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Theorem ogrpinv0le 26178
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 26164 . . . . . 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 26166 . . . . 5  |-  ( G  e. oMnd  ->  G  e.  Mnd )
5 ogrpsub.0 . . . . . 6  |-  B  =  ( Base `  G
)
6 ogrpinv.3 . . . . . 6  |-  .0.  =  ( 0g `  G )
75, 6mndidcl 15438 . . . . 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 754 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  ->  X  e.  B )
101simplbi 460 . . . . . 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 15582 . . . . 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 2442 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
185, 16, 17omndadd 26168 . . . 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 1231 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<_  X )  -> 
(  .0.  ( +g  `  G ) ( I `
 X ) ) 
.<_  ( X ( +g  `  G ) ( I `
 X ) ) )
205, 17, 6grplid 15567 . . . 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 15584 . . . 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 4320 . 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 754 . . . . 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 26168 . . . 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 1231 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  ( ( I `  X
) ( +g  `  G
) X )  .<_  (  .0.  ( +g  `  G
) X ) )
335, 17, 6, 12grplinv 15583 . . . 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 15567 . . . 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 4320 . 2  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<_  .0.  )  ->  .0. 
.<_  X )
3824, 37impbida 828 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 1369    e. wcel 1756   class class class wbr 4291   ` cfv 5417  (class class class)co 6090   Basecbs 14173   +g cplusg 14237   lecple 14244   0gc0g 14377   Mndcmnd 15408   Grpcgrp 15409   invgcminusg 15410  oMndcomnd 26159  oGrpcogrp 26160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4402  ax-sep 4412  ax-nul 4420  ax-pow 4469  ax-pr 4530
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2719  df-rex 2720  df-reu 2721  df-rmo 2722  df-rab 2723  df-v 2973  df-sbc 3186  df-csb 3288  df-dif 3330  df-un 3332  df-in 3334  df-ss 3341  df-nul 3637  df-if 3791  df-sn 3877  df-pr 3879  df-op 3883  df-uni 4091  df-iun 4172  df-br 4292  df-opab 4350  df-mpt 4351  df-id 4635  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5380  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-riota 6051  df-ov 6093  df-0g 14379  df-mnd 15414  df-grp 15544  df-minusg 15545  df-omnd 26161  df-ogrp 26162
This theorem is referenced by: (None)
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