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Theorem mndodconglem 15179
Description: Lemma for mndodcong 15180. (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 )
mndodconglem.1  |-  ( ph  ->  G  e.  Mnd )
mndodconglem.2  |-  ( ph  ->  A  e.  X )
mndodconglem.3  |-  ( ph  ->  ( O `  A
)  e.  NN )
mndodconglem.4  |-  ( ph  ->  M  e.  NN0 )
mndodconglem.5  |-  ( ph  ->  N  e.  NN0 )
mndodconglem.6  |-  ( ph  ->  M  <  ( O `
 A ) )
mndodconglem.7  |-  ( ph  ->  N  <  ( O `
 A ) )
mndodconglem.8  |-  ( ph  ->  ( M  .x.  A
)  =  ( N 
.x.  A ) )
Assertion
Ref Expression
mndodconglem  |-  ( (
ph  /\  M  <_  N )  ->  M  =  N )

Proof of Theorem mndodconglem
StepHypRef Expression
1 mndodconglem.2 . . . . . . 7  |-  ( ph  ->  A  e.  X )
2 mndodconglem.3 . . . . . . . . . . 11  |-  ( ph  ->  ( O `  A
)  e.  NN )
32nnred 10015 . . . . . . . . . 10  |-  ( ph  ->  ( O `  A
)  e.  RR )
43recnd 9114 . . . . . . . . 9  |-  ( ph  ->  ( O `  A
)  e.  CC )
5 mndodconglem.4 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN0 )
65nn0red 10275 . . . . . . . . . 10  |-  ( ph  ->  M  e.  RR )
76recnd 9114 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
8 mndodconglem.5 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN0 )
98nn0red 10275 . . . . . . . . . 10  |-  ( ph  ->  N  e.  RR )
109recnd 9114 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
114, 7, 10addsubassd 9431 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 A )  +  M )  -  N
)  =  ( ( O `  A )  +  ( M  -  N ) ) )
122nnzd 10374 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  A
)  e.  ZZ )
135nn0zd 10373 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ZZ )
1412, 13zaddcld 10379 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  A )  +  M
)  e.  ZZ )
1514zred 10375 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  A )  +  M
)  e.  RR )
16 mndodconglem.7 . . . . . . . . . 10  |-  ( ph  ->  N  <  ( O `
 A ) )
17 nn0addge1 10266 . . . . . . . . . . 11  |-  ( ( ( O `  A
)  e.  RR  /\  M  e.  NN0 )  -> 
( O `  A
)  <_  ( ( O `  A )  +  M ) )
183, 5, 17syl2anc 643 . . . . . . . . . 10  |-  ( ph  ->  ( O `  A
)  <_  ( ( O `  A )  +  M ) )
199, 3, 15, 16, 18ltletrd 9230 . . . . . . . . 9  |-  ( ph  ->  N  <  ( ( O `  A )  +  M ) )
208nn0zd 10373 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
21 znnsub 10322 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( ( O `  A )  +  M
)  e.  ZZ )  ->  ( N  < 
( ( O `  A )  +  M
)  <->  ( ( ( O `  A )  +  M )  -  N )  e.  NN ) )
2220, 14, 21syl2anc 643 . . . . . . . . 9  |-  ( ph  ->  ( N  <  (
( O `  A
)  +  M )  <-> 
( ( ( O `
 A )  +  M )  -  N
)  e.  NN ) )
2319, 22mpbid 202 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 A )  +  M )  -  N
)  e.  NN )
2411, 23eqeltrrd 2511 . . . . . . 7  |-  ( ph  ->  ( ( O `  A )  +  ( M  -  N ) )  e.  NN )
