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Theorem mndodconglem 16692
Description: Lemma for mndodcong 16693. (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 10571 . . . . . . . . . 10  |-  ( ph  ->  ( O `  A
)  e.  RR )
43recnd 9639 . . . . . . . . 9  |-  ( ph  ->  ( O `  A
)  e.  CC )
5 mndodconglem.4 . . . . . . . . . . 11  |-  ( ph  ->  M  e.  NN0 )
65nn0red 10874 . . . . . . . . . 10  |-  ( ph  ->  M  e.  RR )
76recnd 9639 . . . . . . . . 9  |-  ( ph  ->  M  e.  CC )
8 mndodconglem.5 . . . . . . . . . . 11  |-  ( ph  ->  N  e.  NN0 )
98nn0red 10874 . . . . . . . . . 10  |-  ( ph  ->  N  e.  RR )
109recnd 9639 . . . . . . . . 9  |-  ( ph  ->  N  e.  CC )
114, 7, 10addsubassd 9970 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 A )  +  M )  -  N
)  =  ( ( O `  A )  +  ( M  -  N ) ) )
122nnzd 10989 . . . . . . . . . . . 12  |-  ( ph  ->  ( O `  A
)  e.  ZZ )
135nn0zd 10988 . . . . . . . . . . . 12  |-  ( ph  ->  M  e.  ZZ )
1412, 13zaddcld 10994 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  A )  +  M
)  e.  ZZ )
1514zred 10990 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  A )  +  M
)  e.  RR )
16 mndodconglem.7 . . . . . . . . . 10  |-  ( ph  ->  N  <  ( O `
 A ) )
17 nn0addge1 10863 . . . . . . . . . . 11  |-  ( ( ( O `  A
)  e.  RR  /\  M  e.  NN0 )  -> 
( O `  A
)  <_  ( ( O `  A )  +  M ) )
183, 5, 17syl2anc 661 . . . . . . . . . 10  |-  ( ph  ->  ( O `  A
)  <_  ( ( O `  A )  +  M ) )
199, 3, 15, 16, 18ltletrd 9759 . . . . . . . . 9  |-  ( ph  ->  N  <  ( ( O `  A )  +  M ) )
208nn0zd 10988 . . . . . . . . . 10  |-  ( ph  ->  N  e.  ZZ )
21 znnsub 10931 . . . . . . . . . 10  |-  ( ( N  e.  ZZ  /\  ( ( O `  A )  +  M
)  e.  ZZ )  ->  ( N  < 
( ( O `  A )  +  M
)  <->  ( ( ( O `  A )  +  M )  -  N )  e.  NN ) )
2220, 14, 21syl2anc 661 . . . . . . . . 9  |-  ( ph  ->  ( N  <  (
( O `  A
)  +  M )  <-> 
( ( ( O `
 A )  +  M )  -  N
)  e.  NN ) )
2319, 22mpbid 210 . . . . . . . 8  |-  ( ph  ->  ( ( ( O `
 A )  +  M )  -  N
)  e.  NN )
2411, 23eqeltrrd 2546 . . . . . . 7  |-  ( ph  ->  ( ( O `  A )  +  ( M  -  N ) )  e.  NN )
254, 7, 10addsub12d 9973 . . . . . . . . 9  |-  ( ph  ->  ( ( O `  A )  +  ( M  -  N ) )  =  ( M  +  ( ( O `
 A )  -  N ) ) )
2625oveq1d 6311 . . . . . . . 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 6311 . . . . . . . . . 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 10931 . . . . . . . . . . . . . 14  |-  ( ( N  e.  ZZ  /\  ( O `  A )  e.  ZZ )  -> 
( N  <  ( O `  A )  <->  ( ( O `  A
)  -  N )  e.  NN ) )
3120, 12, 30syl2anc 661 . . . . . . . . . . . . 13  |-  ( ph  ->  ( N  <  ( O `  A )  <->  ( ( O `  A
)  -  N )  e.  NN ) )
3216, 31mpbid 210 . . . . . . . . . . . 12  |-  ( ph  ->  ( ( O `  A )  -  N
)  e.  NN )
3332nnnn0d 10873 . . . . . . . . . . 11  |-  ( ph  ->  ( ( O `  A )  -  N
)  e.  NN0 )
34 odcl.1 . . . . . . . . . . . 12  |-  X  =  ( Base `  G
)
35 odid.3 . . . . . . . . . . . 12  |-  .x.  =  (.g
`  G )
36 eqid 2457 . . . . . . . . . . . 12  |-  ( +g  `  G )  =  ( +g  `  G )
3734, 35, 36mulgnn0dir 16292 . . . . . . . . . . 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 1230 . . . . . . . . . 10  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( M  .x.  A ) ( +g  `  G
) ( ( ( O `  A )  -  N )  .x.  A ) ) )
