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Theorem mndodconglem 14872
Description: Lemma for mndodcong 14873. (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 9777 . . . . . . . . . 10  |-  ( ph  ->  ( O `  A
)  e.  RR )
43recnd 8877 . . . . . . . . 9  |-  ( ph  ->  ( O `  A
)  e.  CC )
5 mndodconglem.4 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN0 )
65nn0red 10035 . . . . . . . . . 10  |-  ( ph  ->  M  e.  RR )
76recnd 8877 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
8 mndodconglem.5 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN0 )
98nn0red 10035 . . . . . . . . . 10  |-  ( ph  ->  N  e.  RR )
109recnd 8877 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
114, 7, 10addsubassd 9193 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 A )  +  M )  -  N
)  =  ( ( O `  A )  +  ( M  -  N ) ) )
122nnzd 10132 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  A
)  e.  ZZ )
135nn0zd 10131 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ZZ )
1412, 13zaddcld 10137 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  A )  +  M
)  e.  ZZ )
1514zred 10133 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  A )  +  M
)  e.  RR )
16 mndodconglem.7 . . . . . . . . . 10  |-  ( ph  ->  N  <  ( O `
 A ) )
17 nn0addge1 10026 . . . . . . . . . . 11  |-  ( ( ( O `  A
)  e.  RR  /\  M  e.  NN0 )  -> 
( O `  A
)  <_  ( ( O `  A )  +  M ) )
183, 5, 17syl2anc 642 . . . . . . . . . 10  |-  ( ph  ->  ( O `  A
)  <_  ( ( O `  A )  +  M ) )
199, 3, 15, 16, 18ltletrd 8992 . . . . . . . . 9  |-  ( ph  ->  N  <  ( ( O `  A )  +  M ) )
208nn0zd 10131 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
21 znnsub 10080 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( ( O `  A )  +  M
)  e.  ZZ )  ->  ( N  < 
( ( O `  A )  +  M
)  <->  ( ( ( O `  A )  +  M )  -  N )  e.  NN ) )
2220, 14, 21syl2anc 642 . . . . . . . . 9  |-  ( ph  ->  ( N  <  (
( O `  A
)  +  M )  <-> 
( ( ( O `
 A )  +  M )  -  N
)  e.  NN ) )
2319, 22mpbid 201 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 A )  +  M )  -  N
)  e.  NN )
2411, 23eqeltrrd 2371 . . . . . . 7  |-  ( ph  ->  ( ( O `  A )  +  ( M  -  N ) )  e.  NN )
254, 7, 10addsub12d 9196 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  A )  +  ( M  -  N ) )  =  ( M  +  ( ( O `
 A )  -  N ) ) )
2625oveq1d 5889 . . . . . . . 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 5889 . . . . . . . . . 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 10080 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  ( O `  A )  e.  ZZ )  -> 
( N  <  ( O `  A )  <->  ( ( O `  A
)  -  N )  e.  NN ) )
3120, 12, 30syl2anc 642 . . . . . . . . . . . . 13  |-  ( ph  ->  ( N  <  ( O `  A )  <->  ( ( O `  A
)  -  N )  e.  NN ) )
3216, 31mpbid 201 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( O `  A )  -  N
)  e.  NN )
3332nnnn0d 10034 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  A )  -  N
)  e.  NN0 )
34 odcl.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
35 odid.3 . . . . . . . . . . . 12  |-  .x.  =  (.g
`  G )
36 eqid 2296 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
3734, 35, 36mulgnn0dir 14606 . . . . . . . . . . 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 1184 . . . . . . . . . 10  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( M  .x.  A ) ( +g  `  G
) ( ( ( O `  A )  -  N )  .x.  A ) ) )
3934, 35, 36mulgnn0dir 14606 . . . . . . . . . . 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 1184 . . . . . . . . . 10  |-  ( ph  ->  ( ( N  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( N  .x.  A ) ( +g  `  G
) ( ( ( O `  A )  -  N )  .x.  A ) ) )
4128, 38, 403eqtr4d 2338 . . . . . . . . 9  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( N  +  ( ( O `  A )  -  N ) ) 
.x.  A ) )
4210, 4pncan3d 9176 . . . . . . . . . . 11  |-  ( ph  ->  ( N  +  ( ( O `  A
)  -  N ) )  =  ( O `
 A ) )
4342oveq1d 5889 . . . . . . . . . 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 14869 . . . . . . . . . . 11  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  .0.  )
471, 46syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  A )  .x.  A
)  =  .0.  )
4843, 47eqtrd 2328 . . . . . . . . 9  |-  ( ph  ->  ( ( N  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  .0.  )
4941, 48eqtrd 2328 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  .0.  )
5026, 49eqtrd 2328 . . . . . . 7  |-  ( ph  ->  ( ( ( O `
 A )  +  ( M  -  N
) )  .x.  A
)  =  .0.  )
5134, 44, 35, 45odlem2 14870 . . . . . . 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 1182 . . . . . 6  |-  ( ph  ->  ( O `  A
)  e.  ( 1 ... ( ( O `
 A )  +  ( M  -  N
) ) ) )
53 elfzle2 10816 . . . . . 6  |-  ( ( O `  A )  e.  ( 1 ... ( ( O `  A )  +  ( M  -  N ) ) )  ->  ( O `  A )  <_  ( ( O `  A )  +  ( M  -  N ) ) )
5452, 53syl 15 . . . . 5  |-  ( ph  ->  ( O `  A
)  <_  ( ( O `  A )  +  ( M  -  N ) ) )
5513, 20zsubcld 10138 . . . . . . 7  |-  ( ph  ->  ( M  -  N
)  e.  ZZ )
5655zred 10133 . . . . . 6  |-  ( ph  ->  ( M  -  N
)  e.  RR )
573, 56addge01d 9376 . . . . 5  |-  ( ph  ->  ( 0  <_  ( M  -  N )  <->  ( O `  A )  <_  ( ( O `
 A )  +  ( M  -  N
) ) ) )
5854, 57mpbird 223 . . . 4  |-  ( ph  ->  0  <_  ( M  -  N ) )
596, 9subge0d 9378 . . . 4  |-  ( ph  ->  ( 0  <_  ( M  -  N )  <->  N  <_  M ) )
6058, 59mpbid 201 . . 3  |-  ( ph  ->  N  <_  M )
616, 9letri3d 8977 . . . 4  |-  ( ph  ->  ( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
6261biimprd 214 . . 3  |-  ( ph  ->  ( ( M  <_  N  /\  N  <_  M
)  ->  M  =  N ) )
6360, 62mpan2d 655 . 2  |-  ( ph  ->  ( M  <_  N  ->  M  =  N ) )
6463imp 418 1  |-  ( (
ph  /\  M  <_  N )  ->  M  =  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   class class class wbr 4039   ` cfv 5271  (class class class)co 5874   RRcr 8752   0cc0 8753   1c1 8754    + caddc 8756    < clt 8883    <_ cle 8884    - cmin 9053   NNcn 9762   NN0cn0 9981   ZZcz 10040   ...cfz 10798   Basecbs 13164   +g cplusg 13224   0gc0g 13416   Mndcmnd 14377  .gcmg 14382   odcod 14856
This theorem is referenced by:  mndodcong  14873
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
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-nn 9763  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-seq 11063  df-0g 13420  df-mnd 14383  df-mulg 14508  df-od 14860
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