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Theorem odadd 16332
Description: The order of a product is the product of the orders, if the factors have coprime order. (Contributed by Mario Carneiro, 20-Oct-2015.)
Hypotheses
Ref Expression
odadd1.1  |-  O  =  ( od `  G
)
odadd1.2  |-  X  =  ( Base `  G
)
odadd1.3  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
odadd  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  ( A  .+  B ) )  =  ( ( O `  A )  x.  ( O `  B ) ) )

Proof of Theorem odadd
StepHypRef Expression
1 simpl1 991 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  G  e.  Abel )
2 ablgrp 16282 . . . . 5  |-  ( G  e.  Abel  ->  G  e. 
Grp )
31, 2syl 16 . . . 4  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  G  e.  Grp )
4 simpl2 992 . . . 4  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  A  e.  X
)
5 simpl3 993 . . . 4  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  B  e.  X
)
6 odadd1.2 . . . . 5  |-  X  =  ( Base `  G
)
7 odadd1.3 . . . . 5  |-  .+  =  ( +g  `  G )
86, 7grpcl 15551 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .+  B
)  e.  X )
93, 4, 5, 8syl3anc 1218 . . 3  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( A  .+  B )  e.  X
)
10 odadd1.1 . . . 4  |-  O  =  ( od `  G
)
116, 10odcl 16039 . . 3  |-  ( ( A  .+  B )  e.  X  ->  ( O `  ( A  .+  B ) )  e. 
NN0 )
129, 11syl 16 . 2  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  ( A  .+  B ) )  e.  NN0 )
136, 10odcl 16039 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
144, 13syl 16 . . 3  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  A )  e.  NN0 )
156, 10odcl 16039 . . . 4  |-  ( B  e.  X  ->  ( O `  B )  e.  NN0 )
165, 15syl 16 . . 3  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  B )  e.  NN0 )
1714, 16nn0mulcld 10641 . 2  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 A )  x.  ( O `  B
) )  e.  NN0 )
18 simpr 461 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 A )  gcd  ( O `  B
) )  =  1 )
1918oveq2d 6107 . . . 4  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 ( A  .+  B ) )  x.  ( ( O `  A )  gcd  ( O `  B )
) )  =  ( ( O `  ( A  .+  B ) )  x.  1 ) )
2012nn0cnd 10638 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  ( A  .+  B ) )  e.  CC )
2120mulid1d 9403 . . . 4  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 ( A  .+  B ) )  x.  1 )  =  ( O `  ( A 
.+  B ) ) )
2219, 21eqtrd 2475 . . 3  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 ( A  .+  B ) )  x.  ( ( O `  A )  gcd  ( O `  B )
) )  =  ( O `  ( A 
.+  B ) ) )
2310, 6, 7odadd1 16330 . . . 4  |-  ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  ->  (
( O `  ( A  .+  B ) )  x.  ( ( O `
 A )  gcd  ( O `  B
) ) )  ||  ( ( O `  A )  x.  ( O `  B )
) )
2423adantr 465 . . 3  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 ( A  .+  B ) )  x.  ( ( O `  A )  gcd  ( O `  B )
) )  ||  (
( O `  A
)  x.  ( O `
 B ) ) )
2522, 24eqbrtrrd 4314 . 2  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  ( A  .+  B ) )  ||  ( ( O `  A )  x.  ( O `  B ) ) )
2610, 6, 7odadd2 16331 . . . 4  |-  ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  ->  (
( O `  A
)  x.  ( O `
 B ) ) 
||  ( ( O `
 ( A  .+  B ) )  x.  ( ( ( O `
 A )  gcd  ( O `  B
) ) ^ 2 ) ) )
2726adantr 465 . . 3  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 A )  x.  ( O `  B
) )  ||  (
( O `  ( A  .+  B ) )  x.  ( ( ( O `  A )  gcd  ( O `  B ) ) ^
2 ) ) )
2818oveq1d 6106 . . . . . 6  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( ( O `  A )  gcd  ( O `  B ) ) ^
2 )  =  ( 1 ^ 2 ) )
29 sq1 11960 . . . . . 6  |-  ( 1 ^ 2 )  =  1
3028, 29syl6eq 2491 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( ( O `  A )  gcd  ( O `  B ) ) ^
2 )  =  1 )
3130oveq2d 6107 . . . 4  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 ( A  .+  B ) )  x.  ( ( ( O `
 A )  gcd  ( O `  B
) ) ^ 2 ) )  =  ( ( O `  ( A  .+  B ) )  x.  1 ) )
3231, 21eqtrd 2475 . . 3  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 ( A  .+  B ) )  x.  ( ( ( O `
 A )  gcd  ( O `  B
) ) ^ 2 ) )  =  ( O `  ( A 
.+  B ) ) )
3327, 32breqtrd 4316 . 2  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 A )  x.  ( O `  B
) )  ||  ( O `  ( A  .+  B ) ) )
34 dvdseq 13580 . 2  |-  ( ( ( ( O `  ( A  .+  B ) )  e.  NN0  /\  ( ( O `  A )  x.  ( O `  B )
)  e.  NN0 )  /\  ( ( O `  ( A  .+  B ) )  ||  ( ( O `  A )  x.  ( O `  B ) )  /\  ( ( O `  A )  x.  ( O `  B )
)  ||  ( O `  ( A  .+  B
) ) ) )  ->  ( O `  ( A  .+  B ) )  =  ( ( O `  A )  x.  ( O `  B ) ) )
3512, 17, 25, 33, 34syl22anc 1219 1  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  ( A  .+  B ) )  =  ( ( O `  A )  x.  ( O `  B ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   1c1 9283    x. cmul 9287   2c2 10371   NN0cn0 10579   ^cexp 11865    || cdivides 13535    gcd cgcd 13690   Basecbs 14174   +g cplusg 14238   Grpcgrp 15410   odcod 16028   Abelcabel 16278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-inf2 7847  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359  ax-pre-sup 9360
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-1st 6577  df-2nd 6578  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-sup 7691  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-div 9994  df-nn 10323  df-2 10380  df-3 10381  df-n0 10580  df-z 10647  df-uz 10862  df-rp 10992  df-fz 11438  df-fzo 11549  df-fl 11642  df-mod 11709  df-seq 11807  df-exp 11866  df-cj 12588  df-re 12589  df-im 12590  df-sqr 12724  df-abs 12725  df-dvds 13536  df-gcd 13691  df-0g 14380  df-mnd 15415  df-grp 15545  df-minusg 15546  df-sbg 15547  df-mulg 15548  df-od 16032  df-cmn 16279  df-abl 16280
This theorem is referenced by:  gexexlem  16334
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