Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ogrpinvlt Structured version   Unicode version

Theorem ogrpinvlt 27538
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 )
Assertion
Ref Expression
ogrpinvlt  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( X  .<  Y  <->  ( I `  Y )  .<  (
I `  X )
) )

Proof of Theorem ogrpinvlt
StepHypRef Expression
1 simp1l 1020 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  G  e. oGrp )
2 simp2 997 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  X  e.  B )
3 simp3 998 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  Y  e.  B )
4 ogrpgrp 27517 . . . . . 6  |-  ( G  e. oGrp  ->  G  e.  Grp )
51, 4syl 16 . . . . 5  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  G  e.  Grp )
6 ogrpinvlt.0 . . . . . 6  |-  B  =  ( Base `  G
)
7 ogrpinvlt.2 . . . . . 6  |-  I  =  ( invg `  G )
86, 7grpinvcl 15967 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( I `  Y
)  e.  B )
95, 3, 8syl2anc 661 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( I `  Y )  e.  B
)
10 ogrpinvlt.1 . . . . 5  |-  .<  =  ( lt `  G )
11 eqid 2467 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
126, 10, 11ogrpaddltbi 27533 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  ( I `  Y
)  e.  B ) )  ->  ( X  .<  Y  <->  ( X ( +g  `  G ) ( I `  Y
) )  .<  ( Y ( +g  `  G
) ( I `  Y ) ) ) )
131, 2, 3, 9, 12syl13anc 1230 . . 3  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( X  .<  Y  <->  ( X ( +g  `  G ) ( I `  Y
) )  .<  ( Y ( +g  `  G
) ( I `  Y ) ) ) )
14 eqid 2467 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
156, 11, 14, 7grprinv 15969 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( Y ( +g  `  G ) ( I `
 Y ) )  =  ( 0g `  G ) )
165, 3, 15syl2anc 661 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( Y
( +g  `  G ) ( I `  Y
) )  =  ( 0g `  G ) )
1716breq2d 4465 . . 3  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( ( X ( +g  `  G
) ( I `  Y ) )  .< 
( Y ( +g  `  G ) ( I `
 Y ) )  <-> 
( X ( +g  `  G ) ( I `
 Y ) ) 
.<  ( 0g `  G
) ) )
18 simp1r 1021 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  (oppg
`  G )  e. oGrp
)
196, 11grpcl 15935 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( I `  Y
)  e.  B )  ->  ( X ( +g  `  G ) ( I `  Y
) )  e.  B
)
205, 2, 9, 19syl3anc 1228 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( X
( +g  `  G ) ( I `  Y
) )  e.  B
)
216, 14grpidcl 15950 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
221, 4, 213syl 20 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( 0g `  G )  e.  B
)
236, 7grpinvcl 15967 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( I `  X
)  e.  B )
245, 2, 23syl2anc 661 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( I `  X )  e.  B
)
256, 10, 11, 1, 18, 20, 22, 24ogrpaddltrbid 27535 . . 3  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( ( X ( +g  `  G
) ( I `  Y ) )  .< 
( 0g `  G
)  <->  ( ( I `
 X ) ( +g  `  G ) ( X ( +g  `  G ) ( I `
 Y ) ) )  .<  ( (
I `  X )
( +g  `  G ) ( 0g `  G
) ) ) )
2613, 17, 253bitrd 279 . 2  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( X  .<  Y  <->  ( ( I `
 X ) ( +g  `  G ) ( X ( +g  `  G ) ( I `
 Y ) ) )  .<  ( (
I `  X )
( +g  `  G ) ( 0g `  G
) ) ) )
276, 11, 14, 7grplinv 15968 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( I `  X ) ( +g  `  G ) X )  =  ( 0g `  G ) )
