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Theorem mndodcongi 16061
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 16060 . . . . 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 207 . . . 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 1189 . . 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 1182 . 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 10684 . . . . . . 7  |-  ( M  e.  NN0  ->  M  e.  ZZ )
10 nn0z 10684 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  ZZ )
11 zsubcl 10702 . . . . . . 7  |-  ( ( M  e.  ZZ  /\  N  e.  ZZ )  ->  ( M  -  N
)  e.  ZZ )
129, 10, 11syl2an 477 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( M  -  N
)  e.  ZZ )
13123ad2ant3 1011 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( M  -  N )  e.  ZZ )
14 0dvds 13568 . . . . 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 10604 . . . . . . 7  |-  ( M  e.  NN0  ->  M  e.  CC )
17 nn0cn 10604 . . . . . . 7  |-  ( N  e.  NN0  ->  N  e.  CC )
18 subeq0 9650 . . . . . . 7  |-  ( ( M  e.  CC  /\  N  e.  CC )  ->  ( ( M  -  N )  =  0  <-> 
M  =  N ) )
1916, 17, 18syl2an 477 . . . . . 6  |-  ( ( M  e.  NN0  /\  N  e.  NN0 )  -> 
( ( M  -  N )  =  0  <-> 
M  =  N ) )
20193ad2ant3 1011 . . . . 5  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( M  -  N )  =  0  <->  M  =  N ) )
21 oveq1 6113 . . . . 5  |-  ( M  =  N  ->  ( M  .x.  A )  =  ( N  .x.  A
) )
2220, 21syl6bi 228 . . . 4  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( M  -  N )  =  0  ->  ( M  .x.  A )  =  ( N  .x.  A
) ) )
2315, 22sylbid 215 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( 0 
||  ( M  -  N )  ->  ( M  .x.  A )  =  ( N  .x.  A
) ) )
24 breq1 4310 . . . 4  |-  ( ( O `  A )  =  0  ->  (
( O `  A
)  ||  ( M  -  N )  <->  0  ||  ( M  -  N
) ) )
2524imbi1d 317 . . 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 222 . 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 16054 . . . 4  |-  ( A  e.  X  ->  ( O `  A )  e.  NN0 )
28273ad2ant2 1010 . . 3  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( O `  A )  e.  NN0 )
29 elnn0 10596 . . 3  |-  ( ( O `  A )  e.  NN0  <->  ( ( O `
 A )  e.  NN  \/  ( O `
 A )  =  0 ) )
3028, 29sylib 196 . 2  |-  ( ( G  e.  Mnd  /\  A  e.  X  /\  ( M  e.  NN0  /\  N  e.  NN0 )
)  ->  ( ( O `  A )  e.  NN  \/  ( O `
 A )  =  0 ) )
318, 26, 30mpjaod 381 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756   class class class wbr 4307   ` cfv 5433  (class class class)co 6106   CCcc 9295   0cc0 9297    - cmin 9610   NNcn 10337   NN0cn0 10594   ZZcz 10661    || cdivides 13550   Basecbs 14189   0gc0g 14393   Mndcmnd 15424  .gcmg 15429   odcod 16043
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 4418  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387  ax-inf2 7862  ax-cnex 9353  ax-resscn 9354  ax-1cn 9355  ax-icn 9356  ax-addcl 9357  ax-addrcl 9358  ax-mulcl 9359  ax-mulrcl 9360  ax-mulcom 9361  ax-addass 9362  ax-mulass 9363  ax-distr 9364  ax-i2m1 9365  ax-1ne0 9366  ax-1rid 9367  ax-rnegex 9368  ax-rrecex 9369  ax-cnre 9370  ax-pre-lttri 9371  ax-pre-lttrn 9372  ax-pre-ltadd 9373  ax-pre-mulgt0 9374  ax-pre-sup 9375
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 2577  df-ne 2622  df-nel 2623  df-ral 2735  df-rex 2736  df-reu 2737  df-rmo 2738  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-pss 3359  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-tp 3897  df-op 3899  df-uni 4107  df-iun 4188  df-br 4308  df-opab 4366  df-mpt 4367  df-tr 4401  df-eprel 4647  df-id 4651  df-po 4656  df-so 4657  df-fr 4694  df-we 4696  df-ord 4737  df-on 4738  df-lim 4739  df-suc 4740  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-rn 4866  df-res 4867  df-ima 4868  df-iota 5396  df-fun 5435  df-fn 5436  df-f 5437  df-f1 5438  df-fo 5439  df-f1o 5440  df-fv 5441  df-riota 6067  df-ov 6109  df-oprab 6110  df-mpt2 6111  df-om 6492  df-1st 6592  df-2nd 6593  df-recs 6847  df-rdg 6881  df-er 7116  df-en 7326  df-dom 7327  df-sdom 7328  df-sup 7706  df-pnf 9435  df-mnf 9436  df-xr 9437  df-ltxr 9438  df-le 9439  df-sub 9612  df-neg 9613  df-div 10009  df-nn 10338  df-n0 10595  df-z 10662  df-uz 10877  df-rp 11007  df-fz 11453  df-fl 11657  df-mod 11724  df-seq 11822  df-dvds 13551  df-0g 14395  df-mnd 15430  df-mulg 15563  df-od 16047
This theorem is referenced by: (None)
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