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Theorem rpexp12i 14645
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 14644 . . 3  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  M  e.  NN0 )  ->  (
( A  gcd  B
)  =  1  -> 
( ( A ^ M )  gcd  B
)  =  1 ) )
213adant3r 1261 . 2  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A  gcd  B )  =  1  -> 
( ( A ^ M )  gcd  B
)  =  1 ) )
3 simp2 1006 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  B  e.  ZZ )
4 simp1 1005 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  A  e.  ZZ )
5 simp3l 1033 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  M  e.  NN0 )
6 zexpcl 12284 . . . . 5  |-  ( ( A  e.  ZZ  /\  M  e.  NN0 )  -> 
( A ^ M
)  e.  ZZ )
74, 5, 6syl2anc 665 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( A ^ M
)  e.  ZZ )
8 simp3r 1034 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  ->  N  e.  NN0 )
9 rpexp1i 14644 . . . 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 1264 . . 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 14458 . . . . 5  |-  ( ( ( A ^ M
)  e.  ZZ  /\  B  e.  ZZ )  ->  ( ( A ^ M )  gcd  B
)  =  ( B  gcd  ( A ^ M ) ) )
127, 3, 11syl2anc 665 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A ^ M )  gcd  B
)  =  ( B  gcd  ( A ^ M ) ) )
1312eqeq1d 2431 . . 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 12284 . . . . . 6  |-  ( ( B  e.  ZZ  /\  N  e.  NN0 )  -> 
( B ^ N
)  e.  ZZ )
153, 8, 14syl2anc 665 . . . . 5  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( B ^ N
)  e.  ZZ )
16 gcdcom 14458 . . . . 5  |-  ( ( ( A ^ M
)  e.  ZZ  /\  ( B ^ N )  e.  ZZ )  -> 
( ( A ^ M )  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd  ( A ^ M
) ) )
177, 15, 16syl2anc 665 . . . 4  |-  ( ( A  e.  ZZ  /\  B  e.  ZZ  /\  ( M  e.  NN0  /\  N  e.  NN0 ) )  -> 
( ( A ^ M )  gcd  ( B ^ N ) )  =  ( ( B ^ N )  gcd  ( A ^ M
) ) )
1817eqeq1d 2431 . . 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 271 . 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 45 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 370    /\ w3a 982    = wceq 1437    e. wcel 1870  (class class class)co 6305   1c1 9539   NN0cn0 10869   ZZcz 10937   ^cexp 12269    gcd cgcd 14442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597  ax-cnex 9594  ax-resscn 9595  ax-1cn 9596  ax-icn 9597  ax-addcl 9598  ax-addrcl 9599  ax-mulcl 9600  ax-mulrcl 9601  ax-mulcom 9602  ax-addass 9603  ax-mulass 9604  ax-distr 9605  ax-i2m1 9606  ax-1ne0 9607  ax-1rid 9608  ax-rnegex 9609  ax-rrecex 9610  ax-cnre 9611  ax-pre-lttri 9612  ax-pre-lttrn 9613  ax-pre-ltadd 9614  ax-pre-mulgt0 9615  ax-pre-sup 9616
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-reu 2789  df-rmo 2790  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-tp 4007  df-op 4009  df-uni 4223  df-int 4259  df-iun 4304  df-br 4427  df-opab 4485  df-mpt 4486  df-tr 4521  df-eprel 4765  df-id 4769  df-po 4775  df-so 4776  df-fr 4813  df-we 4815  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-pred 5399  df-ord 5445  df-on 5446  df-lim 5447  df-suc 5448  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609  df-riota 6267  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-om 6707  df-1st 6807  df-2nd 6808  df-wrecs 7036  df-recs 7098  df-rdg 7136  df-1o 7190  df-2o 7191  df-oadd 7194  df-er 7371  df-en 7578  df-dom 7579  df-sdom 7580  df-fin 7581  df-sup 7962  df-inf 7963  df-pnf 9676  df-mnf 9677  df-xr 9678  df-ltxr 9679  df-le 9680  df-sub 9861  df-neg 9862  df-div 10269  df-nn 10610  df-2 10668  df-3 10669  df-n0 10870  df-z 10938  df-uz 11160  df-rp 11303  df-fz 11783  df-fl 12025  df-mod 12094  df-seq 12211  df-exp 12270  df-cj 13141  df-re 13142  df-im 13143  df-sqrt 13277  df-abs 13278  df-dvds 14284  df-gcd 14443  df-prm 14594
This theorem is referenced by:  ablfac1b  17638  jm2.20nn  35558
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