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Theorem rpexp12i 13112
Description: Relative primality passes to symmetric powers. (Contributed by Stefan O'Rear, 27-Sep-2014.)
Assertion
Ref Expression
rpexp12i  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A  gcd  B )  =  1  -> 
( ( A ^ M )  gcd  ( B ^ N ) )  =  1 ) )

Proof of Theorem rpexp12i
StepHypRef Expression
1 rpexp1i 13111 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN0 )  ->  (
( A  gcd  B
)  =  1  -> 
( ( A ^ M )  gcd  B
)  =  1 ) )
213adant3r 1181 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A  gcd  B )  =  1  -> 
( ( A ^ M )  gcd  B
)  =  1 ) )
3 simp2 958 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  B  e.  ZZ )
4 simp1 957 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  A  e.  ZZ )
5 simp3l 985 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  M  e.  NN0 )
6 zexpcl 11386 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  ZZ )
74, 5, 6syl2anc 643 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( A ^ M
)  e.  ZZ )
8 simp3r 986 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  N  e.  NN0 )
9 rpexp1i 13111 . . . 4  |-  ( ( B  e.  ZZ  /\  ( A ^ M )  e.  ZZ  /\  N  e.  NN0 )  ->  (
( B  gcd  ( A ^ M ) )  =  1  ->  (
( B ^ N
)  gcd  ( A ^ M ) )  =  1 ) )
103, 7, 8, 9syl3anc 1184 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( B  gcd  ( A ^ M ) )  =  1  -> 
( ( B ^ N )  gcd  ( A ^ M ) )  =  1 ) )
11 gcdcom 13010 . . . . 5  |-  ( ( ( A ^ M
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^ M )  gcd  B
)  =  ( B  gcd  ( A ^ M ) ) )
127, 3, 11syl2anc 643 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A ^ M )  gcd  B
)  =  ( B  gcd  ( A ^ M ) ) )
1312eqeq1d 2443 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( ( A ^ M )  gcd 
B )  =  1  <-> 
( B  gcd  ( A ^ M ) )  =  1 ) )
14 zexpcl 11386 . . . . . 6  |-  ( ( B  e.  ZZ  /\  N  e.  NN0 )  -> 
( B ^ N
)  e.  ZZ )
153, 8, 14syl2anc 643 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( B ^ N
)  e.  ZZ )
16 gcdcom 13010 . . . . 5  |-  ( ( ( A ^ M
)  e.  ZZ  /\  ( B ^ N )  e.  ZZ )  -> 
( ( A ^ M )  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd  ( A ^ M
) ) )
177, 15, 16syl2anc 643 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A ^ M )  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd  ( A ^ M
) ) )
1817eqeq1d 2443 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( ( A ^ M )  gcd  ( B ^ N
) )  =  1  <-> 
( ( B ^ N )  gcd  ( A ^ M ) )  =  1 ) )
1910, 13, 183imtr4d 260 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( ( A ^ M )  gcd 
B )  =  1  ->  ( ( A ^ M )  gcd  ( B ^ N
) )  =  1 ) )
202, 19syld 42 1  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A  gcd  B )  =  1  -> 
( ( A ^ M )  gcd  ( B ^ N ) )  =  1 ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    /\ w3a 936    = wceq 1652    e. wcel 1725  (class class class)co 6073   1c1 8981   NN0cn0 10211   ZZcz 10272   ^cexp 11372    gcd cgcd 12996
This theorem is referenced by:  ablfac1b  15618  jm2.20nn  27022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693  ax-cnex 9036  ax-resscn 9037  ax-1cn 9038  ax-icn 9039  ax-addcl 9040  ax-addrcl 9041  ax-mulcl 9042  ax-mulrcl 9043  ax-mulcom 9044  ax-addass 9045  ax-mulass 9046  ax-distr 9047  ax-i2m1 9048  ax-1ne0 9049  ax-1rid 9050  ax-rnegex 9051  ax-rrecex 9052  ax-cnre 9053  ax-pre-lttri 9054  ax-pre-lttrn 9055  ax-pre-ltadd 9056  ax-pre-mulgt0 9057  ax-pre-sup 9058
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-reu 2704  df-rmo 2705  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-tp 3814  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-suc 4579  df-om 4838  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fn 5449  df-f 5450  df-f1 5451  df-fo 5452  df-f1o 5453  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-1st 6341  df-2nd 6342  df-riota 6541  df-recs 6625  df-rdg 6660  df-1o 6716  df-2o 6717  df-oadd 6720  df-er 6897  df-en 7102  df-dom 7103  df-sdom 7104  df-fin 7105  df-sup 7438  df-pnf 9112  df-mnf 9113  df-xr 9114  df-ltxr 9115  df-le 9116  df-sub 9283  df-neg 9284  df-div 9668  df-nn 9991  df-2 10048  df-3 10049  df-n0 10212  df-z 10273  df-uz 10479  df-rp 10603  df-fz 11034  df-fl 11192  df-mod 11241  df-seq 11314  df-exp 11373  df-cj 11894  df-re 11895  df-im 11896  df-sqr 12030  df-abs 12031  df-dvds 12843  df-gcd 12997  df-prm 13070
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