MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mndodconglem Unicode version

Theorem mndodconglem 14856
Description: Lemma for mndodcong 14857. (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 9761 . . . . . . . . . 10  |-  ( ph  ->  ( O `  A
)  e.  RR )
43recnd 8861 . . . . . . . . 9  |-  ( ph  ->  ( O `  A
)  e.  CC )
5 mndodconglem.4 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN0 )
65nn0red 10019 . . . . . . . . . 10  |-  ( ph  ->  M  e.  RR )
76recnd 8861 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
8 mndodconglem.5 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN0 )
98nn0red 10019 . . . . . . . . . 10  |-  ( ph  ->  N  e.  RR )
109recnd 8861 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
114, 7, 10addsubassd 9177 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 A )  +  M )  -  N
)  =  ( ( O `  A )  +  ( M  -  N ) ) )
122nnzd 10116 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  A
)  e.  ZZ )
135nn0zd 10115 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ZZ )
1412, 13zaddcld 10121 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  A )  +  M
)  e.  ZZ )
1514zred 10117 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  A )  +  M
)  e.  RR )
16 mndodconglem.7 . . . . . . . . . 10  |-  ( ph  ->  N  <  ( O `
 A ) )
17 nn0addge1 10010 . . . . . . . . . . 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 8976 . . . . . . . . 9  |-  ( ph  ->  N  <  ( ( O `  A )  +  M ) )
208nn0zd 10115 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
21 znnsub 10064 . . . . . . . . . 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 2358 . . . . . . 7  |-  ( ph  ->  ( ( O `  A )  +  ( M  -  N ) )  e.  NN )
254, 7, 10addsub12d 9180 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  A )  +  ( M  -  N ) )  =  ( M  +  ( ( O `
 A )  -  N ) ) )
2625oveq1d 5873 . . . . . . . 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 5873 . . . . . . . . . 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 10064 . . . . . . . . . . . . . 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 10018 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  A )  -  N
)  e.  NN0 )
34 odcl.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
35 odid.3 . . . . . . . . . . . 12  |-  .x.  =  (.g
`  G )
36 eqid 2283 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
3734, 35, 36mulgnn0dir 14590 . . . . . . . . . . 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 14590 . . . . . . . . . . 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 2325 . . . . . . . . 9  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( N  +  ( ( O `  A )  -  N ) ) 
.x.  A ) )
4210, 4pncan3d 9160 . . . . . . . . . . 11  |-  ( ph  ->  ( N  +  ( ( O `  A
)  -  N ) )  =  ( O `
 A ) )
4342oveq1d 5873 . . . . . . . . . 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 14853 . . . . . . . . . . 11  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  .0.  )
471, 46syl 15 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  A )  .x.  A
)  =  .0.  )
4843, 47eqtrd 2315 . . . . . . . . 9  |-  ( ph  ->  ( ( N  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  .0.  )
4941, 48eqtrd 2315 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  .0.  )
5026, 49eqtrd 2315 . . . . . . 7  |-  ( ph  ->  ( ( ( O `
 A )  +  ( M  -  N
) )  .x.  A
)  =  .0.  )
5134, 44, 35, 45odlem2 14854 . . . . . . 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 10800 . . . . . 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 10122 . . . . . . 7  |-  ( ph  ->  ( M  -  N
)  e.  ZZ )
5655zred 10117 . . . . . 6  |-  ( ph  ->  ( M  -  N
)  e.  RR )
573, 56addge01d 9360 . . . . 5  |-  ( ph  ->  ( 0  <_  ( M  -  N )  <->  ( O `  A )  <_  ( ( O `
 A )  +  ( M  -  N
) ) ) )
5854, 57mpbird 223 . . . 4  |-  ( ph  ->  0  <_  ( M  -  N ) )
596, 9subge0d 9362 . . . 4  |-  ( ph  ->  ( 0  <_  ( M  -  N )  <->  N  <_  M ) )
6058, 59mpbid 201 . . 3  |-  ( ph  ->  N  <_  M )
616, 9letri3d 8961 . . . 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 1623    e. wcel 1684   class class class wbr 4023   ` cfv 5255  (class class class)co 5858   RRcr 8736   0cc0 8737   1c1 8738    + caddc 8740    < clt 8867    <_ cle 8868    - cmin 9037   NNcn 9746   NN0cn0 9965   ZZcz 10024   ...cfz 10782   Basecbs 13148   +g cplusg 13208   0gc0g 13400   Mndcmnd 14361  .gcmg 14366   odcod 14840
This theorem is referenced by:  mndodcong  14857
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512  ax-inf2 7342  ax-cnex 8793  ax-resscn 8794  ax-1cn 8795  ax-icn 8796  ax-addcl 8797  ax-addrcl 8798  ax-mulcl 8799  ax-mulrcl 8800  ax-mulcom 8801  ax-addass 8802  ax-mulass 8803  ax-distr 8804  ax-i2m1 8805  ax-1ne0 8806  ax-1rid 8807  ax-rnegex 8808  ax-rrecex 8809  ax-cnre 8810  ax-pre-lttri 8811  ax-pre-lttrn 8812  ax-pre-ltadd 8813  ax-pre-mulgt0 8814
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 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rmo 2551  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  df-suc 4398  df-om 4657  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-riota 6304  df-recs 6388  df-rdg 6423  df-er 6660  df-en 6864  df-dom 6865  df-sdom 6866  df-sup 7194  df-pnf 8869  df-mnf 8870  df-xr 8871  df-ltxr 8872  df-le 8873  df-sub 9039  df-neg 9040  df-nn 9747  df-n0 9966  df-z 10025  df-uz 10231  df-fz 10783  df-seq 11047  df-0g 13404  df-mnd 14367  df-mulg 14492  df-od 14844
  Copyright terms: Public domain W3C validator