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Theorem ogrpsub 27580
Description: In an ordered group, the ordering is compatible with group subtraction (Contributed by Thierry Arnoux, 30-Jan-2018.)
Hypotheses
Ref Expression
ogrpsub.0  |-  B  =  ( Base `  G
)
ogrpsub.1  |-  .<_  =  ( le `  G )
ogrpsub.2  |-  .-  =  ( -g `  G )
Assertion
Ref Expression
ogrpsub  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  ( X  .-  Z )  .<_  ( Y 
.-  Z ) )

Proof of Theorem ogrpsub
StepHypRef Expression
1 isogrp 27565 . . . . 5  |-  ( G  e. oGrp 
<->  ( G  e.  Grp  /\  G  e. oMnd ) )
21simprbi 464 . . . 4  |-  ( G  e. oGrp  ->  G  e. oMnd )
323ad2ant1 1018 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  G  e. oMnd )
4 simp21 1030 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  X  e.  B
)
5 simp22 1031 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  Y  e.  B
)
6 ogrpgrp 27566 . . . . 5  |-  ( G  e. oGrp  ->  G  e.  Grp )
763ad2ant1 1018 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  G  e.  Grp )
8 simp23 1032 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  Z  e.  B
)
9 ogrpsub.0 . . . . 5  |-  B  =  ( Base `  G
)
10 eqid 2443 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
119, 10grpinvcl 15969 . . . 4  |-  ( ( G  e.  Grp  /\  Z  e.  B )  ->  ( ( invg `  G ) `  Z
)  e.  B )
127, 8, 11syl2anc 661 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  ( ( invg `  G ) `
 Z )  e.  B )
13 simp3 999 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  X  .<_  Y )
14 ogrpsub.1 . . . 4  |-  .<_  =  ( le `  G )
15 eqid 2443 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
169, 14, 15omndadd 27569 . . 3  |-  ( ( G  e. oMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  ( ( invg `  G ) `  Z
)  e.  B )  /\  X  .<_  Y )  ->  ( X ( +g  `  G ) ( ( invg `  G ) `  Z
) )  .<_  ( Y ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) )
173, 4, 5, 12, 13, 16syl131anc 1242 . 2  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  ( X ( +g  `  G ) ( ( invg `  G ) `  Z
) )  .<_  ( Y ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) )
18 ogrpsub.2 . . . 4  |-  .-  =  ( -g `  G )
199, 15, 10, 18grpsubval 15967 . . 3  |-  ( ( X  e.  B  /\  Z  e.  B )  ->  ( X  .-  Z
)  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) )
204, 8, 19syl2anc 661 . 2  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  ( X  .-  Z )  =  ( X ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) )
219, 15, 10, 18grpsubval 15967 . . 3  |-  ( ( Y  e.  B  /\  Z  e.  B )  ->  ( Y  .-  Z
)  =  ( Y ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) )
225, 8, 21syl2anc 661 . 2  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  ( Y  .-  Z )  =  ( Y ( +g  `  G
) ( ( invg `  G ) `
 Z ) ) )
2317, 20, 223brtr4d 4467 1  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  ( X  .-  Z )  .<_  ( Y 
.-  Z ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 974    = wceq 1383    e. wcel 1804   class class class wbr 4437   ` cfv 5578  (class class class)co 6281   Basecbs 14509   +g cplusg 14574   lecple 14581   Grpcgrp 15927   invgcminusg 15928   -gcsg 15929  oMndcomnd 27560  oGrpcogrp 27561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1605  ax-4 1618  ax-5 1691  ax-6 1734  ax-7 1776  ax-8 1806  ax-9 1808  ax-10 1823  ax-11 1828  ax-12 1840  ax-13 1985  ax-ext 2421  ax-rep 4548  ax-sep 4558  ax-nul 4566  ax-pow 4615  ax-pr 4676  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 976  df-tru 1386  df-ex 1600  df-nf 1604  df-sb 1727  df-eu 2272  df-mo 2273  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2593  df-ne 2640  df-ral 2798  df-rex 2799  df-reu 2800  df-rmo 2801  df-rab 2802  df-v 3097  df-sbc 3314  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3771  df-if 3927  df-pw 3999  df-sn 4015  df-pr 4017  df-op 4021  df-uni 4235  df-iun 4317  df-br 4438  df-opab 4496  df-mpt 4497  df-id 4785  df-xp 4995  df-rel 4996  df-cnv 4997  df-co 4998  df-dm 4999  df-rn 5000  df-res 5001  df-ima 5002  df-iota 5541  df-fun 5580  df-fn 5581  df-f 5582  df-f1 5583  df-fo 5584  df-f1o 5585  df-fv 5586  df-riota 6242  df-ov 6284  df-oprab 6285  df-mpt2 6286  df-1st 6785  df-2nd 6786  df-0g 14716  df-mgm 15746  df-sgrp 15785  df-mnd 15795  df-grp 15931  df-minusg 15932  df-sbg 15933  df-omnd 27562  df-ogrp 27563
This theorem is referenced by:  ogrpsublt  27585  archiabllem1a  27608  archiabllem2c  27612  ornglmulle  27668  orngrmulle  27669
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