254, 7, 10addsub12d 9434 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  A )  +  ( M  -  N ) )  =  ( M  +  ( ( O `
 A )  -  N ) ) )
2625oveq1d 6096 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 A )  +  ( M  -  N
) )  .x.  A
)  =  ( ( M  +  ( ( O `  A )  -  N ) ) 
.x.  A ) )
27 mndodconglem.8 . . . . . . . . . . 11  |-  ( ph  ->  ( M  .x.  A
)  =  ( N 
.x.  A ) )
2827oveq1d 6096 . . . . . . . . . 10  |-  ( ph  ->  ( ( M  .x.  A ) ( +g  `  G ) ( ( ( O `  A
)  -  N ) 
.x.  A ) )  =  ( ( N 
.x.  A ) ( +g  `  G ) ( ( ( O `
 A )  -  N )  .x.  A
) ) )
29 mndodconglem.1 . . . . . . . . . . 11  |-  ( ph  ->  G  e.  Mnd )
30 znnsub 10322 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  ( O `  A )  e.  ZZ )  -> 
( N  <  ( O `  A )  <->  ( ( O `  A
)  -  N )  e.  NN ) )
3120, 12, 30syl2anc 643 . . . . . . . . . . . . 13  |-  ( ph  ->  ( N  <  ( O `  A )  <->  ( ( O `  A
)  -  N )  e.  NN ) )
3216, 31mpbid 202 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( O `  A )  -  N
)  e.  NN )
3332nnnn0d 10274 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  A )  -  N
)  e.  NN0 )
34 odcl.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
35 odid.3 . . . . . . . . . . . 12  |-  .x.  =  (.g
`  G )
36 eqid 2436 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
3734, 35, 36mulgnn0dir 14913 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( M  e.  NN0  /\  ( ( O `  A )  -  N
)  e.  NN0  /\  A  e.  X )
)  ->  ( ( M  +  ( ( O `  A )  -  N ) )  .x.  A )  =  ( ( M  .x.  A
) ( +g  `  G
) ( ( ( O `  A )  -  N )  .x.  A ) ) )
3829, 5, 33, 1, 37syl13anc 1186 . . . . . . . . . 10  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( M  .x.  A ) ( +g  `  G
) ( ( ( O `  A )  -  N )  .x.  A ) ) )
3934, 35, 36mulgnn0dir 14913 . . . . . . . . . . 11  |-  ( ( G  e.  Mnd  /\  ( N  e.  NN0  /\  ( ( O `  A )  -  N
)  e.  NN0  /\  A  e.  X )
)  ->  ( ( N  +  ( ( O `  A )  -  N ) )  .x.  A )  =  ( ( N  .x.  A
) ( +g  `  G
) ( ( ( O `  A )  -  N )  .x.  A ) ) )
4029, 8, 33, 1, 39syl13anc 1186 . . . . . . . . . 10  |-  ( ph  ->  ( ( N  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( N  .x.  A ) ( +g  `  G
) ( ( ( O `  A )  -  N )  .x.  A ) ) )
4128, 38, 403eqtr4d 2478 . . . . . . . . 9  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( N  +  ( ( O `  A )  -  N ) ) 
.x.  A ) )
4210, 4pncan3d 9414 . . . . . . . . . . 11  |-  ( ph  ->  ( N  +  ( ( O `  A
)  -  N ) )  =  ( O `
 A ) )
4342oveq1d 6096 . . . . . . . . . 10  |-  ( ph  ->  ( ( N  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( O `  A ) 
.x.  A ) )
44 odcl.2 . . . . . . . . . . . 12  |-  O  =  ( od `  G
)
45 odid.4 . . . . . . . . . . . 12  |-  .0.  =  ( 0g `  G )
4634, 44, 35, 45odid 15176 . . . . . . . . . . 11  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  .0.  )
471, 46syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  A )  .x.  A
)  =  .0.  )
