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Theorem ogrpinv0lt 26205
Description: In an ordered group, the ordering is compatible with group inverse. (Contributed by Thierry Arnoux, 3-Sep-2018.)
Hypotheses
Ref Expression
ogrpinvlt.0  |-  B  =  ( Base `  G
)
ogrpinvlt.1  |-  .<  =  ( lt `  G )
ogrpinvlt.2  |-  I  =  ( invg `  G )
ogrpinv0lt.3  |-  .0.  =  ( 0g `  G )
Assertion
Ref Expression
ogrpinv0lt  |-  ( ( G  e. oGrp  /\  X  e.  B )  ->  (  .0.  .<  X  <->  ( I `  X )  .<  .0.  )
)

Proof of Theorem ogrpinv0lt
StepHypRef Expression
1 simpll 753 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  G  e. oGrp )
2 ogrpgrp 26185 . . . . . 6  |-  ( G  e. oGrp  ->  G  e.  Grp )
31, 2syl 16 . . . . 5  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  G  e.  Grp )
4 ogrpinvlt.0 . . . . . 6  |-  B  =  ( Base `  G
)
5 ogrpinv0lt.3 . . . . . 6  |-  .0.  =  ( 0g `  G )
64, 5grpidcl 15585 . . . . 5  |-  ( G  e.  Grp  ->  .0.  e.  B )
73, 6syl 16 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  .0.  e.  B )
8 simplr 754 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  X  e.  B )
9 ogrpinvlt.2 . . . . . 6  |-  I  =  ( invg `  G )
104, 9grpinvcl 15602 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( I `  X
)  e.  B )
113, 8, 10syl2anc 661 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  ( I `  X
)  e.  B )
12 simpr 461 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  .0.  .<  X )
13 ogrpinvlt.1 . . . . 5  |-  .<  =  ( lt `  G )
14 eqid 2443 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
154, 13, 14ogrpaddlt 26200 . . . 4  |-  ( ( G  e. oGrp  /\  (  .0.  e.  B  /\  X  e.  B  /\  (
I `  X )  e.  B )  /\  .0.  .<  X )  ->  (  .0.  ( +g  `  G
) ( I `  X ) )  .< 
( X ( +g  `  G ) ( I `
 X ) ) )
161, 7, 8, 11, 12, 15syl131anc 1231 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  (  .0.  ( +g  `  G ) ( I `
 X ) ) 
.<  ( X ( +g  `  G ) ( I `
 X ) ) )
174, 14, 5grplid 15587 . . . 4  |-  ( ( G  e.  Grp  /\  ( I `  X
)  e.  B )  ->  (  .0.  ( +g  `  G ) ( I `  X ) )  =  ( I `
 X ) )
183, 11, 17syl2anc 661 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  (  .0.  ( +g  `  G ) ( I `
 X ) )  =  ( I `  X ) )
194, 14, 5, 9grprinv 15604 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( X ( +g  `  G ) ( I `
 X ) )  =  .0.  )
203, 8, 19syl2anc 661 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  ( X ( +g  `  G ) ( I `
 X ) )  =  .0.  )
2116, 18, 203brtr3d 4340 . 2  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  ( I `  X
)  .<  .0.  )
22 simpll 753 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  G  e. oGrp )
2322, 2syl 16 . . . . 5  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  G  e.  Grp )
24 simplr 754 . . . . 5  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  X  e.  B )
2523, 24, 10syl2anc 661 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  ( I `  X )  e.  B )
2622, 2, 63syl 20 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  .0. 
e.  B )
27 simpr 461 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  ( I `  X ) 
.<  .0.  )
284, 13, 14ogrpaddlt 26200 . . . 4  |-  ( ( G  e. oGrp  /\  (
( I `  X
)  e.  B  /\  .0.  e.  B  /\  X  e.  B )  /\  (
I `  X )  .<  .0.  )  ->  (
( I `  X
) ( +g  `  G
) X )  .< 
(  .0.  ( +g  `  G ) X ) )
2922, 25, 26, 24, 27, 28syl131anc 1231 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  ( ( I `  X
) ( +g  `  G
) X )  .< 
(  .0.  ( +g  `  G ) X ) )
304, 14, 5, 9grplinv 15603 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( I `  X ) ( +g  `  G ) X )  =  .0.  )
3123, 24, 30syl2anc 661 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  ( ( I `  X
) ( +g  `  G
) X )  =  .0.  )
324, 14, 5grplid 15587 . . . 4  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  (  .0.  ( +g  `  G ) X )  =  X )
3323, 24, 32syl2anc 661 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  (  .0.  ( +g  `  G
) X )  =  X )
3429, 31, 333brtr3d 4340 . 2  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  .0. 
.<  X )
3521, 34impbida 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 4311   ` cfv 5437  (class class class)co 6110   Basecbs 14193   +g cplusg 14257   0gc0g 14397   ltcplt 15130   Grpcgrp 15429   invgcminusg 15430  oGrpcogrp 26180
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 4422  ax-sep 4432  ax-nul 4440  ax-pow 4489  ax-pr 4550
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 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2739  df-rex 2740  df-reu 2741  df-rmo 2742  df-rab 2743  df-v 2993  df-sbc 3206  df-csb 3308  df-dif 3350  df-un 3352  df-in 3354  df-ss 3361  df-nul 3657  df-if 3811  df-sn 3897  df-pr 3899  df-op 3903  df-uni 4111  df-iun 4192  df-br 4312  df-opab 4370  df-mpt 4371  df-id 4655  df-xp 4865  df-rel 4866  df-cnv 4867  df-co 4868  df-dm 4869  df-rn 4870  df-res 4871  df-ima 4872  df-iota 5400  df-fun 5439  df-fn 5440  df-f 5441  df-f1 5442  df-fo 5443  df-f1o 5444  df-fv 5445  df-riota 6071  df-ov 6113  df-0g 14399  df-plt 15147  df-mnd 15434  df-grp 15564  df-minusg 15565  df-omnd 26181  df-ogrp 26182
This theorem is referenced by:  archirngz  26225
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