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Theorem ogrpinvlt 26120
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 1007 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  G  e. oGrp )
2 simp2 984 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  X  e.  B )
3 simp3 985 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  Y  e.  B )
4 ogrpgrp 26099 . . . . . 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 15576 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( I `  Y
)  e.  B )
95, 3, 8syl2anc 656 . . . 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 2441 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
126, 10, 11ogrpaddltbi 26115 . . . 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 1215 . . 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 2441 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
156, 11, 14, 7grprinv 15578 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( Y ( +g  `  G ) ( I `
 Y ) )  =  ( 0g `  G ) )
165, 3, 15syl2anc 656 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( Y
( +g  `  G ) ( I `  Y
) )  =  ( 0g `  G ) )
1716breq2d 4301 . . 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 1008 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  (oppg
`  G )  e. oGrp
)
196, 11grpcl 15544 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( I `  Y
)  e.  B )  ->  ( X ( +g  `  G ) ( I `  Y
) )  e.  B
)
205, 2, 9, 19syl3anc 1213 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( X
( +g  `  G ) ( I `  Y
) )  e.  B
)
216, 14grpidcl 15559 . . . . 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 15576 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( I `  X
)  e.  B )
245, 2, 23syl2anc 656 . . . 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 26117 . . 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 15577 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( I `  X ) ( +g  `  G ) X )  =  ( 0g `  G ) )
285, 2, 27syl2anc 656 . . . . 5  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
I `  X )
( +g  `  G ) X )  =  ( 0g `  G ) )
2928oveq1d 6105 . . . 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 15545 . . . . 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 1215 . . . 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 15561 . . . . 5  |-  ( ( G  e.  Grp  /\  ( I `  Y
)  e.  B )  ->  ( ( 0g
`  G ) ( +g  `  G ) ( I `  Y
) )  =  ( I `  Y ) )
335, 9, 32syl2anc 656 . . . 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 2481 . . 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 15562 . . . 4  |-  ( ( G  e.  Grp  /\  ( I `  X
)  e.  B )  ->  ( ( I `
 X ) ( +g  `  G ) ( 0g `  G
) )  =  ( I `  X ) )
365, 24, 35syl2anc 656 . . 3  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
I `  X )
( +g  `  G ) ( 0g `  G
) )  =  ( I `  X ) )
3734, 36breq12d 4302 . 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 960    = wceq 1364    e. wcel 1761   class class class wbr 4289   ` cfv 5415  (class class class)co 6090   Basecbs 14170   +g cplusg 14234   0gc0g 14374   ltcplt 15107   Grpcgrp 15406   invgcminusg 15407  oppgcoppg 15853  oGrpcogrp 26094
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371  ax-cnex 9334  ax-resscn 9335  ax-1cn 9336  ax-icn 9337  ax-addcl 9338  ax-addrcl 9339  ax-mulcl 9340  ax-mulrcl 9341  ax-mulcom 9342  ax-addass 9343  ax-mulass 9344  ax-distr 9345  ax-i2m1 9346  ax-1ne0 9347  ax-1rid 9348  ax-rnegex 9349  ax-rrecex 9350  ax-cnre 9351  ax-pre-lttri 9352  ax-pre-lttrn 9353  ax-pre-ltadd 9354  ax-pre-mulgt0 9355
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 961  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2261  df-mo 2262  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-nel 2607  df-ral 2718  df-rex 2719  df-reu 2720  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-pss 3341  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-tp 3879  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-tr 4383  df-eprel 4628  df-id 4632  df-po 4637  df-so 4638  df-fr 4675  df-we 4677  df-ord 4718  df-on 4719  df-lim 4720  df-suc 4721  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-riota 6049  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-om 6476  df-tpos 6744  df-recs 6828  df-rdg 6862  df-er 7097  df-en 7307  df-dom 7308  df-sdom 7309  df-pnf 9416  df-mnf 9417  df-xr 9418  df-ltxr 9419  df-le 9420  df-sub 9593  df-neg 9594  df-nn 10319  df-2 10376  df-3 10377  df-4 10378  df-5 10379  df-6 10380  df-7 10381  df-8 10382  df-9 10383  df-10 10384  df-ndx 14173  df-slot 14174  df-base 14175  df-sets 14176  df-plusg 14247  df-ple 14254  df-0g 14376  df-plt 15124  df-mnd 15411  df-grp 15538  df-minusg 15539  df-oppg 15854  df-omnd 26095  df-ogrp 26096
This theorem is referenced by:  archirngz  26139  archiabllem2c  26145  archiabllem2b  26146
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