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Theorem ogrpinvlt 26187
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 1012 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  G  e. oGrp )
2 simp2 989 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  X  e.  B )
3 simp3 990 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  Y  e.  B )
4 ogrpgrp 26166 . . . . . 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 15583 . . . . 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 2443 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
126, 10, 11ogrpaddltbi 26182 . . . 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 1220 . . 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 2443 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
156, 11, 14, 7grprinv 15585 . . . . 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 4304 . . 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 1013 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  (oppg
`  G )  e. oGrp
)
196, 11grpcl 15551 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( I `  Y
)  e.  B )  ->  ( X ( +g  `  G ) ( I `  Y
) )  e.  B
)
205, 2, 9, 19syl3anc 1218 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( X
( +g  `  G ) ( I `  Y
) )  e.  B
)
216, 14grpidcl 15566 . . . . 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 15583 . . . . 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 26184 . . 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 15584 . . . . . 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 6106 . . . 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 15552 . . . . 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 1220 . . . 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 15568 . . . . 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 2483 . . 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 15569 . . . 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 4305 . 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 965    = wceq 1369    e. wcel 1756   class class class wbr 4292   ` cfv 5418  (class class class)co 6091   Basecbs 14174   +g cplusg 14238   0gc0g 14378   ltcplt 15111   Grpcgrp 15410   invgcminusg 15411  oppgcoppg 15860  oGrpcogrp 26161
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 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372  ax-cnex 9338  ax-resscn 9339  ax-1cn 9340  ax-icn 9341  ax-addcl 9342  ax-addrcl 9343  ax-mulcl 9344  ax-mulrcl 9345  ax-mulcom 9346  ax-addass 9347  ax-mulass 9348  ax-distr 9349  ax-i2m1 9350  ax-1ne0 9351  ax-1rid 9352  ax-rnegex 9353  ax-rrecex 9354  ax-cnre 9355  ax-pre-lttri 9356  ax-pre-lttrn 9357  ax-pre-ltadd 9358  ax-pre-mulgt0 9359
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2568  df-ne 2608  df-nel 2609  df-ral 2720  df-rex 2721  df-reu 2722  df-rmo 2723  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-pss 3344  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-tp 3882  df-op 3884  df-uni 4092  df-iun 4173  df-br 4293  df-opab 4351  df-mpt 4352  df-tr 4386  df-eprel 4632  df-id 4636  df-po 4641  df-so 4642  df-fr 4679  df-we 4681  df-ord 4722  df-on 4723  df-lim 4724  df-suc 4725  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-riota 6052  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-om 6477  df-tpos 6745  df-recs 6832  df-rdg 6866  df-er 7101  df-en 7311  df-dom 7312  df-sdom 7313  df-pnf 9420  df-mnf 9421  df-xr 9422  df-ltxr 9423  df-le 9424  df-sub 9597  df-neg 9598  df-nn 10323  df-2 10380  df-3 10381  df-4 10382  df-5 10383  df-6 10384  df-7 10385  df-8 10386  df-9 10387  df-10 10388  df-ndx 14177  df-slot 14178  df-base 14179  df-sets 14180  df-plusg 14251  df-ple 14258  df-0g 14380  df-plt 15128  df-mnd 15415  df-grp 15545  df-minusg 15546  df-oppg 15861  df-omnd 26162  df-ogrp 26163
This theorem is referenced by:  archirngz  26206  archiabllem2c  26212  archiabllem2b  26213
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