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Theorem odadd 15158
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 958 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  G  e.  Abel )
2 ablgrp 15110 . . . . 5  |-  ( G  e.  Abel  ->  G  e. 
Grp )
31, 2syl 15 . . . 4  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  G  e.  Grp )
4 simpl2 959 . . . 4  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  A  e.  X
)
5 simpl3 960 . . . 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 14511 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .+  B
)  e.  X )
93, 4, 5, 8syl3anc 1182 . . 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 14867 . . 3  |-  ( ( A  .+  B )  e.  X  ->  ( O `  ( A  .+  B ) )  e. 
NN0 )
129, 11syl 15 . 2  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  ( A  .+  B ) )  e.  NN0 )
136, 10odcl 14867 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
144, 13syl 15 . . 3  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  A )  e.  NN0 )
156, 10odcl 14867 . . . 4  |-  ( B  e.  X  ->  ( O `  B )  e.  NN0 )
165, 15syl 15 . . 3  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  B )  e.  NN0 )
1714, 16nn0mulcld 10039 . 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 447 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 A )  gcd  ( O `  B
) )  =  1 )
1918oveq2d 5890 . . . 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 10036 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  ( A  .+  B ) )  e.  CC )
2120mulid1d 8868 . . . 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 2328 . . 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 15156 . . . 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 451 . . 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 4061 . 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 15157 . . . 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 451 . . 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 5889 . . . . . 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 11214 . . . . . 6  |-  ( 1 ^ 2 )  =  1
3028, 29syl6eq 2344 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( ( O `  A )  gcd  ( O `  B ) ) ^
2 )  =  1 )
3130oveq2d 5890 . . . 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 2328 . . 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 4063 . 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 12592 . 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 1183 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 set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   1c1 8754    x. cmul 8758   2c2 9811   NN0cn0 9981   ^cexp 11120    || cdivides 12547    gcd cgcd 12701   Basecbs 13164   +g cplusg 13224   Grpcgrp 14378   odcod 14856   Abelcabel 15106
This theorem is referenced by:  gexexlem  15160
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-inf2 7358  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830  ax-pre-sup 8831
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-sup 7210  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-div 9440  df-nn 9763  df-2 9820  df-3 9821  df-n0 9982  df-z 10041  df-uz 10247  df-rp 10371  df-fz 10799  df-fzo 10887  df-fl 10941  df-mod 10990  df-seq 11063  df-exp 11121  df-cj 11600  df-re 11601  df-im 11602  df-sqr 11736  df-abs 11737  df-dvds 12548  df-gcd 12702  df-0g 13420  df-mnd 14383  df-grp 14505  df-minusg 14506  df-sbg 14507  df-mulg 14508  df-od 14860  df-cmn 15107  df-abl 15108
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