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

Theorem ogrpinvlt 27948
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 1018 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  G  e. oGrp )
2 simp2 995 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  X  e.  B )
3 simp3 996 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  Y  e.  B )
4 ogrpgrp 27927 . . . . . 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 16294 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( I `  Y
)  e.  B )
95, 3, 8syl2anc 659 . . . 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 2454 . . . . 5  |-  ( +g  `  G )  =  ( +g  `  G )
126, 10, 11ogrpaddltbi 27943 . . . 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 1228 . . 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 2454 . . . . . 6  |-  ( 0g
`  G )  =  ( 0g `  G
)
156, 11, 14, 7grprinv 16296 . . . . 5  |-  ( ( G  e.  Grp  /\  Y  e.  B )  ->  ( Y ( +g  `  G ) ( I `
 Y ) )  =  ( 0g `  G ) )
165, 3, 15syl2anc 659 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( Y
( +g  `  G ) ( I `  Y
) )  =  ( 0g `  G ) )
1716breq2d 4451 . . 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 1019 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  (oppg
`  G )  e. oGrp
)
196, 11grpcl 16262 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B  /\  ( I `  Y
)  e.  B )  ->  ( X ( +g  `  G ) ( I `  Y
) )  e.  B
)
205, 2, 9, 19syl3anc 1226 . . . 4  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( X
( +g  `  G ) ( I `  Y
) )  e.  B
)
216, 14grpidcl 16277 . . . . 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 16294 . . . . 5  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( I `  X
)  e.  B )
245, 2, 23syl2anc 659 . . . 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 27945 . . 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 16295 . . . . . 6  |-  ( ( G  e.  Grp  /\  X  e.  B )  ->  ( ( I `  X ) ( +g  `  G ) X )  =  ( 0g `  G ) )
285, 2, 27syl2anc 659 . . . . 5  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
I `  X )
( +g  `  G ) X )  =  ( 0g `  G ) )
2928oveq1d 6285 . . . 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 16263 . . . . 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 1228 . . . 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 16279 . . . . 5  |-  ( ( G  e.  Grp  /\  ( I `  Y
)  e.  B )  ->  ( ( 0g
`  G ) ( +g  `  G ) ( I `  Y
) )  =  ( I `  Y ) )
335, 9, 32syl2anc 659 . . . 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 2503 . . 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 16280 . . . 4  |-  ( ( G  e.  Grp  /\  ( I `  X
)  e.  B )  ->  ( ( I `
 X ) ( +g  `  G ) ( 0g `  G
) )  =  ( I `  X ) )
365, 24, 35syl2anc 659 . . 3  |-  ( ( ( G  e. oGrp  /\  (oppg `  G )  e. oGrp )  /\  X  e.  B  /\  Y  e.  B
)  ->  ( (
I `  X )
( +g  `  G ) ( 0g `  G
) )  =  ( I `  X ) )
3734, 36breq12d 4452 . 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 367    /\ w3a 971    = wceq 1398    e. wcel 1823   class class class wbr 4439   ` cfv 5570  (class class class)co 6270   Basecbs 14716   +g cplusg 14784   0gc0g 14929   ltcplt 15769   Grpcgrp 16252   invgcminusg 16253  oppgcoppg 16579  oGrpcogrp 27922
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rmo 2812  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-tpos 6947  df-recs 7034  df-rdg 7068  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-2 10590  df-3 10591  df-4 10592  df-5 10593  df-6 10594  df-7 10595  df-8 10596  df-9 10597  df-10 10598  df-ndx 14719  df-slot 14720  df-base 14721  df-sets 14722  df-plusg 14797  df-ple 14804  df-0g 14931  df-plt 15787  df-mgm 16071  df-sgrp 16110  df-mnd 16120  df-grp 16256  df-minusg 16257  df-oppg 16580  df-omnd 27923  df-ogrp 27924
This theorem is referenced by:  archirngz  27967  archiabllem2c  27973
  Copyright terms: Public domain W3C validator