3934, 35, 36mulgnn0dir 16292 . . . . . . . . . . 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 1230 . . . . . . . . . 10  |-  ( ph  ->  ( ( N  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( N  .x.  A ) ( +g  `  G
) ( ( ( O `  A )  -  N )  .x.  A ) ) )
4128, 38, 403eqtr4d 2508 . . . . . . . . 9  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  ( ( N  +  ( ( O `  A )  -  N ) ) 
.x.  A ) )
4210, 4pncan3d 9953 . . . . . . . . . . 11  |-  ( ph  ->  ( N  +  ( ( O `  A
)  -  N ) )  =  ( O `
 A ) )
4342oveq1d 6311 . . . . . . . . . 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 16689 . . . . . . . . . . 11  |-  ( A  e.  X  ->  (
( O `  A
)  .x.  A )  =  .0.  )
471, 46syl 16 . . . . . . . . . 10  |-  ( ph  ->  ( ( O `  A )  .x.  A
)  =  .0.  )
4843, 47eqtrd 2498 . . . . . . . . 9  |-  ( ph  ->  ( ( N  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  .0.  )
4941, 48eqtrd 2498 . . . . . . . 8  |-  ( ph  ->  ( ( M  +  ( ( O `  A )  -  N
) )  .x.  A
)  =  .0.  )
5026, 49eqtrd 2498 . . . . . . 7  |-  ( ph  ->  ( ( ( O `
 A )  +  ( M  -  N
) )  .x.  A
)  =  .0.  )
5134, 44, 35, 45odlem2 16690 . . . . . . 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 1228 . . . . . 6  |-  ( ph  ->  ( O `  A
)  e.  ( 1 ... ( ( O `
 A )  +  ( M  -  N
) ) ) )
53 elfzle2 11715 . . . . . 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 10995 . . . . . . 7  |-  ( ph  ->  ( M  -  N
)  e.  ZZ )
5655zred 10990 . . . . . 6  |-  ( ph  ->  ( M  -  N
)  e.  RR )
573, 56addge01d 10161 . . . . 5  |-  ( ph  ->  ( 0  <_  ( M  -  N )  <->  ( O `  A )  <_  ( ( O `
 A )  +  ( M  -  N
) ) ) )
5854, 57mpbird 232 . . . 4  |-  ( ph  ->  0  <_  ( M  -  N ) )
596, 9subge0d 10163 . . . 4  |-  ( ph  ->  ( 0  <_  ( M  -  N )  <->  N  <_  M ) )
6058, 59mpbid 210 . . 3  |-  ( ph  ->  N  <_  M )
616, 9letri3d 9744 . . . 4  |-  ( ph  ->  ( M  =  N  <-> 
( M  <_  N  /\  N  <_  M ) ) )
6261biimprd 223 . . 3  |-  ( ph  ->  ( ( M  <_  N  /\  N  <_  M
)  ->  M  =  N ) )
6360, 62mpan2d 674 . 2  |-  ( ph  ->  ( M  <_  N  ->  M  =  N ) )
6463imp 429 1  |-  ( (
ph  /\  M  <_  N )  ->  M  =  N )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1395    e. wcel 1819   class class class wbr 4456   ` cfv 5594  (class class class)co 6296   RRcr 9508   0cc0 9509   1c1 9510    + caddc 9512    < clt 9645    <_ cle 9646    - cmin 9824   NNcn 10556   NN0cn0 10816   ZZcz 10885   ...cfz 11697   Basecbs 14644   +g cplusg 14712   0gc0g 14857   Mndcmnd 16046  .gcmg 16183   odcod 16676
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591  ax-inf2 8075  ax-cnex 9565  ax-resscn 9566  ax-1cn 9567  ax-icn 9568  ax-addcl 9569  ax-addrcl 9570  ax-mulcl 9571  ax-mulrcl 9572  ax-mulcom 9573  ax-addass 9574  ax-mulass 9575  ax-distr 9576  ax-i2m1 9577  ax-1ne0 9578  ax-1rid 9579  ax-rnegex 9580  ax-rrecex 9581  ax-cnre 9582  ax-pre-lttri 9583  ax-pre-lttrn 9584  ax-pre-ltadd 9585  ax-pre-mulgt0 9586
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rmo 2815  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-1st 6799  df-2nd 6800  df-recs 7060  df-rdg 7094  df-er 7329  df-en 7536  df-dom 7537  df-sdom 7538  df-sup 7919  df-pnf 9647  df-mnf 9648  df-xr 9649  df-ltxr 9650  df-le 9651  df-sub 9826  df-neg 9827  df-nn 10557  df-n0 10817  df-z 10886  df-uz 11107  df-fz 11698  df-seq 12111  df-0g 14859  df-mgm 15999  df-sgrp 16038  df-mnd 16048  df-mulg 16187  df-od 16680
This theorem is referenced by:  mndodcong  16693
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