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Theorem odadd 15467
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 961 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  G  e.  Abel )
2 ablgrp 15419 . . . . 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 962 . . . 4  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  A  e.  X
)
5 simpl3 963 . . . 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 14820 . . . 4  |-  ( ( G  e.  Grp  /\  A  e.  X  /\  B  e.  X )  ->  ( A  .+  B
)  e.  X )
93, 4, 5, 8syl3anc 1185 . . 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 15176 . . 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 15176 . . . 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 15176 . . . 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 10281 . 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 449 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( O `
 A )  gcd  ( O `  B
) )  =  1 )
1918oveq2d 6099 . . . 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 10278 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( O `  ( A  .+  B ) )  e.  CC )
2120mulid1d 9107 . . . 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 2470 . . 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 15465 . . . 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 453 . . 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 4236 . 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 15466 . . . 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 453 . . 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 6098 . . . . . 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 11478 . . . . . 6  |-  ( 1 ^ 2 )  =  1
3028, 29syl6eq 2486 . . . . 5  |-  ( ( ( G  e.  Abel  /\  A  e.  X  /\  B  e.  X )  /\  ( ( O `  A )  gcd  ( O `  B )
)  =  1 )  ->  ( ( ( O `  A )  gcd  ( O `  B ) ) ^
2 )  =  1 )
3130oveq2d 6099 . . . 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 2470 . . 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 4238 . 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 12899 . 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 1186 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 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4214   ` cfv 5456  (class class class)co 6083   1c1 8993    x. cmul 8997   2c2 10051   NN0cn0 10223   ^cexp 11384    || cdivides 12854    gcd cgcd 13008   Basecbs 13471   +g cplusg 13531   Grpcgrp 14687   odcod 15165   Abelcabel 15415
This theorem is referenced by:  gexexlem  15469
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703  ax-inf2 7598  ax-cnex 9048  ax-resscn 9049  ax-1cn 9050  ax-icn 9051  ax-addcl 9052  ax-addrcl 9053  ax-mulcl 9054  ax-mulrcl 9055  ax-mulcom 9056  ax-addass 9057  ax-mulass 9058  ax-distr 9059  ax-i2m1 9060  ax-1ne0 9061  ax-1rid 9062  ax-rnegex 9063  ax-rrecex 9064  ax-cnre 9065  ax-pre-lttri 9066  ax-pre-lttrn 9067  ax-pre-ltadd 9068  ax-pre-mulgt0 9069  ax-pre-sup 9070
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 938  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-pss 3338  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-tp 3824  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-tr 4305  df-eprel 4496  df-id 4500  df-po 4505  df-so 4506  df-fr 4543  df-we 4545  df-ord 4586  df-on 4587  df-lim 4588  df-suc 4589  df-om 4848  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-1st 6351  df-2nd 6352  df-riota 6551  df-recs 6635  df-rdg 6670  df-er 6907  df-en 7112  df-dom 7113  df-sdom 7114  df-sup 7448  df-pnf 9124  df-mnf 9125  df-xr 9126  df-ltxr 9127  df-le 9128  df-sub 9295  df-neg 9296  df-div 9680  df-nn 10003  df-2 10060  df-3 10061  df-n0 10224  df-z 10285  df-uz 10491  df-rp 10615  df-fz 11046  df-fzo 11138  df-fl 11204  df-mod 11253  df-seq 11326  df-exp 11385  df-cj 11906  df-re 11907  df-im 11908  df-sqr 12042  df-abs 12043  df-dvds 12855  df-gcd 13009  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-sbg 14816  df-mulg 14817  df-od 15169  df-cmn 15416  df-abl 15417
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