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Theorem odadd 16729
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 999 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  G  e.  Abel )
2 ablgrp 16676 . . . . 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 1000 . . . 4  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  A  e.  X
)
5 simpl3 1001 . . . 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 15935 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .+  B
)  e.  X )
93, 4, 5, 8syl3anc 1228 . . 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 16433 . . 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 16433 . . . 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 16433 . . . 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 10869 . 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 6311 . . . 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 10866 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  ( A  .+  B ) )  e.  CC )
2120mulid1d 9625 . . . 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 2508 . . 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 16727 . . . 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 4475 . 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 16728 . . . 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 6310 . . . . . 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 12242 . . . . . 6  |-  ( 1 ^ 2 )  =  1
3028, 29syl6eq 2524 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( ( O `  A )  gcd  ( O `  B ) ) ^
2 )  =  1 )
3130oveq2d 6311 . . . 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 2508 . . 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 4477 . 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 13909 . 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 1229 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   1c1 9505    x. cmul 9509   2c2 10597   NN0cn0 10807   ^cexp 12146    || cdivides 13864    gcd cgcd 14020   Basecbs 14507   +g cplusg 14572   Grpcgrp 15925   odcod 16422   Abelcabl 16672
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-inf2 8070  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581  ax-pre-sup 9582
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-1st 6795  df-2nd 6796  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-sup 7913  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-div 10219  df-nn 10549  df-2 10606  df-3 10607  df-n0 10808  df-z 10877  df-uz 11095  df-rp 11233  df-fz 11685  df-fzo 11805  df-fl 11909  df-mod 11977  df-seq 12088  df-exp 12147  df-cj 12912  df-re 12913  df-im 12914  df-sqrt 13048  df-abs 13049  df-dvds 13865  df-gcd 14021  df-0g 14714  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-sbg 15931  df-mulg 15932  df-od 16426  df-cmn 16673  df-abl 16674
This theorem is referenced by:  gexexlem  16731
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