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Theorem rpexp12i 13925
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 13924 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN0 )  ->  (
( A  gcd  B
)  =  1  -> 
( ( A ^ M )  gcd  B
)  =  1 ) )
213adant3r 1216 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A  gcd  B )  =  1  -> 
( ( A ^ M )  gcd  B
)  =  1 ) )
3 simp2 989 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  B  e.  ZZ )
4 simp1 988 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  A  e.  ZZ )
5 simp3l 1016 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  M  e.  NN0 )
6 zexpcl 11996 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  ZZ )
74, 5, 6syl2anc 661 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( A ^ M
)  e.  ZZ )
8 simp3r 1017 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  N  e.  NN0 )
9 rpexp1i 13924 . . . 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 1219 . . 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 13821 . . . . 5  |-  ( ( ( A ^ M
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^ M )  gcd  B
)  =  ( B  gcd  ( A ^ M ) ) )
127, 3, 11syl2anc 661 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A ^ M )  gcd  B
)  =  ( B  gcd  ( A ^ M ) ) )
1312eqeq1d 2456 . . 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 11996 . . . . . 6  |-  ( ( B  e.  ZZ  /\  N  e.  NN0 )  -> 
( B ^ N
)  e.  ZZ )
153, 8, 14syl2anc 661 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( B ^ N
)  e.  ZZ )
16 gcdcom 13821 . . . . 5  |-  ( ( ( A ^ M
)  e.  ZZ  /\  ( B ^ N )  e.  ZZ )  -> 
( ( A ^ M )  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd  ( A ^ M
) ) )
177, 15, 16syl2anc 661 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A ^ M )  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd  ( A ^ M
) ) )
1817eqeq1d 2456 . . 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 268 . 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 44 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 setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1370    e. wcel 1758  (class class class)co 6199   1c1 9393   NN0cn0 10689   ZZcz 10756   ^cexp 11981    gcd cgcd 13807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481  ax-cnex 9448  ax-resscn 9449  ax-1cn 9450  ax-icn 9451  ax-addcl 9452  ax-addrcl 9453  ax-mulcl 9454  ax-mulrcl 9455  ax-mulcom 9456  ax-addass 9457  ax-mulass 9458  ax-distr 9459  ax-i2m1 9460  ax-1ne0 9461  ax-1rid 9462  ax-rnegex 9463  ax-rrecex 9464  ax-cnre 9465  ax-pre-lttri 9466  ax-pre-lttrn 9467  ax-pre-ltadd 9468  ax-pre-mulgt0 9469  ax-pre-sup 9470
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-nel 2650  df-ral 2803  df-rex 2804  df-reu 2805  df-rmo 2806  df-rab 2807  df-v 3078  df-sbc 3293  df-csb 3395  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-pss 3451  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-tp 3989  df-op 3991  df-uni 4199  df-int 4236  df-iun 4280  df-br 4400  df-opab 4458  df-mpt 4459  df-tr 4493  df-eprel 4739  df-id 4743  df-po 4748  df-so 4749  df-fr 4786  df-we 4788  df-ord 4829  df-on 4830  df-lim 4831  df-suc 4832  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-f1 5530  df-fo 5531  df-f1o 5532  df-fv 5533  df-riota 6160  df-ov 6202  df-oprab 6203  df-mpt2 6204  df-om 6586  df-1st 6686  df-2nd 6687  df-recs 6941  df-rdg 6975  df-1o 7029  df-2o 7030  df-oadd 7033  df-er 7210  df-en 7420  df-dom 7421  df-sdom 7422  df-fin 7423  df-sup 7801  df-pnf 9530  df-mnf 9531  df-xr 9532  df-ltxr 9533  df-le 9534  df-sub 9707  df-neg 9708  df-div 10104  df-nn 10433  df-2 10490  df-3 10491  df-n0 10690  df-z 10757  df-uz 10972  df-rp 11102  df-fz 11554  df-fl 11758  df-mod 11825  df-seq 11923  df-exp 11982  df-cj 12705  df-re 12706  df-im 12707  df-sqr 12841  df-abs 12842  df-dvds 13653  df-gcd 13808  df-prm 13881
This theorem is referenced by:  ablfac1b  16692  jm2.20nn  29493
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