285, 2, 27syl2anc 661 . . . . 5  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
I `  X )
( +g  `  G ) X )  =  ( 0g `  G ) )
2928oveq1d 6310 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
( I `  X
) ( +g  `  G
) X ) ( +g  `  G ) ( I `  Y
) )  =  ( ( 0g `  G
) ( +g  `  G
) ( I `  Y ) ) )
306, 11grpass 15936 . . . . 5  |-  ( ( G  e.  Grp  /\  ( ( I `  X )  e.  B  /\  X  e.  B  /\  ( I `  Y
)  e.  B ) )  ->  ( (
( I `  X
) ( +g  `  G
) X ) ( +g  `  G ) ( I `  Y
) )  =  ( ( I `  X
) ( +g  `  G
) ( X ( +g  `  G ) ( I `  Y
) ) ) )
315, 24, 2, 9, 30syl13anc 1230 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
( I `  X
) ( +g  `  G
) X ) ( +g  `  G ) ( I `  Y
) )  =  ( ( I `  X
) ( +g  `  G
) ( X ( +g  `  G ) ( I `  Y
) ) ) )
326, 11, 14grplid 15952 . . . . 5  |-  ( ( G  e.  Grp  /\  ( I `  Y
)  e.  B )  ->  ( ( 0g
`  G ) ( +g  `  G ) ( I `  Y
) )  =  ( I `  Y ) )
335, 9, 32syl2anc 661 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( ( 0g `  G ) ( +g  `  G ) ( I `  Y
) )  =  ( I `  Y ) )
3429, 31, 333eqtr3d 2516 . . 3  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
I `  X )
( +g  `  G ) ( X ( +g  `  G ) ( I `
 Y ) ) )  =  ( I `
 Y ) )
356, 11, 14grprid 15953 . . . 4  |-  ( ( G  e.  Grp  /\  ( I `  X
)  e.  B )  ->  ( ( I `
 X ) ( +g  `  G ) ( 0g `  G
) )  =  ( I `  X ) )
365, 24, 35syl2anc 661 . . 3  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
I `  X )
( +g  `  G ) ( 0g `  G
) )  =  ( I `  X ) )
3734, 36breq12d 4466 . 2  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
( I `  X
) ( +g  `  G
) ( X ( +g  `  G ) ( I `  Y
) ) )  .< 
( ( I `  X ) ( +g  `  G ) ( 0g
`  G ) )  <-> 
( I `  Y
)  .<  ( I `  X ) ) )
3826, 37bitrd 253 1  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( X  .<  Y  <->  ( I `  Y )  .<  (
I `  X )
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   class class class wbr 4453   ` cfv 5594  (class class class)co 6295   Basecbs 14507   +g cplusg 14572   0gc0g 14712   ltcplt 15445   Grpcgrp 15925   invgcminusg 15926  oppgcoppg 16252  oGrpcogrp 27512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587  ax-cnex 9560  ax-resscn 9561  ax-1cn 9562  ax-icn 9563  ax-addcl 9564  ax-addrcl 9565  ax-mulcl 9566  ax-mulrcl 9567  ax-mulcom 9568  ax-addass 9569  ax-mulass 9570  ax-distr 9571  ax-i2m1 9572  ax-1ne0 9573  ax-1rid 9574  ax-rnegex 9575  ax-rrecex 9576  ax-cnre 9577  ax-pre-lttri 9578  ax-pre-lttrn 9579  ax-pre-ltadd 9580  ax-pre-mulgt0 9581
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  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 6256  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-tpos 6967  df-recs 7054  df-rdg 7088  df-er 7323  df-en 7529  df-dom 7530  df-sdom 7531  df-pnf 9642  df-mnf 9643  df-xr 9644  df-ltxr 9645  df-le 9646  df-sub 9819  df-neg 9820  df-nn 10549  df-2 10606  df-3 10607  df-4 10608  df-5 10609  df-6 10610  df-7 10611  df-8 10612  df-9 10613  df-10 10614  df-ndx 14510  df-slot 14511  df-base 14512  df-sets 14513  df-plusg 14585  df-ple 14592  df-0g 14714  df-plt 15462  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15929  df-minusg 15930  df-oppg 16253  df-omnd 27513  df-ogrp 27514
This theorem is referenced by:  archirngz  27557  archiabllem2c  27563  archiabllem2b  27564
  Copyright terms: Public domain W3C validator