ap 11702	. . . . . . . . 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 10873	. . . . . . 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 1275	. . . . . 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 2274	. . . . 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 725	. . 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 1227	. 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 1312	1 |- ( ( A e. CC /\ A # 0 /\ N e. ZZ ) -> ( abs ` ( A ^ N ) ) = ( ( abs ` A ) ^ N ) )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   /\ w3a 1005   = wceq 1398   e. wcel 2202   class class class wbr 4093   ` cfv 5333  (class class class)co 6028   CCcc 8090   RRcr 8091   0cc0 8092   1c1 8093   -ucneg 8410   # cap 8820   / cdiv 8911   NNcn 9202   NN0cn0 9461   ZZcz 9540   ^cexp 10863   abscabs 11637

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692  ax-cnex 8183  ax-resscn 8184  ax-1cn 8185  ax-1re 8186  ax-icn 8187  ax-addcl 8188  ax-addrcl 8189  ax-mulcl 8190  ax-mulrcl 8191  ax-addcom 8192  ax-mulcom 8193  ax-addass 8194  ax-mulass 8195  ax-distr 8196  ax-i2m1 8197  ax-0lt1 8198  ax-1rid 8199  ax-0id 8200  ax-rnegex 8201  ax-precex 8202  ax-cnre 8203  ax-pre-ltirr 8204  ax-pre-ltwlin 8205  ax-pre-lttrn 8206  ax-pre-apti 8207  ax-pre-ltadd 8208  ax-pre-mulgt0 8209  ax-pre-mulext 8210  ax-arch 8211  ax-caucvg 8212

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-nel 2499  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-tr 4193  df-id 4396  df-po 4399  df-iso 4400  df-iord 4469  df-on 4471  df-ilim 4472  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-recs 6514  df-frec 6600  df-pnf 8275  df-mnf 8276  df-xr 8277  df-ltxr 8278  df-le 8279  df-sub 8411  df-neg 8412  df-reap 8814  df-ap 8821  df-div 8912  df-inn 9203  df-2 9261  df-3 9262  df-4 9263  df-n0 9462  df-z 9541  df-uz 9817  df-rp 9950  df-seqfrec 10773  df-exp 10864  df-cj 11482  df-re 11483  df-im 11484  df-rsqrt 11638  df-abs 11639

This theorem is referenced by:  lgseisen  15893