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Theorem mndodcongi 15182
Description: If two multipliers are congruent relative to the base point's order, the corresponding multiples are the same. For monoids, the reverse implication is false for elements with infinite order. For example, the powers of  2 mod  10 are 1,2,4,8,6,2,4,8,6,... so that the identity 1 never repeats, which is infinite order by our definition, yet other numbers like 6 appear many times in the sequence. (Contributed by Mario Carneiro, 23-Sep-2015.)
Hypotheses
Ref Expression
odcl.1  |-  X  =  ( Base `  G
)
odcl.2  |-  O  =  ( od `  G
)
odid.3  |-  .x.  =  (.g
`  G )
odid.4  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
mndodcongi  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( O `  A )  ||  ( M  -  N
)  ->  ( M  .x.  A )  =  ( N  .x.  A ) ) )

Proof of Theorem mndodcongi
StepHypRef Expression
1 odcl.1 . . . . . 6  |-  X  =  ( Base `  G
)
2 odcl.2 . . . . . 6  |-  O  =  ( od `  G
)
3 odid.3 . . . . . 6  |-  .x.  =  (.g
`  G )
4 odid.4 . . . . . 6  |-  .0.  =  ( 0g `  G )
51, 2, 3, 4mndodcong 15181 . . . . 5  |-  ( ( ( G  e.  Mnd  /\  A  e.  X )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( O `
 A )  e.  NN )  ->  (
( O `  A
)  ||  ( M  -  N )  <->  ( M  .x.  A )  =  ( N  .x.  A ) ) )
65biimpd 200 . . . 4  |-  ( ( ( G  e.  Mnd  /\  A  e.  X )  /\  ( M  e. 
NN0  /\  N  e.  NN0 )  /\  ( O `
 A )  e.  NN )  ->  (
( O `  A
)  ||  ( M  -  N )  ->  ( M  .x.  A )  =  ( N  .x.  A
) ) )
763expia 1156 . . 3  |-  ( ( ( G  e.  Mnd  /\  A  e.  X )  /\  ( M  e. 
NN0  /\  N  e.  NN0 ) )  ->  (
( O `  A
)  e.  NN  ->  ( ( O `  A
)  ||  ( M  -  N )  ->  ( M  .x.  A )  =  ( N  .x.  A
) ) ) )
873impa 1149 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( O `  A )  e.  NN  ->  ( ( O `  A )  ||  ( M  -  N
)  ->  ( M  .x.  A )  =  ( N  .x.  A ) ) ) )
9 nn0z 10305 . . . . . . 7  |-  ( M  e.  NN0  ->  M  e.  ZZ )
10 nn0z 10305 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
11 zsubcl 10320 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N
)  e.  ZZ )
129, 10, 11syl2an 465 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  -  N
)  e.  ZZ )
13123ad2ant3 981 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( M  -  N )  e.  ZZ )
14 0dvds 12871 . . . . 5  |-  ( ( M  -  N )  e.  ZZ  ->  (
0  ||  ( M  -  N )  <->  ( M  -  N )  =  0 ) )
1513, 14syl 16 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( 0 
||  ( M  -  N )  <->  ( M  -  N )  =  0 ) )
16 nn0cn 10232 . . . . . . 7  |-  ( M  e.  NN0  ->  M  e.  CC )
17 nn0cn 10232 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  CC )
18 subeq0 9328 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  -  N )  =  0  <-> 
M  =  N ) )
1916, 17, 18syl2an 465 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( M  -  N )  =  0  <-> 
M  =  N ) )
20193ad2ant3 981 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( M  -  N )  =  0  <->  M  =  N ) )
21 oveq1 6089 . . . . 5  |-  ( M  =  N  ->  ( M  .x.  A )  =  ( N  .x.  A
) )
2220, 21syl6bi 221 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( M  -  N )  =  0  ->  ( M  .x.  A )  =  ( N  .x.  A
) ) )
2315, 22sylbid 208 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( 0 
||  ( M  -  N )  ->  ( M  .x.  A )  =  ( N  .x.  A
) ) )
24 breq1 4216 . . . 4  |-  ( ( O `  A )  =  0  ->  (
( O `  A
)  ||  ( M  -  N )  <->  0  ||  ( M  -  N
) ) )
2524imbi1d 310 . . 3  |-  ( ( O `  A )  =  0  ->  (
( ( O `  A )  ||  ( M  -  N )  ->  ( M  .x.  A
)  =  ( N 
.x.  A ) )  <-> 
( 0  ||  ( M  -  N )  ->  ( M  .x.  A
)  =  ( N 
.x.  A ) ) ) )
2623, 25syl5ibrcom 215 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( O `  A )  =  0  ->  (
( O `  A
)  ||  ( M  -  N )  ->  ( M  .x.  A )  =  ( N  .x.  A
) ) ) )
271, 2odcl 15175 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
28273ad2ant2 980 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( O `  A )  e.  NN0 )
29 elnn0 10224 . . 3  |-  ( ( O `  A )  e.  NN0  <->  ( ( O `
 A )  e.  NN  \/  ( O `
 A )  =  0 ) )
3028, 29sylib 190 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( O `  A )  e.  NN  \/  ( O `
 A )  =  0 ) )
318, 26, 30mpjaod 372 1  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( O `  A )  ||  ( M  -  N
)  ->  ( M  .x.  A )  =  ( N  .x.  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    \/ wo 359    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   class class class wbr 4213   ` cfv 5455  (class class class)co 6082   CCcc 8989   0cc0 8991    - cmin 9292   NNcn 10001   NN0cn0 10222   ZZcz 10283    || cdivides 12853   Basecbs 13470   0gc0g 13724   Mndcmnd 14685  .gcmg 14690   odcod 15164
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 2418  ax-rep 4321  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404  ax-un 4702  ax-inf2 7597  ax-cnex 9047  ax-resscn 9048  ax-1cn 9049  ax-icn 9050  ax-addcl 9051  ax-addrcl 9052  ax-mulcl 9053  ax-mulrcl 9054  ax-mulcom 9055  ax-addass 9056  ax-mulass 9057  ax-distr 9058  ax-i2m1 9059  ax-1ne0 9060  ax-1rid 9061  ax-rnegex 9062  ax-rrecex 9063  ax-cnre 9064  ax-pre-lttri 9065  ax-pre-lttrn 9066  ax-pre-ltadd 9067  ax-pre-mulgt0 9068  ax-pre-sup 9069
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-reu 2713  df-rmo 2714  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-pss 3337  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-tp 3823  df-op 3824  df-uni 4017  df-iun 4096  df-br 4214  df-opab 4268  df-mpt 4269  df-tr 4304  df-eprel 4495  df-id 4499  df-po 4504  df-so 4505  df-fr 4542  df-we 4544  df-ord 4585  df-on 4586  df-lim 4587  df-suc 4588  df-om 4847  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fn 5458  df-f 5459  df-f1 5460  df-fo 5461  df-f1o 5462  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-1st 6350  df-2nd 6351  df-riota 6550  df-recs 6634  df-rdg 6669  df-er 6906  df-en 7111  df-dom 7112  df-sdom 7113  df-sup 7447  df-pnf 9123  df-mnf 9124  df-xr 9125  df-ltxr 9126  df-le 9127  df-sub 9294  df-neg 9295  df-div 9679  df-nn 10002  df-n0 10223  df-z 10284  df-uz 10490  df-rp 10614  df-fz 11045  df-fl 11203  df-mod 11252  df-seq 11325  df-dvds 12854  df-0g 13728  df-mnd 14691  df-mulg 14816  df-od 15168
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