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Theorem ogrpsub 27366
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 27351 . . . . 5  |-  ( G  e. oGrp 
<->  ( G  e.  Grp  /\  G  e. oMnd ) )
21simprbi 464 . . . 4  |-  ( G  e. oGrp  ->  G  e. oMnd )
323ad2ant1 1017 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  G  e. oMnd )
4 simp21 1029 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  X  e.  B
)
5 simp22 1030 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  Y  e.  B
)
61simplbi 460 . . . . 5  |-  ( G  e. oGrp  ->  G  e.  Grp )
763ad2ant1 1017 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  G  e.  Grp )
8 simp23 1031 . . . 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 2467 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
119, 10grpinvcl 15893 . . . 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 998 . . 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 2467 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
169, 14, 15omndadd 27355 . . 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 1241 . 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 15891 . . 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 15891 . . 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 4477 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 973    = wceq 1379    e. wcel 1767   class class class wbr 4447   ` cfv 5586  (class class class)co 6282   Basecbs 14483   +g cplusg 14548   lecple 14555   Grpcgrp 15720   invgcminusg 15721   -gcsg 15723  oMndcomnd 27346  oGrpcogrp 27347
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-1st 6781  df-2nd 6782  df-0g 14690  df-mnd 15725  df-grp 15855  df-minusg 15856  df-sbg 15857  df-omnd 27348  df-ogrp 27349
This theorem is referenced by:  ogrpsublt  27371  archiabllem1a  27394  archiabllem2c  27398  ornglmulle  27455  orngrmulle  27456
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