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Theorem ogrpinv0lt 27579
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 27559 . . . . . 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 15947 . . . . 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 15964 . . . . 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 2441 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
154, 13, 14ogrpaddlt 27574 . . . 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 1240 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  .0.  .<  X )  ->  (  .0.  ( +g  `  G ) ( I `
 X ) ) 
.<  ( X ( +g  `  G ) ( I `
 X ) ) )
174, 14, 5grplid 15949 . . . 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 15966 . . . 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 4462 . 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 27574 . . . 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 1240 . . 3  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  ( ( I `  X
) ( +g  `  G
) X )  .< 
(  .0.  ( +g  `  G ) X ) )
304, 14, 5, 9grplinv 15965 . . . 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 15949 . . . 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 4462 . 2  |-  ( ( ( G  e. oGrp  /\  X  e.  B )  /\  ( I `  X
)  .<  .0.  )  ->  .0. 
.<  X )
3521, 34impbida 830 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 1381    e. wcel 1802   class class class wbr 4433   ` cfv 5574  (class class class)co 6277   Basecbs 14504   +g cplusg 14569   0gc0g 14709   ltcplt 15439   Grpcgrp 15922   invgcminusg 15923  oGrpcogrp 27554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rmo 2799  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-riota 6238  df-ov 6280  df-0g 14711  df-plt 15457  df-mgm 15741  df-sgrp 15780  df-mnd 15790  df-grp 15926  df-minusg 15927  df-omnd 27555  df-ogrp 27556
This theorem is referenced by:  archirngz  27599
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