4843, 47eqtrd 2468 . . . . . . . . 9  |-  ( ph  ->  ( ( N  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  .0.  )
4941, 48eqtrd 2468 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  .0.  )
5026, 49eqtrd 2468 . . . . . . 7  |-  ( ph  ->  ( ( ( O `
 A )  +  ( M  -  N
) )  .x.  A
)  =  .0.  )
5134, 44, 35, 45odlem2 15177 . . . . . . 7  |-  ( ( A  e.  X  /\  ( ( O `  A )  +  ( M  -  N ) )  e.  NN  /\  ( ( ( O `
 A )  +  ( M  -  N
) )  .x.  A
)  =  .0.  )  ->  ( O `  A
)  e.  ( 1 ... ( ( O `
 A )  +  ( M  -  N
) ) ) )
521, 24, 50, 51syl3anc 1184 . . . . . 6  |-  ( ph  ->  ( O `  A
)  e.  ( 1 ... ( ( O `
 A )  +  ( M  -  N
) ) ) )
53 elfzle2 11061 . . . . . 6  |-  ( ( O `  A )  e.  ( 1 ... ( ( O `  A )  +  ( M  -  N ) ) )  ->  ( O `  A )  <_  ( ( O `  A )  +  ( M  -  N ) ) )
5452, 53syl 16 . . . . 5  |-  ( ph  ->  ( O `  A
)  <_  ( ( O `  A )  +  ( M  -  N ) ) )
5513, 20zsubcld 10380 . . . . . . 7  |-  ( ph  ->  ( M  -  N
)  e.  ZZ )
5655zred 10375 . . . . . 6  |-  ( ph  ->  ( M  -  N
)  e.  RR )
573, 56addge01d 9614 . . . . 5  |-  ( ph  ->  ( 0  <_  ( M  -  N )  <->  ( O `  A )  <_  ( ( O `
 A )  +  ( M  -  N
) ) ) )
5854, 57mpbird 224 . . . 4  |-  ( ph  ->  0  <_  ( M  -  N ) )
596, 9subge0d 9616 . . . 4  |-  ( ph  ->  ( 0  <_  ( M  -  N )  <->  N  <_  M ) )
6058, 59mpbid 202 . . 3  |-  ( ph  ->  N  <_  M )
616, 9letri3d 9215 . . . 4  |-  ( ph  ->  ( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
6261biimprd 215 . . 3  |-  ( ph  ->  ( ( M  <_  N  /\  N  <_  M
)  ->  M  =  N ) )
6360, 62mpan2d 656 . 2  |-  ( ph  ->  ( M  <_  N  ->  M  =  N ) )
6463imp 419 1  |-  ( (
ph  /\  M  <_  N )  ->  M  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   class class class wbr 4212   ` cfv 5454  (class class class)co 6081   RRcr 8989   0cc0 8990   1c1 8991    + caddc 8993    < clt 9120    <_ cle 9121    - cmin 9291   NNcn 10000   NN0cn0 10221   ZZcz 10282   ...cfz 11043   Basecbs 13469   +g cplusg 13529   0gc0g 13723   Mndcmnd 14684  .gcmg 14689   odcod 15163
This theorem is referenced by:  mndodcong  15180
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701  ax-inf2 7596  ax-cnex 9046  ax-resscn 9047  ax-1cn 9048  ax-icn 9049  ax-addcl 9050  ax-addrcl 9051  ax-mulcl 9052  ax-mulrcl 9053  ax-mulcom 9054  ax-addass 9055  ax-mulass 9056  ax-distr 9057  ax-i2m1 9058  ax-1ne0 9059  ax-1rid 9060  ax-rnegex 9061  ax-rrecex 9062  ax-cnre 9063  ax-pre-lttri 9064  ax-pre-lttrn 9065  ax-pre-ltadd 9066  ax-pre-mulgt0 9067
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-recs 6633  df-rdg 6668  df-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-sup 7446  df-pnf 9122  df-mnf 9123  df-xr 9124  df-ltxr 9125  df-le 9126  df-sub 9293  df-neg 9294  df-nn 10001  df-n0 10222  df-z 10283  df-uz 10489  df-fz 11044  df-seq 11324  df-0g 13727  df-mnd 14690  df-mulg 14815  df-od 15167
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