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Theorem ogrpaddlt 27861
Description: In an ordered group, strict ordering is compatible with group addition. (Contributed by Thierry Arnoux, 20-Jan-2018.)
Hypotheses
Ref Expression
ogrpaddlt.0  |-  B  =  ( Base `  G
)
ogrpaddlt.1  |-  .<  =  ( lt `  G )
ogrpaddlt.2  |-  .+  =  ( +g  `  G )
Assertion
Ref Expression
ogrpaddlt  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( X  .+  Z )  .<  ( Y  .+  Z ) )

Proof of Theorem ogrpaddlt
StepHypRef Expression
1 isogrp 27845 . . . . 5  |-  ( G  e. oGrp 
<->  ( G  e.  Grp  /\  G  e. oMnd ) )
21simprbi 462 . . . 4  |-  ( G  e. oGrp  ->  G  e. oMnd )
323ad2ant1 1015 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  G  e. oMnd )
4 simp2 995 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B ) )
5 simp1 994 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  G  e. oGrp )
6 simp21 1027 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  X  e.  B
)
7 simp22 1028 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  Y  e.  B
)
8 simp3 996 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  X  .<  Y )
9 eqid 2382 . . . . . 6  |-  ( le
`  G )  =  ( le `  G
)
10 ogrpaddlt.1 . . . . . 6  |-  .<  =  ( lt `  G )
119, 10pltle 15708 . . . . 5  |-  ( ( G  e. oGrp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X
( le `  G
) Y ) )
1211imp 427 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X ( le
`  G ) Y )
135, 6, 7, 8, 12syl31anc 1229 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  X ( le
`  G ) Y )
14 ogrpaddlt.0 . . . 4  |-  B  =  ( Base `  G
)
15 ogrpaddlt.2 . . . 4  |-  .+  =  ( +g  `  G )
1614, 9, 15omndadd 27849 . . 3  |-  ( ( G  e. oMnd  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X ( le `  G ) Y )  ->  ( X  .+  Z ) ( le
`  G ) ( Y  .+  Z ) )
173, 4, 13, 16syl3anc 1226 . 2  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( X  .+  Z ) ( le
`  G ) ( Y  .+  Z ) )
1810pltne 15709 . . . . 5  |-  ( ( G  e. oGrp  /\  X  e.  B  /\  Y  e.  B )  ->  ( X  .<  Y  ->  X  =/=  Y ) )
1918imp 427 . . . 4  |-  ( ( ( G  e. oGrp  /\  X  e.  B  /\  Y  e.  B )  /\  X  .<  Y )  ->  X  =/=  Y
)
205, 6, 7, 8, 19syl31anc 1229 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  X  =/=  Y
)
21 ogrpgrp 27846 . . . . . 6  |-  ( G  e. oGrp  ->  G  e.  Grp )
2214, 15grprcan 16200 . . . . . . 7  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Z
)  =  ( Y 
.+  Z )  <->  X  =  Y ) )
2322biimpd 207 . . . . . 6  |-  ( ( G  e.  Grp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B
) )  ->  (
( X  .+  Z
)  =  ( Y 
.+  Z )  ->  X  =  Y )
)
2421, 23sylan 469 . . . . 5  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( ( X  .+  Z )  =  ( Y  .+  Z
)  ->  X  =  Y ) )
2524necon3d 2606 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )
)  ->  ( X  =/=  Y  ->  ( X  .+  Z )  =/=  ( Y  .+  Z ) ) )
26253impia 1191 . . 3  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  =/=  Y
)  ->  ( X  .+  Z )  =/=  ( Y  .+  Z ) )
275, 4, 20, 26syl3anc 1226 . 2  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( X  .+  Z )  =/=  ( Y  .+  Z ) )
28 ovex 6224 . . . 4  |-  ( X 
.+  Z )  e. 
_V
29 ovex 6224 . . . 4  |-  ( Y 
.+  Z )  e. 
_V
309, 10pltval 15707 . . . 4  |-  ( ( G  e. oGrp  /\  ( X  .+  Z )  e. 
_V  /\  ( Y  .+  Z )  e.  _V )  ->  ( ( X 
.+  Z )  .< 
( Y  .+  Z
)  <->  ( ( X 
.+  Z ) ( le `  G ) ( Y  .+  Z
)  /\  ( X  .+  Z )  =/=  ( Y  .+  Z ) ) ) )
3128, 29, 30mp3an23 1314 . . 3  |-  ( G  e. oGrp  ->  ( ( X 
.+  Z )  .< 
( Y  .+  Z
)  <->  ( ( X 
.+  Z ) ( le `  G ) ( Y  .+  Z
)  /\  ( X  .+  Z )  =/=  ( Y  .+  Z ) ) ) )
32313ad2ant1 1015 . 2  |-  ( ( G  e. oGrp  /\  ( X  e.  B  /\  Y  e.  B  /\  Z  e.  B )  /\  X  .<  Y )  ->  ( ( X 
.+  Z )  .< 
( Y  .+  Z
)  <->  ( ( X 
.+  Z ) ( le `  G ) ( Y  .+  Z
)  /\  ( X  .+  Z )  =/=  ( Y  .+  Z ) ) ) )
3317, 27, 32mpbir2and 920 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    <-> wb 184    /\ wa 367    /\ w3a 971    = wceq 1399    e. wcel 1826    =/= wne 2577   _Vcvv 3034   class class class wbr 4367   ` cfv 5496  (class class class)co 6196   Basecbs 14634   +g cplusg 14702   lecple 14709   ltcplt 15687   Grpcgrp 16170  oMndcomnd 27840  oGrpcogrp 27841
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-reu 2739  df-rmo 2740  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-mpt 4427  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-riota 6158  df-ov 6199  df-0g 14849  df-plt 15705  df-mgm 15989  df-sgrp 16028  df-mnd 16038  df-grp 16174  df-omnd 27842  df-ogrp 27843
This theorem is referenced by:  ogrpaddltbi  27862  ogrpaddltrd  27863  ogrpinv0lt  27866  isarchi3  27884  archirngz  27886  archiabllem1b  27889  archiabllem2c  27892  ofldchr  27958
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