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Theorem ogrpinv0le 27858
Description: In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
Hypotheses
Ref Expression
ogrpsub.0  |-  B  =  ( Base `  G
)
ogrpsub.1  |-  .<_  =  ( le `  G )
ogrpinv.2  |-  I  =  ( invg `  G )
ogrpinv.3  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
ogrpinv0le  |-  ( ( G  e. oGrp  /\  X  e.  B )  ->  (  .0.  .<_  X  <->  ( I `  X )  .<_  .0.  )
)

Proof of Theorem ogrpinv0le
StepHypRef Expression
1 isogrp 27844 . . . . . 6  |-  ( G  e. oGrp 
<->  ( G  e.  Grp  /\  G  e. oMnd ) )
21simprbi 464 . . . . 5  |-  ( G  e. oGrp  ->  G  e. oMnd )
32ad2antrr 725 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  ->  G  e. oMnd )
4 omndmnd 27846 . . . . 5  |-  ( G  e. oMnd  ->  G  e.  Mnd )
5 ogrpsub.0 . . . . . 6  |-  B  =  ( Base `  G
)
6 ogrpinv.3 . . . . . 6  |-  .0.  =  ( 0g `  G )
75, 6mndidcl 16064 . . . . 5  |-  ( G  e.  Mnd  ->  .0.  e.  B )
83, 4, 73syl 20 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  e.  B )
9 simplr 755 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  ->  X  e.  B )
10 ogrpgrp 27845 . . . . . 6  |-  ( G  e. oGrp  ->  G  e.  Grp )
1110ad2antrr 725 . . . . 5  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  ->  G  e.  Grp )
12 ogrpinv.2 . . . . . 6  |-  I  =  ( invg `  G )
135, 12grpinvcl 16221 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( I `  X
)  e.  B )
1411, 9, 13syl2anc 661 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  -> 
( I `  X
)  e.  B )
15 simpr 461 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  ->  .0.  .<_  X )
16 ogrpsub.1 . . . . 5  |-  .<_  =  ( le `  G )
17 eqid 2457 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
185, 16, 17omndadd 27848 . . . 4  |-  ( ( G  e. oMnd  /\  (  .0.  e.  B  /\  X  e.  B  /\  (
I `  X )  e.  B )  /\  .0.  .<_  X )  ->  (  .0.  ( +g  `  G
) ( I `  X ) )  .<_  ( X ( +g  `  G
) ( I `  X ) ) )
193, 8, 9, 14, 15, 18syl131anc 1241 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  -> 
(  .0.  ( +g  `  G ) ( I `
 X ) ) 
.<_  ( X ( +g  `  G ) ( I `
 X ) ) )
205, 17, 6grplid 16206 . . . 4  |-  ( ( G  e.  Grp  /\  ( I `  X
)  e.  B )  ->  (  .0.  ( +g  `  G ) ( I `  X ) )  =  ( I `
 X ) )
2111, 14, 20syl2anc 661 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  -> 
(  .0.  ( +g  `  G ) ( I `
 X ) )  =  ( I `  X ) )
225, 17, 6, 12grprinv 16223 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G ) ( I `
 X ) )  =  .0.  )
2311, 9, 22syl2anc 661 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  -> 
( X ( +g  `  G ) ( I `
 X ) )  =  .0.  )
2419, 21, 233brtr3d 4485 . 2  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  -> 
( I `  X
)  .<_  .0.  )
252ad2antrr 725 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  G  e. oMnd )
2610ad2antrr 725 . . . . 5  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  G  e.  Grp )
27 simplr 755 . . . . 5  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  X  e.  B )
2826, 27, 13syl2anc 661 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  ( I `  X )  e.  B )
2925, 4, 73syl 20 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  .0. 
e.  B )
30 simpr 461 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  ( I `  X ) 
.<_  .0.  )
315, 16, 17omndadd 27848 . . . 4  |-  ( ( G  e. oMnd  /\  (
( I `  X
)  e.  B  /\  .0.  e.  B  /\  X  e.  B )  /\  (
I `  X )  .<_  .0.  )  ->  (
( I `  X
) ( +g  `  G
) X )  .<_  (  .0.  ( +g  `  G
) X ) )
3225, 28, 29, 27, 30, 31syl131anc 1241 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  ( ( I `  X
) ( +g  `  G
) X )  .<_  (  .0.  ( +g  `  G
) X ) )
335, 17, 6, 12grplinv 16222 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( I `  X ) ( +g  `  G ) X )  =  .0.  )
3426, 27, 33syl2anc 661 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  ( ( I `  X
) ( +g  `  G
) X )  =  .0.  )
355, 17, 6grplid 16206 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  ( +g  `  G ) X )  =  X )
3626, 27, 35syl2anc 661 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  (  .0.  ( +g  `  G
) X )  =  X )
3732, 34, 363brtr3d 4485 . 2  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  .0. 
.<_  X )
3824, 37impbida 832 1  |-  ( ( G  e. oGrp  /\  X  e.  B )  ->  (  .0.  .<_  X  <->  ( I `  X )  .<_  .0.  )
)
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   Basecbs 14643   +g cplusg 14711   lecple 14718   0gc0g 14856   Mndcmnd 16045   Grpcgrp 16179   invgcminusg 16180  oMndcomnd 27839  oGrpcogrp 27840
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
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  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-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-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  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-0g 14858  df-mgm 15998  df-sgrp 16037  df-mnd 16047  df-grp 16183  df-minusg 16184  df-omnd 27841  df-ogrp 27842
This theorem is referenced by: (None)
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