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Theorem ogrpsub 26326
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 26311 . . . . 5  |-  ( G  e. oGrp 
<->  ( G  e.  Grp  /\  G  e. oMnd ) )
21simprbi 464 . . . 4  |-  ( G  e. oGrp  ->  G  e. oMnd )
323ad2ant1 1009 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  G  e. oMnd )
4 simp21 1021 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  X  e.  B
)
5 simp22 1022 . . 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 1009 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<_  Y )  ->  G  e.  Grp )
8 simp23 1023 . . . 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 2454 . . . . 5  |-  ( invg `  G )  =  ( invg `  G )
119, 10grpinvcl 15703 . . . 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 990 . . 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 2454 . . . 4  |-  ( +g  `  G )  =  ( +g  `  G )
169, 14, 15omndadd 26315 . . 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 1232 . 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 15701 . . 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 15701 . . 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 4431 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 965    = wceq 1370    e. wcel 1758   class class class wbr 4401   ` cfv 5527  (class class class)co 6201   Basecbs 14293   +g cplusg 14358   lecple 14365   Grpcgrp 15530   invgcminusg 15531   -gcsg 15533  oMndcomnd 26306  oGrpcogrp 26307
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4512  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rmo 2807  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3397  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-iun 4282  df-br 4402  df-opab 4460  df-mpt 4461  df-id 4745  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-riota 6162  df-ov 6204  df-oprab 6205  df-mpt2 6206  df-1st 6688  df-2nd 6689  df-0g 14500  df-mnd 15535  df-grp 15665  df-minusg 15666  df-sbg 15667  df-omnd 26308  df-ogrp 26309
This theorem is referenced by:  ogrpsublt  26331  archiabllem1a  26354  archiabllem2c  26358  ornglmulle  26419  orngrmulle  26420
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