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Theorem ogrpinvlt 28561
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 1054 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  G  e. oGrp )
2 simp2 1031 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  X  e.  B )
3 simp3 1032 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  Y  e.  B )
4 ogrpgrp 28540 . . . . . 6  |-  ( G  e. oGrp  ->  G  e.  Grp )
51, 4syl 17 . . . . 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 16789 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( I `  Y
)  e.  B )
95, 3, 8syl2anc 673 . . . 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 2471 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
126, 10, 11ogrpaddltbi 28556 . . . 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 1294 . . 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 2471 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
156, 11, 14, 7grprinv 16791 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( Y ( +g  `  G ) ( I `
 Y ) )  =  ( 0g `  G ) )
165, 3, 15syl2anc 673 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( Y
( +g  `  G ) ( I `  Y
) )  =  ( 0g `  G ) )
1716breq2d 4407 . . 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 1055 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  (oppg
`  G )  e. oGrp
)
196, 11grpcl 16757 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( I `  Y
)  e.  B )  ->  ( X ( +g  `  G ) ( I `  Y
) )  e.  B
)
205, 2, 9, 19syl3anc 1292 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( X
( +g  `  G ) ( I `  Y
) )  e.  B
)
216, 14grpidcl 16772 . . . . 5  |-  ( G  e.  Grp  ->  ( 0g `  G )  e.  B )
221, 4, 213syl 18 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( 0g `  G )  e.  B
)
236, 7grpinvcl 16789 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( I `  X
)  e.  B )
245, 2, 23syl2anc 673 . . . 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 28558 . . 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 287 . 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 16790 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( I `  X ) ( +g  `  G ) X )  =  ( 0g `  G ) )
285, 2, 27syl2anc 673 . . . . 5  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
I `  X )
( +g  `  G ) X )  =  ( 0g `  G ) )
2928oveq1d 6323 . . . 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 16758 . . . . 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 1294 . . . 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 16774 . . . . 5  |-  ( ( G  e.  Grp  /\  ( I `  Y
)  e.  B )  ->  ( ( 0g
`  G ) ( +g  `  G ) ( I `  Y
) )  =  ( I `  Y ) )
335, 9, 32syl2anc 673 . . . 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 2513 . . 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 16775 . . . 4  |-  ( ( G  e.  Grp  /\  ( I `  X
)  e.  B )  ->  ( ( I `
 X ) ( +g  `  G ) ( 0g `  G
) )  =  ( I `  X ) )
365, 24, 35syl2anc 673 . . 3  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
I `  X )
( +g  `  G ) ( 0g `  G
) )  =  ( I `  X ) )
3734, 36breq12d 4408 . 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 261 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 189    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   class class class wbr 4395   ` cfv 5589  (class class class)co 6308   Basecbs 15199   +g cplusg 15268   0gc0g 15416   ltcplt 16264   Grpcgrp 16747   invgcminusg 16748  oppgcoppg 17074  oGrpcogrp 28535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  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-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-tpos 6991  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-3 10691  df-4 10692  df-5 10693  df-6 10694  df-7 10695  df-8 10696  df-9 10697  df-10 10698  df-ndx 15202  df-slot 15203  df-base 15204  df-sets 15205  df-plusg 15281  df-ple 15288  df-0g 15418  df-plt 16282  df-mgm 16566  df-sgrp 16605  df-mnd 16615  df-grp 16751  df-minusg 16752  df-oppg 17075  df-omnd 28536  df-ogrp 28537
This theorem is referenced by:  archirngz  28580  archiabllem2c  28586
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